
This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order ϝ. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.
Citation: Ismail Gad Ameen, Dumitru Baleanu, Hussien Shafei Hussien. Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations[J]. AIMS Mathematics, 2024, 9(6): 15819-15836. doi: 10.3934/math.2024764
[1] | Yuwei Cao, Bing Li . Existence and global exponential stability of compact almost automorphic solutions for Clifford-valued high-order Hopfield neutral neural networks with D operator. AIMS Mathematics, 2022, 7(4): 6182-6203. doi: 10.3934/math.2022344 |
[2] | Nina Huo, Bing Li, Yongkun Li . Global exponential stability and existence of almost periodic solutions in distribution for Clifford-valued stochastic high-order Hopfield neural networks with time-varying delays. AIMS Mathematics, 2022, 7(3): 3653-3679. doi: 10.3934/math.2022202 |
[3] | Xiaofang Meng, Yongkun Li . Pseudo almost periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays and leakage delays on time scales. AIMS Mathematics, 2021, 6(9): 10070-10091. doi: 10.3934/math.2021585 |
[4] | Jin Gao, Lihua Dai . Anti-periodic synchronization of quaternion-valued high-order Hopfield neural networks with delays. AIMS Mathematics, 2022, 7(8): 14051-14075. doi: 10.3934/math.2022775 |
[5] | Abdulaziz M. Alanazi, R. Sriraman, R. Gurusamy, S. Athithan, P. Vignesh, Zaid Bassfar, Adel R. Alharbi, Amer Aljaedi . System decomposition method-based global stability criteria for T-S fuzzy Clifford-valued delayed neural networks with impulses and leakage term. AIMS Mathematics, 2023, 8(7): 15166-15188. doi: 10.3934/math.2023774 |
[6] | Qian Cao, Xiaojin Guo . Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays. AIMS Mathematics, 2020, 5(6): 5402-5421. doi: 10.3934/math.2020347 |
[7] | Xiaojin Guo, Chuangxia Huang, Jinde Cao . Nonnegative periodicity on high-order proportional delayed cellular neural networks involving D operator. AIMS Mathematics, 2021, 6(3): 2228-2243. doi: 10.3934/math.2021135 |
[8] | Yongkun Li, Xiaoli Huang, Xiaohui Wang . Weyl almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks with time-varying delays. AIMS Mathematics, 2022, 7(4): 4861-4886. doi: 10.3934/math.2022271 |
[9] | Qi Shao, Yongkun Li . Almost periodic solutions for Clifford-valued stochastic shunting inhibitory cellular neural networks with mixed delays. AIMS Mathematics, 2024, 9(5): 13439-13461. doi: 10.3934/math.2024655 |
[10] | Hedi Yang . Weighted pseudo almost periodicity on neutral type CNNs involving multi-proportional delays and D operator. AIMS Mathematics, 2021, 6(2): 1865-1879. doi: 10.3934/math.2021113 |
This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order ϝ. The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.
As is well known, in the practical application of neural networks (NNs), it is often necessary to use and design NN models with different dynamic characteristics for different application scenarios and purposes. Therefore, the study of the dynamic behavior of NNs has become an important issue that is widely concerned in both theoretical research and practical applications of NNs. Therefore, the dynamics of various types of NNs, especially numerous classical NNs such as recurrent NNs [1,2], bidirectional associative memory NNs [3], inertial NNs [4,5], Hopfield NNs [6], Cohen-Grossberg NNs [7], etc., have been widely studied. It should be mentioned here that due to the stronger approximation, faster convergence speed, larger storage capacity, and higher fault tolerance of high-order NNs compared to low-order NNs, the dynamics research of high-order NNs has received widespread attention [8,9,10,11,12].
Meanwhile, owing to the fact that algebra-valued NNs, such as complex-valued [13,14], quaternion-valued [15,16,17,18], Clifford-valued [19,20,21,22,23], and octonion-valued NNs [24,25,26], are extensions of real-valued NNs and have more advantages than real-valued NNs in many application scenarios, research on the dynamics of algebra-valued NNs has gradually become a new hotspot in the field of NN research in recent years. It is worth mentioning here that the Clifford-valued high-order Hopfield fuzzy NN represents a sophisticated integration of Clifford algebra, high-order synaptic connections, and fuzzy logic, enabling it to achieve advanced applications in multidimensional data processing and complex system modeling. For example, it has applications in the fields of multidimensional signal processing, secure communication and image encryption, optimization and control systems, neuroscience, and cognitive modeling [27,28,29,30].
On the one hand, from both theoretical and practical perspectives, NN models with time-varying connection weights and time-varying external inputs are more realistic than those with constant connection weights and constant external inputs. Meanwhile, time delay effects are inevitable. As a result, the rate of change in the state of a neuron depends not only on its current state but also on its historical state, and even more so, on the rate of change in its historical state. It is precisely for these reasons that researchers have proposed various neutral-type NN models with D operators and conducted extensive research on their dynamics [31,32,33,34]. In addition, fuzzy logic and NNs complement each other: fuzzy systems provide interpretability and handle uncertainty, while NNs offer powerful learning from data. Their integration bridges the gap between data-driven machine learning and human-like reasoning, making systems more adaptable, transparent, and robust in real-world applications. Indeed, fuzzy NNs have been successfully applied in many fields such as signal processing, pattern recognition, associative memory, and image processing [35,36,37,38,39].
On the other hand, as is well known, almost periodic oscillation is an important dynamic of NNs with time-varying connection weights and time-varying external inputs. In the past few decades, the almost periodic oscillations of various NNs have been studied by countless scholars [23,31,33,34]. We know that besides Bohr's concept of almost periodicity, there are also Stepanov almost periodicity, Weyl almost periodicity, Besicovitch almost periodicity, and so on [40]. It should be pointed out here that Besicovitch almost periodicity is the most complex almost periodicity among Bohr almost periodicity, Stepanov almost periodicity, and Weyl almost periodicity, and that Stepanov almost periodicity, Weyl almost periodicity, and Besicovitch almost periodicity are referred to as generalized almost periodicity. Meanwhile, it should be noted that the product of two generalized almost periodic functions in the same sense may not necessarily be a generalized almost periodic function in that sense. Because of this reason, the emergence of high-order terms in high-order NNs poses difficulties for studying the generalized almost periodic oscillations of high-order NNs. As a consequence, the results of generalized almost periodic oscillations for high-order NNs are still very rare. Thereupon, it is necessary to further study the generalized almost periodic oscillation problem of high-order NNs.
Inspired by the above observations, this paper considers a class of Clifford-valued high-order Hopfield fuzzy NNs with time-varying delays and D operators as follows:
[xi(t)−ai(t)xi(t−τi(t))]′=−bi(t)xi(t)+n∑j=1cij(t)fj(xj(t))+n∑j=1uij(t)fj(xj(t−σij(t)))+n∑j=1γij(t)μj(t)+v∑j=1n∑k=1θijk(t)gj(xj(t−δijk(t)))gk(xk(t−δijk(t)))+n⋀j=1αij(t)fj(xj(t−ηij(t)))+n⋁j=1βij(t)fj(xj(t−ηij(t)))+n⋀j=1n⋀k=1qijk(t)gj(xj(t−δijk(t)))gk(xk(t−δijk(t)))+n⋁j=1n⋁k=1νijk(t)gj(xj(t−δijk(t)))gk(xk(t−δijk(t)))+n⋀j=1Tij(t)μj(t)+n⋁j=1Sij(t)μj(t)+Ii(t), | (1.1) |
where i∈J:={1,2,⋯,n}, xi(t)∈A indicates the state of the ith unit at time t; A is a real Clifford algebra; bi(t)∈A represents the self feedback coefficient at time t; αij(t),βij(t),Tij(t),Sij(t)∈A stand for the elements of the fuzzy feedback MIN template and fuzzy feed forward MAX template, respectively; ai(t),cij(t),uij(t) and θijk(t),qijk(t),νijk(t)∈A represent the first-order and second-order connection weights of the NN; γij(t) stands for the element of the feed forward template; ⋀ and ⋁ denote the fuzzy AND and OR operations, respectively; μj(t)∈A represents the input of the jth neuron; Ii(t)∈A corresponds to the external input to the ith unit; fj and gj:A→A signify the nonlinear activation functions; and τi(t),σij(t),ηij(t),δijk(t)∈R+ denote the transmission delays.
The initial value condition associated with (1.1) is given as
xi(s)=φi(s),s∈[−ϱ,0],i∈J, | (1.2) |
where φi∈BC([−ϱ,0],A),ϱ=maxi,j,k∈J{supt∈Rτi(t),supt∈Rσij(t),supt∈Rηij(t),supt∈Rδijk(t)}.
The main purpose of this paper is to investigate the existence and stability of Besicovitch almost periodic solutions for system (1.1). The main contributions of this paper are as follows:
1. This paper is the first one to investigate the existence of Besicovitch almost periodic solutions for system (1.1), and the results of this paper still hold true and are new in the following special cases of system (1.1).
(ⅰ) System (1.1) is a real-valued, complex-valued, or quaternion-valued system.
(ⅱ) System (1.1) is a real-valued, complex-valued, or quaternion-valued system without D operators, i.e. ai(t)=0.
(ⅲ) System (1.1) is a real-valued system without D operators and fuzzy terms, i.e., ai(t)=αij(t)=qijk(t)=νijk(t)=Tij(t)=Sij(t)≡0.
2. The research method proposed in the paper can be used to study the generalized almost periodic dynamics for other high-order NNs.
Remark 1.1. The method we propose can be summarized as follows: First, we use the fixed point theorem to prove the existence of solutions for system (1.1) that are bounded and continuous with respect to the Besicovitch seminorm on a closed subset of an appropriate Banach space. Then, we apply the definition and inequality techniques to prove that this solution is Besicovitch almost periodic.
The remaining part of the paper is arranged as follows: In the second section, we review some relevant concepts, introduce some symbols used in this article, cite a useful lemma, and state and prove the completeness of the space we will use. In the third section, we investigate the existence and stability of Besicovitch almost periodic solutions for system (1.1). In the fourth section, we provide an example to demonstrate the correctness of our results. Finally, in the fifth section, we provide a brief conclusion.
Let A={∑A∈ΩxAeA∈R} indicate a real Clifford algebra over Rm [41], where Ω={∅,1,2,⋯,A,⋯,12,⋯,m}, eA=eh1eh2⋯ehv, 1≤h1<h2<⋯<hv≤m, and in addition, e∅=e0=1, and eh, h=1,2,⋯,m are said to be Clifford generators and satisfy ep=1,p=0,1,2,⋯,s,e2p=−1,p=s+1,s+2,⋯,m, where s<m, and epeq+eqep=0,p≠q,p,q=1,2,⋯,m. For every x=∑A∈ΩxAeA∈A and y=(y1,y2,⋯,yn)T∈An, we define |x|1=maxA∈Ω{|xA|} and |y|n=maxi∈J{|yi|1}, respectively, and then the spaces (A,|⋅|1) and (An,|⋅|) are Banach ones.
Since there is no order relation among Clifford numbers, as in [42], for x=∑A∈ΩxAeA,y=∑A∈ΩyAeA, we define x⋀y=∑A∈Ω(min{xA,yA})eA and x⋁y=∑A∈Ω(max{xA,yA})eA. According to this regulation, for example, regarding the 6th and 7th terms on the right-hand side of Equation (1.1), we have
n⋀j=1αij(t)fj(xj(t−ηij(t)))=∑A∈Ω(min1≤j≤n{αAij(t)fAj(xj(t−ηij(t)))})eA |
and
n⋁j=1βij(t)fj(xj(t−ηij(t)))=∑A∈Ω(max1≤j≤n{αAij(t)fAj(xj(t−ηij(t)))})eA. |
For x=∑A∈ΩxAeA∈A, we indicate xc=x−x∅.
For the sake of generality in the subsequent discussion of this section, let (X,‖⋅‖) be a Banach space and Lploc(R,X) with 1≤p<+∞ be the space consisting of measurable and locally p-integrable functions from R into X. In the next section, we will take X=R, X=A, or X=An.
Definition 2.1. [40] A bounded continuous function φ:R→X is said to be almost periodic, if for every ε>0, there exists a number ℓ(ε)>0 such that for each a∈R, there exists a point σ∈[a,a+ℓ] satisfying
‖φ(t+σ)−φ(t)‖<ε. |
The family of such functions will be signified by AP(R,X).
For φ∈Lploc(R,X), the Besicovitch seminorm is defined as the following:
‖φ‖Bp={¯liml→∞12l∫l−l‖φ(t)‖pdt}1p. |
Definition 2.2. [43] A function φ∈Lploc(R,X) is called Bp-continuous if limh→0‖φ(⋅+h)−φ(⋅)‖Bp=0 and is called Bp-bounded if ‖φ‖Bp<∞.
Henceforth, we will denote the set of all functions that are Bp-continuous and Bp-bounded by BCBp(R,X).
Definition 2.3. [40] A function φ∈Lploc(R,X) is said to be Besicovitch almost periodic, if for every ε>0, there exists a positive number ℓ>0 such that for each a∈R, there exists a point σ∈[a,a+ℓ] satisfying
‖φ(⋅+σ)−φ(⋅)‖Bp<ε. |
Denote by BpAP(R,X) the class of such functions and, for simplicity, call them Bp-almost periodic functions.
Lemma 2.1. [44] If αij,βij∈C(R,A),gj∈C(A,A),i,j∈J, then one has
|n⋀i=1αij(t)gj(x)−n⋀i=1αij(t)gj(y)|1≤n∑i=1|αij(t)‖‖gj(x)−gj(y)|1,|n⋁i=1βij(t)gj(x)−n⋁i=1βij(t)gj(y)|1≤n∑i=1‖βij(t)‖|gj(x)−gj(y)|1. |
Let L∞(R,X) be the set of all essentially bounded measurable functions from R to X, then (L∞(R,X),‖⋅‖∞) is a Banach space, where ‖⋅‖∞:=esssupt∈R‖⋅‖ denotes the essential supremum norm.
Denote
Z={x|x∈L∞(R,X)∩BCBp(R,X)}. |
Then we have the following lemma which is crucial for the proof of our main result of this paper.
Lemma 2.2. The space (Z,‖⋅‖∞) is a Banach space.
Proof. Let {ϕn}⊂Z be a Cauchy sequence, and then for every ε>0, there is a positive integer N1 such that for n,m>N1,
‖ϕn(⋅)−ϕm(⋅)‖∞<ε3. |
Since {ϕn}⊂Z⊂L∞(R,X) and (L∞(R,X),‖⋅‖∞) is a Banach space, there exists ϕ∈L∞(R,An) such that ϕn→ϕ as n→∞ with respect to the norm ‖⋅‖∞. To complete the proof, it suffices to prove that ϕ∈BCBp(R,X). From limn→∞ϕn=ϕ in regard to the essential supremum norm, it follows that there exists a positive integer N2 such that for n>N2,
‖ϕn(⋅)−ϕ(⋅)‖∞<ε3. |
Now, take N0=max{N1,N2}, and then, due to the fact that ϕN0+1∈BCBp(R,X), there exists a δ=δ(ε)>0 such that for any h∈R with |h|<ε, it holds that
‖ϕN0+1(⋅+h)−ϕN0+1(⋅)‖Bp<ε3. |
Consequently,
‖ϕ(⋅+h)−ϕ(⋅)‖Bp≤‖ϕ(⋅+h)−ϕN0+1(⋅+h)‖Bp+‖ϕN0+1(⋅+h)−ϕN0+1(⋅)‖Bp+‖ϕN0+1(⋅)−ϕ(⋅)‖Bp≤‖ϕ(⋅+h)−ϕN0+1(⋅+h)‖∞+‖ϕN0+1(⋅+h)−ϕN0+1(⋅)‖Bp+‖ϕN0+1(⋅)−ϕ(⋅)‖∞≤ε3+ε3+ε3=ε, |
which implies ϕ∈BCBp(R,X). The proof is completed.
In this section, for x∈L∞(R,A), we denote |x|∞=maxA∈Ω{esssupt∈R|xA(t)|} and for z=(z1,z2,⋯,zn)T=(∑AzA1eA,∑AzA2eA,⋯,∑AzAneA)T∈L∞(R,An), we denote ‖z‖∞=maxi∈J{|zi|∞}. Let Z={z|z∈L∞(R,An)∩BCBp(R,An)}, and then, according to Lemma 2.2, (Z,‖⋅‖∞) is a Banach space. For x∈BpAP(R,A) and z∈BpAP(R,An), we will use |x|Bp and ‖z‖Bp to represent the seminorms of x and z, respectively.
In what follows, we will employ the following symbols:
ˉg=supt∈R‖g(t)‖Yandg_=inft∈R‖g(t)‖Y, |
where g:R→Y is a bounded function and (Y,‖⋅‖Y) is a normed space. Moreover, we will use the following assumptions:
(A1) For i,j,k∈J, functions b∅i∈AP(R,R+) with b_∅i>0, ai,bci,μj,cij,uij,αij,βij,θijk,qijk,νijk∈AP(R,A),τi,σij,ηij,δijk∈AP(R,R)∩C1(R,R+) with τ′i(t)≤ˉτ′i<1,σ′ij(t)≤ˉσ′ij<1,η′ij(t)≤ˉη′ij<1,δ′ijk(t)≤ˉδ′ijk<1, where ˉτ′,ˉσ′,ˉη′,ˉδ′ are constants, and γij,Tij,Sij,Ii∈L∞(R,A)∩BpAP(R,A).
(A2) For all j∈J, functions fj,gj∈C(A,A) with fj(0)=0,gj(0)=0, and there exist positive constants Lfj,Lgj,Mgj, and Mgk such that for any u,v∈A,
|fj(u)−fj(v)|1≤Lfj|u−v|1,|gj(u)−gj(v)|1≤Lgj|u−v|1,|gj(u)|1≤Mgj,|gk(u)|1≤Mgk. |
(A3) For i∈J, there exist positive constants ϑi such that
ρ:=maxi∈J{ˉai+1b_∅i[ˉb∅iˉai+ˉbci+ϑ−1i(n∑j=1ˉcijLfjϑj+n∑j=1ˉuijLfjϑj+n∑i=1n∑k=1ˉθijkLgjMgkϑj+n∑j=1ˉαijLfjϑj+n∑j=1ˉβijLfjϑj+n∑i=1n∑k=1ˉqijkLgjMgkϑj+n∑i=1n∑k=1ˉνijkLgjMgkϑj)]}<1. |
(A4) For the constants ϑi,i∈J, mentioned in (A3), and p,q>1 with 1p+1q=1, it holds that
P:=2p−1maxi∈J{4p−1(ˉai)p11−ˉτ′i+70p−1(1b_∅i)p+qq(ˉaiˉb∅i)peb_∅iˉτi1−ˉτ′i+70p−1(1b_∅i)p+qq(ˉbci)p+35p−1ϑ−pi(1b_∅i)p+qq(n∑j=1(ˉcij)q)pqn∑j=1(Lfjϑj)p+70p−1ϑ−pi(1b_∅i)p+1q(n∑j=1(ˉuij)q)pq×n∑j=1(Lfjϑj)peb_∅iˉσij1−ˉσ′ij+140p−1ϑ−pi(1b_∅i)p+qqn2pq[n∑j=1n∑k=1(ˉθijkMgkLgjϑj)p+n∑j=1n∑k=1(ˉθijkMgjLgkϑk)p]eb_∅iˉδijk1−ˉδ′ijk+70p−1ϑ−pi(1b_∅i)p+qq(n∑j=1(ˉαij)q)pq×n∑j=1(Lfjϑj)peb_∅iˉηij1−ˉη′ij+70p−1ϑ−pi(1b_∅i)p+qq(n∑j=1(ˉβij)q)pqn∑j=1(Lfjϑj)peb_∅iˉηij1−ˉη′ij+140p−1ϑ−pi(1b_∅i)p+qqn2pq(n∑j=1n∑k=1(ˉqijkMgkLgjϑj)p+n∑j=1n∑k=1(ˉqijkMgjLgkϑk)p)×eb_∅iˉδijk1−ˉδ′ijk+140p−1ϑ−pi(1b_∅i)p+qqn2pq(n∑j=1n∑k=1(ˉνijkMgkLgjϑj)p+n∑j=1n∑k=1(ˉνijkMgjLgkϑk)p)epqb_∅iˉδijk1−ˉδ′ijk}<1. |
For i∈J, let yi(t)=ϑ−1ixi(t),Zi(t)=yi(t)−ai(t)yi(t−τi(t)), where ϑi>0 are constants, and then system (1.1) turns into
Z′i(t)=−b∅i(t)Zi(t)−b∅i(t)ai(t)yi(t−τi(t))−bci(t)yi(t)+ϑ−1i[n∑j=1cij(t)fj(ϑjyj(t)+n∑j=1uij(t)fj(ϑjyj(t−σij(t)))+n∑j=1γij(t)μj(t)+n∑j=1n∑k=1θijk(t)gj(ϑjyj(t−δijk(t)))gk(ϑkyk(t−γijk(t)))+n⋀j=1αij(t)fj(ϑjyj(t−ηij(t)))+n⋁j=1βij(t)fj(ϑjyj(t−ηij(t)))+n⋀j=1n⋀k=1qijk(t)gj(ϑjyj(t−δijk(t)))gk(ϑkyk(t−δijk(t)))+n⋁j=1n⋁k=1νijk(t)gj(ϑjyj(t−δijk(t)))gk(ϑkyk(t−δijk(t)))+n⋀j=1Tij(t)μi(t)+n⋁j=1Sij(t)μi(t)+Ii(t)],i∈J. | (3.1) |
Multiplying both sides of (3.1) with e∫tt0b∅i(u)du and integrating over the interval [t0,t], then it holds that
yi(t)=ai(t)yi(t−τi(t))+[yi(t0)−ai(t0)yi(t0−τi(t0))]e−∫tt0b∅i(u)du+∫tt0e−∫tsb∅i(u)du(Ny)i(s)ds,i∈J, | (3.2) |
where
(Ny)i(s)=−b∅i(s)ai(s)yi(s−τi(s))−bci(s)yi(s)+ϑ−1i(n∑j=1cij(s)fj(ϑjyj(s))+n∑j=1uij(s)fj(ϑjyj(s−σij(s)))+n∑j=1γij(s)μj(s)+n∑j=1n∑k=1θijk(s)gj(ϑjyj(s−δijk(s)))gk(ϑkyk(s−δijk(s)))+n⋀j=1αij(s)fj(ϑjyj(s−ηij(s)))+n⋁j=1βij(s)fj(ϑjyj(s−ηij(s)))+n⋀j=1n⋀k=1qijk(s)gj(ϑjyj(s−δijk(s)))gk(ϑkyk(s−δijk(s)))+n⋁j=1n⋁k=1νijk(s)gj(ϑjyj(s−δijk(s)))gk(ϑkyk(s−δijk(s)))+n⋀j=1Tij(s)μi(s)+n⋁j=1Sij(s)μi(s)+Ii(s)). |
It is easy to verify that if y∗(t)=(y∗1(t),y∗2(t),⋯,y∗n(t)) solves system (3.1), then x∗(t)=(x∗1(t),x∗2(t),⋯,x∗n(t))=(ϑ−11y∗1(t),ϑ−12y∗2(t),⋯,ϑ−1ny∗n(t)) solves system (1.1).
Definition 3.1. A function x=(x1,x2,⋯,xn):R→An is called a solution of (1.1) provided that there exist positive numbers ϑi such that yi(t)=ϑ−1ixi(t),i∈J fulfill (3.2).
Set
ˆφ=(ˆφ1(t),ˆφ2(t),⋯,ˆφn(t))T, |
where
ˆφi(t)=∫t−∞e−∫tsb∅i(u)du(n∑j=1γij(s)μj(s)+n⋀j=1Tij(s)μj(s)+n⋁j=1Sij(s)μj(s)+Ii(s))ds,i∈J. |
It is easy to see that ˆφ is well defined under condition (A1). Choose a positive constant r with r>‖ˆφ‖∞.
Then, we are now in a position to present and prove our existence result.
Theorem 3.1. Assume that (A1)−(A4) hold. Then, system (1.1) admits a unique Bp-almost periodic solution in Z∗:={φ|φ∈Z,‖φ−ˆφ‖∞≤ρr1−ρ}.
Proof. Letting t0→∞, from (3.2), one gets
yi(t)=ai(t)yi(t−τi(t))+∫t−∞e−∫tsb∅i(u)du(Ny)i(s)ds,i∈J. |
Define a mapping T:Z∗→Z∗ by setting (Tφ)(t)=((Tφ)1(t),(Tφ)2(t),⋯,(Tφ)n(t))T for φ∈Z and t∈R, where (Tφ)i(t)=ai(t)φi(t−τi(t))+∫t−∞e−∫tsb∅i(u)du(Nφ)i(s)ds,i∈J.
To begin with, we show that T(Z∗)⊂Z∗.
Note that, for any φ∈Z∗, it holds that
‖φ‖∞≤‖φ−ˆφ‖∞+‖ˆφ‖∞≤ρr1−ρ+r=r1−ρ |
and ‖Nφ‖∞<∞.
For every φ∈Z∗, we infer that
‖Tφ−ˆφ‖∞=maxi∈J{esssupt∈R|ai(t)φi(t−τi(t))+∫t−∞e−∫tsb∅i(u)du[−b∅i(s)ai(s)φi(s−τi(s))−bci(s)φi(s)+ϑ−1i(n∑j=1cij(s)fj(ϑjφj(s))+n∑j=1uij(s)fj(ϑjφj(s−σij(s)))+n∑i=1n∑k=1θijk(s)gj(ϑjφj(s−δijk(s)))gk(ϑkφk(s−δijk(s)))+n⋀j=1αij(s)fj(ϑjφj(s−ηij(s)))+n⋁j=1βij(s)fj(ϑjφj(s−ηij(s)))+n⋀j=1n⋀k=1qijk(s)gj(ϑjφj(s−δijk(s)))gk(ϑkφk(s−δijk(s)))+n⋁j=1n⋀k=1νijk(s)gj(ϑjφj(s−δijk(s)))gk(ϑkφk(s−δijk(s))))]ds|1}≤maxi∈J{ˉai‖φ‖∞+∫t−∞e−b_∅i(t−s)[ˉb∅iˉai+ˉbci+ϑ−1i(n∑j=1ˉcijLfjϑj+n∑j=1ˉuijLfjϑj+n∑i=1n∑k=1ˉθijkLgjMgkϑjϑk+n∑j=1ˉαijLfjϑj+n∑j=1ˉβijLfjϑj+n∑j=1n∑k=1ˉqijkLgjMgkϑjϑk+n∑j=1n∑k=1ˉνijkLgjMgkϑjϑk)]‖φ‖∞ds}≤maxi∈J{ˉai+1b_∅i(ˉb∅iˉai+ˉbci+n∑j=1ˉcijLfj+n∑j=1ˉuijLfj+n∑i=1n∑k=1ˉθijkLgjMgk+n∑j=1ˉαijLfj+n∑j=1ˉβijLfj+n∑i=1n∑k=1ˉqijkLgjMgk+n∑i=1n∑k=1ˉνijkLgjMgk)}‖φ‖∞=ρ‖φ‖∞<r1−ρ,i∈J. |
In addition, by condition (A1) and the fact that φ∈Z, we have that for every ε>0, there exists a positive number δ=δ(ε)(<ε) such that for any h∈R with |h|<δ, it holds
|ai(t+h)−ai(t)|1<ε,|φi(⋅+h)−φi(⋅)|Bp<ε,|τi(t+h)−τi(t)|<δ,i∈J. |
Without loss of generality, in the sequel, we assume that h>0, then we deduce that
‖Tφ(⋅+h)−Tφ(⋅)‖pBp≤2p−1maxi∈J{¯liml→∞12l∫l−l|ai(t+h)φi(t+h−τi(t+h))−ai(t)φi(t−τi(t))|p1dt}+2p−1maxi∈J{¯liml→∞12l∫l−l|∫t+h−∞e−∫t+hsb∅i(u)du(Nφ)i(s)ds−∫t−∞e−∫tsb∅i(u)du(Nφ)i(s)ds|p1dt}≤6p−1maxi∈J{¯liml→∞12l∫l−l|ai(t+h)−ai(t)|p1|φi(t+h−τi(t+h))|p1dt}+6p−1maxi∈J{¯liml→∞12l∫l−l|ai(t)|p1|φi(t+h−τi(t+h))−φi(t−τi(t+h))|p1dt}+6p−1maxi∈J{¯liml→∞12l∫l−l|ai(t)|p1|φi(t−τi(t+h))−φi(t−τi(t))|p1dt}+4p−1maxi∈J{¯liml→∞12l∫l−l|∫t−∞|e−∫t+hsb∅i(u)du−e−∫tsb∅i(u)du|(Nφ)i(s)ds|p1dt}+4p−1maxi∈J{¯liml→∞12l∫l−l|∫t+hte−∫t+hsb∅i(u)du(Nφ)i(s)ds|p1dt}≤6p−1maxi∈J{¯liml→∞12l∫l−l|ai(t+h)−ai(t)|p1|φi(t+h−τi(t+h))|p1dt}+6p−1maxi∈J{¯liml→∞12l∫l−l|ai(t)|p1|φi(t+h−τi(t+h))−φi(t−τi(t+h))|p1dt}+6p−1maxi∈J{¯liml→∞12l∫l−l|ai(t)|p1|φi(t−τi(t+h))−φi(t−τi(t))|p1dt}+4p−1maxi∈J{¯liml→∞12l∫l−l|∫t−∞e−b_∅i(t−s)|∫t+hsb∅i(u)du−∫tsb∅i(u)du|(Nφ(s))ids|p1}+4p−1maxi∈J{¯liml→∞12l∫l−l|∫t+hte−∫t+hsb∅i(u)du(Nφ)i(s)ds|p1dt}≤maxi∈J{6p−1εp‖φ‖p∞+6p−1ˉapiεp+6p−1ˉapiεp+4p−1hp[(ˉb∅ib_∅i)p+(1b_∅i)p]‖Nφ‖p∞}≤maxi∈J{6p−1‖φ‖p∞+6p−1ˉai+6p−1ˉai+4p−1[(ˉb∅ib_∅i)p+(1b_∅i)p]‖Nφ‖p∞}εp, |
which implies Tφ∈BCBp(R,An). Therefore, T(Z∗)⊂Z∗.
Next, we will prove that T is a contraction mapping. In fact, for any φ,ψ∈Z∗,i∈J, we have
‖Tφ−Tψ‖∞≤esssupt∈R|ai(t)(φi(t−τi(t))−ψi(t−τi(t)))|1+esssupt∈R|∫t−∞e−∫tsb∅i(u)du[−b∅i(s)ai(s)(φi(s−τi(s))−ψi(s−τi(s)))−bci(s)(φi(s)−ψi(s))+ϑ−1i(n∑j=1cij(s)(fj(ϑjφj(s))−fj(ϑjψj(s)))+n∑j=1uij(s)(fj(ϑjφj(s−σij(s)))−fj(ϑjψj(s−σij(s))))+n∑i=1n∑k=1θijk(s)(gj(ϑjφj(s−δijk(s)))gk(ϑkφk(s−δijk(s)))−gj(ϑjψj(s−δijk(s))))gk(ϑkψk(s−δijk(s))))+n∑j=1αij(s)(fj(ϑjφj(s−ηij(s)))−fj(ϑjψj(s−ηij(s))))+n∑j=1βij(s)(fj(ϑjφj(s−ηij(s)))−fj(ϑjψj(s−ηij(s))))+n∑i=1n∑k=1qijk(s)(gj(ϑjφj(s−δijk(s)))gk(ϑkφk(s−δijk(s)))−gj(ϑjψj(s−δijk(s))))gk(ϑkψk(s−δijk(s))))+n∑i=1n∑k=1νijk(s)(gj(ϑjφj(s−δijk(s)))gk(ϑkφk(s−δijk(s)))−gj(ϑjψj(s−δijk(s))))gk(ϑkψk(s−δijk(s)))]ds≤ˉai‖φ−ψ‖∞+∫t−∞eb_∅i(t−s)[ˉb∅iˉai+ˉbci+ϑ−1i(n∑j=1ˉcijLfjϑj+n∑j=1ˉuijLfjϑj+n∑j=1n∑k=1ˉθijkLgjMgkϑj+n∑j=1ˉαijLfjϑj+n∑j=1ˉβijLfjϑj+n∑j=1n∑k=1ˉqijkLgjMgkϑj+n∑j=1n∑k=1ˉνijkLgjMgkϑj)]‖φ−ψ‖∞ds≤maxi∈J{ˉai+1b_∅i[ˉb∅iˉai+ˉbci+ϑ−1i(n∑j=1ˉcijLfjϑj+n∑j=1ˉuijLfjϑj+n∑i=1n∑k=1ˉθijkLgjMgkϑj+n∑j=1ˉαijLfjϑj+n∑j=1ˉβijLfjϑj+n∑i=1n∑k=1ˉqijkLgjMgkϑj+n∑i=1n∑k=1ˉνijkLgjMgkϑj)]}‖φ−ψ‖∞=ρ‖φ−ψ‖∞,i∈J, |
which combined with condition (A3) means that T is a contraction mapping. Thereupon, T has a unique fixed point φ∗∈Z∗.
Finally, we will examine that φ∗ is Bp-almost periodic.
Since φ∗∈Z∗⊂BCBp(R,An), for any ε>0, there exists a σ>0(σ<ε) such that, for any ℏ∈R with |ℏ|<σ,
|φ∗i(⋅+ℏ)−φ∗i(⋅)|Bp<ε,i∈J. |
Based on this and (A1), there exists ♭ such that, for all i\in\mathcal{J} ,
\begin{align} |a_{i}(\cdot+{\flat} )-a_{i}(\cdot)|_{1} < \varepsilon, \, \, \, \, |b_{i}^{\emptyset }(\cdot+{\flat})-b_{i}^{\emptyset }(\cdot)| < \varepsilon, \, \, \, \, |b_{i}^{c }(\cdot+{\flat})-b_{i}^{c}(\cdot)|_{1} < \varepsilon, \end{align} | (3.3) |
\begin{align} |c_{ij}(\cdot+{\flat} )-c_{ij}(\cdot)|_{1} < \varepsilon, \, \, \, \, |u_{ij}(\cdot+{\flat} )-u_{ij}(\cdot)|_{1} < \varepsilon, \, \, \, \, |\gamma_{ij}(\cdot+{\flat} )-\gamma_{ij}(\cdot)|_{B^{p}} < \varepsilon, \end{align} | (3.4) |
\begin{align} |\theta _{ijk}(\cdot+{\flat} )-\theta _{ijk}(\cdot)|_{1} < \varepsilon, \, \, \, \, |\alpha _{ij}(\cdot+{\flat} )-\alpha _{ij}(\cdot)|_{1} < \varepsilon, \, \, \, \, |\beta_{ij}(\cdot+{\flat} )-\beta_{ij}(\cdot)|_{1} < \varepsilon, \end{align} | (3.5) |
\begin{align} |q_{ijk}(\cdot+{\flat} )-q_{ijk}(\cdot)|_{1} < \varepsilon, \, \, \, \, |\nu_{ijk}(\cdot+{\flat} )-\nu_{ijk}(\cdot)|_{1} < \varepsilon, \, \, \, \, |T_{ij}(\cdot+{\flat} )-T_{ij}(\cdot)|_{B^{p}} < \varepsilon, \end{align} | (3.6) |
\begin{align} |\tau_i(t+\flat)-\tau(t)| < \varepsilon, \, \, |\sigma_{ij}(t+\flat)-\sigma_{ij}(t)| < \varepsilon, \, \, |\delta_{ijk}(t+\flat)-\delta_{ijk}(t)| < \varepsilon, \end{align} | (3.7) |
\begin{align} |\eta_{ij}(t+\flat)-\eta_{ij}(t)| < \varepsilon, \, \, |S_{ij}(\cdot+{\flat} )-S_{ij}(\cdot)|_{B^{p}} < \varepsilon, \, \, \, \, |I_{i}(\cdot+{\flat} )-I_{i}(\cdot)|_{B^{p}} < \varepsilon, \end{align} | (3.8) |
\begin{align} |\varphi_{i}^{*}(\cdot-\tau_{i}(\cdot+{\flat}))-\varphi_{i}^{*}(\cdot-\tau_{i}(\cdot))|_{B^p} < \varepsilon, \, \, \, \, |\varphi_{j}^{*}(\cdot-\sigma_{ij}(\cdot+{\flat}))-\varphi_{j}^{*}(\cdot-\sigma_{ij}(\cdot))|_{B^p} < \varepsilon, \end{align} | (3.9) |
\begin{align} |\varphi_{i}^{*}(\cdot-\eta_{ij}(\cdot+{\flat}))-\varphi_{i}^{*}(\cdot-\eta_{ij}(\cdot))|_{B^p} < \varepsilon, \, \, \, \, |\varphi_{j}^{*}(\cdot-\delta_{ijk}(\cdot+{\flat}))-\varphi_{j}^{*}(\cdot-\delta_{ijk}(\cdot))|_{B^p} < \varepsilon, \end{align} | (3.10) |
\begin{align} |\varphi_{k}^{*}(\cdot-\delta_{ijk}(\cdot+{\flat}))-\varphi_{k}^{*}(\cdot-\delta_{ijk}(\cdot))|_{B^p} < \varepsilon. \end{align} | (3.11) |
Then, we deduce that
\begin{align*} & \|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^p}^p \\ \leq\, & 2^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l | a_i(t+{\flat}) \varphi_i^*(t+{\flat}-\tau_i(t+{\flat}))-a_i(t) \varphi_i^*(t-\tau_i(t))|_1 ^p d t\bigg\} \\ & +2^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg|\int_{-\infty}^{t+{\flat}} e^{-\int_s^{t+{\flat}} b_i^{\emptyset}(u+{\flat}) d u}(N^{\varphi^*}) i(s) d s \\ & -\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u) d u}(N^{\varphi^*})_i(s) d s\bigg|_1 ^p d t\bigg\} \\ \leq\, & 4^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l|a_i(t+{\flat})(\varphi_i^*(t+{\flat}-\tau_i(t+{\flat}))-\varphi_i^*(t-\tau_i(t)))|_1^p d t\bigg\} \\ & +4^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l|(a_i(t+{\flat})-a_i(t)) \varphi_i^*(t-\tau_i(t))|_1^p d t\bigg\} \\ & +2^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} (N^{\varphi^*})_i(s+{\flat}) d s \\ &-\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u) d u}(N^{\varphi^*})_i(s) d s\bigg|_1 ^p d t\bigg\} \\ \leq\, & 4^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^{l}|a_i(t+{\flat})( \varphi_i^*(t+{\flat}-\tau_i(t+{\flat}))-\varphi_i^*(t-\tau_i(t)))|_1 ^p d t\bigg\} \\ & +4^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^{l}|(a_i(t+{\flat})-a_i(t)) \varphi_i^*(t-\tau_i(t))|_1^p d t\bigg\} \\ &+70^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^{l}\bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} a_i(s+{\flat}) b_i^{\emptyset}(s+{\flat})\\ &\times(\varphi_i^*(s+{\flat}-\tau_i(s+{\flat}))-\varphi_i^*(s-\tau_i(s))) d s\bigg|_1^p d t\bigg\}\\ & +70^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg|\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}(a_i(s+{\flat})-a_i(s)) b_i^{\emptyset}(s+{\flat})\\ &\times\varphi_i^*(s-\tau_i(s)) d s\bigg|_1 ^p d t\bigg\} \\ & +70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} a_i(s)(b_i^{\emptyset}(s+{\flat})\\ &-b_i^{\emptyset}(s))\varphi_i^*(s-\tau_i(s)) d s\bigg|_1 ^p d t\bigg\} \\ &+70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-1}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset} (u) du}\Big|\\ &\times a_i(s) b_i^{\emptyset}(s) \varphi_i^*(s-\tau_{i}(s)) d s\bigg|_1 ^p d t\bigg\} \\ & +70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}(b_i^c(s+{\flat}) \varphi_i^*(s+{\flat})-b_i^c(s) \varphi_i^*(s))d s\bigg|_1 ^p d t\bigg\} \\ & +70^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|b_i^c(s) \varphi_i^*(s) d s\bigg|_1 ^p d t \bigg\}\\ \end{align*} |
\begin{align*} &\;\;\;\;+70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \sum\limits_{j = 1}^n c_{i j}(s+{\flat})(f_j(\vartheta_j \varphi_j^*(s+{\flat}))\\ &\;\;\;\;-f_j(\vartheta_j \varphi_i^*(s)))d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \sum\limits_{j = 1}^n(c_{i j}(s+{\flat})\\ &\;\;\;\;-c_{i j}(s))f_j(\vartheta_j \varphi_j^*(s)) d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big| \\ &\;\;\;\;\times\sum\limits_{j = 1}^n c_{i j}(s) f_j(\vartheta_j \varphi_j^*(s)) d s\bigg|_1 ^p d t\bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg|\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \sum\limits_{j = 1}^n u_{i j}(s+{\flat})\\ &\;\;\;\;\times(f_j(\vartheta_j \varphi_j^*(s+{\flat}-\sigma_{i j}(s+{\flat})))-f_j(\vartheta_j \varphi_j^*(s-\sigma_{i j}(s)))) d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \sum\limits_{j = 1}^n(u_{i j}(s+{\flat})-u_{i j}(s))\\ &\;\;\;\;\times f_j(\vartheta_j \varphi_j^*(s-\sigma_{i j}(s))) d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|\\ &\;\;\;\;\times\sum\limits_{j = 1}^n u_{i j}(s) f_j(\vartheta_j \varphi_j^*(s-\sigma_{i j}(s))) d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \bigg(\sum\limits_{j = 1}^n \gamma_{i j}(s+{\flat}) \mu_j(s+{\flat}) \\ &\;\;\;\;-\sum\limits_{j = 1}^n \gamma_{i j}(s) \mu_j(s)\bigg)d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t \Big|e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} -e^{-\int_{s}^{t}b_{i}^{\emptyset}(u)du}\Big|\sum\limits_{j = 1}^n \gamma_{i j}(s) \mu_j(s)d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg |\int_{-\infty}^t e^{-\int _s^t b_i^{\emptyset}( u+{\flat}) d u} \sum\limits_{j = 1}^n \sum\limits_{k = 1}^n \theta_{i j k}(s+{\flat})\\ &\;\;\;\;\times(g_j(\vartheta_j \varphi_j^*(s+{\flat}-\delta_{i j k}(s+{\flat}))) g_k ( \vartheta_k \varphi_k^*(s+{\flat}-\delta_{i j k}(s+{\flat}))) \\ &\;\;\;\; -g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s )))) d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg|\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \sum\limits_{j = 1}^n \sum\limits_{k = 1}^n(\theta_{i j k}(s+{\flat})\\ &\;\;\;\; -\theta_{i j k}(s)) g_j(\vartheta_j \varphi_j^*(s-\delta_{ijk} k(s))) g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s) )) d s\bigg|_1 ^p d t\bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big| \\ &\;\;\;\; \times \sum\limits_{j = 1}^n \sum\limits_{k = 1}^n \theta_{i j k}(s) g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s))) d s\bigg|_1 ^p d t\bigg\} \\ \end{align*} |
\begin{align*} &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \sum\limits_{j = 1}^n \alpha_{i j}(s+{\flat})\\ &\;\;\;\;\times(f _ { j } (\vartheta _ { j } \varphi _ { j } ^ { * } (s+{\flat}-\eta_{i j}(s+{\flat})))-f_j(\vartheta_j \varphi_j^* ( s-\eta_{ij}(s))))d s\bigg|_1 ^p d t \bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset} ( u+{\flat}) d u} \sum\limits_{j = 1}^n(\alpha_{i j} ( s+{\flat})-\alpha_{i j}(s)) \\ &\;\;\;\;\times f_j(\vartheta_{j}\varphi_j^*(s-\eta_{j j}(s))) d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|\\ &\;\;\;\;\times\sum\limits_{j = 1}^n \alpha_{i j}(s) f_j(\vartheta_{j}\varphi_j^*(s-\eta_{j j}(s))) d s\bigg|_1^p d t\bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \sum\limits_{j = 1}^n \beta_{i j}(s+{\flat})\\ &\;\;\;\;\times(f _ { j } (\vartheta _ { j } \varphi _ { j } ^ { * } (s+{\flat}-\eta_{i j}(s+{\flat})))-f_j(\vartheta_j \varphi_j^* ( s-\eta_{ij}(s))))d s\bigg|_1 ^p d t \bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset} ( u+{\flat}) d u} \sum\limits_{j = 1}^n(\beta_{i j} ( s+{\flat})-\beta_{i j}(s)) \\ &\;\;\;\;\times f_j(\vartheta_{j}\varphi_j^*(s-\eta_{j j}(s))) d s\bigg|_1 ^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|\\ &\;\;\;\;\times\sum\limits_{j = 1}^n \beta_{i j}(s) f_j(\vartheta_{j}\varphi_j^*(s-\eta_{j j}(s))) d s\bigg|_1^p d t\bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset} ( u+{\flat}) d u} \sum\limits_{j = 1}^n \sum\limits_{k = 1}^n q_{i j k}(s+{\flat})\\ &\;\;\;\;\times(g _ { j } (\vartheta_j \varphi_j^*(s+{\flat}-\delta_{i j k}(s+{\flat})) g_k(\vartheta_k \varphi_k^*(s+{\flat}-\delta_{i j k}(s + {\flat})))\\ &\;\;\;\;-g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k (\vartheta_k \varphi_k^*(s-\delta_{i j k}(s)))) d s\bigg|_1^p d t\bigg\} \\ \end{align*} |
\begin{align*} &\;\;\;\;+70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_{i}^{\emptyset}(u+{\flat}) d u}\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n(q_{i j k}(s+{\flat})\\ &\;\;\;\;-q_{i j k}(s)) g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s))) d s\bigg|_1^p d t\bigg\} \\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in \mathcal{J}}\bigg\{\limsup _ { l \rightarrow \infty }(2l)^{-1} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|\\ &\;\;\;\;\times \sum\limits_{j = 1}^n \sum\limits_{k = 1}^n q_{i j k}(s)g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s))) d s\bigg|_1 ^p d t\bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset} ( u+{\flat}) d u} \sum\limits_{j = 1}^n \sum\limits_{k = 1}^n \nu_{i j k}(s+{\flat})\\ &\;\;\;\;\times(g _ { j } (\vartheta_j \varphi_j^*(s+{\flat}-\delta_{i j k}(s+{\flat})) g_k(\vartheta_k \varphi_k^*(s+{\flat}-\delta_{i j k}(s + {\flat})))\\ &\;\;\;\;-g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k (\vartheta_k \varphi_k^*(s-\delta_{i j k}(s)))) d s\bigg|_1^p d t\bigg\} \\ &\;\;\;\;+70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg| \int_{-\infty}^t e^{-\int_s^t b_{i}^{\emptyset}(u+{\flat}) d u}\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n(\nu_{i j k}(s+{\flat})\\ &\;\;\;\;-\nu_{i j k}(s)) g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s))) d s\bigg|_1^p d t\bigg\} \\ &\;\;\;\;+70^{p-1} \vartheta_i^{-p} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|\\ &\;\;\;\;\times \sum\limits_{j = 1}^n \sum\limits_{k = 1}^n \nu_{i j k}(s)g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s))) g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s))) d s\bigg|_1 ^p d t\bigg\}\\ \end{align*} |
\begin{align*} &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \bigg(\sum\limits_{j = 1}^n T_{i j}(s+{\flat}) \mu_j(s+{\flat}) \\ &\;\;\;\;-\sum\limits_{j = 1}^n T_{i j}(s) \mu_j(s)\bigg)d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t \Big|e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} -e^{-\int_{s}^{t}b_{i}^{\emptyset}(u)du}\Big|\sum\limits_{j = 1}^n T _{i j}(s) \mu_j(s)d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \bigg(\sum\limits_{j = 1}^n S_{i j}(s+{\flat}) \mu_j(s+{\flat}) \\ &\;\;\;\;-\sum\limits_{j = 1}^n S_{i j}(s) \mu_j(s)\bigg)d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t \Big|e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} -e^{-\int_{s}^{t}b_{i}^{\emptyset}(u)du}\Big|\sum\limits_{j = 1}^n S _{i j}(s) \mu_j(s)d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} (I_{i}(s+{\flat})- I_{i}(s))d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\; +70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg|\int_{-\infty}^t \Big|e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} -e^{-\int_{s}^{t}b_{i}^{\emptyset}(u)du}\Big| I _{i}(s) d s\bigg|_1 ^p d t \bigg\}\\ &\;\;\;\;: = \sum\limits_{i = 1}^{37}K_{i}. \end{align*} |
Furthermore, based on inequalities (3.3)–(3.11), the Hölder inequality, and the Fubini theorem, we can deduce that
\begin{align*} K_{1}\leq\, &8^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \bigg[\int_{-l}^{l}|a_i(t+{\flat})|^{p}(| \varphi_i^*(t+{\flat}-\tau_i(t+{\flat}))\\ &-\varphi_{i}^{*}(t-\tau_{i}(t+{\flat}))|_{1}^{p}+|\varphi_{i}^{*}(t-\tau_{i}(t+{\flat}))-\varphi_i^*(t-\tau_i(t))|_{1}^{p}) d t\bigg\}\\ \leq\, &8^{p-1} \max _{i \in \mathcal{J}}\bigg\{(\bar{a}_{i})^{p}\frac{1}{1-\bar{\tau^{\prime}}_{i}}| \varphi_i^*(t+{\flat})-\varphi_{i}^{*}(t)|_{B^{p}}^{p}+(\bar{a}_{i})^{p}\varepsilon^{p}\bigg\}, \\ K_{2}\leq\, &4^{p-1} \max _{i \in \mathcal{J}}\{\|\varphi^*\|_{\infty}^{p}\varepsilon^{p}\}, \\ K_{3} \leq\, &140^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^{l}\bigg[\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}ds\bigg]^{\frac{p}{q}}\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}\\ &\times|a_i(s+{\flat})b_i^{\emptyset}(s+{\flat})|_1(|\varphi_i^*(s+{\flat}-\tau_i(s+{\flat}))-\varphi_i^*(s-\tau_i(s+{\flat}))|_1^p\\ &+|\varphi_i^*(s-\tau_i(s+{\flat}))-\varphi_i^*(s-\tau_i(s))|_{1}^{p}) d s d t\bigg\}\\ \leq\, &140^{p-1}\max\limits_{i\in\mathcal{J}}\bigg\{\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p}{q}}(\bar{a}_{i}\bar{b}_{i}^{\emptyset})^{p}\bigg[\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^{l}\int_{-\infty}^{t-\tau_{i}(t+{\flat})}\frac{e^{-\underline{b}_{i}^{\emptyset}(t-s-\bar{\tau}_{i})}}{1-\bar{\tau}_{i}^{\prime}}\\ &\times|\varphi_{i}^{*}(s+{\flat})-\varphi_{i}^{*}(s)|_{1}^{p}dsdt\bigg]+\frac{\varepsilon^{p}}{\underline{b}_{i}^{\emptyset}}\bigg\}\\ \leq\, &140^{p-1}\max\limits_{i\in\mathcal{J}}\bigg\{\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p}{q}}(\bar{a}_{i}\bar{b}_{i}^{\emptyset})^{p} \bigg[\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\tau}_{i}}}{1-\bar{\tau^{\prime}}_{i}}\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^{l}\int_{-\infty}^{t}e^{-\underline{b}_{i}^{\emptyset}(t-s)}\\ &\times|\varphi_{i}^{*}(s+{\flat})-\varphi_{i}^{*}(s)|_{1}^{p}dsdt\bigg]+\frac{\varepsilon^{p}}{\underline{b}_{i}^{\emptyset}}\bigg\}\\ \leq\, &140^{p-1}\max\limits_{i\in\mathcal{J}}\bigg\{\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p}{q}}(\bar{a}_{i}\bar{b}_{i}^{\emptyset})^{p} \bigg[\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\tau}_{i}}}{1-\bar{\tau^{\prime}}_{i}}\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-\infty}^{l}(2l)^{-1}\int_{s-2l}^{s}e^{-\underline{b}_{i}^{\emptyset}(l-s)}\\ &\times|\varphi_{i}^{*}(t+{\flat})-\varphi_{i}^{*}(t)|_{1}^{p}dtds\bigg]+\frac{\varepsilon^{p}}{\underline{b}_{i}^{\emptyset}}\bigg\}\\ \leq\, &140^{p-1}\max\limits_{i\in\mathcal{J}}\bigg\{\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}(\bar{a}_{i}\bar{b}_{i}^{\emptyset})^{p} \bigg(\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\tau}_{i}}}{1-\bar{\tau^{\prime}}_{i}}\|\varphi^{*}(t+{\flat})-\varphi^{*}(t)\|_{B^{p}}^{p}+\varepsilon^{p}\bigg)\bigg\}, \\ K_{4}\leq\, &70^{p-1} \max _{i \in \mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg(\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}ds\bigg)^{\frac{p}{q}}\\ &\times\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}|(a_i(s+{\flat})-a_i(s)) b_i^{\emptyset}(s+{\flat})\varphi_i^*(s-\tau_i(s)) |_{1}^{p}d s d t\bigg\} \\ \leq\, & 70^{p-1} \max _{i \in \mathcal{J}}\bigg\{\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}(\bar{b}_{i}^{\emptyset})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{p}\bigg\}.\\ K_{5}\leq\, &70^{p-1} \max _{i \in \mathcal{J}}\bigg\{\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}(\bar{a}_{i})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{p}\bigg\}, \\ K_{6} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-1}^l\bigg[\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset} (u) du}\Big|ds\bigg]^{\frac{p}{q}}\\ &\times \int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset} (u) du}\Big||a_i(s) b_i^{\emptyset}(s) \varphi_i^*(s-\tau_{i}(s)) |_1 ^p d s d t\bigg\} \\ \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} (\bar{a}_{i}\bar{b}_{i}^{\emptyset})^{p}\|\varphi^*\|^{p}_{\infty}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{7} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg[\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}ds\bigg]^{\frac{p}{q}}\\ &\times\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}|b_i^c(s+{\flat}) \varphi_i^*(s+{\flat})-b_i^c(s) \varphi_i^*(s)|_{1}^{p}d sd t\bigg\} \\ \leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{\bigg( \frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p}{q}}\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-\infty}^l e^{-\underline{b}_i^{\emptyset}(l-s) }\int_{s-2l}^{s}((\bar{b}_i^c)^{p} \\ \end{align*} |
\begin{align*} &\times|\varphi_i^*(t+{\flat})-\varphi_i^*(t)|_{1}^{p}+|b_i^c(t+{\flat})-b_i^c(t)|_{1}^{p}\|\varphi^*\|_\infty^{p}) d td s\bigg\} \\ \leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{\bigg( \frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}((\bar{b}_i^c)^{p} \|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p}+\|\varphi^*\|_{\infty}^{p}\varepsilon^{p})\bigg\}, \\ K_{8}\leq\, & 70^{p-1} \max _{i \in\mathcal{J}}\bigg\{(\frac{1}{\underline{b}_{i}^{\emptyset}})^{\frac{2(p+q)}{q}} (\bar{b}_{i}^{\emptyset})^{p}\|\varphi^*\|^{p}_{\infty}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{9}\leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg[ \int_{-\infty}^te^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}ds \bigg]^{\frac{p}{q}}\\ &\times\int_{-\infty}^t e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} \bigg|\sum\limits_{j = 1}^n c_{i j}(s+{\flat})(f_j(\vartheta_j \varphi_j^*(s+{\flat}))-f_j(\vartheta_j \varphi_i^*(s)))\bigg|_1 ^pd sd t\bigg\} \\ \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{ \vartheta_i^{-p} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p}{q}}\bigg(\sum\limits_{j = 1}^{n}(\bar{c}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-\infty}^l e^{- \underline{b}_i^{\emptyset}(l-s)}\\ &\times\int_{s-2l}^{s} |\varphi_j^*(t+{\flat}))- \varphi_i^*(t)|_1 ^p d t d s\bigg\} \\ \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{ \vartheta_i^{-p} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg(\sum\limits_{j = 1}^{n}(\bar{c}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*(t+{\flat}))- \varphi^*(t)\|_{B^{p}} ^p\bigg\}, \\ K_{10}\leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{p}\bigg\}, \\ K_{11}\leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{c}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{12} \leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+1}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{u}_{ij})^{q}\bigg)^{\frac{p}{q}} \sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\\ &\times\bigg[\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\sigma}_{ij}}}{1-\bar{\sigma^{\prime}}_{ij}}\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p}+\varepsilon^{p}\bigg]\bigg\}, \\ \end{align*} |
\begin{align*} K_{13} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}n^{\frac{p}{q}}\|\varphi^*\|_{\infty}^{p}\varepsilon^{p}\bigg\}, \\ K_{14}\leq\, &70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg[\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|^{\frac{q}{p}}ds\bigg]^{\frac{p}{q}}\\ &\times\int_{-\infty}^t\Big| e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u}-e^{-\int_s^t b_i^{\emptyset}(u) d u} \Big|^{\frac{p}{q}}\bigg|\sum\limits_{j = 1}^n u_{i j}(s) f_j(\vartheta_j \varphi_j^*(s-\sigma_{i j}(s))) \bigg|_1 ^pd s d t\bigg\} \\ \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}}\bigg(\sum\limits_{j = 1}^{n}(\bar{u}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{15} \leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{ \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg[\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{p}+\sum\limits_{j = 1}^{n}|\gamma_{ij}|^{p}_\infty\bigg]\varepsilon^{p} \bigg\}, \\ K_{16}\leq\, &70^{p-1} \vartheta_i^{-p} \max _{i \in\mathcal{J}}\bigg\{\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l\bigg[\int_{-\infty}^t \Big|e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} -e^{-\int_{s}^{t}b_{i}^{\emptyset}(u)du}\Big|^{\frac{q}{p}}ds\bigg]^{\frac{p}{q}}\\ &\times\int_{-\infty}^t \Big|e^{-\int_s^t b_i^{\emptyset}(u+{\flat}) d u} -e^{-\int_{s}^{t}b_{i}^{\emptyset}(u)du}\Big|^{\frac{p}{q}}\bigg|\sum\limits_{j = 1}^n \gamma_{i j}(s) \mu_j(s)\bigg|_1 ^pd s d t \bigg\}\\ \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}|\gamma_{ij}|^{q}_\infty\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{17}\leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\overline{\lim\limits_{l\to\infty}}{ \frac{1}{2l}} \int_{-l}^l \bigg(\int_{-\infty}^t e^{-\int _s^t b_i^{\emptyset}( u+{\flat}) d u} ds\bigg)^{\frac{p}{q}}\int_{-\infty}^t e^{-\int _s^t b_i^{\emptyset}( u+{\flat}) d u} \\ &\times\bigg(\sum\limits_{j = 1}^n\sum\limits_{k = 1}^n \bar{\theta}_{i j k}(M^g_k|g_j(\vartheta_j \varphi_j^*(s+{\flat}-\delta_{i j k}(s+{\flat})))-g_j(\vartheta_j \varphi_j^*(s-\delta_{i j k}(s)))|_{1} \\ &+ M_j^g|g_k ( \vartheta_k \varphi_k^*(s+{\flat}-\delta_{i j k}(s+{\flat})))- g_k(\vartheta_k \varphi_k^*(s-\delta_{i j k}(s )))) |_{1})d sd t\bigg)^p\bigg\} \\ \leq\, &280^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\bigg[\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n( \bar{\theta}_{i j k}M_k^gL_j^g\vartheta_j)^p+\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{\theta}_{ijk}M_j^gL_{k}^{g}\vartheta_{k})^{p}\bigg]\\ &\times\bigg(\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\delta}_{ijk}}}{1-\bar{\delta^{\prime}}_{ijk}}\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p}+\varepsilon^{p}\bigg)\bigg\}, \\ K_{18} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\sum\limits_{k = 1}^{n} \sum\limits_{j = 1}^{n}(M_j^gM_{k}^{g})^{p}\varepsilon^{p}\bigg\}, \\ K_{19}\leq\, &70^{p-1}\max\limits_{i\in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{p}}n^{\frac{2p}{q}} \sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{\theta}_{ijk}M_j^gM_k^g)^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{20}\leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg(\sum\limits_{j = 1}^n( \bar{\alpha}_{i j})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p} \\ &\times\bigg[\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\eta}_{ij}}}{1-\bar{\eta^{\prime}}_{ij}}\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p}+\varepsilon^{p}\bigg]\bigg\}, \\ K_{21} \leq\, &70^{p-1} \max _{i \in \mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{p}\bigg\}, \\ \end{align*} |
\begin{align*} K_{22} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{\alpha}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{23} \leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg(\sum\limits_{j = 1}^n (\bar{\beta}_{i j})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\\ &\times\bigg[\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\eta}_{ij}}}{1-\bar{\eta^{\prime}}_{ij}}\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p} + \varepsilon^{p}\bigg] \bigg\}, \\ K_{24} \leq\, &70^{p-1} \max _{i \in \mathcal{J}}\bigg\{ \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{p}\bigg\}, \\ K_{25} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{\beta}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{26} \leq\, &280^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\bigg(\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n(\bar{q}_{ijk}M_k^gL_{j}^{g}\vartheta_{j})^{p}+\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n(\bar{q}_{ijk}M_j^gL_k^g\vartheta_k)^p\bigg)\\ &\times\bigg(\frac{e^{\underline{b}_{i}^{\emptyset}\bar{\delta}_{ijk}}}{1-\bar{\delta}_{ijk}^{\prime}}\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p}+\varepsilon^{p}\bigg)\bigg\}, \\ K_{27} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\sum\limits_{k = 1}^{n} \sum\limits_{j = 1}^{n}(M_j^gM_{k}^{g})^{p}\varepsilon^{p}\bigg\}, \\ K_{28} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}(\frac{1}{\underline{b}_{i}^{\emptyset}})^{\frac{2(p+q)}{p}}n^{\frac{2p}{q}}\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{q}_{ijk}M_j^gM_k^g)^p\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{29} \leq\, &280^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\bigg(\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n( \bar{\nu}_{ijk}M_{k}^{g}L_{j}^{g}\vartheta_{j})^{p}+\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n( \bar{\nu}_{ijk}M_{j}^{g}L_{k}^{g}\vartheta_{k})^{p}\bigg)\\ &\times\bigg(\frac{e^{\frac{p}{q}\underline{b}_{i}^{\emptyset}\bar{\delta}_{ijk}}}{1-\bar{\delta^{\prime}}_{ijk}}\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p}+\varepsilon^{p}\bigg)\bigg\}, \\ K_{30} \leq\, &70^{p-1}\max\limits_{i\in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}} n^{\frac{2p}{q}}\sum\limits_{k = 1}^{n}\sum\limits_{j = 1}^{n}(M_j^gM_k^g )^p\varepsilon^{p}\bigg\}, \\ K_{31}\leq\, & 70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{p}}n^{\frac{2p}{q}}\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{\nu}_{ijk}M_j^gM_k^g)^p\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{32} \leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg[\bigg(\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{q}\bigg)^{\frac{p}{q}} +\bigg(\sum\limits_{j = 1}^{n}(|T_{ij}|_\infty)^{q}\bigg)^{\frac{p}{q}}\bigg]\varepsilon^{p} \bigg\}, \\ K_{33} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(|T_{ij}|_\infty\bar{\mu}_{j})^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{34} \leq\, &140^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+1}{q}}\bigg[\bigg(\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{q}\bigg)^{\frac{p}{q}} +\bigg(\sum\limits_{j = 1}^{n}(|S_{ij}|_\infty)^{q}\bigg)^{\frac{p}{q}}\bigg]\varepsilon^{p} \bigg\}, \\ K_{35} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{ \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(|S_{ij}|_\infty\bar{\mu}_{j})^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}, \\ K_{36} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\varepsilon^{p} \bigg\}, \\ K_{37} \leq\, &70^{p-1} \max _{i \in\mathcal{J}}\bigg\{\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}}|I_{i}|_\infty^{p}\varepsilon^{\frac{p+q}{q}}\bigg\}. \end{align*} |
From the above estimates, it follows that
\begin{align} &\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p} \leq P\|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p}+Q\varepsilon^{p}, \end{align} | (3.12) |
where P is defined in condition (A_4) and
\begin{align*} Q = \, & 2^{p-1}\max _{i \in \mathcal{J}}\bigg\{4^{p-1}(\bar{a}_{i})^{p}\varepsilon^{p-1}+2^{p-1} \bigg( \frac{r}{1-\rho}\bigg)^p\varepsilon^{p-1}+70^{p-1}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}(\bar{a}_{i}\bar{b}_{i}^{\emptyset})^{p}\varepsilon^{p-1}\\ &+35^{p-1} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}(\bar{b}_{i}^{\emptyset})^{p}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{p-1}+35^{p-1} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}(\bar{a}_{i})^{p}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{p-1}\\ &+35^{p-1} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} (\bar{a}_{i}\bar{b}_{i}^{\emptyset})^{p}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{\frac{p}{q}}+70^{p-1} \bigg( \frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{p-1}\\ &+35^{p-1} (\frac{1}{\underline{b}_{i}^{\emptyset}})^{\frac{2(p+q)}{q}} (\bar{b}_{i}^{\emptyset})^{p}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{\frac{p}{q}}+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\\ &\times\bigg(\frac{r}{1-\rho}\bigg)^{p} \varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{c}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{\frac{p}{q}}\\ &+ 70^{p-1}\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+1}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{u}_{ij})^{q}\bigg)^{\frac{p}{q}} \sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\\ &\times \sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}n^{\frac{p}{q}}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{u}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\\ &\times \bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{\frac{p}{q}}+70^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg[\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{p}+\sum\limits_{j = 1}^{n}|\gamma_{ij}|^{p}_\infty\bigg]\varepsilon^{p-1} \\ &+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}|\gamma_{ij}|^{q}_\infty\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{p}\varepsilon^{\frac{p}{q}}+140^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\\ &\times\bigg[\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n( \bar{\theta}_{i j k}M_k^gL_j^g\vartheta_j)^p+\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{\theta}_{ijk}M_j^gL_{k}^{g}\vartheta_{k})^{p}\bigg]\varepsilon^{p-1}\bigg\} +35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\\ &\times n^{\frac{2p}{q}}\sum\limits_{k = 1}^{n} \sum\limits_{j = 1}^{n}(M_j^gM_{k}^{g})^{p}\varepsilon^{p-1}+35^{p-1}\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{p}}n^{\frac{2p}{q}} \sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{\theta}_{ijk}M_j^gM_k^g)^{p}\varepsilon^{\frac{p}{q}}\\ &+70^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg(\sum\limits_{j = 1}^n( \bar{\alpha}_{i j})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\varepsilon^{p-1}\bigg]+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\\ &\times n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} \bigg(\sum\limits_{j = 1}^{n}(\bar{\alpha}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\\ &\times \bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{\frac{p}{q}}+700^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg(\sum\limits_{j = 1}^n (\bar{\beta}_{i j})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\varepsilon^{p-1} \end{align*} |
\begin{align*} &\;\;\;\;\;\;+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\bigg(\frac{r}{1-\rho}\bigg)^{p}\varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}}\\ &\;\;\;\;\;\;\times\bigg(\sum\limits_{j = 1}^{n}(\bar{\beta}_{ij})^{q}\bigg)^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(L_{j}^{f}\vartheta_{j})^{p}\|\varphi^*\|_{\infty}^{p}\varepsilon^{\frac{p}{q}}+140^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\\ &\;\;\;\;\;\;\times\bigg(\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n(\bar{q}_{ijk}M_k^gL_{j}^{g}\vartheta_{j})^{p}+\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n(\bar{q}_{ijk}M_j^gL_k^g\vartheta_k)^p\bigg)\varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\\ &\;\;\;\;\;\;\times n^{\frac{2p}{q}} \sum\limits_{k = 1}^{n}\sum\limits_{j = 1}^{n}(M_j^gM_{k}^{g})^{p}\varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p}(\frac{1}{\underline{b}_{i}^{\emptyset}})^{\frac{2(p+q)}{p}}n^{\frac{2p}{q}}\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{q}_{ijk}M_j^gM_k^g)^p\varepsilon^{\frac{p}{q}}\\ &\;\;\;\;\;\;+140^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}n^{\frac{2p}{q}}\bigg(\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n( \bar{\nu}_{ijk}M_{k}^{g}L_{j}^{g}\vartheta_{j})^{p}+\sum\limits_{j = 1}^n \sum\limits_{k = 1}^n( \bar{\nu}_{ijk}M_{j}^{g}L_{k}^{g}\vartheta_{k})^{p}\bigg)\varepsilon^{p-1}\\ &\;\;\;\;\;\;+35^{p-1}\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}} n^{\frac{2p}{q}}\sum\limits_{k = 1}^{n}\sum\limits_{j = 1}^{n}(M_j^gM_k^g )^p\varepsilon^{p-1}+35^{p-1}\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{p}}n^{\frac{2p}{q}}\\ &\;\;\;\;\;\;\times\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{n}(\bar{\nu}_{ijk}M_j^gM_k^g)^p\varepsilon^{\frac{p}{q}}+70^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\bigg[\bigg(\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{q}\bigg)^{\frac{p}{q}}\\ &\;\;\;\;\;\;+\bigg(\sum\limits_{j = 1}^{n}(|T_{ij}|_\infty)^{q}\bigg)^{\frac{p}{q}}\bigg]\varepsilon^{p-1}+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(|T_{ij}|_\infty\bar{\mu}_{j})^{p}\varepsilon^{\frac{p}{q}}\\ &\;\;\;\;\;\;+70^{p-1} \vartheta_i^{-p} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+1}{q}}\bigg[\bigg(\sum\limits_{j = 1}^{n}(\bar{\mu}_{j})^{q}\bigg)^{\frac{p}{q}} +\bigg(\sum\limits_{j = 1}^{n}(|S_{ij}|_\infty)^{q}\bigg)^{\frac{p}{q}}\bigg]\varepsilon^{p-1} \\ &\;\;\;\;\;\;+35^{p-1} \vartheta_i^{-p} \bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}} n^{\frac{p}{q}}\sum\limits_{j = 1}^{n}(|S_{ij}|_\infty\bar{\mu}_{j})^{p}\varepsilon^{\frac{p}{q}}+35^{p-1}\vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{p+q}{q}}\varepsilon^{p-1} \\ &\;\;\;\;\;\;+35^{p-1} \vartheta_i^{-p}\bigg(\frac{1}{\underline{b}_{i}^{\emptyset}}\bigg)^{\frac{2(p+q)}{q}}|I_{i}|_\infty^{p}\varepsilon^{\frac{p}{q}}\bigg\}. \end{align*} |
Hence, by (3.12) and (A_4) , it holds that
\begin{align*} \|\varphi^*(t+{\flat})-\varphi^*(t)\|_{B^{p}}^{p} \leq\frac{Q\varepsilon^{p}}{1-P}, \end{align*} |
which implies that \varphi^{*} is B^p -almost periodic. The proof is finished.
Remark 3.1. Although we can prove that W = (L^\infty(\mathbb{R}, \mathcal{A})\cap B_{AP}^p(\mathbb{R}, \mathcal{A}), \|\cdot\|_\infty) is a Banach space, we still cannot directly use the fixed point theorem to determine the existence of almost periodic solutions for (1.1). Because there are higher-order terms in system (1.1), and W is not an algebra, we cannot prove that operator T is a self mapping.
It is easy to prove the following stability results using the same method as the proof of Theorem 4.1 in [33] or the proof of Theorem 15 in [31].
Theorem 3.2. Assume that (A_{1})-(A_{4}) hold. Then system (1.1) possesses a unique Besicovitch almost periodic solution, which is globally exponentially stable, i.e., if \bar{x} is the Besecovitch almost periodic solution with initial value \bar{\varphi} and x(t) is an arbitrary solution of system (1.1) with initial value \varphi , then there exist positive numbers \zeta > 0 and N > 0 satisfying
|x(t)-\bar{x}(t)|_{1}\leq N\|\varphi-\bar{\varphi}\|_{\varrho}e^{-\zeta t}, \, \, \, t > 0, |
in which \|\varphi-\bar{\varphi}\|_{\varrho} = \max\limits_{i\in\mathcal{J}}\bigg\{\sup\limits_{t\in[- \varrho, 0]}|\varphi_{i}(t)-a_{i}(t)\varphi_{i}(t)-(\bar{\varphi}_{i}(t)-a_{i}(t)\bar{\varphi}_{i}(t))|_{1}\bigg\} .
In this section, we provide an example to demonstrate the validity of the results obtained in this paper.
Example 4.1. In system (1.1), let m = 3, n = 2 , and for i, j, k = 1, 2 , take the coefficients are as follows:
\begin{align*} x_i(t) = \, &e_0x_i^0(t)+e_1x_i^1(t)+e_2x_i^2(t)+e_3x_i^3(t)+e_{12}x_i^{12}(t)+e_{13}x_i^{13}(t)+e_{23}x_i^{23}(t)+e_{123}x_i^{123}(t), \\ f_j(x) = \, &\frac{1}{100}e_{0}\sin{x_j^{12}}+\frac{3}{250}\sin(x_j^{12}+x_j^{123})e_{1}+\frac{1}{168}\sin(x_j^{1}+x_j^{13})e_{2}+\frac{1}{53}\sin(x_j^{2}+x_j^{123})e_{3}\\ &+\frac{1}{125}e_{12}\arctan{x_j^{3}}+\frac{1}{156}\sin(x_j^{3}+x_j^{123})e_{13}+\frac{1}{150}e_{23}\tanh{x_j^{12}}+\frac{1}{40}\sin(x_j^{1}+x_j^{12}\\ &+x_j^{123})e_{123}, \\ g_j(x) = \, &\frac{1}{48}e_{0}\sin{x_j^{13}}+\frac{1}{153}\sin(x_j^{12}+x_j^{23})e_{1}+\frac{1}{150}e_{2}\arctan{x_j^{13}}+\frac{1}{120}\sin(x_j^{12}+x_j^{123})e_{3}\\ &+\frac{1}{150}\sin(x_j^{12}-x_j^{23})e_{12}+\frac{1}{250}\sin(x_j^{2}+x_j^{13})e_{13}+\frac{1}{57}e_{23}\sin{x_j^{13}}+\frac{1}{60}\sin(x_j^{0}+x_j^{3}\\ &+x_j^{23})e_{123}, \\ a_1(t) = \, &(0.01+0.004\sin{t})e_0+(0.01+0.001\sin{\sqrt{6}t})e_1+(0.01+0.001\cos{\sqrt{3}t})e_2\\ &+(0.01+0.002\sin{\sqrt{5}t})e_3+(0.01+0.003\sin{t})e_{12}+(0.01+0.002\cos{t})e_{13}\\ &+(0.01+0.002\cos{\sqrt{2}t})e_{23}+(0.01+0.002\sin{\sqrt{2}t})e_{123}, \\ a_2(t) = \, &(0.01+0.002\sin{t})e_0+(0.01+0.002\cos{3t})e_1+(0.01+0.001\sin{\sqrt{2}t})e_2\\ &+(0.01+0.001\sin{\sqrt{7}t})e_3+(0.01+0.001\sin{\sqrt{5}t})e_{12}+(0.01+0.002\cos{t})e_{13}\\ &+(0.01+0.002\cos{\sqrt{5}t})e_{23}+(0.01+0.002\sin{\sqrt{3}t})e_{123}, \\ b_{1}(t) = \, & (10+0.05\sin t)e_0+(0.2+0.01\cos\sqrt{2}t)e_1+(0.2+0.02\sin t)e_2\\ &+(0.2+0.01\sin\sqrt{3}t)e_3+(0.2+0.06\sin3t)e_{12}+(0.2+0.05\sin2t)e_{13}\\ &+(0.2+0.01\sin t)e_{23}+(0.2+0.01\cos\sqrt{3}t)e_{123}, \\ b_2(t) = \, & (0.2+0.01\cos\sqrt{3}t)e_0+(0.2+0.02\sin t)e_1+(0.2+0.07\cos t)e_2\\ &+(0.2+0.05\cos\sqrt{5}t)e_3+(10+0.05\sin\sqrt{5}t)e_{12}+(0.2+0.02\cos\sqrt{5}t)e_{13}\\ &+(0.2+0.01\sin t)e_{23}+(0.2+0.01\sin7t)e_{123}, \\ c_{11}(t) = \, &0.01e_{0}\sin{2t}+0.02e_{3}\sin{2t}+0.02e_{23}\cos{\sqrt{2}t}+0.03e_{123}\cos{11t}, \\ c_{12}(t) = \, &0.01e_{0}\sin{\sqrt{5}t}+0.02e_{2}\cos^{2}{3t}+0.01e_{3}\sin{5t}+0.03e_{12}\sin{\sqrt{3}t}, \\ c_{21}(t) = \, &0.01e_{0}\sin{6t}+0.02e_{3}\cos{\sqrt{2}t}+0.03e_{23}\cos{\sqrt{3}t}+0.03e_{123}\cos^{2}{2t}, \\ c_{22}(t) = \, &0.01e_{0}\sin^{2}{7t}+0.04e_{2}\cos{6t}+0.04e_{3}\sin{\sqrt{5}t}+0.03e_{12}\cos{7t}, \\ u_{11}(t) = \, &0.02e_{0}\sin{4t}+0.01e_{3}\cos{\sqrt{2}t}+0.04e_{23}\cos{\sqrt{3}t}+0.03e_{123}\sin^{2}{2t}, \\ u_{12}(t) = \, &0.02e_{0}\cos{9t}+0.03e_{2}\cos^{2}{3t}+0.04e_{3}\sin{5t}+0.01e_{12}\sin{3t}, \\ u_{21}(t) = \, &0.02e_{0}\sin{3t}+0.03e_{3}\cos{\sqrt{3}t}+0.01e_{23}\cos{\sqrt{3}t}+0.02e_{123}\sin^{2}{7t}, \\ u_{22}(t) = \, &0.02e_{0}\cos{t}+0.03e_{2}\sin^{2}{5t}+0.03e_{3}\sin{3t}+0.04e_{12}\cos{2t}, \\ \alpha_{11}(t) = \, &0.01e_{3}\sin{3t}+0.04e_{12}\cos{5t}+0.02e_{13}\cos{\sqrt{5}t}+0.03e_{123}\sin^{2}{3t}, \\ \alpha_{12}(t) = \, &0.01e_{2}\sin{\sqrt{3}t}+0.02e_{3}\sin{4t}+0.03e_{12}\sin{7t}+0.01e_{123}\cos^{2}{5t}, \\ \alpha_{21}(t) = \, &0.03e_{3}\cos{4t}+0.01e_{12}\sin{\sqrt{5}t}+0.03e_{23}\cos{4t}+0.01e_{123}\cos^{2}{5t}, \\ \alpha_{22}(t) = \, &0.01e_{0}\cos{\sqrt{5}t}+0.04e_{3}\sin{3t}+0.03e_{12}\sin^{2}{3t}+0.02e_{123}\cos{4t}, \\ \beta_{11}(t) = \, &0.01e_{0}\cos{5t}+0.02e_{1}\sin{3t}+0.01e_{2}\sin{\sqrt{7}t}+0.03e_{123}\cos{3t}, \\ \beta_{12}(t) = \, &0.03e_{0}\sin{7t}+0.02e_{1}\sin{t}+0.03e_{2}\sin{\sqrt{3}t}+0.02e_{23}\sin{3t}, \\ \beta_{21}(t) = \, &0.03e_{0}\cos{7t}+0.02e_{1}\sin{\sqrt{5}t}+0.04e_{2}\sin{\sqrt{6}t}+0.01e_{123}\sin{5t}, \\ \beta_{22}(t) = \, &0.03e_{0}\cos{2t}+0.02e_{1}\cos{3t}+0.04e_{2}\cos{\sqrt{2}t}+0.02e_{23}\cos{3t}, \\ \theta_{111}(t) = \, &0.03e_{0}\sin{\sqrt{2}t}+0.02e_{1}\cos{\sqrt{5}t}+0.01e_{2}\sin{\sqrt{7}t}+0.02e_{12}\cos{2t}, \\ \theta_{112}(t) = \, &0.04e_{0}\cos{\sqrt{5}t}+0.02e_{1}\sin{3t}+0.03e_{2}\sin{\sqrt{3}t}+0.03e_{13}\cos{\sqrt{3}t}, \\ \theta_{121}(t) = \, &0.02e_{0}\sin{\sqrt{3}t}+0.02e_{2}\cos{3t}+0.04e_{12}\sin{\sqrt{6}t}+0.01e_{23}\cos{3t}, \\ \theta_{122}(t) = \, &0.03e_{0}\sin{\sqrt{5}t}+0.04e_{2}\cos{5t}+0.03e_{12}\sin{\sqrt{5}t}+0.02e_{23}\cos{4t}, \\ \theta_{211}(t) = \, &0.03e_{0}\sin{3t}+0.03e_{1}\cos{t}+0.01e_{2}\cos{4t}+0.01e_{12}\sin{5t}, \\ \theta_{212}(t) = \, &0.04e_{0}\cos{3t}+0.02e_{1}\sin{t}+0.01e_{2}\cos{3t}+0.01e_{13}\sin{3t}, \\ \theta_{221}(t) = \, &0.03e_{0}\sin{4t}+0.03e_{2}\cos{\sqrt{2}t}+0.02e_{12}\sin{5t}+0.03e_{23}\cos{3t}, \\ \theta_{222}(t) = \, &0.01e_{0}\sin{4t}+0.01e_{2}\cos{3t}+0.01e_{12}\sin{4t}+0.04e_{123}\cos{2t}, \\ q_{111}(t) = \, &0.06e_{0}\sin{5t}+0.04e_{1}\cos{6t}+0.03e_{12}\sin{\sqrt{3}t}+0.03e_{23}\sin^{2}{2t}, \\ q_{112}(t) = \, &0.05e_{0}\sin{2t}+0.02e_{1}\cos{\sqrt{2}t}+0.04e_{2}\sin{3t}+0.02e_{23}\cos{3t}, \\ q_{121}(t) = \, &0.02e_{0}\cos{4t}+0.02e_{1}\cos^{2}{3t}+0.05e_{2}\sin{4t}+0.03e_{23}\cos{4t}, \\ q_{122}(t) = \, &0.05e_{0}\sin{4t}+0.03e_{1}\cos{\sqrt{3}t}+0.03e_{12}\cos{2t}+0.04e_{23}\sin{3t}, \\ q_{211}(t) = \, &0.01e_{0}\cos{\sqrt{5}t}+0.02e_{1}\sin{3t}+0.04e_{12}\cos{\sqrt{3}t}+0.02e_{23}\sin{t}, \\ q_{212}(t) = \, &0.01e_{0}\cos{4t}+0.06e_{1}\cos{\sqrt{2}t}+0.03e_{12}\sin{5t}+0.04e_{23}\cos{2t}, \\ q_{221}(t) = \, &0.01e_{0}\cos{3t}+0.02e_{1}\sin{t}+0.02e_{2}\cos{\sqrt{3}t}+0.01e_{23}\sin{2t}, \\ q_{222}(t) = \, &0.01e_{0}\cos{9t}+0.05e_{1}\cos^{2}{3t}+0.03e_{12}\sin{t}+0.05e_{23}\cos{t}, \\ \nu_{111}(t) = \, &0.02e_{0}\sin{3t}+0.03e_{1}\cos{\sqrt{3}t}+0.01e_{2}\cos{5t}+0.01e_{12}\sin{4t}, \\ \nu_{112}(t) = \, &0.02e_{0}\cos{t}+0.03e_{1}\sin^{2}{5t}+0.02e_{12}\cos{3t}+0.03e_{23}\sin{2t}, \\ \nu_{121}(t) = \, &0.05e_{0}\sin{3t}+0.04e_{2}\sin{3t}+0.03e_{12}\cos{2t}+0.02e_{23}\sin{3t}, \\ \nu_{122}(t) = \, &0.03e_{0}\cos{2t}+0.04e_{2}\cos{\sqrt{3}t}+0.05e_{12}\sin{\sqrt{2}t}+0.04e_{23}\cos{\sqrt{3}t}, \\ \nu_{211}(t) = \, &0.03e_{0}\sin{7t}+0.04e_{1}\sin{3t}+0.02e_{2}\cos{\sqrt{3}t}+0.03e_{12}\sin{2t}, \\ \nu_{212}(t) = \, &0.05e_{0}\cos{5t}+0.02e_{2}\cos{4t}+0.03e_{12}\sin{7t}+0.04e_{23}\cos{t}, \\ \nu_{221}(t) = \, &0.02e_{0}\sin{3t}+0.03e_{2}\cos{2t}+0.04e_{12}\sin{t}+0.01e_{23}\cos{2t}, \\ \nu_{222}(t) = \, &0.03e_{0}\sin{6t}+0.03e_{2}\cos{t}+0.01e_{12}\sin{2t}+0.04e_{23}\cos{3t}, \\ I_1(t) = \, &0.32e_{0}(\sin{\sqrt{5}t}+\frac{1}{1+t^2})+0.5e_{1}\cos{\sqrt{2}t}+0.36e_{2}\sin{\sqrt{3}t}+0.23e_{3}\cos{\sqrt{3}t}\\ &+0.49e_{12}\sin{\sqrt{2}t}+0.25e_{13}\cos{\sqrt{5}t}+0.42e_{23}\cos{\sqrt{5}t}+0.15e_{123}(\sin{\sqrt{3}t}+e^{-|t|}), \\ I_2(t) = \, &0.25e_{0}\cos{\sqrt{3}t}+0.42e_{1}(\sin{\sqrt{2}t}+e^{-|t|})+0.28e_{2}\sin{\sqrt{3}t}+0.45e_{3}\cos{\sqrt{3}t}\\ &+0.32e_{12}(\cos{\sqrt{2}t}+\frac{1}{1+t^{2}})+0.46e_{13}\sin{\sqrt{3}t}+0.15e_{23}\sin{\sqrt{5}t}+0.26e_{123}\cos{\sqrt{3}t}, \\ \gamma_{ij}(t) = \, &0.07e_{0}\cos{2t}+0.03e_{1}\sin{2t}+0.04e_{2}(\sin{2t}+\frac{1}{1+t^{2}})+0.06e_{3}\cos{t}\\ &+0.08e_{12}(\sin{t}+e^{-|t|})+0.02e_{13}\cos{t}+0.05e_{23}\sin{2t}+0.04e_{123}\cos{t}, \\ \mu_{j}(t) = \, &0.2e_{0}\sin{\sqrt{2}t}+0.3e_{1}\sin{\sqrt{3}t}+0.4e_{2}\sin{\sqrt{2}t}+0.6e_{3}\cos{\sqrt{2}t}\\ &+0.8e_{12}\cos{\sqrt{3}t}+0.3e_{13}\cos{\sqrt{5}t}+0.4e_{23}\sin{\sqrt{3}t}+0.2e_{123}\cos{\sqrt{6}t}, \\ T_{ij}(t) = \, &0.006e_{0}\sin{t}+0.004e_{1}\sin{t}+0.003e_{2}(\sin{t}+\frac{1}{1+t^{2}})+0.001e_{3}\cos{t}\\ &+0.002e_{12}\sin{t}+0.003e_{13}(\cos{t}+e^{-|t|})+0.002e_{23}\cos{t}+0.005e_{123}\sin{t}, \\ S_{ij}(t) = \, &0.002e_{0}\cos{t}+0.003e_{1}\sin{t}+0.001e_{2}\cos{t}+0.001e_{3}(\sin{t}+\frac{1}{2+t^{2}})\\ &+0.003e_{12}(\sin{t}+e^{-|t+1|})+0.002e_{13}\cos{t}+0.001e_{23}\sin{t}+0.002e_{123}\cos{t}, \\ \sigma_{ij}(t) = \, &1-0.3\sin{t}, \quad \eta_{ij}(t) = 1-0.8\cos{3t}, \quad \delta_{ijk}(t) = 1-0.6\sin{2t}, \\ \tau_1(t) = \, &1-0.1\sin{t}, \quad \tau_2(t) = 1-0.3\cos{t} . \end{align*} |
Then, it is easy to see that conditions (A_1) and (A_2) are satisfied.
Moreover, take \vartheta_{1} = \vartheta_{2} = 1, p = 3, q = \frac{3}{2} , then through simple calculations, we obtain
\begin{align*} &\bar{a}_{1} = 0.014, \bar{a}_{2} = 0.012, \bar{b}_{1}^{\emptyset} = 10.05, \bar{b}_{1}^{c} = 0.26, \underline{b}_{1}^{\emptyset} = 9.95, \bar{b}_{2}^{\emptyset} = 10.05, \bar{b}_{2}^{c} = 0.27, \underline{b}_{2}^{\emptyset} = 9.95, \\ &\bar{c}_{11} = 0.03, \bar{c}_{12} = 0.03, \bar{c}_{21} = 0.03, \bar{c}_{22} = 0.04, \bar{u}_{11} = 0.04, \bar{u}_{12} = 0.04, \bar{u}_{21} = 0.03, \bar{u}_{22} = 0.04, \\ &\bar{\alpha}_{11} = 0.04, \bar{\alpha}_{12} = 0.03, \bar{\alpha}_{21} = 0.03, \bar{\alpha}_{22} = 0.04, \bar{\beta}_{11} = 0.03, \bar{\beta}_{12} = 0.03, \bar{\beta}_{21} = 0.04, \bar{\beta}_{22} = 0.04, \\ &\bar{\theta}_{111} = 0.03, \bar{\theta}_{112} = 0.04, \bar{\theta}_{121} = 0.04, \bar{\theta}_{122} = 0.04, \bar{\theta}_{211} = 0.03, \bar{\theta}_{212} = 0.04, \bar{\theta}_{221} = 0.03, \\ &\bar{\theta}_{222} = 0.04, \bar{q}_{111} = 0.06, \bar{q}_{112} = 0.05, \bar{q}_{121} = 0.05, \bar{q}_{122} = 0.05, \bar{q}_{211} = 0.04, \bar{q}_{212} = 0.06, \\ &\bar{q}_{221} = 0.02, \bar{q}_{222} = 0.05, \bar{\nu}_{111} = 0.03, \bar{\nu}_{112} = 0.03, \bar{\nu}_{121} = 0.05, \bar{\nu}_{122} = 0.05, \bar{\nu}_{211} = 0.04, \\ &\bar{\nu}_{212} = 0.05, \bar{\nu}_{221} = 0.04, \bar{\nu}_{222} = 0.04, \bar{\tau}_{i} = \bar{\tau}'_{i} = 0.1, \bar{\sigma}_{ij} = \bar{\sigma}'_{ij} = 0.3, \bar{\eta}_{ij} = \bar{\eta}'_{ij} = 0.8, \\ &\bar{\delta}_{ijk} = \bar{\delta}'_{ijk} = 0.6, L_{1}^{f} = L_{2}^{f} = \frac{1}{40}, L_{1}^{g} = L_{2}^{g} = \frac{1}{48}, M_{1}^{g} = M_{2}^{g} = \frac{1}{48} , \rho \approx0.054972 < 1, \\ &P\approx0.518973 < 1 . \end{align*} |
Hence, (A_{3}) and (A_{4}) are also satisfied. Consequently, in view of Theorem 3.2, we know that system (1.1) has a unique Besicovitch almost periodic solution that is globally exponentially stable (see Figures 1–4).
Remark 4.1. Even when the system considered in Example 4.1 degenerates into a real-valued system, there are no existing results to derive the results of Example 4.1.
This article introduces a new method to establish the existence and global exponential stability of Besicovitch almost periodic solutions for Clifford-valued high-order Hopfield fuzzy NNs with D operators. The methods and results of this article can be applied to study the generalized almost periodic and almost automorphic dynamics of high-order NNs.
Bing Li: Methodology, Conceptualization, Writing - review and editing; Yuan Ning: Writing - original draft, Visualization; Yongkun Li: Methodology, Conceptualization, Funding acquisition, Writing - review and editing. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was supported by the National Natural Science Foundation of China, grant number 12261098.
The authors declare no conflict of interest.
[1] |
G. S. Teodoro, J. A. T. Machado, E. C. de Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195–208. https://doi.org/10.1016/j.jcp.2019.03.008 doi: 10.1016/j.jcp.2019.03.008
![]() |
[2] |
H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
![]() |
[3] | J. F. Gómez-Aguilar, A. Atangana, Applications of fractional calculus to modeling in dynamics and chaos, Boca Raton: Chapman & Hall/CRC Press, 2022. https://doi.org/10.1201/9781003006244 |
[4] | J. A. T. Machado, Fractional calculus: fundamentals and applications, In: Acoustics and vibration of mechanical structures—AVMS-2017: Proceedings of the 14th AVMS Conference, 2018, 3–11. https://doi.org/10.1007/978-3-319-69823-6_1 |
[5] | S. Chakraverty, R. M. Jena, S. K. Jena, Computational fractional dynamical systems: fractional differential equations and applications, John Wiley & Sons, 2023. |
[6] |
C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: a review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141–159. https://doi.org/10.1016/j.cnsns.2017.04.001 doi: 10.1016/j.cnsns.2017.04.001
![]() |
[7] |
J. A. T. Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, et al., Some applications of fractional calculus in engineering, Math. Probl. Eng., 2010 (2010), 1–34. https://doi.org/10.1155/2010/639801 doi: 10.1155/2010/639801
![]() |
[8] | S. Das, Observation of fractional calculus in physical system description, In: Functional fractional calculus, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-20545-3_3 |
[9] | R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000. |
[10] | H. Jafari, B. Mehdinejadiani, D. Baleanu, Fractional calculus for modeling unconfined groundwater, Berlin, Boston: De Gruyter, 2019. https://doi.org/10.1515/9783110571905-007 |
[11] | N. Su, Fractional calculus for hydrology, soil science and geomechanics, Boca Raton: CRC Press, 2020. https://doi.org/10.1201/9781351032421 |
[12] |
C. P. Li, Y. Q. Chen, J. Kurths, Fractional calculus and its applications, Phil. Trans. R. Soc. A, 371 (2013), 20130037. http://dx.doi.org/10.1098/rsta.2013.0037 doi: 10.1098/rsta.2013.0037
![]() |
[13] |
Z. J. Meng, L. F. Wang, H. Li, W. Zhang, Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math., 92 (2015), 1275–1291. https://doi.org/10.1080/00207160.2014.932909 doi: 10.1080/00207160.2014.932909
![]() |
[14] |
K. Kumar, R. K. Pandey, S. Sharma, Comparative study of three numerical schemes for fractional integro-differential equations, J. Comput. Appl. Math., 315 (2017), 287–302. https://doi.org/10.1016/j.cam.2016.11.013 doi: 10.1016/j.cam.2016.11.013
![]() |
[15] |
M. B. Almatrafi, A. R. Alharbi, A. R. Seadawy, Structure of analytical and numerical wave solutions for the Ito integro-differential equation arising in shallow water waves, J. King Saud Univ. Sci., 33 (2021), 101375. https://doi.org/10.1016/j.jksus.2021.101375 doi: 10.1016/j.jksus.2021.101375
![]() |
[16] |
K. Agilan, V. Parthiban, Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations, J. Appl. Math. Comput., 69 (2023), 1765–1793. https://doi.org/10.1007/s12190-022-01810-2 doi: 10.1007/s12190-022-01810-2
![]() |
[17] |
M. Derakhshan, M. Jahanshahi, H. K. demneh, Investigation the boundary and initial value problems including fractional integro-differential equations with singular kernels, J. Adv. Math. Model., 11 (2021), 97–108. https://doi.org/10.22055/JAMM.2021.34670.1848 doi: 10.22055/JAMM.2021.34670.1848
![]() |
[18] |
X. H. Yang, Z. M. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
![]() |
[19] |
J. W. Wang, X. X. Jiang, X. H. Yang, H. X. Zhang, A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers' type nonlinearity, J. Appl. Math. Comput., 70 (2024), 489–511. https://doi.org/10.1007/s12190-023-01975-4 doi: 10.1007/s12190-023-01975-4
![]() |
[20] |
J. W. Wang, X. X. Jiang, H. X. Zhang, A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers' equation, Appl. Math. Lett., 151 (2024), 109002. https://doi.org/10.1016/j.aml.2024.109002 doi: 10.1016/j.aml.2024.109002
![]() |
[21] |
L. J. Wu, H. X. Zhang, X. H. Yang, F. R. Wang, A second-order finite difference method for the multi-term fourth-order integral-differential equations on graded meshes, Comput. Appl. Math., 41 (2022), 313. https://doi.org/10.1007/s40314-022-02026-7 doi: 10.1007/s40314-022-02026-7
![]() |
[22] |
X. H. Yang, W. L. Qiu, H. F. Chen, H. X. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497–513. https://doi.org/10.1016/j.apnum.2021.11.004 doi: 10.1016/j.apnum.2021.11.004
![]() |
[23] |
F. R. Wang, X. H. Yang, H. X. Zhang, L. J. Wu, A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel, Math. Comput. Simul., 199 (2022), 38–59. https://doi.org/10.1016/j.matcom.2022.03.004 doi: 10.1016/j.matcom.2022.03.004
![]() |
[24] | H. X. Zhang, X. X. Jiang, F. R. Wang, X. H. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., 2024, 1–25. https://doi.org/10.1007/s12190-024-02000-y |
[25] |
F. Safari, An accurate RBF-based meshless technique for the inverse multi-term time-fractional integro-differential equation, Eng. Anal. Bound. Elem., 153 (2023), 116–125. https://doi.org/10.1016/j.enganabound.2023.05.015 doi: 10.1016/j.enganabound.2023.05.015
![]() |
[26] |
S. Z. Rida, H. S. Hussien, Efficient Mittag-Leffler collocation method for solving linear and nonlinear fractional differential equations, Mediterr. J. Math., 15 (2018), 1–15. https://doi.org/10.1007/s00009-018-1174-0 doi: 10.1007/s00009-018-1174-0
![]() |
[27] |
S. Z. Rida, H. S. Hussien, A. H. Noreldeen, M. M. Farag, Effective fractional technical for some fractional initial value problems, Int. J. Appl. Comput. Math., 8 (2022), 149. https://doi.org/10.1007/s40819-022-01346-w doi: 10.1007/s40819-022-01346-w
![]() |
[28] |
M. S. Akel, H. S. Hussein, Numerical treatment of solving singular integral equations by using Sinc approximations, Appl. Math. Comput., 218 (2011), 3565–3573. https://doi.org/10.1016/j.amc.2011.08.102 doi: 10.1016/j.amc.2011.08.102
![]() |
[29] |
S. Behera, S. S. Ray, On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels, Comput. Appl. Math., 41 (2022), 211. https://doi.org/10.1007/s40314-022-01897-0 doi: 10.1007/s40314-022-01897-0
![]() |
[30] |
G. D. Shi, Y. L. Gong, M. X. Yi, Alternative Legendre polynomials method for nonlinear fractional integro-differential equations with weakly singular kernel, J. Math., 2021 (2021), 1–13. https://doi.org/10.1155/2021/9968237 doi: 10.1155/2021/9968237
![]() |
[31] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006. |
[32] | D. Baleanu, Z. B. Guvenc, J. A. T. Machado, New trends in nanotechnology and fractional calculus applications, Dordrecht: Springer, 2010. https://doi.org/10.1007/978-90-481-3293-5 |
[33] |
M. Bahmanpour, M. T. Kajani, M. Maleki, Solving Fredholm integral equations of the first kind using Muntz wavelets, Appl. Numer. Math., 143 (2019), 159–171. https://doi.org/10.1016/j.apnum.2019.04.007 doi: 10.1016/j.apnum.2019.04.007
![]() |
[34] |
S. C. Shiralashetti, S. Kumbinarasaiah, Laguerre wavelets exact Parseval frame-based numerical method for the solution of system of differential equations, Int. J. Comput. Math., 6 (2020), 101. https://doi.org/10.1007/s40819-020-00848-9 doi: 10.1007/s40819-020-00848-9
![]() |
[35] |
B. B. Tavasani, A. H. R. Sheikhani, H. Aminikhah, Numerical scheme to solve a class of variable-order Hilfer-Prabhakar fractional differential equations with Jacobi wavelets polynomials, Appl. Math. J. Chinese Univ., 37 (2022), 35–51. https://doi.org/10.1007/s11766-022-4241-z doi: 10.1007/s11766-022-4241-z
![]() |
[36] | D. Hong, J. Z. Wang, R. Gardner, Real analysis with an introduction to wavelets and applications, Elsevier, 2005. |
[37] | A. M. Mathai, H. J. Haubold, Special functions for applied scientists, New York: Springer, 2008. https://doi.org/10.1007/978-0-387-75894-7 |
[38] |
J. Shahni, R. Singh, Laguerre wavelet method for solving Thomas-Fermi type equations, Eng. Comput., 38 (2022), 2925–2935. https://doi.org/10.1007/s00366-021-01309-7 doi: 10.1007/s00366-021-01309-7
![]() |
[39] |
B. Q. Tang, X. F. Li, Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput., 199 (2008), 406–413. https://doi.org/10.1016/j.amc.2007.09.058 doi: 10.1016/j.amc.2007.09.058
![]() |
[40] | P. K. Kythe, P. Puri, Computational methods for linear integral equations, Boston: Birkhauser, 2002. |
[41] |
M. X. Yi, J. Huang, CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel, Int. J. Comput. Math., 92 (2015), 1715–1728. https://doi.org/10.1080/00207160.2014.964692 doi: 10.1080/00207160.2014.964692
![]() |
[42] |
V. V. Zozulya, P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, J. Chin. Inst. Eng., 22 (1999), 763–775. https://doi.org/10.1080/02533839.1999.9670512 doi: 10.1080/02533839.1999.9670512
![]() |
[43] |
S. Nemati, S. Sedaghat, I. Mohammadi, A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels, J. Comput. Appl. Math., 308 (2016), 231–242. https://doi.org/10.1016/j.cam.2016.06.012 doi: 10.1016/j.cam.2016.06.012
![]() |
[44] |
Y. X. Wang, L. Zhu, Z. Wang, Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel, Adv. Differ. Equ., 2018 (2018), 1–13. https://doi.org/10.1186/s13662-018-1699-3 doi: 10.1186/s13662-018-1699-3
![]() |
[45] |
S. Nemati, P. M. Lima, Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions, Appl. Math. Comput., 327 (2018), 79–92. https://doi.org/10.1016/j.amc.2018.01.030 doi: 10.1016/j.amc.2018.01.030
![]() |