The synchronization problem and the dynamics analysis of neural networks have been thoroughly explored, and there have been many interesting results. This paper presents a review of the issues of synchronization problem, the periodic solution and the stability/stabilization with emphasis on the memristive neural networks and reaction-diffusion neural networks. First, this paper introduces the origin and development of neural networks. Then, based on different types of neural networks, some synchronization problems and the design of the controllers are introduced and summarized in detail. Some results of the periodic solution are discussed according to different neural networks, including bi-directional associative memory (BAM) neural networks and cellular neural networks. From the perspective of memristive neural networks and reaction-diffusion neural networks, some results of stability and stabilization are reviewed comprehensively with latest progress. Based on a review of dynamics analysis of neural networks, some applications in creation psychology are also introduced. Finally, the conclusion and the future research directions are provided.
Citation: Xiangwen Yin. A review of dynamics analysis of neural networks and applications in creation psychology[J]. Electronic Research Archive, 2023, 31(5): 2595-2625. doi: 10.3934/era.2023132
[1] | Ishfaq Mallah, Idris Ahmed, Ali Akgul, Fahd Jarad, Subhash Alha . On ψ-Hilfer generalized proportional fractional operators. AIMS Mathematics, 2022, 7(1): 82-103. doi: 10.3934/math.2022005 |
[2] | Mohammed A. Almalahi, Satish K. Panchal, Fahd Jarad, Mohammed S. Abdo, Kamal Shah, Thabet Abdeljawad . Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative. AIMS Mathematics, 2022, 7(9): 15994-16016. doi: 10.3934/math.2022876 |
[3] | Dumitru Baleanu, Babak Shiri . Generalized fractional differential equations for past dynamic. AIMS Mathematics, 2022, 7(8): 14394-14418. doi: 10.3934/math.2022793 |
[4] | Marimuthu Mohan Raja, Velusamy Vijayakumar, Anurag Shukla, Kottakkaran Sooppy Nisar, Wedad Albalawi, Abdel-Haleem Abdel-Aty . A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order r∈(1,2) with impulses. AIMS Mathematics, 2023, 8(5): 10802-10821. doi: 10.3934/math.2023548 |
[5] | Sumbal Ahsan, Rashid Nawaz, Muhammad Akbar, Saleem Abdullah, Kottakkaran Sooppy Nisar, Velusamy Vijayakumar . Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method. AIMS Mathematics, 2022, 7(7): 13169-13191. doi: 10.3934/math.2022726 |
[6] | Shuang-Shuang Zhou, Saima Rashid, Saima Parveen, Ahmet Ocak Akdemir, Zakia Hammouch . New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators. AIMS Mathematics, 2021, 6(5): 4507-4525. doi: 10.3934/math.2021267 |
[7] | Muhammad Tariq, Sotiris K. Ntouyas, Hijaz Ahmad, Asif Ali Shaikh, Bandar Almohsen, Evren Hincal . A comprehensive review of Grüss-type fractional integral inequality. AIMS Mathematics, 2024, 9(1): 2244-2281. doi: 10.3934/math.2024112 |
[8] | Sajid Iqbal, Muhammad Samraiz, Gauhar Rahman, Kottakkaran Sooppy Nisar, Thabet Abdeljawad . Some new Grüss inequalities associated with generalized fractional derivative. AIMS Mathematics, 2023, 8(1): 213-227. doi: 10.3934/math.2023010 |
[9] | Lakhlifa Sadek, Tania A Lazǎr . On Hilfer cotangent fractional derivative and a particular class of fractional problems. AIMS Mathematics, 2023, 8(12): 28334-28352. doi: 10.3934/math.20231450 |
[10] | Saima Rashid, Abdulaziz Garba Ahmad, Fahd Jarad, Ateq Alsaadi . Nonlinear fractional differential equations and their existence via fixed point theory concerning to Hilfer generalized proportional fractional derivative. AIMS Mathematics, 2023, 8(1): 382-403. doi: 10.3934/math.2023018 |
The synchronization problem and the dynamics analysis of neural networks have been thoroughly explored, and there have been many interesting results. This paper presents a review of the issues of synchronization problem, the periodic solution and the stability/stabilization with emphasis on the memristive neural networks and reaction-diffusion neural networks. First, this paper introduces the origin and development of neural networks. Then, based on different types of neural networks, some synchronization problems and the design of the controllers are introduced and summarized in detail. Some results of the periodic solution are discussed according to different neural networks, including bi-directional associative memory (BAM) neural networks and cellular neural networks. From the perspective of memristive neural networks and reaction-diffusion neural networks, some results of stability and stabilization are reviewed comprehensively with latest progress. Based on a review of dynamics analysis of neural networks, some applications in creation psychology are also introduced. Finally, the conclusion and the future research directions are provided.
Recently, fractional calculus has attained assimilated bounteous flow and significant importance due to its rife utility in the areas of technology and applied analysis. Fractional derivative operators have given a new rise to mathematical models such as thermodynamics, fluid flow, mathematical biology, and virology, see [1,2,3]. Previously, several researchers have explored different concepts related to fractional derivatives, such as Riemann-Liouville, Caputo, Riesz, Antagana-Baleanu, Caputo-Fabrizio, etc. As a result, this investigation has been directed at various assemblies of arbitrary order differential equations framed by numerous analysts, (see [4,5,6,7,8,9,10]). It has been perceived that the supreme proficient technique for deliberating such an assortment of diverse operators that attracted incredible presentation in research-oriented fields, for example, quantum mechanics, chaos, thermal conductivity, and image processing, is to manage widespread configurations of fractional operators that include many other operators, see the monograph and research papers [11,12,13,14,15,16,17,18,19,20,21,22].
In [23], the author proposed a novel idea of fractional operators, which is called GPF operator, that recaptures the Riemann-Liouville fractional operators into a solitary structure. In [24], the authors analyzed the existence of the FDEs as well as demonstrated the uniqueness of the GPF derivative by utilizing Kransnoselskii's fixed point hypothesis and also dealt with the equivalency of the mixed type Volterra integral equation.
Fractional calculus can be applied to a wide range of engineering and applied science problems. Physical models of true marvels frequently have some vulnerabilities which can be reflected as originating from various sources. Additionally, fuzzy sets, fuzzy real-valued functions, and fuzzy differential equations seem like a suitable mechanism to display the vulnerabilities marked out by elusiveness and dubiousness in numerous scientific or computer graphics of some deterministic certifiable marvels. Here we broaden it to several research areas where the vulnerability lies in information, for example, ecological, clinical, practical, social, and physical sciences [25,26,27].
In 1965, Zadeh [28] proposed fuzziness in set theory to examine these issues. The fuzzy structure has been used in different pure and applied mathematical analyses, such as fixed-point theory, control theory, topology, and is also helpful for fuzzy automata and so forth. In [29], authors also broadened the idea of a fuzzy set and presented fuzzy functions. This concept has been additionally evolved and the bulk of the utilization of this hypothesis has been deliberated in [30,31,32,33,34,35] and the references therein. The concept of HD has been correlated with fuzzy Riemann-Liouville differentiability by employing the Hausdorff measure of non-compactness in [36,37].
Numerous researchers paid attention to illustrating the actual verification of certain fuzzy integral equations by employing the appropriate compactness type assumptions. Different methodologies and strategies, in light of HD or generalized HD (see [38]) have been deliberated in several credentials in the literature (see for instance [39,40,41,42,43,44,45,46,47,48,49]) and we presently sum up quickly a portion of these outcomes. In [50], the authors proved the existence of solutions to fuzzy FDEs considering Hukuhara fractional Riemann-Liouville differentiability as well as the uniqueness of the aforesaid problem. In [51,52], the authors investigated the generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions. Bede and Stefanini [39] investigated and discovered novel ideas for fuzzy-valued mappings that correlate with generalized differentiability. In [43], Hoa introduced the subsequent fuzzy FDE with order ϑ∈(0,1):
{(cDϑσ+1Φ)(ζ)=F(ζ,Φ(ζ)),Φ(σ1)=Φ0∈E, | (1.1) |
where a fuzzy function is F:[σ1,σ2]×E→E with a nontrivial fuzzy constant Φ0∈E. The article addressed certain consequences on clarification of the fractional fuzzy differential equations and showed that the aforesaid equations in both cases (differential/integral) are not comparable in general. A suitable assumption was provided so that this correspondence would be effective. Hoa et al. [53] proposed the Caputo-Katugampola FDEs fuzzy set having the initial condition:
{(cDϑ,ρσ+1Φ)(ζ)=F(ζ,Φ(ζ)),Φ(σ1)=Φ0, | (1.2) |
where 0<σ1<ζ≤σ2, cDϑ,ρσ+1 denotes the fuzzy Caputo-Katugampola fractional generalized Hukuhara derivative and a fuzzy function is F:[σ1,σ2]×E→E. An approach of continual estimates depending on generalized Lipschitz conditions was employed to discuss the actual as well as the uniqueness of the solution. Owing to the aforementioned phenomena, in this article, we consider a novel fractional derivative (merely identified as Hilfer GPF-derivative). Consequently, in the framework of the proposed derivative, we establish the basic mathematical tools for the investigation of GPF-FFHD which associates with a fractional order fuzzy derivative. We investigated the actuality and uniqueness consequences of the clarification to a fuzzy fractional IVP by employing GPF generalized HD by considering an approach of continual estimates via generalized Lipschitz condition. Moreover, we derived the FVFIE using a generalized fuzzy GPF derivative is presented. Finally, we demonstrate the problems of actual and uniqueness of the clarification of this group of equations. The Hilfer-GPF differential equation is presented as follows:
{Dϑ,q,βσ+1Φ(ζ)=F(ζ,Φ(ζ)),ζ∈[σ1,T],0≤σ1<TI1−γ,βσ1Φ(σ1)=m∑j=1RjΦ(νj),ϑ≤γ=ϑ+q−ϑq,νj∈(σ1,T], | (1.3) |
where Dϑ,q,βσ+1(.) is the Hilfer GPF-derivative of order ϑ∈(0,1),I1−γ,βσ1(.) is the GPF integral of order 1−γ>0,Rj∈R, and a continuous function F:[σ1,T]×R→R with νj∈[σ1,T] fulfilling σ<ν1<...<νm<T for j=1,...,m. To the furthest extent that we might actually know, nobody has examined the existence and uniqueness of solution (1.3) regarding FVFIEs under generalized fuzzy Hilfer-GPF-HD with fuzzy initial conditions. An illustrative example of fractional-order in the complex domain is proposed and provides the exact solution in terms of the Fox-Wright function.
The following is the paper's summary. Notations, hypotheses, auxiliary functions, and lemmas are presented in Section 2. In Section 3, we establish the main findings of our research concerning the existence and uniqueness of solutions to Problem 1.3 by means of the successive approximation approach. We developed the fuzzy GPF Volterra-Fredholm integrodifferential equation in Section 4. Section 5 consists of concluding remarks.
Throughout this investigation, E represents the space of all fuzzy numbers on R. Assume the space of all Lebsegue measureable functions with complex values F on a finite interval [σ1,σ2] is identified by χrc(σ1,σ2) such that
‖F‖χrc<∞,c∈R,1≤r≤∞. |
Then, the norm
‖F‖χrc=(σ2∫σ1|ζcF(ζ)|rdζζ)1/r∞. |
Definition 2.1. ([53]) A fuzzy number is a fuzzy set Φ:R→[0,1] which fulfills the subsequent assumptions:
(1) Φ is normal, i.e., there exists ζ0∈R such that Φ(ζ0)=1;
(2) Φ is fuzzy convex in R, i.e, for δ∈[0,1],
Φ(δζ1+(1−δ)ζ2)≥min{Φ(ζ1),Φ(ζ2)}foranyζ1,ζ2∈R; |
(3) Φ is upper semicontinuous on R;
(4) [z]0=cl{z1∈R|Φ(z1)>0} is compact.
C([σ1,σ2],E) indicates the set of all continuous functions and set of all absolutely continuous fuzzy functions signifys by AC([σ1,σ2],E) on the interval [σ1,σ2] having values in E.
Let γ∈(0,1), we represent the space of continuous mappings by
Cγ[σ1,σ2]={F:(σ1,σ2]→E:eβ−1β(ζ−σ1)(ζ−σ1)1−γF(ζ)∈C[σ1,σ2]}. |
Assume that a fuzzy set Φ:R↦[0,1] and all fuzzy mappings Φ:[σ1,σ2]→E defined on L([σ1,σ2],E) such that the mappings ζ→ˉD0[Φ(ζ),ˆ0] lies in L1[σ1,σ2].
There is a fuzzy number Φ on R, we write [Φ]ˇq={z1∈R|Φ(z1)≥ˇq} the ˇq-level of Φ, having ˇq∈(0,1].
From assertions (1) to (4); it is observed that the ˇq-level set of Φ∈E, [Φ]ˇq is a nonempty compact interval for any ˇq∈(0,1]. The ˇq-level of a fuzzy number Φ is denoted by [Φ_(ˇq),ˉΦ(ˇq)].
For any δ∈R and Φ1,Φ2∈E, then the sum Φ1+Φ2 and the product δΦ1 are demarcated as: [Φ1+Φ2]ˇq=[Φ1]ˇq+[Φ2]ˇq and [δ.Φ1]ˇq=δ[Φ1]ˇq, for all ˇq∈[0,1], where [Φ1]ˇq+[Φ2]ˇq is the usual sum of two intervals of R and δ[Φ1]ˇq is the scalar multiplication between δ and the real interval.
For any Φ∈E, the diameter of the ˇq-level set of Φ is stated as diam[μ]ˇq=ˉμ(ˇq)−μ_(ˇq).
Now we demonstrate the notion of Hukuhara difference of two fuzzy numbers which is mainly due to [54].
Definition 2.2. ([54]) Suppose Φ1,Φ2∈E. If there exists Φ3∈E such that Φ1=Φ2+Φ3, then Φ3 is known to be the Hukuhara difference of Φ1 and Φ2 and it is indicated by Φ1⊖Φ2. Observe that Φ1⊖Φ2≠Φ1+(−)Φ2.
Definition 2.3. ([54]) We say that ¯D0[Φ1,Φ2] is the distance between two fuzzy numbers if
¯D0[Φ1,Φ2]=supˇq∈[0,1]H([Φ1]ˇq,[Φ2]ˇq),∀Φ1,Φ2∈E, |
where the Hausdroff distance between [Φ1]ˇq and [Φ2]ˇq is defined as
H([Φ1]ˇq,[Φ2]ˇq)=max{|Φ_(ˇq)−ˉΦ(ˇq)|,|ˉΦ(ˇq)−Φ_(ˇq)|}. |
Fuzzy sets in E is also refereed as triangular fuzzy numbers that are identified by an ordered triple Φ=(σ1,σ2,σ3)∈R3 with σ1≤σ2≤σ3 such that [Φ]ˇq=[Φ_(ˇq),ˉΦ(ˇq)] are the endpoints of ˇq-level sets for all ˇq∈[0,1], where Φ_(ˇq)=σ1+(σ2−σ1)ˇq and ˉΦ(ˇq)=σ3−(σ3−σ2)ˇq.
Generally, the parametric form of a fuzzy number Φ is a pair [Φ]ˇq=[Φ_(ˇq),ˉΦ(ˇq)] of functions Φ_(ˇq),ˉΦ(ˇq),ˇq∈[0,1], which hold the following assumptions:
(1) μ_(ˇq) is a monotonically increasing left-continuous function;
(2) ˉμ(ˇq) is a monotonically decreasing left-continuous function;
(3) μ_(ˇq)≤ˉμ(ˇq),ˇq∈[0,1].
Now we mention the generalized Hukuhara difference of two fuzzy numbers which is proposed by [38].
Definition 2.4. ([38]) The generalized Hukuhara difference of two fuzzy numbers Φ1,Φ2∈E (gH-difference in short) is stated as follows
Φ1⊖gHΦ2=Φ3⇔Φ1=Φ2+Φ3orΦ2=Φ1+(−1)Φ3. |
A function Φ:[σ1,σ2]→E is said to be d-increasing (d-decreasing) on [σ1,σ2] if for every ˇq∈[0,1]. The function ζ→diam[Φ(ζ)]ˇq is nondecreasing (nonincreasing) on [σ1,σ2]. If Φ is d-increasing or d-decreasing on [σ1,σ2], then we say that Φ is d-monotone on [σ1,σ2].
Definition 2.5. ([39])The generalized Hukuhara derivative of a fuzzy-valued function F:(σ1,σ2)→E at ζ0 is defined as
F′gH(ζ0)=limh→0F(ζ0+h)⊖gHF(ζ0)h, |
if (F)′gH(ζ0)∈E, we say that F is generalized Hukuhara differentiable (gH-differentiable) at ζ0.
Moreover, we say that F is [(i)−gH]-differentiable at ζ0 if
[F′gH(ζ0)]ˇq=[[limh→0F_(ζ0+h)⊖gHF_(ζ0)h]ˇq,[limh→0ˉF(ζ0+h)⊖gHˉF(ζ0)h]ˇq]=[(F_)′(ˇq,ζ0),(ˉF)′(ˇq,ζ0)], | (2.1) |
and that F is [(ii)−gH]-differentiable at ζ0 if
[F′gH(ζ0)]ˇq=[(ˉF)′(ˇq,ζ0),(F_)′(ˇq,ζ0)]. | (2.2) |
Definition 2.6. ([49]) We state that a point ζ0∈(σ1,σ2), is a switching point for the differentiability of F, if in any neighborhood U of ζ0 there exist points ζ1<ζ0<ζ2 such that
Type Ⅰ. at ζ1 (2.1) holds while (2.2) does not hold and at ζ2 (2.2) holds and (2.1) does not hold, or
Type Ⅱ. at ζ1 (2.2) holds while (2.1) does not hold and at ζ2 (2.1) holds and (2.2) does not hold.
Definition 2.7. ([23]) For β∈(0,1] and let the left-sided GPF-integral operator of order ϑ of F is defined as follows
Iϑ,βσ+1F(ζ)=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1F(ν)dν,ζ>σ1, | (2.3) |
where β∈(0,1], ϑ∈C, Re(ϑ)>0 and Γ(.) is the Gamma function.
Definition 2.8. ([23]) For β∈(0,1] and let the left-sided GPF-derivative operator of order ϑ of F is defined as follows
Dϑ,βσ+1F(ζ)=Dn,ββn−ϑΓ(n−ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)n−ϑ−1F(ν)dν, | (2.4) |
where β∈(0,1], ϑ∈C,Re(ϑ)>0,n=[ϑ]+1 and Dn,β represents the nth-derivative with respect to proportionality index β.
Definition 2.9. ([23]) For β∈(0,1] and let the left-sided GPF-derivative in the sense of Caputo of order ϑ of F is defined as follows
cDϑ,βσ+1F(ζ)=1βn−ϑΓ(n−ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)n−ϑ−1(Dn,βF)(ν)dν, | (2.5) |
where β∈(0,1], ϑ∈C,Re(ϑ)>0 and n=[ϑ]+1.
Let Φ∈L([σ1,σ2],E), then the GPF integral of order ϑ of the fuzzy function Φ is stated as:
Φβϑ(ζ)=(Iϑ,βσ+1Φ)(ζ)=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1Φ(ν)dν,ζ>σ1. | (2.6) |
Since [Φ(ζ)]ˇq=[Φ_(ˇq,ζ),ˉΦ(ˇq,ζ)] and 0<ϑ<1, we can write the fuzzy GPF-integral of the fuzzy mapping Φ depend on lower and upper mappingss, that is,
[(Iϑ,βσ+1Φ)(ζ)]ˇq=[(Iϑ,βσ+1Φ_)(ˇq,ζ),(Iϑ,βσ+1ˉΦ)(ˇq,ζ)],ζ≥σ1, | (2.7) |
where
(Iϑ,βσ+1Φ_)(ˇq,ζ)=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1Φ_(ˇq,ν)dν, | (2.8) |
and
(Iϑ,βσ+1ˉΦ)(ˇq,ζ)=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1ˉΦ(ˇq,ν)dν. | (2.9) |
Definition 2.10. For n∈N, order ϑ and type q hold n−1<ϑ≤n with 0≤q≤1. The left-sided fuzzy Hilfer-proportional gH-fractional derivative, with respect to ζ having β∈(0,1] of a function ζ∈Cβ1−γ[σ1,σ2], is stated as
(Dϑ,q,βσ+1Φ)(ζ)=(Iq(1−ϑ),βσ+1Dβ(I(1−q)(1−ϑ),βσ+1Φ))(ζ), |
where DβΦ(ν)=(1−β)Φ(ν)+βΦ′(ν) and if the gH-derivative Φ′(1−ϑ),β(ζ) exists for ζ∈[σ1,σ2], where
Φβ(1−ϑ)(ζ):=(I(1−ϑ),βσ+1Φ)(ζ)=1β1−ϑΓ(1−ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑΦ(ν)dν,ζ≥σ1. |
Definition 2.11. Let Φ′∈L([σ1,σ2],E) and the fractional generalized Hukuhara GPF-derivative of fuzzy-valued function Φ is stated as:
(gHDϑ,βσ+1Φ)(ζ)=I1−ϑ,βσ+1(Φ′gH)(ζ)=1β1−ϑΓ(1−ϑ)ϑ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑΦ′gH(ν)dν,ν∈(σ1,ζ). | (2.10) |
Furthermore, we say that Φ is GPF[(i)−gH]-differentiable at ζ0 if
[(gHDϑ,βσ+1)]ˇq=[[1β1−ϑΓ(1−ϑ)ϑ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑΦ_′gH(ν)dν]ˇq,[1β1−ϑΓ(1−ϑ)ϑ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑˉΦ′gH(ν)dν]ˇq]=[(gHD_ϑ,βσ+1)(ˇq,ζ),(gHˉDϑ,βσ+1)(ˇq,ζ)] | (2.11) |
and that Φ is GPF[(i)−gH]-differentiable at ζ0 if
[(gHDϑ,βσ+1)]ˇq=[(gHˉDϑ,βσ+1)(ˇq,ζ),(gHD_ϑ,βσ+1)(ˇq,ζ)]. | (2.12) |
Definition 2.12. We say that a point ζ0∈(σ1,σ2), is a switching point for the differentiability of F, if in any neighborhood U of ζ0 there exist points ζ1<ζ0<ζ2 such that
Type Ⅰ. at ζ1 (2.11) holds while (2.12) does not hold and at ζ2 (2.12) holds and (2.11) does not hold, or
Type Ⅱ. at ζ1 (2.12) holds while (2.11) does not hold and at ζ2 (2.11) holds and (2.12) does not hold.
Proposition 1. ([23]) Let ϑ,ϱ∈C such that Re(ϑ)>0 and Re(ϱ)>0. Then for any β∈(0,1], we have
(Iϑ,βσ+1eβ−1β(s−σ1)ϱ−1)(ζ)=Γ(ϱ)βϑΓ(ϱ+ϑ)eβ−1β(ζ−σ1)(ζ−σ1)ϱ+ϑ−1,(Dϑ,βσ+1eβ−1β(s−σ1)ϱ−1)(ζ)=Γ(ϱ)βϑΓ(ϱ−ϑ)eβ−1β(ζ−σ1)(ζ−σ1)ϱ−ϑ−1,(Iϑ,βσ+1eβ−1β(σ2−s)ϱ−1)(ζ)=Γ(ϱ)βϑΓ(ϱ+ϑ)eβ−1β(σ2−s)(σ2−ζ)ϱ+ϑ−1,(Dϑ,βσ+1eβ−1β(σ2−s)ϱ−1)(ζ)=Γ(ϱ)βϑΓ(ϱ−ϑ)eβ−1β(σ2−s)(σ2−s)ϱ−ϑ−1. |
Lemma 2.13. ([24])For β∈(0,1], ϑ>0, 0≤γ<1. If Φ∈Cγ[σ1,σ2] and I1−ϑσ+1Φ∈C1γ[σ1,σ2], then
(Iϑ,βσ+1Dϑ,βσ+1Φ)(ζ)=Φ(ζ)−eβ−1β(ζ−σ1)(ζ−σ1)ϑ−1βϑ−1Γ(ϑ)(I1−ϑ,βσ+1Φ)(σ1). |
Lemma 2.14. ([24]) Let Φ∈L1(σ1,σ2). If Dq(1−ϑ),βσ+1Φ exists on L1(σ1,σ2), then
Dϑ,q,βσ+1Iϑ,βσ+1Φ=Iq(1−ϑ),βσ+1Dq(1−ϑ),βσ+1Φ. |
Lemma 2.15. Suppose there is a d-monotone fuzzy mapping Φ∈AC([σ1,σ2],E), where [Φ(ζ)]ˇq=[Φ_(ˇq,ζ),ˉΦ(ˇq,ζ)] for 0≤ˇq≤1,σ1≤ζ≤σ2, then for 0<ϑ<1 and β∈(0,1], we have
(i)[(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Dϑ,q,βσ+1Φ_(ˇq,ζ),Dϑ,q,βσ+1ˉΦ(ˇq,ζ)] for ζ∈[σ1,σ2], if Φ is d-increasing;
(ii)[(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Dϑ,q,βσ+1ˉΦ(ˇq,ζ),Dϑ,q,βσ+1Φ_(ˇq,ζ)] for ζ∈[σ1,σ2], if Φ is d-decreasing.
Proof. It is to be noted that if Φ is d-increasing, then [Φ′(ζ)]ˇq=[ddζΦ_(ˇq,ζ),ddζˉΦ(ˇq,ζ)]. Taking into account Definition 2.10, we have
[(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Iq(1−ϑ),βσ+1Dβ(I(1−q)(1−ϑ),βσ+1Φ_)(ˇq,ζ),Iq(1−ϑ),βσ+1Dβ(I(1−q)(1−ϑ),βσ+1ˉΦ)(ˇq,ζ)]=[Dϑ,q,βσ+1Φ_(ˇq,ζ),Dϑ,q,βσ+1ˉΦ(ˇq,ζ)]. |
If Φ is d-decreasing, then [Φ′(ζ)]ˇq=[ddζˉΦ(ˇq,ζ),ddζΦ_(ˇq,ζ)], we have
[(Dϑ,q,βσ+1Φ)(ζ)]ˇq=[Iq(1−ϑ),βσ+1Dβ(I(1−q)(1−ϑ),βσ+1ˉΦ)(ˇq,ζ),Iq(1−ϑ),βσ+1Dβ(I(1−q)(1−ϑ),βσ+1Φ_)(ˇq,ζ)]=[Dϑ,q,βσ+1ˉΦ(ˇq,ζ),Dϑ,q,βσ+1Φ_(ˇq,ζ)]. |
This completes the proof.
Lemma 2.16. For β∈(0,1],ϑ∈(0,1). If Φ∈AC([σ1,σ2],E) is a d-monotone fuzzy function. We take
z1(ζ):=(Iϑ,βσ+1Φ)(ζ)=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1Φ(ν)dν, |
and
z(1−ϑ),β1:=(I(1−ϑ),βσ+1Φ)(ζ)=1β1−ϑΓ(1−ϑ)ϑ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑΦ′gH(ν)dν, |
is d-increasing on (σ1,σ2], then
(Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=Φ(ζ)⊖m∑j=1RjΦ(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1, |
and
(Dϑ,q,βσ+1Iϑ,βσ+1Φ)(ζ)=Φ(ζ). |
Proof. If z1(ζ) is d-increasing on [σ1,σ2] or z1(ζ) is d-decreasing on [σ1,σ2] and z(1−ϑ),β1(ζ) is d-increasing on (σ1,σ2].
Utilizing the Definitions 2.6, 2.10 and Lemma 2.13 with the initial condition (I1−γ,βσ+1Φ)(σ1)=0, we have
(Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=(Iϑ,βσ+1Iq(1−ϑ),βσ+1DβI(1−q)(1−ϑ),βσ+1Φ)(ζ)=(Iγ,βσ+1DβI1−γ,βσ+1Φ)(ζ)=(Iγ,βσ+1Dγ,βσ+1Φ)(ζ)=Φ(ζ)⊖I1−γ,βσ+1Φβγ−1Γ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1. | (2.13) |
Now considering Proposition 1, Lemma 2.13 and Lemma 2.14, we obtain
(Dϑ,q,βσ+1Iϑ,βσ+1Φ)(ζ)=(Iq(1−ϑ),βσ+1Dq(1−ϑ),βσ+1Φ)(ζ)=Φ(ζ)⊖(I1−q(1−ϑ),βσ+1Φ)(σ1)eβ−1β(ζ−σ1)βq(1−ϑ)Γ(q(1−ϑ))(ζ−σ1)q(1−ϑ)−1=Φ(ζ). |
On contrast, since Φ∈AC([σ1,σ2],E), there exists a constant K such that K=supζ∈[σ1,σ2]¯D0[Φ(ζ),ˆ0].
Then
¯D0[Iϑ,βσ+1Φ(ζ),ˆ0]≤K1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1dν≤K1βϑΓ(ϑ)ζ∫σ1|eβ−1β(ζ−ν)|(ζ−ν)ϑ−1dν=KβϑΓ(ϑ+1)(ζ−σ1)ϑ, |
where we have used the fact |eβ−1βζ|<1 and Iϑ,βσ+1Φ(ζ)=0 and ζ=σ1.
This completes the proof.
Lemma 2.17. Let there be a continuous mapping Φ:[σ1,σ2]→R+ on [σ1,σ2] and hold Dϑ,q,βσ+1Φ(ζ)≤F(ξ,Φ(ξ)),ξ≥σ1, where F∈C([σ1,σ1]×R+,R+). Assume that m(ζ)=m(ζ,σ1,ξ0) is the maximal solution of the IVP
Dϑ,q,βσ+1ξ(ζ)=F(ζ,ξ),(I1−γ,βσ+1ξ)(σ1)=ξ0≥0, | (2.14) |
on [σ1,σ2]. Then, if Φ(σ1)≤ξ0, we have Φ(ζ)≤m(ζ),ζ∈[σ1,σ2].
Proof. The proof is simple and can be derived as parallel to Theorem 2.2 in [53].
Lemma 2.18. Assume the IVP described as:
Dϑ,q,βσ+1Φ(ζ)=F(ζ,Φ(ζ)),(I1−γ,βσ+1Φ)(σ1)=Φ0=0,ζ∈[σ1,σ2]. | (2.15) |
Let α>0 be a given constant and B(Φ0,α)={Φ∈R:|Φ−Φ0|≤α}. Assume that the real-valued functions F:[σ1,σ2]×[0,α]→R+ satisfies the following assumptions:
(i) F∈C([σ1,σ2]×[0,α],R+),F(ζ,0)≡0,0≤F(ζ,Φ)≤MF for all (ζ,Φ)∈[σ1,σ2]×[0,α];
(ii) F(ζ,Φ) is nondecreasing in Φ for every ζ∈[σ1,σ2]. Then the problem (2.15) has at least one solution defined on [σ1,σ2] and Φ(ζ)∈B(Φ0,α).
Proof. The proof is simple and can be derived as parallel to Theorem 2.3 in [53].
In this investigation, we find the existence and uniqueness of solution to problem 1.3 by utilizing the successive approximation technique by considering the generalized Lipschitz condition of the right-hand side.
Lemma 3.1. For γ=ϑ+q(1−ϑ),ϑ∈(0,1),q∈[0,1] with β∈(0,1], and let there is a fuzzy function F:(σ1,σ2]×E→E such that ζ→F(ζ,Φ) belongs to Cβγ([σ1,σ2],E) for any Φ∈E. Then a d-monotone fuzzy function Φ∈C([σ1,σ2],E) is a solution of IVP (1.3) if and only if Φ satisfies the integral equation
Φ(ζ)⊖gHm∑j=1RjΦ(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1F(ν,Φ(ν))dν,ζ∈[σ1,σ2],j=1,2,...,m. | (3.1) |
and the fuzzy function ζ→I1−γσ+1F(ζ,Φ) is d-increasing on (σ1,σ2].
Proof. Let Φ∈C([σ1,σ2],E) be a d-monotone solution of (1.3), and considering z1(ζ):=Φ(ζ)⊖gH(I1−γ,βσ+1Φ)(σ1),ζ∈(σ1,σ2]. Since Φ is d-monotone on [σ1,σ2], it follows that ζ→z1(ζ) is d-increasing on [σ1,σ2] (see [43]).
From (1.3) and Lemma 2.16, we have
(Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=Φ(ζ)⊖m∑j=1RjΦ(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1,∀ζ∈[σ1,σ2]. | (3.2) |
Since F(ζ,Φ)∈Cγ([σ1,σ2],E) for any Φ∈E, and from (1.3), observes that
(Iϑ,βσ+1Dϑ,q,βσ+1Φ)(ζ)=Iϑ,βσ+1F(ζ,Φ(ζ))=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1F(ν,Φ(ν))dν,∀ζ∈[σ1,σ2]. | (3.3) |
Additionally, since z1(ζ) is d-increasing on (σ1,σ2]. Also, we observe that ζ→Fϑ,β(ζ,Φ) is also d-increasing on (σ1,σ2].
Reluctantly, merging (3.2) and (3.3), we get the immediate consequence.
Further, suppose Φ∈C([σ1,σ2],E) be a d-monotone fuzzy function fulfills (3.1) and such that ζ→Fϑ,β(ζ,Φ) is d-increasing on (σ1,σ2]. By the continuity of the fuzzy mapping F, the fuzzy mapping ζ→Fϑ,β(ζ,Φ) is continuous on (σ1,σ2] with Fϑ,β(σ1,Φ(σ1))=limζ→σ+1Fϑ,β(ζ,Φ)=0. Then
Φ(ζ)=m∑j=1RjΦ(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1+(Iϑ,βσ+1F(ζ,ζ))(ζ),I1−γ,βσ+1Φ(ζ)=m∑j=1RjΦ(ζj)+(I1−q(1−ϑ)σ+1F(ζ,Φ(ζ)))(ζ), |
and
I1−γ,βσ+1Φ(0)=m∑j=1RjΦ(ζj). |
Moreover, since ζ→Fϑ,β(ζ,Φ) is d-increasing on (σ1,σ2]. Applying, the operator Dϑ,q,βσ+1 on (3.1), yields
Dϑ,q,βσ+1(Φ(ζ)⊖gHm∑j=1RjΦ(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1)=Dϑ,q,βσ+1(1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1F(ν,Φ(ν))dν)=F(ζ,Φ(ζ)). |
This completes the proof.
In our next result, we use the following assumption. For a given constant ℏ>0, and let B(Φ0,ℏ)={Φ∈E:¯D0[Φ,Φ0]≤ℏ}.
Theorem 3.2. Let F∈C([σ1,σ2]×B(Φ0,ℏ),E) and suppose that the subsequent assumptions hold:
(i) there exists a positive constant MF such that ¯D0[F(ζ,z1),ˆ0]≤MF, for all (ζ,z1)∈[σ1,σ2]×B(Φ0,ℏ);
(ii) for every ζ∈[σ1,σ2] and every z1,ω∈B(Φ0,ℏ),
¯D0[F(ζ,z1),F(ζ,ω)]≤g(ζ,¯D0[z1,ω]), | (3.4) |
where g(ζ,.)∈C([σ1,σ2]×[0,β],R+) satisfies the assumption in Lemma 2.18 given that problem (2.15) has only the solution ϕ(ζ)≡0 on [σ1,σ2]. Then the subsequent successive approximations given by Φ0(ζ)=Φ0 and for n=1,2,...,
Φn(ζ)⊖gHm∑j=1RjΦ(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1F(ν,Φn−1(ν))dν, |
converges consistently to a fixed point of problem (1.3) on certain interval [σ1,T] for some T∈(σ1,σ2] given that the mapping ζ→Iϑ,βσ+1F(ζ,Φn(ζ)) is d-increasing on [σ1,T].
Proof. Take σ1<ζ∗ such that ζ∗≤[βϑℏ.Γ(1+ϑ)M+σ1]1ϑ, where M=max{Mg,MF} and put T:=min{ζ∗,σ2}. Let S be a set of continuous fuzzy functions Φ such that ω(σ1)=Φ0 and ω(ζ)∈B(Φ0,ℏ) for all ζ∈[σ1,T]. Further, we suppose the sequence of continuous fuzzy function {Φn}∞n=0 given by Φ0(ζ)=Φ0,∀ζ∈[σ1,T] and for n=1,2,..,
Φn(ζ)⊖gHm∑j=1RjΦn−1(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1F(ν,Φn−1(ν))dν. | (3.5) |
Firstly, we show that Φn(ζ)∈C([σ1,T],B(Φ0,ℏ)). For n≥1 and for any ζ1,ζ2∈[σ1,T] with ζ1<ζ2, we have
¯D0(Φn(ζ1)⊖gHm∑j=1RjΦn−1(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1,Φn(ζ2)⊖gHm∑j=1RjΦn−1(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1)≤1βϑΓ(ϑ)ζ1∫σ1[eβ−1β(ζ1−ν)(ζ1−ν)ϑ−1−eβ−1β(ζ2−ν)(ζ2−ν)ϑ−1]¯D0[F(ν,Φn−1(ν)),ˆ0]dν+1βϑΓ(ϑ)ζ2∫ζ1eβ−1β(ζ2−ν)(ζ2−ν)ϑ−1¯D0[F(ν,Φn−1(ν)),ˆ0]dν. |
Using the fact that |eβ−1βζ|<1, then, on the right-hand side from the last inequality, the subsequent integral becomes 1βϑΓ(1+ϑ)(ζ2−ζ1)ϑ. Therefore, with the similar assumption as we did above, the first integral reduces to 1βϑΓ(1+ϑ)[(ζ1−σ1)ϑ−(ζ2−σ1)ϑ+(ζ2−ζ1)ϑ]. Thus, we conclude
¯D0[Φn((ζ1),Φn(ζ2))]≤MFβϑΓ(1+ϑ)[(ζ1−σ1)ϑ−(ζ2−σ1)ϑ+2(ζ2−ζ1)ϑ]≤2MFβϑΓ(1+ϑ)(ζ2−ζ1)ϑ. |
In the limiting case as ζ1→ζ2, then the last expression of the above inequality tends to 0, which shows Φn is a continuous function on [σ1,T] for all n≥1.
Moreover, it follows that Φn∈B(Φ0,ℏ) for all n≥0,ζ∈[σ1,T] if and only if Φn(ζ)⊖gHm∑j=1RjΦ(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1∈B(0,ℏ) for all ζ∈[σ1,T] and for all n≥0.
Also, if we assume that Φn−1(ζ)∈S for all ζ∈[σ1,T],n≥2, then
¯D0[Φn(ζ)⊖gHm∑j=1RjΦn−1(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1,ˆ0]≤1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1¯D0[F(ν,Φn−1(ν)),ˆ0]dν=MF(ζ−σ1)ϑβϑΓ(1+ϑ)≤ℏ. |
It follows that Φn(ζ)∈S,∀∈[σ1,T].
Henceforth, by mathematical induction, we have Φn(ζ)∈S,∀ζ∈[σ1,T] and ∀n≥1.
Further, we show that the sequence Φn(ζ) converges uniformly to a continuous function Φ∈C([σ1,T],B(Φ0,ℏ)). By assertion (ii) and mathematical induction, we have for ζ∈[σ1,T]
¯D0[Φn+1(ζ)⊖gHm∑j=1RjΦn(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1,Φn(ζ)⊖gHm∑j=1RjΦn−1(ζj)βγΓ(γ)eβ−1β(ζ−σ1)(ζ−σ1)γ−1]≤ϕn(ζ),n=0,1,2,..., | (3.6) |
where ϕn(ζ) is defined as follows:
ϕn(ζ)=1βϑΓ(ϑ)ζ∫σ1eβ−1β(ζ−ν)(ζ−ν)ϑ−1g(ν,ϕn−1(ν))dν, | (3.7) |
where we have used the fact that |eβ−1βζ|<1 and ϕ0(ζ)=M(ζ−σ1)ϑβϑΓ(ϑ+1). Thus, we have, for ζ∈[σ1,T] and for n=0,1,2,...,
¯D0[Dϑ,qσ+1Φn+1(ζ),Dϑ,qσ+1Φn(ζ)]≤¯D0[F(ζ,Φn(ζ)),F(ζ,Φn−1(ζ))]≤g(ζ,¯D0[Φn(ζ),Φn−1(ζ)])≤g(ζ,ϕn−1(ζ)). |
Let n≤m and ζ∈[σ1,T], then one obtains
Dϑ,qσ+1¯D0[Φn(ζ),Φm(ζ)]≤¯D0[Dϑ,qσ+1Φn(ζ),Dϑ,qσ+1Φm(ζ)]≤¯D0[Dϑ,qσ+1Φn(ζ),Dϑ,qσ+1Φn+1(ζ)]+¯D0[Dϑ,qσ+1Φn+1(ζ),Dϑ,qσ+1Φm+1(ζ)]+¯D0[Dϑ,qσ+1Φm+1(ζ),Dϑ,qσ+1Φm(ζ)]≤2g(ζ,ϕn−1(ζ))+g(ζ,¯D0[Φn(ζ),Φm(ζ)]). |
From we observe that the solution is a unique solution of problem (2.15) and uniformly converges to , for every there exists a natural number such that
Using the fact that and by using Lemma 2.17, we have for
(3.8) |
where is the maximal solution to the following
Taking into account Lemma 2.17, we deduce that converges uniformly to the maximal solution of (2.15) on as
Therefore, in view of (3.8), we can obtain is large enough such that, for
(3.9) |
Since is a complete metric space and (3.9) holds, thus converges uniformly to Hence
(3.10) |
Because of Lemma 3.1, the function is the solution to (1.3) on
In order to find the unique solution, assume that is another solution of problem (1.3) on We denote Then and for every we have
(3.11) |
Further, using the comaprison Lemma 2.17, we get where is a maximal solution of the IVP By asseration we have and hence
This completes the proof.
Corollary 1. For and let Assume that there exist positive constants such that, for every
Then the subsequent successive approximations given by and for
converges consistently to a fixed point of problem (1.3) on for certain given that the mapping is -increasing on
Example 3.3. For and Assume that the linear fuzzy - under Hilfer--derivative and moreover, the subsequent assumptions hold:
(3.12) |
Applying Lemma 3.1, we have
where and furthermore, assuming the diameter on the right part of the aforementioned equation is increasing. Observing fulfill the suppositions of Corollary 1.
In order to find the analytical view of (3.12), we utilized the technique of successive approximation. Putting and
Letting assuming there is a -increasing mapping then we have
In contrast, if we consider and is -decreasing, then we have
For , we have
if and there is -increasing mapping , we have
and there is -increasing mapping So, continuing inductively and in the limiting case, when we attain the solution
for every and is -increasing, or and is -decreasing, accordingly. Therefore, by means of Mittag-Leffler function the solution of problem (3.12) is expressed by
for every of and is -increasing. Alternately, if and is -decreasing, then we get the solution of problem (3.12)
Consider IVP
(4.1) |
where and is a real number and the operation denote the derivative of order is continuous in which fulfills certain supposition that will be determined later, and
(4.2) |
with such that
Now, we investigate the existence and uniqueness of the solution of problem (4.1). To establish the main consequences, we require the following necessary results.
Theorem 4.1. Let be a fuzzy-valued function on Then
is -differentiable at iff is -differentiable at
is -differentiable at iff is -differentiable at
Proof. In view of Definition 2.18 and Definition 2.11, the proof is straightforward.
Lemma 4.2. ([44]) Let there be a fuzzy valued mapping such that then
(4.3) |
Lemma 4.3. The IVP (4.1) is analogous to subsequent equation
(4.4) |
if be -differentiable,
(4.5) |
if be -differentiable, and
(4.6) |
if there exists a point such that is -differentiable on and -differentiable on and
Proof. By means of the integral operator (2.6) on both sides of (4.1), yields
(4.7) |
Utilizing Lemma 4.2 and Definition 2.6, we gat
(4.8) |
In view of Defnition 2.17 and Theorem 4.1, if be -differentiable,
(4.9) |
and if be -differentiable
(4.10) |
In addition, when we have a switchpoint of type the -differentiability changes from type to type at Then by (4.9) and (4.10) and Definition 2.12, The proof is easy to comprehend.
Also, we proceed with the following assumptions:
is continuous and there exist positive real functions such that
There exist a number such that
and
Theorem 4.4. Let be a bounded continuous functions and holds Then the IVP (4.1) has a unique solution which is -differentiable on given that where is given in
Proof. Assuming is -differentiability and be fixed. Propose a mapping by
(4.11) |
Next we prove that is contraction. For by considering of and by distance properties (2.3), one has
(4.12) |
Now, we find that
(4.13) |
Analogously,
(4.14) |
Then we have
(4.15) |
Consequently, is a contraction mapping on having a fixed point Henceforth, the IVP (4.1) has unique solution.
Theorem 4.5. For and let be a bounded continuous functions and satisfies Let the sequence is given by
(4.16) |
is described for any Then the sequence converges to fixed point of problem (4.1) which is -differentiable on given that where is defined in
Proof. We now prove that the sequence , given in (4.16), is a Cauchy sequence in To do just that, we'll require
(4.17) |
where
Since is Lipschitz continuous, In view of Definition (2.3), we show that
(4.18) |
Since promises that the sequence is a Cauchy sequence in Consequently, there exist such that converges to Thus, we need to illustrate that is a solution of the problem (4.1).
(4.19) |
In the limiting case, when Thus we have
(4.20) |
By Lemma 4.3, we prove that is a solution of the problem (4.1). In order to prove the uniqness of let be another solution of problem (4.1) on Utilizing Lemma 4.3, gets
Analogously, by employing the distance properties and Lipschitiz continuity of consequently, we deduce that since we have for all Hence, the proof is completed.
Example 4.6. Suppose the Cauchy problem by means of differential operator (2.4)
(4.21) |
where is analytic in and is analytic in the unit disk. Therefore, can be written as
Consider Then the solution can be formulated as follows:
(4.22) |
where are constants. Putting (4.22) in (4.21), yields
Since
then the simple computations gives the expression
Consequently, we get
Therefore, we have the subsequent solution
or equivalently
where is assumed to be arbitrary constant, we take
Therefore, for appropriate we have
where
The present investigation deal with an IVP for fuzzy and we employ a new scheme of successive approximations under generalized Lipschitz condition to obtain the existence and uniqueness consequences of the solution to the specified problem. Furthermore, another method to discover exact solutions of fuzzy by utilizing the solutions of integer order differential equations is considered. Additionally, the existence consequences for under - with fuzzy initial conditions are proposed. Also, the uniqueness of the so-called integrodifferential equations is verified. Meanwhile, we derived the equivalent integral forms of the original fuzzy whichis utilized to examine the convergence of these arrangements of conditions. Two examples enlightened the efficacy and preciseness of the fractional-order and the other one presents the exact solution by means of the Fox-Wright function. For forthcoming mechanisms, we will relate the numerical strategies for the estimated solution of nonlinear fuzzy
The authors would like to express their sincere thanks to the support of Taif University Researchers Supporting Project Number (TURSP-2020/217), Taif University, Taif, Saudi Arabia.
The authors declare that they have no competing interests.
[1] | G. Rajchakit, P. Agarwal, S. Ramalingam, Stability Analysis of Neural Networks, Springer, 2022. https://doi.org/10.1007/978-981-16-6534-9 |
[2] |
G. Rajchakit, R. Sriraman, N. Boonsatit, P. Hammachukiattikul, C. P. Lim, P. Agarwal, Global exponential stability of clifford-valued neural networks with time-varying delays and impulsive effects, Adv. Differ. Equations, 2021 (2021), 1–21. https://doi.org/10.1186/s13662-021-03367-z doi: 10.1186/s13662-021-03367-z
![]() |
[3] |
N. Boonsatit, G. Rajchakit, R. Sriraman, C. P. Lim, P. Agarwal, Finite-/fixed-time synchronization of delayed clifford-valued recurrent neural networks, Adv. Differ. Equations, 2021 (2021), 1–25. https://doi.org/10.1186/s13662-021-03438-1 doi: 10.1186/s13662-021-03438-1
![]() |
[4] |
W. S. McCulloch, W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5 (1943), 115–133. https://doi.org/10.1007/BF02478259 doi: 10.1007/BF02478259
![]() |
[5] | D. O. Hebb, The Organization of Behavior: A Neuropsychological Theory, Psychology Press, 2005. https://doi.org/10.4324/9781410612403 |
[6] |
F. Rosenblatt, The perceptron: a probabilistic model for information storage and organization in the brain, Psychol. Rev., 65 (1958), 386–408. https://doi.org/10.1037/h0042519 doi: 10.1037/h0042519
![]() |
[7] | M. Minsky, S. Papert, An introduction to computational geometry, Cambridge tiass., HIT, 479 (1969), 480. |
[8] |
J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, PNAS, 81 (1984), 3088–3092. https://doi.org/10.1073/pnas.81.10.3088 doi: 10.1073/pnas.81.10.3088
![]() |
[9] |
D. E. Rumelhart, G. E. Hinton, R. J. Williams, Learning representations by back-propagating errors, Nature, 323 (1986), 533–536. https://doi.org/10.1038/323533a0 doi: 10.1038/323533a0
![]() |
[10] |
L. O. Chua, L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35 (1988), 1257–1272. https://doi.org/10.1109/31.7600 doi: 10.1109/31.7600
![]() |
[11] |
Y. LeCun, L. Bottou, Y. Bengio, P. Haffner, Gradient-based learning applied to document recognition, Proc. IEEE, 86 (1998), 2278–2324. https://doi.org/10.1109/5.726791 doi: 10.1109/5.726791
![]() |
[12] |
G. E. Hinton, S. Osindero, Y. W. Teh, A fast learning algorithm for deep belief nets, Neural Comput., 18 (2006), 1527–1554. https://doi.org/10.1162/neco.2006.18.7.1527 doi: 10.1162/neco.2006.18.7.1527
![]() |
[13] |
H. Lin, C. Wang, Y. Sun, T. Wang, Generating-scroll chaotic attractors from a memristor-based magnetized hopfield neural network, IEEE Trans. Circuits Syst. II Express Briefs, 70 (2022), 311–315. https://doi.org/10.1109/TCSII.2022.3212394 doi: 10.1109/TCSII.2022.3212394
![]() |
[14] |
H. Liu, L. Ma, Z. Wang, Y. Liu, F. E. Alsaadi, An overview of stability analysis and state estimation for memristive neural networks, Neurocomputing, 391 (2020), 1–12. https://doi.org/10.1016/j.neucom.2020.01.066 doi: 10.1016/j.neucom.2020.01.066
![]() |
[15] |
Z. Zeng, D. S. Huang, Z. Wang, Pattern memory analysis based on stability theory of cellular neural networks, Appl. Math. Modell., 32 (2008), 112–121. https://doi.org/10.1016/j.apm.2006.11.010 doi: 10.1016/j.apm.2006.11.010
![]() |
[16] |
Z. Wang, S. Joshi, S. Savel'ev, W. Song, R. Midya, Y. Li, et al., Fully memristive neural networks for pattern classification with unsupervised learning, Nat. Electron., 1 (2018), 137–145. https://doi.org/10.1038/s41928-018-0023-2 doi: 10.1038/s41928-018-0023-2
![]() |
[17] |
C. Tsioustas, P. Bousoulas, J. Hadfield, T. P. Chatzinikolaou, I. A. Fyrigos, V. Ntinas, et al., Simulation of low power self-selective memristive neural networks for in situ digital and analogue artificial neural network applications, IEEE Trans. Nanotechnol., 21 (2022), 505–513. https://doi.org/10.1109/TNANO.2022.3205698 doi: 10.1109/TNANO.2022.3205698
![]() |
[18] |
B. Seyfi, A. Rassoli, M. Imeni Markhali, N. Fatouraee, Characterization of the nonlinear biaxial mechanical behavior of human ureter using constitutive modeling and artificial neural networks, J. Appl. Comput. Mech., 8 (2022), 1186–1195. https://doi.org/10.22055/JACM.2020.33703.2272 doi: 10.22055/JACM.2020.33703.2272
![]() |
[19] |
M. Aliasghary, H. Mobki, H. M. Ouakad, Pull-in phenomenon in the electrostatically micro-switch suspended between two conductive plates using the artificial neural network, J. Appl. Comput. Mech., 8 (2022), 1222–1235. https://doi.org/10.22055/JACM.2021.38569.3248 doi: 10.22055/JACM.2021.38569.3248
![]() |
[20] |
H. Guo, J. Zhang, Y. Zhao, H. Zhang, J. Zhao, X. Yang, et al., Accelerated key distribution method for endogenously secure optical communication by synchronized chaotic system based on fiber channel feature, Opt. Fiber Technol., 75 (2023), 103162. https://doi.org/10.1016/j.yofte.2022.103162 doi: 10.1016/j.yofte.2022.103162
![]() |
[21] |
C. Zhou, C. Wang, W. Yao, H. Lin, Observer-based synchronization of memristive neural networks under dos attacks and actuator saturation and its application to image encryption, Appl. Math. Comput., 425 (2022), 127080. https://doi.org/10.1016/j.amc.2022.127080 doi: 10.1016/j.amc.2022.127080
![]() |
[22] |
H. L. Li, C. Hu, L. Zhang, H. Jiang, J. Cao, Complete and finite-time synchronization of fractional-order fuzzy neural networks via nonlinear feedback control, Fuzzy Sets Syst., 443 (2022), 50–69. https://doi.org/10.1016/j.fss.2021.11.004 doi: 10.1016/j.fss.2021.11.004
![]() |
[23] |
W. Chen, Y. Yu, X. Hai, G. Ren, Adaptive quasi-synchronization control of heterogeneous fractional-order coupled neural networks with reaction-diffusion, Appl. Math. Comput., 427 (2022), 127145. https://doi.org/10.1016/j.amc.2022.127145 doi: 10.1016/j.amc.2022.127145
![]() |
[24] |
Y. Shen, X. Liu, Generalized synchronization of delayed complex-valued dynamical networks via hybrid control, Commun. Nonlinear Sci. Numer. Simul., 118 (2023), 107057. https://doi.org/10.1016/j.cnsns.2022.107057 doi: 10.1016/j.cnsns.2022.107057
![]() |
[25] |
A. Abdurahman, M. Abudusaimaiti, H. Jiang, Fixed/predefined-time lag synchronization of complex-valued bam neural networks with stochastic perturbations, Appl. Math. Comput., 444 (2023), 127811. https://doi.org/10.1016/j.amc.2022.127811 doi: 10.1016/j.amc.2022.127811
![]() |
[26] |
H. Pu, F. Li, Fixed-time projective synchronization of delayed memristive neural networks via aperiodically semi-intermittent switching control, ISA Trans., 133 (2023), 302–316. https://doi.org/10.1016/j.isatra.2022.07.022 doi: 10.1016/j.isatra.2022.07.022
![]() |
[27] |
J. Luo, S. Qu, Y. Chen, X. Chen, Z. Xiong, Synchronization, circuit and secure communication implementation of a memristor-based hyperchaotic system using single input controller, Chin. J. Phys., 71 (2021), 403–417. https://doi.org/10.1016/j.cjph.2021.03.009 doi: 10.1016/j.cjph.2021.03.009
![]() |
[28] |
V. L. Freitas, S. Yanchuk, M. Zaks, E. E. Macau, Synchronization-based symmetric circular formations of mobile agents and the generation of chaotic trajectories, Commun. Nonlinear Sci. Numer. Simul., 94 (2021), 105543. https://doi.org/10.1016/j.cnsns.2020.105543 doi: 10.1016/j.cnsns.2020.105543
![]() |
[29] |
J. Xiang, J. Ren, M. Tan, Asymptotical synchronization for complex-valued stochastic switched neural networks under the sampled-data controller via a switching law, Neurocomputing, 514 (2022), 414–425. https://doi.org/10.1016/j.neucom.2022.09.152 doi: 10.1016/j.neucom.2022.09.152
![]() |
[30] |
Z. Dong, X. Wang, X. Zhang, M. Hu, T. N. Dinh, Global exponential synchronization of discrete-time high-order switched neural networks and its application to multi-channel audio encryption, Nonlinear Anal. Hybrid Syst., 47 (2023), 101291. https://doi.org/10.1016/j.nahs.2022.101291 doi: 10.1016/j.nahs.2022.101291
![]() |
[31] | S. Gong, Z. Guo, S. Wen, Finite-time synchronization of T-S fuzzy memristive neural networks with time delay, Fuzzy Sets Syst., In press. https://doi.org/10.1016/j.fss.2022.10.013 |
[32] |
C. Zhou, C. Wang, Y. Sun, W. Yao, H. Lin, Cluster output synchronization for memristive neural networks, Inf. Sci., 589 (2022), 459–477. https://doi.org/10.1016/j.ins.2021.12.084 doi: 10.1016/j.ins.2021.12.084
![]() |
[33] |
K. Subramanian, P. Muthukumar, S. Lakshmanan, State feedback synchronization control of impulsive neural networks with mixed delays and linear fractional uncertainties, Appl. Math. Comput., 321 (2018), 267–281. https://doi.org/10.1016/j.amc.2017.10.038 doi: 10.1016/j.amc.2017.10.038
![]() |
[34] |
X. Li, W. Zhang, J. Fang, H. Li, Finite-time synchronization of memristive neural networks with discontinuous activation functions and mixed time-varying delays, Neurocomputing, 340 (2019), 99–109. https://doi.org/10.1016/j.neucom.2019.02.051 doi: 10.1016/j.neucom.2019.02.051
![]() |
[35] |
B. Lu, H. Jiang, C. Hu, A. Abdurahman, Spacial sampled-data control for output synchronization of directed coupled reaction-diffusion neural networks with mixed delays, Neural Networks, 123 (2020), 429–440. https://doi.org/10.1016/j.neunet.2019.12.026 doi: 10.1016/j.neunet.2019.12.026
![]() |
[36] |
W. Tai, Q. Teng, Y. Zhou, J. Zhou, Z. Wang, Chaos synchronization of stochastic reaction-diffusion time-delay neural networks via non-fragile output-feedback control, Appl. Math. Comput., 354 (2019), 115–127. https://doi.org/10.1016/j.amc.2019.02.028 doi: 10.1016/j.amc.2019.02.028
![]() |
[37] |
A. Kazemy, R. Saravanakumar, J. Lam, Master-slave synchronization of neural networks subject to mixed-type communication attacks, Inf. Sci., 560 (2021), 20–34. https://doi.org/10.1016/j.ins.2021.01.063 doi: 10.1016/j.ins.2021.01.063
![]() |
[38] |
W. Zhang, S. Yang, C. Li, W. Zhang, X. Yang, Stochastic exponential synchronization of memristive neural networks with time-varying delays via quantized control, Neural Networks, 104 (2018), 93–103. https://doi.org/10.1016/j.neunet.2018.04.010 doi: 10.1016/j.neunet.2018.04.010
![]() |
[39] |
X. Yang, Z. Cheng, X. Li, T. Ma, Exponential synchronization of coupled neutral-type neural networks with mixed delays via quantized output control, J. Franklin Inst., 356 (2019), 8138–8153. https://doi.org/10.1016/j.jfranklin.2019.07.006 doi: 10.1016/j.jfranklin.2019.07.006
![]() |
[40] |
R. Tang, X. Yang, X. Wan, Y. Zou, Z. Cheng, H. M. Fardoun, Finite-time synchronization of nonidentical bam discontinuous fuzzy neural networks with delays and impulsive effects via non-chattering quantized control, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104893. https://doi.org/10.1016/j.cnsns.2019.104893 doi: 10.1016/j.cnsns.2019.104893
![]() |
[41] |
M. Xu, J. L. Wang, P. C. Wei, Synchronization for coupled reaction-diffusion neural networks with and without multiple time-varying delays via pinning-control, Neurocomputing, 227 (2017), 82–91. https://doi.org/10.1016/j.neucom.2016.10.063 doi: 10.1016/j.neucom.2016.10.063
![]() |
[42] |
Y. Li, B. Luo, D. Liu, Z. Yang, Robust synchronization of memristive neural networks with strong mismatch characteristics via pinning control, Neurocomputing, 289 (2018), 144–154. https://doi.org/10.1016/j.neucom.2018.02.006 doi: 10.1016/j.neucom.2018.02.006
![]() |
[43] |
Q. Tang, J. Jian, Exponential synchronization of inertial neural networks with mixed time-varying delays via periodically intermittent control, Neurocomputing, 338 (2019), 181–190. https://doi.org/10.1016/j.neucom.2019.01.096 doi: 10.1016/j.neucom.2019.01.096
![]() |
[44] |
S. Cai, X. Li, P. Zhou, J. Shen, Aperiodic intermittent pinning control for exponential synchronization of memristive neural networks with time-varying delays, Neurocomputing, 332 (2019), 249–258. https://doi.org/10.1016/j.neucom.2018.12.070 doi: 10.1016/j.neucom.2018.12.070
![]() |
[45] |
Y. Yang, Y. He, M. Wu, Intermittent control strategy for synchronization of fractional-order neural networks via piecewise lyapunov function method, J. Franklin Inst., 356 (2019), 4648–4676. https://doi.org/10.1016/j.jfranklin.2018.12.020 doi: 10.1016/j.jfranklin.2018.12.020
![]() |
[46] |
H. A. Tang, S. Duan, X. Hu, L. Wang, Passivity and synchronization of coupled reaction-cdiffusion neural networks with multiple time-varying delays via impulsive control, Neurocomputing, 318 (2018), 30–42. https://doi.org/10.1016/j.neucom.2018.08.005 doi: 10.1016/j.neucom.2018.08.005
![]() |
[47] |
Z. Xu, D. Peng, X. Li, Synchronization of chaotic neural networks with time delay via distributed delayed impulsive control, Neural Networks, 118 (2019), 332–337. https://doi.org/10.1016/j.neunet.2019.07.002 doi: 10.1016/j.neunet.2019.07.002
![]() |
[48] |
M. Li, X. Li, X. Han, J. Qiu, Leader-following synchronization of coupled time-delay neural networks via delayed impulsive control, Neurocomputing, 357 (2019), 101–107. https://doi.org/10.1016/j.neucom.2019.04.063 doi: 10.1016/j.neucom.2019.04.063
![]() |
[49] |
S. Wu, X. Li, Y. Ding, Saturated impulsive control for synchronization of coupled delayed neural networks, Neural Networks, 141 (2021), 261–269. https://doi.org/10.1016/j.neunet.2021.04.012 doi: 10.1016/j.neunet.2021.04.012
![]() |
[50] |
Y. Zhou, H. Zhang, Z. Zeng, Synchronization of memristive neural networks with unknown parameters via event-triggered adaptive control, Neural Networks, 139 (2021), 255–264. https://doi.org/10.1016/j.neunet.2021.02.029 doi: 10.1016/j.neunet.2021.02.029
![]() |
[51] |
A. Kazemy, J. Lam, X. M. Zhang, Event-triggered output feedback synchronization of master-slave neural networks under deception attacks, IEEE Trans. Neural Networks Learn. Syst., 33 (2022), 952–961. https://doi.org/10.1109/TNNLS.2020.3030638 doi: 10.1109/TNNLS.2020.3030638
![]() |
[52] |
X. Li, D. Peng, J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908–4913. https://doi.org/10.1109/TAC.2020.2964558 doi: 10.1109/TAC.2020.2964558
![]() |
[53] |
M. Wang, X. Li, P. Duan, Event-triggered delayed impulsive control for nonlinear systems with application to complex neural networks, Neural Networks, 150 (2022), 213–221. https://doi.org/10.1016/j.neunet.2022.03.007 doi: 10.1016/j.neunet.2022.03.007
![]() |
[54] |
Y. Fang, T. G. Kincaid, Stability analysis of dynamical neural networks, IEEE Trans. Neural Networks, 7 (1996), 996–1006. https://doi.org/10.1109/72.508941 doi: 10.1109/72.508941
![]() |
[55] |
K. A. Smith, Neural networks for combinatorial optimization: a review of more than a decade of research, Informs J. Comput., 11 (1999), 15–34. https://doi.org/10.1287/ijoc.11.1.15 doi: 10.1287/ijoc.11.1.15
![]() |
[56] |
T. Zhang, J. Zhou, Y. Liao, Exponentially stable periodic oscillation and mittag-leffler stabilization for fractional-order impulsive control neural networks with piecewise caputo derivatives, IEEE Trans. Cybern., 52 (2022), 9670–9683. https://doi.org/10.1109/TCYB.2021.3054946 doi: 10.1109/TCYB.2021.3054946
![]() |
[57] | E. N. Lorenz, The mechanics of vacillation, J. Atmos. Sci., 20 (1963), 448–465. https://doi.org/10.1175/1520-0469(1963)020 < 0448: TMOV > 2.0.CO; 2 |
[58] | K. Aihara, T. Takabe, M. Toyoda, Chaotic neural networks, Phys. Lett. A, 144 (1990), 333–340. https://doi.org/10.1016/0375-9601(90)90136-C |
[59] |
H. Lin, C. Wang, Q. Deng, C. Xu, Z. Deng, C. Zhou, Review on chaotic dynamics of memristive neuron and neural network, Nonlinear Dyn., 106 (2021), 959–973. https://doi.org/10.1007/s11071-021-06853-x doi: 10.1007/s11071-021-06853-x
![]() |
[60] | T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer Science & Business Media, 2012. |
[61] |
Y. Li, X. Wang, Almost periodic solutions in distribution of clifford-valued stochastic recurrent neural networks with time-varying delays, Chaos, Solitons Fractals, 153 (2021), 111536. https://doi.org/10.1016/j.chaos.2021.111536 doi: 10.1016/j.chaos.2021.111536
![]() |
[62] |
B. Kosko, Adaptive bidirectional associative memories, Appl. Opt., 26 (1987), 4947–4960. https://doi.org/10.1364/AO.26.004947 doi: 10.1364/AO.26.004947
![]() |
[63] |
J. Cao, New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. Lett. A, 307 (2003), 136–147. https://doi.org/10.1016/S0375-9601(02)01720-6 doi: 10.1016/S0375-9601(02)01720-6
![]() |
[64] |
J. Cao, J. Wang, Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 50 (2003), 34–44. https://doi.org/10.1109/TCSI.2002.807494 doi: 10.1109/TCSI.2002.807494
![]() |
[65] |
D. Li, Z. Zhang, X. Zhang, Periodic solutions of discrete-time quaternion-valued bam neural networks, Chaos, Solitons Fractals, 138 (2020), 110144. https://doi.org/10.1016/j.chaos.2020.110144 doi: 10.1016/j.chaos.2020.110144
![]() |
[66] |
H. R. Wilson, J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J., 12 (1972), 1–24. https://doi.org/10.1016/S0006-3495(72)86068-5 doi: 10.1016/S0006-3495(72)86068-5
![]() |
[67] |
R. Decker, V. W. Noonburg, A periodically forced wilson–cowan system with multiple attractors, SIAM J. Math. Anal., 44 (2012), 887–905. https://doi.org/10.1137/110823365 doi: 10.1137/110823365
![]() |
[68] |
B. Pollina, D. Benardete, V. W. Noonburg, A periodically forced wilson–cowan system, SIAM J. Appl. Math., 63 (2003), 1585–1603. https://doi.org/10.1137/S003613990240814X doi: 10.1137/S003613990240814X
![]() |
[69] |
V. Painchaud, N. Doyon, P. Desrosiers, Beyond wilson-cowan dynamics: oscillations and chaos without inhibition, Biol. Cybern., 116 (2022), 527–543. https://doi.org/10.1007/s00422-022-00941-w doi: 10.1007/s00422-022-00941-w
![]() |
[70] |
J. Cao, Global exponential stability and periodic solutions of delayed cellular neural networks, J. Comput. Syst. Sci., 60 (2000), 38–46. https://doi.org/10.1006/jcss.1999.1658 doi: 10.1006/jcss.1999.1658
![]() |
[71] |
S. Arik, V. Tavsanoglu, On the global asymptotic stability of delayed cellular neural networks, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., 47 (2000), 571–574. https://doi.org/10.1109/81.841859 doi: 10.1109/81.841859
![]() |
[72] |
Y. Li, J. Qin, Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays, Neurocomputing, 292 (2018), 91–103. https://doi.org/10.1016/j.neucom.2018.02.077 doi: 10.1016/j.neucom.2018.02.077
![]() |
[73] |
D. Békollè, K. Ezzinbi, S. Fatajou, D. E. H. Danga, F. M. Béssémè, Attractiveness of pseudo almost periodic solutions for delayed cellular neural networks in the context of measure theory, Neurocomputing, 435 (2021), 253–263. https://doi.org/10.1016/j.neucom.2020.12.047 doi: 10.1016/j.neucom.2020.12.047
![]() |
[74] |
A. Chen, L. Huang, J. Cao, Existence and stability of almost periodic solution for bam neural networks with delays, Appl. Math. Comput., 137 (2003), 177–193. https://doi.org/10.1016/S0096-3003(02)00095-4 doi: 10.1016/S0096-3003(02)00095-4
![]() |
[75] |
Q. Jiang, Q. R. Wang, Almost periodic solutions for quaternion-valued neural networks with mixed delays on time scales, Neurocomputing, 439 (2021), 363–373. https://doi.org/10.1016/j.neucom.2020.09.063 doi: 10.1016/j.neucom.2020.09.063
![]() |
[76] |
L. Pan, J. Cao, Anti-periodic solution for delayed cellular neural networks with impulsive effects, Nonlinear Anal. Real World Appl., 12 (2011), 3014–3027. https://doi.org/10.1016/j.nonrwa.2011.05.002 doi: 10.1016/j.nonrwa.2011.05.002
![]() |
[77] |
C. Ou, Anti-periodic solutions for high-order hopfield neural networks, Comput. Math. Appl., 56 (2008), 1838–1844. https://doi.org/10.1016/j.camwa.2008.04.029 doi: 10.1016/j.camwa.2008.04.029
![]() |
[78] |
L. Chua, Memristor-the missing circuit element, IEEE Trans. Circuit Theory, 18 (1971), 507–519. https://doi.org/10.1109/TCT.1971.1083337 doi: 10.1109/TCT.1971.1083337
![]() |
[79] | L. S. Zhang, Y. C. Jin, Y. D. Song, An overview of dynamics analysis and control of memristive neural networks with delays, Acta Autom. Sin., 47 (2021), 765–779. |
[80] |
M. Liao, C. Wang, Y. Sun, H. Lin, C. Xu, Memristor-based affective associative memory neural network circuit with emotional gradual processes, Neural Comput. Appl., 34 (2022), 13667–13682. https://doi.org/10.1007/s00521-022-07170-z doi: 10.1007/s00521-022-07170-z
![]() |
[81] | Z. Deng, C. Wang, H. Lin, Y. Sun, A memristive spiking neural network circuit with selective supervised attention algorithm, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., Early Access, 2022. https://doi.org/10.1109/TCAD.2022.3228896 |
[82] |
H. Lin, C. Wang, C. Xu, X. Zhang, H. H. Iu, A memristive synapse control method to generate diversified multi-structure chaotic attractors, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 42 (2023), 942–955. https://doi.org/10.1109/TCAD.2022.3186516 doi: 10.1109/TCAD.2022.3186516
![]() |
[83] |
H. Lin, C. Wang, L. Cui, Y. Sun, X. Zhang, W. Yao, Hyperchaotic memristive ring neural network and application in medical image encryption, Nonlinear Dyn., 110 (2022), 841–855. https://doi.org/10.1007/s11071-022-07630-0 doi: 10.1007/s11071-022-07630-0
![]() |
[84] |
Z. Wen, C. Wang, Q. Deng, H. Lin, Regulating memristive neuronal dynamical properties via excitatory or inhibitory magnetic field coupling, Nonlinear Dyn., 110 (2022), 1–13. https://doi.org/10.1007/s11071-022-07813-9 doi: 10.1007/s11071-022-07813-9
![]() |
[85] |
Z. Guo, J. Wang, Z. Yan, Attractivity analysis of memristor-based cellular neural networks with time-varying delays, IEEE Trans. Neural Networks Learn. Syst., 25 (2013), 704–717. https://doi.org/10.1109/TNNLS.2013.2280556 doi: 10.1109/TNNLS.2013.2280556
![]() |
[86] |
L. Wang, Y. Shen, Finite-time stabilizability and instabilizability of delayed memristive neural networks with nonlinear discontinuous controller, IEEE Trans. Neural Networks Learn. Syst., 26 (2015), 2914–2924. https://doi.org/10.1109/TNNLS.2015.2460239 doi: 10.1109/TNNLS.2015.2460239
![]() |
[87] |
A. Wu, Z. Zeng, Algebraical criteria of stability for delayed memristive neural networks, Adv. Differ. Equations, 2015 (2015), 1–12. https://doi.org/10.1186/s13662-015-0449-z doi: 10.1186/s13662-015-0449-z
![]() |
[88] | J. P. Aubin, A. Cellina, Differential Inclusions: Set-valued Maps and Viability Theory, Springer Science & Business Media, 2012. |
[89] | A. F. Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems, Springer Science & Business Media, 2013. |
[90] | J. Hu, J. Wang, Global uniform asymptotic stability of memristor-based recurrent neural networks with time delays, in The 2010 International Joint Conference on Neural Networks (IJCNN), IEEE, (2010), 1–8. https://doi.org/10.1109/IJCNN.2010.5596359 |
[91] |
S. Wen, Z. Zeng, T. Huang, Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays, Neurocomputing, 97 (2012), 233–240. https://doi.org/10.1016/j.neucom.2012.06.014 doi: 10.1016/j.neucom.2012.06.014
![]() |
[92] |
K. Mathiyalagan, R. Anbuvithya, R. Sakthivel, J. H. Park, P. Prakash, Reliable stabilization for memristor-based recurrent neural networks with time-varying delays, Neurocomputing, 153 (2015), 140–147. https://doi.org/10.1016/j.neucom.2014.11.043 doi: 10.1016/j.neucom.2014.11.043
![]() |
[93] |
G. Zhang, Y. Shen, Q. Yin, J. Sun, Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays, Inf. Sci., 232 (2013), 386–396. https://doi.org/10.1016/j.ins.2012.11.023 doi: 10.1016/j.ins.2012.11.023
![]() |
[94] |
A. Wu, Z. Zeng, Global mittag–leffler stabilization of fractional-order memristive neural networks, IEEE Trans. Neural Networks Learn. Syst., 28 (2015), 206–217. https://doi.org/10.1109/TNNLS.2015.2506738 doi: 10.1109/TNNLS.2015.2506738
![]() |
[95] |
L. Chen, J. Cao, R. Wu, J. T. Machado, A. M. Lopes, H. Yang, Stability and synchronization of fractional-order memristive neural networks with multiple delays, Neural Networks, 94 (2017), 76–85. https://doi.org/10.1016/j.neunet.2017.06.012 doi: 10.1016/j.neunet.2017.06.012
![]() |
[96] |
J. Chen, Z. Zeng, P. Jiang, On the periodic dynamics of memristor-based neural networks with time-varying delays, Inf. Sci., 279 (2014), 358–373. https://doi.org/10.1016/j.ins.2014.03.124 doi: 10.1016/j.ins.2014.03.124
![]() |
[97] |
J. Zhao, Exponential stabilization of memristor-based neural networks with unbounded time-varying delays, Sci. China Inf. Sci., 64 (2021), 1–3. https://doi.org/10.1007/s11432-018-9817-4 doi: 10.1007/s11432-018-9817-4
![]() |
[98] |
Z. Zhang, X. Liu, D. Zhou, C. Lin, J. Chen, H. Wang, Finite-time stabilizability and instabilizability for complex-valued memristive neural networks with time delays, IEEE Trans. Syst. Man Cybern.: Syst., 48 (2017), 2371–2382. https://doi.org/10.1109/TSMC.2017.2754508 doi: 10.1109/TSMC.2017.2754508
![]() |
[99] |
M. Syed Ali, G. Narayanan, Z. Orman, V. Shekher, S. Arik, Finite time stability analysis of fractional-order complex-valued memristive neural networks with proportional delays, Neural Process. Lett., 51 (2020), 407–426. https://doi.org/10.1007/s11063-019-10097-7 doi: 10.1007/s11063-019-10097-7
![]() |
[100] |
Z. Cai, L. Huang, Finite-time stabilization of delayed memristive neural networks: Discontinuous state-feedback and adaptive control approach, IEEE Trans. Neural Networks Learn. Syst., 29 (2017), 856–868. https://doi.org/10.1109/TNNLS.2017.2651023 doi: 10.1109/TNNLS.2017.2651023
![]() |
[101] |
L. Wang, Z. Zeng, M. F. Ge, A disturbance rejection framework for finite-time and fixed-time stabilization of delayed memristive neural networks, IEEE Trans. Syst. Man Cybern.: Syst., 51 (2019), 905–915. https://doi.org/10.1109/TSMC.2018.2888867 doi: 10.1109/TSMC.2018.2888867
![]() |
[102] |
Y. Sheng, H. Zhang, Z. Zeng, Stabilization of fuzzy memristive neural networks with mixed time delays, IEEE Trans. Fuzzy Syst., 26 (2017), 2591–2606. https://doi.org/10.1109/TFUZZ.2017.2783899 doi: 10.1109/TFUZZ.2017.2783899
![]() |
[103] |
Q. Xiao, Z. Zeng, Lagrange stability for T–S fuzzy memristive neural networks with time-varying delays on time scales, IEEE Trans. Fuzzy Syst., 26 (2017), 1091–1103. https://doi.org/10.1109/TFUZZ.2017.2704059 doi: 10.1109/TFUZZ.2017.2704059
![]() |
[104] |
Y. Sheng, F. L. Lewis, Z. Zeng, Exponential stabilization of fuzzy memristive neural networks with hybrid unbounded time-varying delays, IEEE Trans. Neural Networks Learn. Syst., 30 (2018), 739–750. https://doi.org/10.1109/TNNLS.2018.2852497 doi: 10.1109/TNNLS.2018.2852497
![]() |
[105] |
Y. Sheng, F. L. Lewis, Z. Zeng, T. Huang, Lagrange stability and finite-time stabilization of fuzzy memristive neural networks with hybrid time-varying delays, IEEE Trans. Cybern., 50 (2019), 2959–2970. https://doi.org/10.1109/TCYB.2019.2912890 doi: 10.1109/TCYB.2019.2912890
![]() |
[106] |
S. Yang, C. Li, T. Huang, Exponential stabilization and synchronization for fuzzy model of memristive neural networks by periodically intermittent control, Neural Networks, 75 (2016), 162–172. https://doi.org/10.1016/j.neunet.2015.12.003 doi: 10.1016/j.neunet.2015.12.003
![]() |
[107] |
X. Wang, J. H. Park, S. Zhong, H. Yang, A switched operation approach to sampled-data control stabilization of fuzzy memristive neural networks with time-varying delay, IEEE Trans. Neural Networks Learn. Syst., 31 (2019), 891–900. https://doi.org/10.1109/TNNLS.2019.2910574 doi: 10.1109/TNNLS.2019.2910574
![]() |
[108] |
R. Zhang, D. Zeng, J. H. Park, H. K. Lam, S. Zhong, Fuzzy adaptive event-triggered sampled-data control for stabilization of T-S fuzzy memristive neural networks with reaction-diffusion terms, IEEE Trans. Fuzzy Syst., 29 (2020), 1775–1785. https://doi.org/10.1109/TFUZZ.2020.2985334 doi: 10.1109/TFUZZ.2020.2985334
![]() |
[109] |
X. Li, T. Huang, J. A. Fang, Event-triggered stabilization for takagi–sugeno fuzzy complex-valued memristive neural networks with mixed time-varying delays, IEEE Trans. Fuzzy Syst., 29 (2020), 1853–1863. https://doi.org/10.1109/TFUZZ.2020.2986713 doi: 10.1109/TFUZZ.2020.2986713
![]() |
[110] |
H. Wei, R. Li, B. Wu, Dynamic analysis of fractional-order quaternion-valued fuzzy memristive neural networks: Vector ordering approach, Fuzzy Sets Syst., 411 (2021), 1–24. https://doi.org/10.1016/j.fss.2020.02.013 doi: 10.1016/j.fss.2020.02.013
![]() |
[111] |
R. Sakthivel, R. Raja, S. M. Anthoni, Exponential stability for delayed stochastic bidirectional associative memory neural networks with markovian jumping and impulses, J. Optim. Theory Appl., 150 (2011), 166–187. https://doi.org/10.1007/s10957-011-9808-4 doi: 10.1007/s10957-011-9808-4
![]() |
[112] |
J. Li, M. Hu, L. Guo, Exponential stability of stochastic memristor-based recurrent neural networks with time-varying delays, Neurocomputing, 138 (2014), 92–98. https://doi.org/10.1016/j.neucom.2014.02.042 doi: 10.1016/j.neucom.2014.02.042
![]() |
[113] |
Z. Meng, Z. Xiang, Stability analysis of stochastic memristor-based recurrent neural networks with mixed time-varying delays, Neural Comput. Appl., 28 (2017), 1787–1799. https://doi.org/10.1007/s00521-015-2146-y doi: 10.1007/s00521-015-2146-y
![]() |
[114] |
X. Li, J. Fang, H. Li, Exponential stabilisation of stochastic memristive neural networks under intermittent adaptive control, IET Control Theory Appl., 11 (2017), 2432–2439. https://doi.org/10.1049/iet-cta.2017.0021 doi: 10.1049/iet-cta.2017.0021
![]() |
[115] |
D. Liu, S. Zhu, W. Chang, Mean square exponential input-to-state stability of stochastic memristive complex-valued neural networks with time varying delay, Int. J. Syst. Sci., 48 (2017), 1966–1977. https://doi.org/10.1080/00207721.2017.1300706 doi: 10.1080/00207721.2017.1300706
![]() |
[116] |
C. Li, J. Lian, Y. Wang, Stability of switched memristive neural networks with impulse and stochastic disturbance, Neurocomputing, 275 (2018), 2565–2573. https://doi.org/10.1016/j.neucom.2017.11.031 doi: 10.1016/j.neucom.2017.11.031
![]() |
[117] |
H. Liu, Z. Wang, B. Shen, T. Huang, F. E. Alsaadi, Stability analysis for discrete-time stochastic memristive neural networks with both leakage and probabilistic delays, Neural Networks, 102 (2018), 1–9. https://doi.org/10.1016/j.neunet.2018.02.003 doi: 10.1016/j.neunet.2018.02.003
![]() |
[118] |
K. Ding, Q. Zhu, Impulsive method to reliable sampled-data control for uncertain fractional-order memristive neural networks with stochastic sensor faults and its applications, Nonlinear Dyn., 100 (2020), 2595–2608. https://doi.org/10.1007/s11071-020-05670-y doi: 10.1007/s11071-020-05670-y
![]() |
[119] |
S. Duan, H. Wang, L. Wang, T. Huang, C. Li, Impulsive effects and stability analysis on memristive neural networks with variable delays, IEEE Trans. Neural Networks Learn. Syst., 28 (2016), 476–481. https://doi.org/10.1109/TNNLS.2015.2497319 doi: 10.1109/TNNLS.2015.2497319
![]() |
[120] |
W. Zhang, T. Huang, X. He, C. Li, Global exponential stability of inertial memristor-based neural networks with time-varying delays and impulses, Neural Networks, 95 (2017), 102–109. https://doi.org/10.1016/j.neunet.2017.03.012 doi: 10.1016/j.neunet.2017.03.012
![]() |
[121] |
W. Zhu, D. Wang, L. Liu, G. Feng, Event-based impulsive control of continuous-time dynamic systems and its application to synchronization of memristive neural networks, IEEE Trans. Neural Networks Learn. Syst., 29 (2017), 3599–3609. https://doi.org/10.1109/TNNLS.2017.2731865 doi: 10.1109/TNNLS.2017.2731865
![]() |
[122] |
H. Wang, S. Duan, T. Huang, C. Li, L. Wang, Novel stability criteria for impulsive memristive neural networks with time-varying delays, Circuits Syst. Signal Process., 35 (2016), 3935–3956. https://doi.org/10.1007/s00034-015-0240-0 doi: 10.1007/s00034-015-0240-0
![]() |
[123] |
J. Qi, C. Li, T. Huang, Stability of delayed memristive neural networks with time-varying impulses, Cognit. Neurodyn., 8 (2014), 429–436. https://doi.org/10.1007/s11571-014-9286-0 doi: 10.1007/s11571-014-9286-0
![]() |
[124] |
J. G. Lu, Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with dirichlet boundary conditions, Chaos Solitons Fractals, 35 (2008), 116–125. https://doi.org/10.1016/j.chaos.2007.05.002 doi: 10.1016/j.chaos.2007.05.002
![]() |
[125] |
J. L. Wang, H. N. Wu, L. Guo, Passivity and stability analysis of reaction-diffusion neural networks with Dirichlet boundary conditions, IEEE Trans. Neural Networks, 22 (2011), 2105–2116. https://doi.org/10.1109/TNN.2011.2170096 doi: 10.1109/TNN.2011.2170096
![]() |
[126] |
L. Wang, R. Zhang, Y. Wang, Global exponential stability of reaction-diffusion cellular neural networks with S-type distributed time delays, Nonlinear Anal. Real World Appl., 10 (2009), 1101–1113. https://doi.org/10.1016/j.nonrwa.2007.12.002 doi: 10.1016/j.nonrwa.2007.12.002
![]() |
[127] |
L. Wang, M. F. Ge, J. Hu, G. Zhang, Global stability and stabilization for inertial memristive neural networks with unbounded distributed delays, Nonlinear Dyn., 95 (2019), 943–955. https://doi.org/10.1007/s11071-018-4606-2 doi: 10.1007/s11071-018-4606-2
![]() |
[128] |
L. Wang, H. He, Z. Zeng, Intermittent stabilization of fuzzy competitive neural networks with reaction diffusions, IEEE Trans. Fuzzy Syst., 29 (2021), 2361–2372. https://doi.org/10.1109/TFUZZ.2020.2999041 doi: 10.1109/TFUZZ.2020.2999041
![]() |
[129] |
R. Rakkiyappan, S. Dharani, Q. Zhu, Synchronization of reaction-diffusion neural networks with time-varying delays via stochastic sampled-data controller, Nonlinear Dyn., 79 (2015), 485–500. https://doi.org/10.1007/s11071-014-1681-x doi: 10.1007/s11071-014-1681-x
![]() |
[130] |
Z. P. Wang, H. N. Wu, J. L. Wang, H. X. Li, Quantized sampled-data synchronization of delayed reaction-diffusion neural networks under spatially point measurements, IEEE Trans. Cybern., 51 (2021), 5740–5751. https://doi.org/10.1109/TCYB.2019.2960094 doi: 10.1109/TCYB.2019.2960094
![]() |
[131] |
Q. Qiu, H. Su, Sampling-based event-triggered exponential synchronization for reaction-diffusion neural networks, IEEE Trans. Neural Networks Learn. Syst., 34 (2021), 1209–1217. https://doi.org/10.1109/TNNLS.2021.3105126 doi: 10.1109/TNNLS.2021.3105126
![]() |
[132] |
D. Zeng, R. Zhang, J. H. Park, Z. Pu, Y. Liu, Pinning synchronization of directed coupled reaction-diffusion neural networks with sampled-data communications, IEEE Trans. Neural Networks Learn. Syst., 31 (2020), 2092–2103. https://doi.org/10.1109/TNNLS.2019.2928039 doi: 10.1109/TNNLS.2019.2928039
![]() |
[133] |
Z. Guo, S. Wang, J. Wang, Global exponential synchronization of coupled delayed memristive neural networks with reaction–diffusion terms via distributed pinning controls, IEEE Trans. Neural Networks Learn. Syst., 32 (2021), 105–116. https://doi.org/10.1109/TNNLS.2020.2977099 doi: 10.1109/TNNLS.2020.2977099
![]() |
[134] |
Y. Cao, Y. Cao, Z. Guo, T. Huang, S. Wen, Global exponential synchronization of delayed memristive neural networks with reaction-diffusion terms, Neural Networks, 123 (2020), 70–81. https://doi.org/10.1016/j.neunet.2019.11.008 doi: 10.1016/j.neunet.2019.11.008
![]() |
[135] |
L. Shanmugam, P. Mani, R. Rajan, Y. H. Joo, Adaptive synchronization of reaction-diffusion neural networks and its application to secure communication, IEEE Trans. Cybern., 50 (2020), 911–922. https://doi.org/10.1109/TCYB.2018.2877410 doi: 10.1109/TCYB.2018.2877410
![]() |
[136] |
J. L. Wang, Z. Qin, H. N. Wu, T. Huang, Passivity and synchronization of coupled uncertain reaction-diffusion neural networks with multiple time delays, IEEE Trans. Neural Networks Learn. Syst., 30 (2019), 2434–2448. https://doi.org/10.1109/TNNLS.2018.2884954 doi: 10.1109/TNNLS.2018.2884954
![]() |
[137] |
R. Zhang, D. Zeng, J. H. Park, Y. Liu, X. Xie, Adaptive event-triggered synchronization of reaction-diffusion neural networks, IEEE Trans. Neural Networks Learn. Syst., 32 (2021), 3723–3735. https://doi.org/10.1109/TNNLS.2020.3027284 doi: 10.1109/TNNLS.2020.3027284
![]() |
[138] |
J. Pan, X. Liu, S. Zhong, Stability criteria for impulsive reaction-diffusion cohen-grossberg neural networks with time-varying delays, Math. Comput. Modell., 51 (2010), 1037–1050. https://doi.org/10.1016/j.mcm.2009.12.004 doi: 10.1016/j.mcm.2009.12.004
![]() |
[139] |
S. Mongolian, Y. Kao, C. Wang, H. Xia, Robust mean square stability of delayed stochastic generalized uncertain impulsive reaction-diffusion neural networks, J. Franklin Inst., 358 (2021), 877–894. https://doi.org/10.1016/j.jfranklin.2020.04.011 doi: 10.1016/j.jfranklin.2020.04.011
![]() |
[140] |
T. Wei, P. Lin, Y. Wang, L. Wang, Stability of stochastic impulsive reaction-diffusion neural networks with S-type distributed delays and its application to image encryption, Neural Networks, 116 (2019), 35–45. https://doi.org/10.1016/j.neunet.2019.03.016 doi: 10.1016/j.neunet.2019.03.016
![]() |
[141] |
T. Wei, X. Li, V. Stojanovic, Input-to-state stability of impulsive reaction-diffusion neural networks with infinite distributed delays, Nonlinear Dyn., 103 (2021), 1733–1755. https://doi.org/10.1007/s11071-021-06208-6 doi: 10.1007/s11071-021-06208-6
![]() |
[142] |
J. Cao, G. Stamov, I. Stamova, S. Simeonov, Almost periodicity in impulsive fractional-order reaction-diffusion neural networks with time-varying delays, IEEE Trans. Cybern., 51 (2021), 151–161. https://doi.org/10.1109/TCYB.2020.2967625 doi: 10.1109/TCYB.2020.2967625
![]() |
[143] | T. Wei, X. Li, J. Cao, Stability of delayed reaction-diffusion neural-network models with hybrid impulses via vector Lyapunov function, IEEE Trans. Neural Networks Learn. Syst., early access, (2022), 1–12. https://doi.org/10.1109/TNNLS.2022.3143884 |
[144] |
C. Hu, H. Jiang, Z. Teng, Impulsive control and synchronization for delayed neural networks with reaction-diffusion terms, IEEE Trans. Neural Networks, 21 (2010), 67–81. https://doi.org/10.1109/TNN.2009.2034318 doi: 10.1109/TNN.2009.2034318
![]() |
[145] |
X. Yang, J. Cao, Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM J. Control Optim., 51 (2013), 3486–3510. https://doi.org/10.1137/120897341 doi: 10.1137/120897341
![]() |
[146] |
W. H. Chen, S. Luo, W. X. Zheng, Impulsive synchronization of reaction–diffusion neural networks with mixed delays and its application to image encryption, IEEE Trans. Neural Networks Learn. Syst., 27 (2016), 2696–2710. https://doi.org/10.1109/TNNLS.2015.2512849 doi: 10.1109/TNNLS.2015.2512849
![]() |
[147] |
H. Chen, P. Shi, C. C. Lim, Pinning impulsive synchronization for stochastic reaction–diffusion dynamical networks with delay, Neural Networks, 106 (2018), 281–293. https://doi.org/10.1016/j.neunet.2018.07.009 doi: 10.1016/j.neunet.2018.07.009
![]() |
[148] |
Y. Wang, P. Lin, L. Wang, Exponential stability of reaction-diffusion high-order Markovian jump hopfield neural networks with time-varying delays, Nonlinear Anal. Real World Appl., 13 (2012), 1353–1361. https://doi.org/10.1016/j.nonrwa.2011.10.013 doi: 10.1016/j.nonrwa.2011.10.013
![]() |
[149] | R. Zhang, H. Wang, J. H. Park, K. Shi, P. He, Mode-dependent adaptive event-triggered control for stabilization of Markovian memristor-based reaction-diffusion neural networks, IEEE Trans. Neural Networks Learn. Syst., early access, (2021), 1–13. https://doi.org/10.1109/TNNLS.2021.3122143 |
[150] |
X. X. Han, K. N. Wu, Y. Niu, Asynchronous boundary stabilization of stochastic Markov jump reaction-diffusion systems, IEEE Trans. Syst. Man Cybern.: Syst., 52 (2022), 5668–5678. https://doi.org/10.1109/TSMC.2021.3130271 doi: 10.1109/TSMC.2021.3130271
![]() |
[151] |
Q. Zhu, X. Li, X. Yang, Exponential stability for stochastic reaction-diffusion BAM neural networks with time-varying and distributed delays, Appl. Math. Comput., 217 (2011), 6078–6091. https://doi.org/10.1016/j.amc.2010.12.077 doi: 10.1016/j.amc.2010.12.077
![]() |
[152] |
Y. Sheng, H. Zhang, Z. Zeng, Stability and robust stability of stochastic reaction–diffusion neural networks with infinite discrete and distributed delays, IEEE Trans. Syst. Man Cybern.: Syst., 50 (2020), 1721–1732. https://doi.org/10.1109/TSMC.2017.2783905 doi: 10.1109/TSMC.2017.2783905
![]() |
[153] |
X. Z. Liu, K. N. Wu, X. Ding, W. Zhang, Boundary stabilization of stochastic delayed Cohen-Grossberg neural networks with diffusion terms, IEEE Trans. Neural Networks Learn. Syst., 33 (2022), 3227–3237. https://doi.org/10.1109/TNNLS.2021.3051363 doi: 10.1109/TNNLS.2021.3051363
![]() |
[154] |
X. Liang, L. Wang, Y. Wang, R. Wang, Dynamical behavior of delayed reaction-diffusion Hopfield neural networks driven by infinite dimensional Wiener processes, IEEE Trans. Neural Networks Learn. Syst., 27 (2016), 1816–1826. https://doi.org/10.1109/TNNLS.2015.2460117 doi: 10.1109/TNNLS.2015.2460117
![]() |
[155] |
T. Wei, P. Lin, Q. Zhu, L. Wang, Y. Wang, Dynamical behavior of nonautonomous stochastic reaction-diffusion neural-network models, IEEE Trans. Neural Networks Learn. Syst., 30 (2019), 1575–1580. https://doi.org/10.1109/TNNLS.2018.2869028 doi: 10.1109/TNNLS.2018.2869028
![]() |
[156] |
Q. Yao, P. Lin, L. Wang, Y. Wang, Practical exponential stability of impulsive stochastic reaction-diffusion systems with delays, IEEE Trans. Cybern., 52 (2022), 2687–2697. https://doi.org/10.1109/TCYB.2020.3022024 doi: 10.1109/TCYB.2020.3022024
![]() |
[157] |
Q. Ma, S. Xu, Y. Zou, G. Shi, Synchronization of stochastic chaotic neural networks with reaction-diffusion terms, Nonlinear Dyn., 67 (2012), 2183–2196. https://doi.org/10.1007/s11071-011-0138-8 doi: 10.1007/s11071-011-0138-8
![]() |
[158] |
Y. Sheng, Z. Zeng, Impulsive synchronization of stochastic reaction-diffusion neural networks with mixed time delays, Neural Networks, 103 (2018), 83–93. https://doi.org/10.1016/j.neunet.2018.03.010 doi: 10.1016/j.neunet.2018.03.010
![]() |
[159] |
M. S. Ali, L. Palanisamy, J. Yogambigai, L. Wang, Passivity-based synchronization of Markovian jump complex dynamical networks with time-varying delays, parameter uncertainties, reaction–diffusion terms, and sampled-data control, J. Comput. Appl. Math., 352 (2019), 79–92. https://doi.org/10.1016/j.cam.2018.10.047 doi: 10.1016/j.cam.2018.10.047
![]() |
[160] |
X. Yang, Q. Song, J. Cao, J. Lu, Synchronization of coupled Markovian reaction-diffusion neural networks with proportional delays via quantized control, IEEE Trans. Neural Networks Learn. Syst., 30 (2019), 951–958. https://doi.org/10.1109/TNNLS.2018.2853650 doi: 10.1109/TNNLS.2018.2853650
![]() |
[161] |
X. Song, J. Man, S. Song, Z. Wang, Finite-time nonfragile time-varying proportional retarded synchronization for Markovian inertial memristive NNs with reaction–diffusion items, Neural Networks, 123 (2020), 317–330. https://doi.org/10.1016/j.neunet.2019.12.011 doi: 10.1016/j.neunet.2019.12.011
![]() |
[162] |
H. Shen, X. Wang, J. Wang, J. Cao, L. Rutkowski, Robust composite synchronization of Markov jump reaction-diffusion neural networks via a disturbance observer-based method, IEEE Trans. Cybern., 52 (2022), 12712–12721. https://doi.org/10.1109/TCYB.2021.3087477 doi: 10.1109/TCYB.2021.3087477
![]() |
1. | Maysaa Al-Qurashi, Saima Rashid, Fahd Jarad, Madeeha Tahir, Abdullah M. Alsharif, New computations for the two-mode version of the fractional Zakharov-Kuznetsov model in plasma fluid by means of the Shehu decomposition method, 2022, 7, 2473-6988, 2044, 10.3934/math.2022117 | |
2. | Saima Rashid, Rehana Ashraf, Zakia Hammouch, New generalized fuzzy transform computations for solving fractional partial differential equations arising in oceanography, 2023, 8, 24680133, 55, 10.1016/j.joes.2021.11.004 | |
3. | Saima Rashid, Rehana Ashraf, Ahmet Ocak Akdemir, Manar A. Alqudah, Thabet Abdeljawad, Mohamed S. Mohamed, Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels, 2021, 5, 2504-3110, 113, 10.3390/fractalfract5030113 | |
4. | Saima Rashid, Rehana Ashraf, Fatimah S. Bayones, A Novel Treatment of Fuzzy Fractional Swift–Hohenberg Equation for a Hybrid Transform within the Fractional Derivative Operator, 2021, 5, 2504-3110, 209, 10.3390/fractalfract5040209 | |
5. | Saima Rashid, Sobia Sultana, Bushra Kanwal, Fahd Jarad, Aasma Khalid, Fuzzy fractional estimates of Swift-Hohenberg model obtained using the Atangana-Baleanu fractional derivative operator, 2022, 7, 2473-6988, 16067, 10.3934/math.2022880 | |
6. | Shuang-Shuang Zhou, Saima Rashid, Asia Rauf, Khadija Tul Kubra, Abdullah M. Alsharif, Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time, 2021, 6, 2473-6988, 12114, 10.3934/math.2021703 | |
7. | Manar A. Alqudah, Rehana Ashraf, Saima Rashid, Jagdev Singh, Zakia Hammouch, Thabet Abdeljawad, Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators, 2021, 5, 2504-3110, 151, 10.3390/fractalfract5040151 | |
8. | Ravichandran VIVEK, Kangarajan K., Dvivek VİVEK, Elsayed ELSAYED, Dynamics and Stability of -Hilfer Fractional Fuzzy Differential Equations with Impulses, 2023, 6, 2651-4001, 115, 10.33434/cams.1257750 | |
9. | Saima Rashid, Fahd Jarad, Hind Alamri, New insights for the fuzzy fractional partial differential equations pertaining to Katugampola generalized Hukuhara differentiability in the frame of Caputo operator and fixed point technique, 2024, 15, 20904479, 102782, 10.1016/j.asej.2024.102782 |