In order to overcome the low accuracy of traditional Extreme Learning Machine (ELM) network in the performance evaluation of Rotate Vector (RV) reducer, a pattern recognition model of ELM based on Ensemble Empirical Mode Decomposition (EEMD) fusion and Improved artificial Jellyfish Search (IJS) algorithm was proposed for RV reducer fault diagnosis. Firstly, it is theoretically proved that the torque transmission of RV reducer has periodicity during normal operation. The characteristics of data periodicity can be effectively reflected by using the test signal periodicity characteristics of rotating machinery and EEMD. Secondly, the Logistic chaotic mapping of population initialization in JS algorithm is replaced by tent mapping. At the same time, the competition mechanism is introduced to form a new IJS. The simulation results of standard test function show that the new algorithm has the characteristics of faster convergence and higher accuracy. The new algorithm was used to optimize the input layer weight of the ELM, and the pattern recognition model of IJS-ELM was established. The model performance was tested by XJTU-SY bearing experimental data set of Xi'an Jiaotong University. The results show that the new model is superior to JS-ELM and ELM in multi-classification performance. Finally, the new model is applied to the fault diagnosis of RV reducer. The results show that the proposed EEMD-IJS-ELM fault diagnosis model has higher accuracy and stability than other models.
Citation: Xiaoyan Wu, Guowen Ye, Yongming Liu, Zhuanzhe Zhao, Zhibo Liu, Yu Chen. Application of Improved Jellyfish Search algorithm in Rotate Vector reducer fault diagnosis[J]. Electronic Research Archive, 2023, 31(8): 4882-4906. doi: 10.3934/era.2023250
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In order to overcome the low accuracy of traditional Extreme Learning Machine (ELM) network in the performance evaluation of Rotate Vector (RV) reducer, a pattern recognition model of ELM based on Ensemble Empirical Mode Decomposition (EEMD) fusion and Improved artificial Jellyfish Search (IJS) algorithm was proposed for RV reducer fault diagnosis. Firstly, it is theoretically proved that the torque transmission of RV reducer has periodicity during normal operation. The characteristics of data periodicity can be effectively reflected by using the test signal periodicity characteristics of rotating machinery and EEMD. Secondly, the Logistic chaotic mapping of population initialization in JS algorithm is replaced by tent mapping. At the same time, the competition mechanism is introduced to form a new IJS. The simulation results of standard test function show that the new algorithm has the characteristics of faster convergence and higher accuracy. The new algorithm was used to optimize the input layer weight of the ELM, and the pattern recognition model of IJS-ELM was established. The model performance was tested by XJTU-SY bearing experimental data set of Xi'an Jiaotong University. The results show that the new model is superior to JS-ELM and ELM in multi-classification performance. Finally, the new model is applied to the fault diagnosis of RV reducer. The results show that the proposed EEMD-IJS-ELM fault diagnosis model has higher accuracy and stability than other models.
Fractional calculus has proven in recent years to be a helpful method of tackling the complexity of complex systems from various scientific and engineering branches. It involves the generalization of the integer order differentiation and integration of a function to non-integer order, see [1]. In recent years, there has been considerable interest in fractional differential equations, with numerous works dedicated to the topic. Notable examples include the books by Benchohra et al. [2,3]. The authors of [4,5] investigated the qualitative theorems of solutions to diverse fractional differential equations and inclusions about memory effects and predictive behavior arguments.
In a recent publication [6], Diaz introduced novel definitions for the special functions k-gamma and k-beta. Interested readers can refer to additional sources such as [7,8] to delve deeper into this topic. Furthermore, in another work [9], Sousa et al. presented the φ-Hilfer derivative of fractional order and elucidated some crucial properties related to this type of fractional operator. Drawing inspiration from the various papers cited earlier, we have introduced a new extension of the renowned Hilfer fractional derivative [10,11,12].
On the other hand, delay differential equations are a type of functional differential equation that arise in various biological and physical applications and often require consideration of variable or state-dependent delays. The study of functional differential equations with delay has garnered significant attention in recent years due to their crucial applications in mathematical models of real-world phenomena. For examples, see [4,5] and the references therein.
In [13], Krim et al. studied the problem
{(ρDϑ20++ϑ1)(ϱ)=g(ϱ,ϑ1(ϱ),(ρDϑ20++ϑ1)(ϱ)),ϱ∈I:=[0,T],ϑ1(T)=ϑ1T∈R, |
where ρDϑ20+ is the Katugampola derivative of fractional order ϑ2∈(0,1], and
g:I×R×R→R |
is a continuous function.
Using the Picard operator method, Krasnoselskii fixed point approach, and Gronwall's inequality lemma, Almalahi et al. [14] established the existence and stability theories for the problem:
{HDϑ1,ϑ2;φ0+y(ϱ)=g(ϱ,y(ϱ),y(ˆg(ϱ)), ϱ∈(0,a2],I1−ϑ3;φ0+y(0+)=k∑ȷ=1cȷy(kȷ),kȷ∈(0,a2),y(ϱ)=δ(ϱ),ϱ∈[−ϑ2,0], |
where HDϑ1,ϑ2;φ0+(⋅) is the φ-Hilfer derivative of fractional order ϑ1∈(0,1) and type ϑ2∈[0,1],I1−ϑ3,φ0+(⋅) is the φ-Riemann-Liouville integral of fractional order (1−ϑ3),
ϑ3=ϑ1+ϑ2(1−ϑ1),0<ϑ3<1,kȷ,ȷ=1,2,…,k |
are prefixed points satisfying 0<k1≤k2≤…≤kȷ<a2, and cȷ∈R,δ∈C[−ϑ2,0], the function
g:(0,a2]×R×R→R |
is continuous, and
ˆg∈C(0,a2]→[−ϑ2,a2] |
with ˆg(ϱ)≤ϱ,ϑ2>0.
In light of the above studies, we focus on a terminal-valued hybrid problem governed by a nonlinear implicit (k,φ)-Hilfer fractional differential equation with mixed-type arguments (retarded and advanced):
(HkDϑ1,ϑ2;φa1+ψy)(ϱ)=g(ϱ,yϱ(⋅),(HkDϑ1,ϑ2;φa1+ψy)(ϱ)), ϱ∈(a1,a2], | (1.1) |
y(a2)=˜n∑ȷ=1αȷy(ϵȷ), | (1.2) |
y(ϱ)=χ(ϱ), ϱ∈[a1−d,a1], d>0, | (1.3) |
y(ϱ)=˜χ(ϱ), ϱ∈[a2,a2+˜d], ˜d>0, | (1.4) |
where HkDϑ1,ϑ2;φa1+ is the (k,φ)-Hilfer derivative of fractional order ϑ1∈(0,k) and type ϑ2∈[0,1] defined in Section 2. Furthermore,
ϑ3=1k(ϑ2(k−ϑ1)+ϑ1),k>0,g:[a1,a2]×C([−d,˜d],R)×R⟶R,ψ∈C([a1,a2],R∖{0}),ϵȷ,ȷ=1,…,˜n |
are pre-fixed points satisfying a1<ϵ1≤…≤ϵ˜n<a2, and αȷ,ȷ=1,…,˜n are real numbers. For each function y defined on [a1−d,a2+˜d] and for any ϱ∈(a1,a2], we denote by yϱ the element defined by
yϱ(s)=y(ϱ+s), s∈[−d,˜d]. |
The following are the primary novelties of the current paper:
● Given the diverse conditions imposed on problems (1.1)–(1.4), our study can be seen as both a continuation and a generalization of the studies mentioned above, such as the papers [13,14].
● The introduced (k,φ)-Hilfer operator serves as an extension, encompassing previously established fractional derivatives such as the Caputo, Hadamard, and Hilfer fractional derivatives already present in the existing literature.
● The number of papers addressing a nonlocal condition combined with retarded and advanced arguments is very limited. Therefore, our work aims to fill this gap in the literature.
● The introduced (k,φ)-Hilfer operator serves as an extension, encompassing previously established fractional derivatives such as the Caputo, Hadamard, and Hilfer fractional derivatives already present in the existing literature.
The structure of this paper is as follows: Section 2 presents certain notations and preliminaries about the φ-Hilfer fractional derivative, the functions k-gamma and k-beta, and some auxiliary results. Further, we give the definition of the (k,φ)-Hilfer type fractional derivative and some essential theorems and lemmas. In Section 3, we present three existence and uniqueness results for the problems (1.1)–(1.4) that are founded on the Banach contraction principle, the Schauder and Krasnoselskii fixed point theorems. In the last section, illustrative examples are provided in support of the results obtained.
First, we present the weighted spaces, notations, definitions, and preliminary facts that are used in this article. Please refer to [3] for all details on these spaces and notations.
Let
0<a1<a2<∞,T=[a1,a2],ϑ1∈(0,k),ϑ2∈[0,1],k>0 |
and
ϑ3=1k(ϑ2(k−ϑ1)+ϑ1). |
The Banach space of continuous functions is denoted by C(T,R) with the norm
‖y‖∞=sup{|y(ϱ)|:ϱ∈T}. |
Let ACn(T,R), Cn(T,R) be the spaces of continuous functions, n-times absolutely continuous, and n-times continuously differentiable functions on T, respectively.
Let
C([−d,˜d],R),C=C([a1−d,a1],R) |
and
˜C=C([a2,a2+˜d],R) |
be the spaces gifted, respectively, with the norms
‖y‖[−d,˜d]=sup{|y(ϱ)|:ϱ∈[−d,˜d]},‖y‖C=sup{|y(ϱ)|:ϱ∈[a1−d,a1]},‖y‖˜C=sup{|y(ϱ)|:ϱ∈[a2,a2+˜d]}. |
Let φ∈C([a1,a2],R) be an increasing function such that φ′(ϱ)≠0, for all ϱ∈T.
Now, let the weighted Banach space be defined as
Cϑ3,k;φ(T)={y:(a1,a2]→R:ϱ→Ψφϑ3(ϱ,a1)y(ϱ)∈C(T,R)}, |
where
Ψφϑ3(ϱ,a1)=(φ(ϱ)−φ(a1))1−ϑ3 |
with the norm
‖y‖Cϑ3,k;φ=supϱ∈T|Ψφϑ3(ϱ,a1)y(ϱ)|, |
and
Cnϑ3,k;φ(T)={y∈Cn−1(T):y(n)∈Cϑ3,k;φ(T)},n∈N,C0ϑ3,k;φ(T)=Cϑ3,k;φ(T) |
with the norm
‖y‖Cnϑ3,k;φ=n−1∑ȷ=0‖y(ȷ)‖∞+‖y(n)‖Cϑ3,k;φ. |
Next, let us define the Banach space
F={y:[a1−d,a2+˜d]→R:y|[a1−d,a1]∈C,y|[a2,a2+˜d]∈˜C and y|(a1,a2]∈Cϑ3,k;φ(T)} |
with the norm
‖y‖F=max{‖y‖C,‖y‖˜C,‖y‖Cϑ3,k;φ}. |
Denote Xpφ(a1,a2), (1≤p≤∞) to the space of each real-valued Lebesgue measurable functions ˆg on [a1,a2] such that ‖ˆg‖Xpφ<∞, with the norm given as
‖ˆg‖Xpφ=(∫a2a1φ′(ϱ)|ˆg(ϱ)|pdϱ)1p, |
where φ is a non-deceasing and non-negative function on [a1,a2], such that φ′ is continuous on [a1,a2] with φ(0)=0.
Definition 2.1. [6] The k-gamma function is given as
Γk(α)=∫∞0ϱα−1e−ϱkkdϱ, α>0, |
where
Γk(α+k)=αΓk(α),Γk(α)=kαk−1Γ(αk),Γk(k)=Γ(1)=1, |
and for k→1, then
Γ(α)=Γk(α). |
Furthermore the k-beta function is defined as follows:
Bk(α,a)=1k∫10ϱαk−1(1−ϱ)ak−1dϱ, |
so that
Bk(α,a)=1kB(αk,ak) |
and
Bk(α,a)=Γk(α)Γk(a)Γk(α+a). |
Definition 2.2. [15] Let
ˆg∈Xpφ(a1,a2), φ(ϱ)>0 |
be a non-decreasing function on (a1,a2] and φ′(ϱ)>0 be continuous on (a1,a2) and ϑ1>0. The generalized k-fractional integral operators of a function ˆg of order ϑ1 are defined by
Jϑ1,k;φa1+ˆg(ϱ)=∫ϱa1ˉΨk,φϑ1(ϱ,s)φ′(s)ˆg(s)ds,Jϑ1,k;φa2−ˆg(ϱ)=∫a2ϱˉΨk,φϑ1(s,ϱ)φ′(s)ˆg(s)ds |
with k>0 and
ˉΨk,φϑ1(ϱ,s)=(φ(ϱ)−φ(s))ϑ1k−1kΓk(ϑ1). |
Additionally, the authors of the work [16] extended these operators and defined the generalized fractional integrals by
Jϑ1,k;φG,a1+ˆg(ϱ)=1kΓk(ϑ1)∫ϱa1φ′(s)ˆg(s)dsG(φ(ϱ)−φ(s),ϑ1k),Jϑ1,k;φG,a2−ˆg(ϱ)=1kΓk(ϑ1)∫a2ϱφ′(s)ˆg(s)dsG(φ(s)−φ(ϱ),ϑ1k), |
where G(z,ϑ1)∈AC[a1,a2].
Theorem 2.3. [16] Let ϑ1>0, k>0, and consider the integrable function ˆg: [a1,a2]→R. Then Jϑ1,k;φG,a1+ˆg exists for all ϱ∈[a1,a2].
Theorem 2.4. [16] Let ˆg∈Xpφ(a1,a2) and take ϑ1>0 and k>0. Then Jϑ1,k;φG,a1+ˆg∈C([a1,a2],R).
Lemma 2.5. [10,11] Consider ϑ1>0, ϑ2>0, and k>0. Then, one has
Jϑ1,k;φa1+Jϑ2,k;φa1+g(ϱ)=Jϑ1+ϑ2,k;φa1+g(ϱ)=Jϑ2,k;φa1+Jϑ1,k;φa1+g(ϱ) |
and
Jϑ1,k;φa2−Jϑ2,k;φa2−g(ϱ)=Jϑ1+ϑ2,k;φa2−g(ϱ)=Jϑ2,k;φa2−Jϑ1,k;φa2−g(ϱ). |
Lemma 2.6. [10,11] Let ϑ1,ϑ2>0 and k>0. Then, we have
Jϑ1,k;φa1+ˉΨk,φϑ2(ϱ,a1)=ˉΨk,φϑ1+ϑ2(ϱ,a1) |
and
Jϑ1,k;φa2−ˉΨk,φϑ2(a2,ϱ)=ˉΨk,φϑ1+ϑ2(a2,ϱ). |
Theorem 2.7. [10,11] Let 0<a1<a2<∞,ϑ1>0,0≤ϑ3<1, k>0, and y∈Cϑ3,k;φ(T). If
ϑ1k>1−ϑ3, |
then
(Jϑ1,k;φa1+y)(a1)=limϱ→a1+(Jϑ1,k;φa1+y)(ϱ)=0. |
Definition 2.8. ((k,φ)-Hilfer derivative [10,11]) Let
n−1<ϑ1k≤n |
with n∈N, T=[a1,a2] an interval such that
−∞≤a1<a2≤∞ |
and
ˆg,φ∈Cn([a1,a2],R) |
are two functions such that φ is increasing and φ′(ϱ)≠0, for all ϱ∈T. The (k,φ)-Hilfer fractional derivative HkDϑ1,ϑ2;φa1+(⋅) and HkDϑ1,ϑ2;φa2−(⋅) of a function ˆg of order ϑ1 and type 0≤ϑ2≤1, with k>0 is defined by
HkDϑ1,ϑ2;φa1+ˆg(ϱ)=(Jϑ2(kn−ϑ1),k;φa1+(1φ′(ϱ)ddϱ)n(knJ(1−ϑ2)(kn−ϑ1),k;φa1+ˆg))(ϱ)=(Jϑ2(kn−ϑ1),k;φa1+δnφ(knJ(1−ϑ2)(kn−ϑ1),k;φa1+ˆg))(ϱ), |
where
δnφ=(1φ′(ϱ)ddϱ)n. |
Lemma 2.9. [10,11] Let ϱ>a1, ϑ1>0,0≤ϑ2≤1, and k>0. Thus, for
0<ϑ3<1,ϑ3=1k(ϑ2(k−ϑ1)+ϑ1), |
and one has
[HkDϑ1,ϑ2;φa1+(Ψφϑ3(s,a1))−1](ϱ)=0. |
g∈Cnϑ3,k;φ[a1,a2],n−1<ϑ1k<n,0≤ϑ2≤1, |
where n∈N and k>0, then
(Jϑ1,k;φa1+ HkDϑ1,ϑ2;φa1+g)(ϱ)=g(ϱ)−n∑ȷ=1(φ(ϱ)−φ(a1))ϑ3−ȷkȷ−nΓk(k(ϑ3−ȷ+1)){δn−ȷφ(Jk(n−ϑ3),k;φa1+g(a1))}, |
where
ϑ3=1k(ϑ2(kn−ϑ1)+ϑ1). |
Particularly, for n=1, one gets
(Jϑ1,k;φa1+ HkDϑ1,ϑ2;φa1+g)(ϱ)=g(ϱ)−(φ(ϱ)−φ(a1))ϑ3−1Γk(ϑ2(k−ϑ1)+ϑ1)J(1−ϑ2)(k−ϑ1),k;φa1+g(a1). |
Lemma 2.11. [10,11] Let ϑ1>0,0≤ϑ2≤1, and y∈C1ϑ3,k;φ(T), where k>0, then for ϱ∈(a1,a2], we have
(HkDϑ1,ϑ2;φa1+ Jϑ1,k;φa1+y)(ϱ)=y(ϱ). |
We start this section by taking the next fractional differential problem:
(HkDϑ1,ϑ2;φa1+ψy)(ϱ)=δ(ϱ), ϱ∈(a1,a2], | (3.1) |
such that 0<ϑ1<k,0≤ϑ2≤1, subjected to the conditions
y(a2)=˜n∑ȷ=1αȷy(ϵȷ), | (3.2) |
y(ϱ)=χ(ϱ), ϱ∈[a1−d,a1], d>0, | (3.3) |
y(ϱ)=˜χ(ϱ), ϱ∈[a2,a2+˜d], ˜d>0, | (3.4) |
where
ϑ3=ϑ2(k−ϑ1)+ϑ1k, |
k>0, αȷ,ȷ=1,…,˜n, belong to R, α˜n+1=−1 and ϵȷ,ȷ=1,…,˜n+1, are pre-fixed points verifying
a1<ϵ1≤…≤ϵ˜n<a2=ϵ˜n+1, |
such that
˜n+1∑ȷ=1αȷΨφϑ3(ϵȷ,a1)≠0, |
and where δ(⋅)∈C(T,R), χ(⋅)∈C, ψ∈C([a1,a2],R∖{0}), and ˜χ(⋅)∈˜C.
Theorem 3.1. The function y verifies (3.1)–(3.4) if and only if
y(ϱ)={1ψ(ϱ)[−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷΨφϑ3(ϱ,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+δ)(ϱ)],ϱ∈(a1,a2],χ(ϱ), ϱ∈[a1−d,a1],˜χ(ϱ), ϱ∈[a2,a2+˜d]. | (3.5) |
Proof. Assume that y satisfies Eqs (3.1)–(3.4), and by implementing the integral operator Jϑ1,k;φa1+(⋅) of fractional order ϑ1 on both sides of (3.1), we have
(Jϑ1,k;φa1+ HkDϑ1,ϑ2;φa1+ψy)(ϱ)=(Jϑ1,k;φa1+δ)(ϱ). |
Using Theorem 2.10, we get
y(ϱ)=1ψ(ϱ)[Jk(1−ϑ3),k;φa1+y(a1)Ψφϑ3(ϱ,a1)Γk(kϑ3)+(Jϑ1,k;φa1+δ)(ϱ)]. | (3.6) |
In what follows, by putting ϱ=ϵȷ into (3.6), and applying αȷ to both sides, one gets
αȷy(ϵȷ)=1ψ(ϵȷ)[αȷJk(1−ϑ3),k;φa1+y(a1)Ψφϑ3(ϵȷ,a1)Γk(kϑ3)+αȷ(Jϑ1,k;φa1+δ)(ϵȷ)]. |
By (3.2) and (3.6) with ϱ=a2, we have
Jk(1−ϑ3),k;φa1+y(a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)Γk(kϑ3)+˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)=1ψ(a2)[Jk(1−ϑ3),k;φa1+y(a1)Ψφϑ3(a2,a1)Γk(kϑ3)+(Jϑ1,k;φa1+δ)(a2)], |
which implies
Jk(1−ϑ3),k;φa1+y(a1)=−(Jϑ1,k;φa1+δ)(a2)ψ(a2)+˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)1ψ(a2)Ψφϑ3(a2,a1)Γk(kϑ3)−˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)Γk(kϑ3)=−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)Γk(kϑ3). | (3.7) |
Substituting (3.7) into (3.6), we obtain (3.5).
Now, we show that y verifies Eq (3.5), it follows that it also verifies (3.1)–(3.4). Applying HkDϑ1,ϑ2;φa1+(⋅) on both sides of (3.5), we get
(HkDϑ1,ϑ2;φa1+ψy)(ϱ)= HkDϑ1,ϑ2;φa1+(−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷΨφϑ3(ϱ,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1))+(HkDϑ1,ϑ2;φa1+Jϑ1,k;φa1+δ)(ϱ). |
In view of Lemmas 2.9 and 2.11, we find Eq (3.1). Now, taking ϱ=a2 in Eq (3.5), we have
ψ(a2)y(a2)=−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷΨφϑ3(a2,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+δ)(a2). | (3.8) |
Substituting ϱ=ϵȷ into (3.5), we get
ψ(ϵȷ)y(ϵȷ)=−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)Ψφϑ3(ϵȷ,a1)˜n+1∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+δ)(ϵȷ). |
Then, we have
˜n∑ȷ=1αȷy(ϵȷ)=−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ), |
and thus,
˜n∑ȷ=1αȷy(ϵȷ)=(Jϑ1,k;φa1+δ)(a2)ψ(a2)−˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)−1ψ(a2)Ψφϑ3(a2,a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+1+˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)=(Jϑ1,k;φa1+δ)(a2)ψ(a2)−˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)ψ(a2)Ψφϑ3(a2,a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)−1ψ(a2)Ψφϑ3(a2,a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+1=((Jϑ1,k;φa1+δ)(a2)ψ(a2)−˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)ψ(a2)Ψφϑ3(a2,a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1))×(ψ(a2)Ψφϑ3(a2,a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)˜n+1∑ȷ=1αȷψ(a2)Ψφϑ3(a2,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1))=(Jϑ1,k;φa1+δ)(a2)Ψφϑ3(a2,a1)˜n∑ȷ=1αȷψ(ϵȷ)Ψφϑ3(ϵȷ,a1)−˜n∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷψ(a2)Ψφϑ3(a2,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1). |
Then,
˜n∑ȷ=1αȷy(ϵȷ)=−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷψ(a2)Ψφϑ3(a2,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+δ)(a2)ψ(a2). | (3.9) |
From (3.8) and (3.9), we find that
y(a2)=˜n∑ȷ=1αȷy(ϵȷ), |
which implies that argument (3.2) holds.
In sequel, we present the following finding as a consequence of Theorem 3.1.
Lemma 3.2. Let
ϑ3=ϑ2(k−ϑ1)+ϑ1k, |
such that 0<ϑ1<k and 0≤ϑ2≤1, and suppose that χ(⋅)∈C, ˜χ(⋅)∈˜C, and
g:T×C([−d,˜d],R)×R→R |
is a continuous function. Then, y∈F is a solution of problems (1.1)–(1.4) iff y is a fixed point of the mapping k: F→F defined by
(ky)(ϱ)={1ψ(ϱ)[−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)˜n+1∑ȷ=1αȷΨφϑ3(ϱ,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+δ)(ϱ)],ϱ∈(a1,a2],χ(ϱ), ϱ∈[a1−d,a1],˜χ(ϱ), ϱ∈[a2,a2+˜d], | (3.10) |
where δ is a function verifying
δ(ϱ)=g(ϱ,yϱ(⋅),δ(ϱ)) |
and
α˜n+1=−1,ϵ˜n+1=a2. |
Next, we present the following hypotheses for using in the sequel analysis:
(Ax1)
g:T×C([−d,˜d],R)×R→R |
is a continuous function.
(Ax2) There exist real numbers ζ1>0 and 0<ζ2<1, where
|g(ϱ,y1,ˆy1)−g(ϱ,y2,ˆy2)|≤ζ1‖y1−y2‖[−d,˜d]+ζ2|ˆy1−ˆy2| |
for any
y1,y2∈C([−d,˜d],R), ˆy1,ˆy2∈R, |
and ϱ∈(a1,a2].
(Ax3) There exist functions m1,m2,m3∈C(T,R+) with
m∗1=supϱ∈Tm1(ϱ), m∗2=supϱ∈Tm2(ϱ), m∗3=supϱ∈Tm3(ϱ)<1, |
such that
|g(ϱ,y,ˆy)|≤m1(ϱ)+m2(ϱ)‖y‖[−d,˜d]+m3(ϱ)|ˆy| |
for any
y∈C([−d,˜d],R), ˆy∈R |
and ϱ∈(a1,a2].
(Ax4) The function ψ is continuous on T and there exists G>0 such that
|ψ(ϱ)|≥G. |
Now, we will study the uniqueness theorem for problems (1.1)–(1.4) by utilizing the Banach fixed point technique [17].
Theorem 3.3. Suppose that (Ax1), (Ax2), and (Ax4) are satisfied. If
L=2ζ1(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k)(1−ζ2) <1, | (3.11) |
then, problems (1.1)–(1.4) have a unique solution in F.
Proof. In order to prove that the mapping k given in (3.10) possesses one fixed point in F. Let us take y,ˆy∈F, thus for any
ϱ∈[a1−d,a1]∪[a2,a2+˜d], |
we have
|ky(ϱ)−kˆy(ϱ)|=0. |
Thus
‖ky−kˆy‖C=‖ky−kˆy‖˜C=0. | (3.12) |
Further, for ϱ∈(a1,a2], we have
|ky(ϱ)−kˆy(ϱ)|≤1|ψ(ϱ)|[˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)|(Jϑ1,k;φa1+|δ1(s)−δ2(s)|)(ϵȷ)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϱ,a1)|ψ(ϵȷ)|Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+|δ1(s)−δ2(s)|)(ϱ)], |
where δ1 and δ2 are functions satisfying the functional equations
δ1(ϱ)=g(ϱ,yϱ(⋅),δ1(ϱ)),δ2(ϱ)=g(ϱ,ˆyϱ(⋅),δ2(ϱ)). |
By (Ax2), we have
|δ1(ϱ)−δ2(ϱ)|=|g(ϱ,yϱ,δ1(ϱ))−g(ϱ,ˆyϱ,δ2(ϱ))|≤ζ1‖yϱ−ˆyϱ‖[−d,˜d]+ζ2|δ1(ϱ)−δ2(ϱ)|. |
Then,
|δ1(ϱ)−δ2(ϱ)|≤ζ11−ζ2‖yϱ−ˆyϱ‖[−d,˜d]. |
Therefore, for each ϱ∈(a1,a2],
|ky(ϱ)−kˆy(ϱ)|≤ζ1˜n+1∑ȷ=1|αȷ|(Jϑ1,k;φa1+‖ys−ˆys‖[−d,˜d])(ϵȷ)G(1−ζ2)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϱ,a1)Ψφϑ3(ϵȷ,a1)+ζ1G(1−ζ2)(Jϑ1,k;φa1+‖ys−ˆys‖[−d,˜d])(ϱ)≤ζ1‖y−ˆy‖FG(1−ζ2)[˜n+1∑ȷ=1|αȷ|(Jϑ1,k;φa1+(1))(ϵȷ)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϱ,a1)Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+(1))(ϱ)]. |
By Lemma 2.6, we have
|ky(ϱ)−kˆy(ϱ)|≤ζ1‖y−ˆy‖FG(1−ζ2)[˜n+1∑ȷ=1|αȷ|(φ(ϵȷ)−φ(a1))ϑ1kΓk(ϑ1+k)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϱ,a1)Ψφϑ3(ϵȷ,a1)+(φ(ϱ)−φ(a1))ϑ1kΓk(ϑ1+k)]. |
Hence,
|Ψφϑ3(ϱ,a1)(ky(ϱ)−kˆy(ϱ))|≤ζ1‖y−ˆy‖FG(1−ζ2)[˜n+1∑ȷ=1|αȷ|(φ(ϵȷ)−φ(a1))ϑ1kΓk(ϑ1+k)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϵȷ,a1)+(φ(ϱ)−φ(a1))1−ϑ3+ϑ1kΓk(ϑ1+k)], |
which implies that
‖ky−kˆy‖Cϑ3,k;φ≤2ζ1(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k)(1−ζ2)‖y−ˆy‖F. |
Thus,
‖ky−kˆy‖Cϑ3,k;φ≤L‖y−ˆy‖F. | (3.13) |
By (3.12) and (3.13), we obtain
‖ky−kˆy‖F≤L‖y−ˆy‖F. |
Based on (3.11), the mapping k is a contraction on F. Therefore, by the Banach fixed point technique, k owns one fixed point y∈F, which is a unique solution for problems (1.1)–(1.4).
Our subsequent existence theorem for problems (1.1)–(1.4) will be proved by the Schauder fixed point technique [17].
Theorem 3.4. Suppose that (Ax1), (Ax3), and (Ax4) are verified. If
ℓ=2m∗2(φ(a2)−φ(a1))1−ϑ3+ϑ1kG(1−m∗3)Γk(ϑ1+k) <1, | (3.14) |
then, problems (1.1)–(1.4) have at least one solution in F.
Proof.
We will split the proof into several steps.
Step 1. The mapping k is continuous.
Consider {yn} to be a convergent sequence to y in F. For each
ϱ∈[a1−d,a1]∪[a2,a2+˜d], |
we have
|kyn(ϱ)−ky(ϱ)|=0. |
For ϱ∈(a1,a2], we have
|ky(ϱ)−kˆy(ϱ)|≤1|ψ(ϱ)|[˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)|(Jϑ1,k;φa1+|δn(s)−δ(s)|)(ϵȷ)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϱ,a1)|ψ(ϵȷ)|Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+|δn(s)−δ(s)|)(ϱ)], |
where δ and δn are functions satisfying the functional equations
δ(ϱ)=g(ϱ,yϱ(⋅),δ(ϱ)),δn(ϱ)=g(ϱ,ynϱ(⋅),δn(ϱ)). |
Since yn→y, then we get δn(ϱ)→δ(ϱ) as n→∞ for each ϱ∈(a1,a2], and since g is continuous, then we have
‖kyn−ky‖F→0 as n→∞. |
Step 2. We show k(BM)⊂BM.
Consider M to be a positive real number, where
M≥max{m∗1ℓm∗2(1−ℓ),‖χ‖C,‖˜χ‖˜C}. |
Now, we present the next closed bounded ball
BM={y∈F:‖y‖F≤M}. |
Then, for each ϱ∈[a1−d,a1], we have
|ky(ϱ)|≤‖χ‖C, |
and for each ϱ∈[a2,a2+˜d], we have
|ky(ϱ)|≤‖˜χ‖˜C. |
Further, for each ϱ∈(a1,a2], (3.10) implies that
|ky(ϱ)|≤1|ψ(ϱ)|[˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)|(Jϑ1,k;φa1+|g(s,ys,δ(s))|)(ϵȷ)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϱ,a1)|ψ(ϵȷ)|Ψφϑ3(ϵȷ,a1)+(Jϑ1,k;φa1+|g(s,ys,δ(s))|)(ϱ)]. | (3.15) |
By hypothesis (Ax3), for ϱ∈(a1,a2], we have
|δ(ϱ)|=|g(ϱ,yϱ,δ(ϱ))|≤m1(ϱ)+m2(ϱ)‖yϱ‖[−d,˜d]+m3(ϱ)|δ(ϱ)|, |
which implies that
|δ(ϱ)|≤m∗1+m∗2M+m∗3|δ(ϱ)|, |
then
|δ(ϱ)|≤m∗1+m∗2M1−m∗3:=Δ. |
Thus for ϱ∈(a1,a2], from (3.15) we get
|Ψφϑ3(ϱ,a1)ky(ϱ)|≤Δ˜n+1∑ȷ=1|αȷ|(Jϑ1,k;φa1+(1))(ϵȷ)G˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϵȷ,a1)+ΔGΨφϑ3(ϱ,a1)(Jϑ1,k;φa1+(1))(ϱ). |
By Lemma 2.6, we have
|Ψφϑ3(ϱ,a1)ky(ϱ)|≤ΔG[˜n+1∑ȷ=1|αȷ|(φ(ϵȷ)−φ(a1))ϑ1kΓk(ϑ1+k)˜n+1∑ȷ=1|αȷ|Ψφϑ3(ϵȷ,a1)+(φ(ϱ)−φ(a1))1−ϑ3+ϑ1kΓk(ϑ1+k)]. |
Thus
|Ψφϑ3(ϱ,a1)ky(ϱ)|≤2Δ(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k)≤M. |
Then, for each
ϱ∈[a1−d,a2+˜d], |
we obtain
‖ky‖F≤M. |
Step 3. We prove that the set k(BM) is relatively compact.
Let
k1,k2∈(a1,a2], k1<k2 |
and let y∈BM. Then,
|Ψφϑ3(k1,a1)ky(k1)−Ψφϑ3(k2,a1)ky(k2)|≤|1ψ(k1)−1ψ(k2)|×[˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)||(Jϑ1,k;φa1+δ)(ϵȷ)|˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)|Ψφϑ3(ϵȷ,a1)]+|Ψφϑ3(k1,a1)ψ(k1)(Jϑ1,k;φa1+δ(s))(k1)−Ψφϑ3(k2,a1)ψ(k2)(Jϑ1,k;φa1+δ(s))(k2)|≤|1ψ(k1)−1ψ(k2)|×[˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)||(Jϑ1,k;φa1+δ)(ϵȷ)|˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)|Ψφϑ3(ϵȷ,a1)]+∫k1a1|Ψφϑ3(k1,a1)ˉΨk,φϑ1(k1,s)ψ(k1)−Ψφϑ3(k2,a1)ˉΨk,φϑ1(k2,s)ψ(k2)||φ′(s)y(s)|ds+|Ψφϑ3(k2,a1)ψ(k2)(Jϑ1,k;φk+1|y(s)|)(k2)|. |
By Lemma 2.6, we get
|Ψφϑ3(k1,a1)ky(k1)−Ψφϑ3(k2,a1)ky(k2)|≤|1ψ(k1)−1ψ(k2)|×[˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)||(Jϑ1,k;φa1+δ)(ϵȷ)|˜n+1∑ȷ=1|αȷ||ψ(ϵȷ)|Ψφϑ3(ϵȷ,a1)]+Δ∫k1a1|Ψφϑ3(k1,a1)ˉΨk,φϑ1(k1,s)ψ(k1)−Ψφϑ3(k2,a1)ˉΨk,φϑ1(k2,s)ψ(k2)||φ′(s)|ds+ΔΨφϑ3(k2,a1)(φ(k2)−φ(k1))ϑ1kGΓk(ϑ1+k). |
As k1→k2, the right side of the above inequality tends to zero. From Steps 1–3, using the Arzela-Ascoli theorem, we infer that k: F→F is a continuous and compact mapping. Consequently, we deduce that k owns at least one fixed point, which is a solution for problems (1.1)–(1.4).
Our third outcome depends on the Krasnoselskii fixed point technique [17].
Theorem 3.5. Suppose that (Ax1)–(Ax4) are verified. If
ζ1(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k)(1−ζ2)<1, | (3.16) |
then, problems (1.1)–(1.4) have a solution in F.
Proof. Let us assume that the ball
Bω={y∈F:||y||F≤ω},ω≥r1+r2 |
with
r1:=(m∗1+m∗2ω)(φ(a2)−φ(a1))1−ϑ3+ϑ1kG(1−m∗3)Γk(ϑ1+k),r2:=max{‖χ‖C,‖˜χ‖˜C,(m∗1+m∗2ω)(φ(a2)−φ(a1))1−ϑ3+ϑ1kG(1−m∗3)Γk(ϑ1+k)}. |
{Next, we introduce the mappings} ∇1 and ∇2 on Bω as follows:
∇1y(ϱ)={−˜n+1∑ȷ=1αȷψ(ϵȷ)(Jϑ1,k;φa1+δ)(ϵȷ)ψ(ϱ)˜n+1∑ȷ=1αȷΨφϑ3(ϱ,a1)ψ(ϵȷ)Ψφϑ3(ϵȷ,a1),ϱ∈(a1,a2],0, ϱ∈[a1−d,a1],0, ϱ∈[a2,a2+˜d], | (3.17) |
and
∇2y(ϱ)={(Jϑ1,k;φa1+δ)(ϱ)ψ(ϱ),ϱ∈(a1,a2],χ(ϱ), ϱ∈[a1−d,a1],˜χ(ϱ), ϱ∈[a2,a2+˜d], | (3.18) |
where δ is a function verifying
δ(ϱ)=g(ϱ,yϱ(⋅),δ(ϱ)). |
Then (3.10) can be written as
ky(ϱ)=∇1y(ϱ)+∇2y(ϱ), y∈F. |
Step 1. We prove that
∇1y+∇2ˆy∈Bω |
for any y,ˆy∈Bω.
By (Ax3) and from (3.10), for ϱ∈(a1,a2], we have
|δ(ϱ)|=|g(ϱ,yϱ,δ(ϱ))|≤m1(ϱ)+m2(ϱ)‖yϱ‖[−d,˜d]+m3(ϱ)|δ(ϱ)|, |
which implies that
|δ(ϱ)|≤m∗1+m∗2ω+m∗3|δ(ϱ)|, |
and then
|δ(ϱ)|≤m∗1+m∗2ω1−m∗3:=A. |
Thus, for ϱ∈(a1,a2] and by (3.17), we have
|Ψφϑ3(ϱ,a1)∇1y(ϱ)|≤A(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k). |
Then, for each ϱ∈[a1−d,a2+˜d], we obtain
‖∇1y‖F≤A(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k). | (3.19) |
For ϱ∈(a1,a2] and by (3.18), we have
|Ψφϑ3(ϱ,a1)∇2ˆy(ϱ)|≤A(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k). |
For each ϱ∈[a1−d,a1], we have
|∇2ˆy(ϱ)|≤‖χ‖C, |
and for each ϱ∈[a2,a2+˜d], we have
|∇2ˆy(ϱ)|≤‖˜χ‖˜C. |
Then, for each ϱ∈[a1−d,a2+˜d], we get
‖∇2ˆy‖F≤max{‖χ‖C,‖˜χ‖˜C,A(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k)}. | (3.20) |
From (3.19) and (3.20), for each ϱ∈[a1−d,a2+˜d], we have
‖∇1y+∇2ˆy‖F≤‖∇1y‖F+‖∇2ˆy‖F≤r1+r2≤ω, |
which infers that
∇1y+∇2ˆy∈Bω. |
Step 2. The mapping ∇1 is a contraction.
In view of the condition (3.16) and Theorem 3.3, the mapping ∇1 is a contraction on F with the norm ‖⋅‖F.
Step 3. ∇2 is continuous and compact.
Let {yn} be a sequence such that yn⟶y in F. For each
ϱ∈[a1−d,a1]∪[a2,a2+˜d], |
we have
|∇2yn(ϱ)−∇2y(ϱ)|=0. |
For ϱ∈(a1,a2], we have
|∇2yn(ϱ)−∇2y(ϱ)|≤1|ψ(ϱ)|(Jϑ1,k;φa1+|δn(s)−δ(s)|)(ϱ), |
such that δ and δn are functions verifying the functional equations
δ(ϱ)=g(ϱ,yϱ(⋅),δ(ϱ)),δn(ϱ)=g(ϱ,ynϱ(⋅),δn(ϱ)). |
Since yn→y, then we get δn(ϱ)→δ(ϱ) as n→∞ for each ϱ∈(a1,a2], and since g is continuous, then we have
‖∇2yn−∇2y‖F→0 as n→∞. |
Then ∇2 is continuous. Next we prove that ∇2 is uniformly bounded on Bω. For each
ϱ∈[a1−d,a2+˜d] |
and any ˆy∈Bω, we get
‖∇2ˆy‖F≤max{‖χ‖C,‖˜χ‖˜C,A(φ(a2)−φ(a1))1−ϑ3+ϑ1kGΓk(ϑ1+k)}. |
This implies that the mapping ∇2 is uniformly bounded on Bω. In order to show the compactness of ∇2, we take k1,k2∈(a1,a2] such that k1<k2, and y∈Bω. Then
|Ψφϑ3(k1,a1)∇2y(k1)−Ψφϑ3(k2,a1)∇2y(k2)|≤|Ψφϑ3(k1,a1)ψ(k1)(Jϑ1,k;φa1+δ(s))(k1)−Ψφϑ3(k2,a1)ψ(k2)(Jϑ1,k;φa1+δ(s))(k2)|≤∫k1a1|Ψφϑ3(k1,a1)ˉΨk,φϑ1(k1,s)ψ(k1)−Ψφϑ3(k2,a1)ˉΨk,φϑ1(k2,s)ψ(k2)||φ′(s)y(s)|ds+|Ψφϑ3(k2,a1)ψ(k2)(Jϑ1,k;φk+1|y(s)|)(k2)|. |
By Lemma 2.6, we get
|Ψφϑ3(k1,a1)∇2y(k1)−Ψφϑ3(k2,a1)∇2y(k2)|≤A∫k1a1|Ψφϑ3(k1,a1)ˉΨk,φϑ1(k1,s)ψ(k1)−Ψφϑ3(k2,a1)ˉΨk,φϑ1(k2,s)ψ(k2)||φ′(s)|ds+AΨφϑ3(k2,a1)(φ(k2)−φ(k1))ϑ1kGΓk(ϑ1+k). |
Note that
|Ψφϑ3(k1,a1)∇2y(k1)−Ψφϑ3(k2,a1)∇2y(k2)|→0 as k1→k2. |
This proves that ∇2Bω is equicontinuous on (a1,a2]. Therefore, ∇2 is compact. Thus, based on the Krasnoselskii fixed point technique, we conclude that k possesses a fixed point, which satisfies problems (1.1)–(1.4).
We give various examples of (1.1)–(1.4), with
T=[1,π],ϑ3=1k(ϑ2(k−ϑ1)+ϑ1),g(ϱ,y,ˆy)=1105+125eπ−ϱ[1+ˆy3+|ˆy|−y1+y],ψ(ϱ)=313e−5(ϱ+sin(ϱ)+2), |
where ϱ∈T, y∈C([−d,˜d],R), and ˆy∈R.
Example 4.1. Taking ϑ2→12, ϑ1=12, k=1, φ(ϱ)=πϱ, α1=1, α2=2, α3=3, ϵ1=54, ϵ2=43, ϵ3=32, ˜n=3, d=˜d=13, and ϑ3=34, we have the system below:
(H1D12,12;φ1+ψy)(ϱ)=g(ϱ,yϱ(⋅),(H1D12,12;φ1+ψy)(ϱ)), ϱ∈(1,π], | (4.1) |
y(π)=y(54)+2x(43)+3x(32), | (4.2) |
y(ϱ)=χ(ϱ), ϱ∈[23,1], | (4.3) |
y(ϱ)=˜χ(ϱ), ϱ∈[π,π+13]. | (4.4) |
We have
Cϑ3,k;φ(T)=C34,1;φ(T)={y:(1,π]→R:(πϱ−π)14y∈C(T,R)}, |
and then
F={y:[23,π+13]→R:y|[23,1]∈C, y|[π,π+13]∈˜C and y|(1,π]∈C34,1;φ(T)}. |
By continuity of the function g, the hypothesis (Ax1) holds. For every
y∈C([−13,13],R),ˆy∈Randϱ∈T, |
one has
|g(ϱ,y,ˆy)|≤1105+125eπ−ϱ(1+‖y‖[−d,˜d]+|ˆy|). |
Then, the condition (Ax3) is satisfied with
m1(ϱ)=m2(ϱ)=m3(ϱ)=1105+125eπ−ϱ |
and
m∗1=m∗2=m∗3=1230. |
The condition (Ax4) is verified since we have that
|ψ(ϱ)|≥613e−5. |
We have
ℓ=52e−5(ππ−π)341374√π ≈0.001995278633 <1. |
Hence, in view of Theorem 3.4, we infer that problems (4.1)–(4.4) possess a solution in F.
Example 4.2. Considering ϑ2→0, ϑ1=12, k=1, φ(ϱ)=ϱρ, α1=1, α2=1, α3=5, ϵ1=32, ϵ2=2, ϵ3=52, ˜n=3, d=˜d=12, ρ=12, and ϑ3=12, we have the next system:
(H1D12,0;φ1+ψy)(ϱ)=(ρD12,01+y)(ϱ)=g(ϱ,yϱ(⋅),(ρD12,01+ψy)(ϱ)), ϱ∈(1,3], | (4.5) |
y(3)=y(32)+y(2)+5x(52), | (4.6) |
y(ϱ)=eϱ, ϱ∈[12,1], | (4.7) |
y(ϱ)=eϱ, ϱ∈[3,72]. | (4.8) |
We have
Cϑ3,k;φ(T)=C12,1;φ(T)={y:(1,3]→R:√(√ϱ−1)y∈C(T,R)}, |
and then
F={y:[12,72]→R:y|[12,1]∈C, y|[3,72]∈˜C and y|(1,3]∈C34,1;φ(T)}. |
Additionally, for every
y1,ˆy1∈C([−12,12],R), y2,ˆy2∈R,andϱ∈T, |
one has
|g(ϱ,y1,y2)−g(ϱ,ˆy1,ˆy2)|≤1105+125eπ−ϱ(‖y1−ˆy1‖[−d,˜d]+|y2−ˆy2|). |
Then, the condition (Ax2) holds with
ζ1=ζ2=1230. |
Since
L≈0.000105319912 <1. |
Hence, all of the hypotheses of Theorem 3.3 are verified. It follows that problems (4.5)–(4.8) possess one solution in F.
Our research considered a class of problems involving nonlinear implicit (k,φ)-Hilfer hybrid fractional differential equations with nonlocal terminal conditions. We achieved this by proving the existence and uniqueness of solutions for these equations. Our strategy hinged on powerful mathematical tools: the Banach contraction principle, Schauder's fixed point theorem, and Krasnoselskii's fixed point techniques. To showcase the practical applications of our findings and the ease of using our theorems, we presented some illustrative examples. These illustrations effectively highlight the flexibility and wide-reaching impact of the studied operator across various cases. It is noteworthy that the introduced (k,φ)-Hilfer operator operates as an extension, encompassing previously established fractional derivatives such as the Caputo, Hadamard, and Hilfer fractional derivative already present in the existing literature. This broader conceptual framework substantially contributes to the ongoing advancement of fractional calculus, thus laying the groundwork for promising directions of future exploration within this ever-evolving and dynamic domain.
A. Salim: conceptualization, data curation, formal analysis, investigation, methodology, writing-original draft; S. T. M. Thabet: conceptualization, data curation, formal analysis, methodology, writing-original draft; I. Kedim: data curation, formal analysis, investigation, methodology, writing-review and editing; M. Vivas-Cortez: investigation, writing-review and editing. All authors have read and agreed to the published version of the article.
The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.
Pontificia Universidad Católica del Ecuador, Proyecto Título: "Algunos resultados Cualitativos sobre Ecuaciones diferenciales fraccionales y desigualdades integrales" Cod UIO2022. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1446).
The authors declare that they have no conflicts of interest.
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