Multiple periodic solutions of delayed competitive neural networks via functional differential inclusions

Neurocomputing 168 (2015) 777–789

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Multiple periodic solutions of delayed competitive neural networks via functional differential inclusions$ Dongshu Wang n, Daozhong Luo School of Mathematical Sciences, Huaqiao University, 362021 Quanzhou, Fujian, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 10 November 2014 Received in revised form 10 May 2015 Accepted 12 May 2015 Communicated by Hui Zhang Available online 4 June 2015

In this paper, a class of general competitive neural networks with discontinuous right-hand sides and mixed time delays is investigated. Based on functional differential inclusions theory and fixed point theorem of set-valued maps, the existence of one and multiple positive periodic solutions for the competitive neural networks are obtained. Without assuming the boundedness or satisfying a growth condition of discontinuous neuron activation functions, the results on the existence of one and multiple positive periodic solutions will also be valid. Some numerical examples are given to show the applicability and effectiveness of our main results. & 2015 Elsevier B.V. All rights reserved.

Keywords: Periodic solution Multiple periodic solutions Discontinuous activation Functional differential inclusions

1. Introduction Recently, the coexistence of multiple equilibrium points or multiple periodic solutions are interesting in many practical applications, such as associative memory storage, decision making, digital selection, pattern recognition and analogy amplification [1–5]. In [6], the multistability of neural networks with smooth sigmoidal activation functions was investigated via geometrical observation. In [7], the multistability of competitive neural networks with different time scales was studied under the assumption that the activation functions are unsaturated piecewise linear functions. In [8], by formulating parameter conditions and using inequality technique, the authors investigated the multistability of delayed competitive neural networks with two classed of general activation functions. For secondorder competitive neural networks with nondecreasing saturated activation functions, the multistability and multiperiodicity was also discussed in [9,10]. Note that periodic oscillations have been found many applications in neural networks [11–16], such as associative memories, pattern recognition, machine learning, and robot motion control. Moreover, an equilibrium point can be regarded as a special case of periodic solution for neural networks with arbitrary period.

☆ Research supported by the National Natural Science Foundation of China (11371127, 11401228), the Natural Science Foundation of Fujian Province (2015J01584), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY201). n Corresponding author at: School of Mathematical Sciences, Huaqiao University, 362021,Quanzhou, Fujian, PR China. E-mail addresses: [email protected] (D. Wang), [email protected] (D. Luo).

http://dx.doi.org/10.1016/j.neucom.2015.05.046 0925-2312/& 2015 Elsevier B.V. All rights reserved.

Therefore, the analysis of periodic solutions for neural networks is more general and interesting. In [17], the multiperiodicity of delayed neural networks was discussed under the assumption that the activation functions are piecewise linear functions. For a class of activation functions consist of nondecreasing functions with saturation's including piecewise linear functions with two corner points and standard activation functions as its special case, Cao et al. discussed the multiperiodicity of delayed neural networks [18]. In [19], under the assumption that the activation functions are increasing and bounded, Zhou et al. studied the multiperiodicity of multidirectional associative memory neural networks via Poincaré mapping. However, all of the above results were based on the assumption that the activation functions are continuous. Note that neural networks with discontinuous neuron activations have been found useful to address a number of interesting engineering tasks, such as dry friction, impacting machines, systems oscillating under the effect of an earthquake, power circuits, switching in electronic circuits, linear complimentarily systems. Therefore, neural networks with discontinuous neuron activations have received a great deal of attention in the literature [20–31] since the pioneering work of Forti and Nistri [20]. By using the theory of fixed point in differential inclusions, the authors analyzed the problems of periodic solutions for various neural network systems with discontinuous activations [22–24,27–30], respectively. Huang et al. studied the multiperiodicity of delayed bidirectional associative memory neural networks with r-level discontinuous activation functions [32,33]. However, all the above-mentioned results were based on the assumption that the discontinuous activation

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D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

functions are bounded or satisfy a growth condition (g.c.) [26]. As pointed out by Gonzalez [34], to truly exploit the potential of neural networks, a nonlinear activation function must be used. When dealing with a dependent variable that is not bounded, one could choose an unbounded nonlinear activation function such as f ðxÞ ¼ x3 . Thus, it is interesting to study the periodicity and multiperiodicity of neural networks with general discontinuous activations which maybe unbounded or nonlinear growth condition. It is well known that time delays are often inevitable, since the finite switching speed of amplifiers and communication time, in many practical applications of neural networks [35–37] like communication systems, electric power systems with lossless transmission lines, control, image processing, pattern recognition, signal processing and associative memory, etc. In reality, discrete (timevarying) delay and distributed delay always occur simultaneously, therefore, much effort has been devoted to analyzing the dynamical behavior of neural networks with both discrete (time-varying) delay and distributed delay [38,39]. Motivated by the above works, in this paper, we consider the following competitive neural network model with different time scales and mixed time delays of the form: N N X X τ dx ðtÞ STM : ϵ i ¼  a^ i ðtÞxi ðtÞ þ D^ ij ðtÞf j ðxj ðtÞÞ þ D^ ij ðtÞf j ðxj ðt  τij ðtÞÞÞ dt j¼1 j¼1

þ

N X

^ k ðtÞ D ij

Z

j¼1

þ B^ i ðtÞ

P X

þ1 0

f j ðxj ðt  sÞÞkij ðsÞ ds

dmij ðtÞ ¼  ci ðtÞmij ðtÞ þ yj ðtÞf i ðxi ðtÞÞ; dt

where xi(t) is the neuron current activity level, f j ðxj ðtÞÞ is the output of neurons, mij(t) is the synaptic efficiency, yj(t) is the external ^ ij ðtÞ, D^ τ ðtÞ and D^ k ðtÞ stimulus, ci(t) is the disposable scaling, D ij ij represent the connection weight, the time-varying delay connection strength, and the distributed delay connection strength between the i-th neuron and the j-th neuron, respectively; B^ i ðtÞ is the strength of the external stimulus, τij ðtÞ is the time-varying delay; kij(s) is the probability kernel of the distributed delay; a^ i ðtÞ is the self-inhibition of the i-th neuron, I^i ðtÞ denotes the external input to the ith neuron, and ϵ 4 0 is the time scale of STM state. For more details on competitive neural networks systems, one can see [9,10,31,40–44] and the references cited therein. Assuming the input stimulus yðtÞ is a constant vector and is P normalized with unit magnitude jyðtÞj2 ¼ Pj¼ 1 y2j ðtÞ ¼ 1 and setPP T ting Si ðtÞ ¼ j ¼ 1 mij ðtÞyj ðtÞ ¼ y ðtÞmi ðtÞ, where yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; …; yP ðtÞÞT , mi ðtÞ ¼ ðmi1 ðtÞ; mi2 ðtÞ; …; miP ðtÞÞT , then the above networks system (1.1) can be simplified as N N X X dxi ðtÞ ^ ij ðtÞf ðxj ðtÞÞ þ ^ τ ðtÞf ðxj ðt  τij ðtÞÞÞ ¼  a^ i ðtÞxi ðtÞ þ D D j j ij dt j¼1 j¼1

þ

N X

^ k ðtÞ D ij

Z

þ1 0

j¼1

i ¼ 1; 2; …; N:

τ k D^ ðtÞ D^ ðtÞ D^ ðtÞ ^ Denote ai ðtÞ ¼ a iϵðtÞ, Dij ðtÞ ¼ ijϵ , Dτij ðtÞ ¼ ijϵ , Dkij ðtÞ ¼ ijϵ , ^ I i ðtÞ ¼ I i ϵðtÞ, then the system (1.2) can be written as

ð1:2Þ

j¼1

Z Dkij ðtÞ

þ1 0

^

Bi ðtÞ ¼ B iϵðtÞ,

N N X X dxi ðtÞ ¼  ai ðtÞxi ðtÞ þ Dij ðtÞf j ðxj ðtÞÞ þ Dτij ðtÞf j ðxj ðt  τij ðtÞÞÞ dt j¼1 j¼1 N X

To study the periodicity of the system (1.3), the following assumptions for parameters are also necessary: for i; j ¼ 1; 2; …; n, ai ðtÞ; Dij ðtÞ; Dτij ðtÞ; Dkij ðtÞ; τij ðtÞ; Bi ðtÞ; I i ðtÞ; ci ðtÞ are continuous ω-periodic functions and such that τij ðtÞ Z 0, ai ðtÞ 4 0, ci ðtÞ 4 0; the delay kernels kij : ½0; þ 1Þ-R are continuous, integrable, and there exist R þ1 constants kij such that 0 j kij ðsÞj ds r kij . For the discontinuous neuron activation functions in (1.3), one can make the following assumptions: (H1) For every i ¼ 1; 2; …; n, fi is continuous expect on a countable set of isolate points ρik, where there exist finite right limits þ  limxi -ðρi Þ þ f i ðxi Þ 9f i ðρik Þ and left limits limxi -ðρi Þ  f i ðxi Þ 9f i ðρik Þ, k k respectively. Moreover, fi has a finite number of discontinuous points on any compact interval of R.  þ  þ Denote co½f ðxi Þ ¼ ½minff i ðxi Þ; f i ðxi Þg; maxff i ðxi Þ; f i ðxi Þg, then it is easy to see that co½f i ðxi Þ is an interval with non-empty interior when fi is discontinuous at xi, while co½f i ðxi Þ ¼ f i ðxi Þ is a singleton when fi is continuous at xi. The main contributions of this paper are: First, by using the fixed point theorem of multi-valued maps and functional differential inclusions theory, we study the existence of one and multiple periodic solutions of delayed competitive neural networks with different time scales, and the discontinuous activations in which maybe unbounded or nonlinear growth condition. Second, some numerical examples are given to show the applicability and effectiveness of our main results.

f j ðxj ðt  sÞÞkij ðsÞ ds þ Bi ðtÞSi ðtÞ þ I i ðtÞ;

Firstly, we introduce the notion of Filippov solution [45,46]. Let us consider the following differential equation in vector notation: dx ¼ f ðt; xÞ; dt

ð2:1Þ

where f : R  Rn -Rn and x A Rn , f ðt; xÞ is discontinuous with respect to x. Definition 2.1 (see Filippov [45, p. 85] and Huang et al. [46, p. 28]). Consider the multi-valued map F : R  Rn -Rn defined as Fðt; xÞ ¼ ⋂δ 4 0 ⋂measðN Þ ¼ 0 co½f ðt; Bðx; δÞ⧹N Þ;

ð2:2Þ

where f ðt; Bðx; δÞ⧹N Þ ¼ ⋃y A Bðx;δÞ⧹N f ðt; yÞ, co½D is the closure of the convex hull of some set D; Bðx; δÞ ¼ fy A Rn : J y  x J r δg is the ball of center x and radius δ; intersection is taken over all sets N of measure zero and over all δ 40; meas N is the Lebesgue measure of set N . A solution in the sense of Filippov (or Filippov solution) of the Cauchy problem for Eq. (2.1) with initial condition xðt 0 Þ ¼ x0 is an absolutely continuous vector-value function x(t) on any compact subinterval of ½0; T Þ, which satisfies xðt 0 Þ ¼ x0 and differential inclusions: dx A Fðt; xÞ dt

f j ðxj ðt  sÞÞkij ðsÞ ds þ B^ i ðtÞSi ðtÞ þ I^i ðtÞ;

dSi ðtÞ ¼  ci ðtÞSi ðtÞ þ f i ðxi ðtÞÞ; dt

þ

ð1:3Þ

2. Preliminaries i ¼ 1; 2; …; N; j ¼ 1; 2; …; P;

ð1:1Þ

ϵ

i ¼ 1; 2; …; N:

mij ðtÞyj ðtÞ þ I^ i ðtÞ;

j¼1

LTM :

dSi ðtÞ ¼  ci ðtÞSi ðtÞ þ f i ðxi ðtÞÞ; dt

a:e: t A ½0; T Þ:

ð2:3Þ

In many engineering applications, a solution in the sense of Filippov (or Filippov Solution) is particularly useful because it is good approximation of solutions of actual systems that possess nonlinearities with very high slop. Under the framework of solution in the sense of Filippov (or Filippov Solution), we will discuss the solution in the sense of Filippov (or Filippov Solution) to the functional differential equation (1.3) in the next step. Note that the competitive neural network (1.3) is defined as a piecewise continuous vector function and the classical definition of solution has been shown to be invalid for the differential equation with discontinuous right-hand side, so we need to specify what is exact meant by a solution of the functional differential equation (1.3)

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

with a discontinuous right-hand side. For this purpose, we extend the definition of solutions in the sense of Filippov (or Filippov Solution) to the functional differential equation (1.3) as follows: Definition 2.2 (Filippov Solution). A vector function z ¼ ðxT ; ST ÞT ¼ ðx1 ; x2 ; …; xN ; S1 ; S2 ; …; SN ÞT : ð  1; T Þ-R2N ; T A ð0; þ 1, is a state solution of the discontinuous system (1.3) on ð  1; T Þ if (1) x and S are continuous on ð  1; T Þ and absolutely continuous on any compact interval of ½0; T Þ; (2) there exists a measurable function γ ¼ ðγ 1 ; γ 2 ; …; γ N ÞT : ð  1; T Þ-RN such that γ j ðtÞ A co½f j ðxj ðtÞÞ for a.e. t A ð  1; T Þ and N N X X dxi ðtÞ ¼  ai ðtÞxi ðtÞ þ Dij ðtÞγ j ðtÞ þ Dτij ðtÞγ j ðt  τij ðtÞÞ dt j¼1 j¼1 Z þ1 N X k Dij ðtÞ γ j ðt  sÞkij ðsÞ ds þ Bi ðtÞSi ðtÞ þ I i ðtÞ; þ

dSi ðtÞ ¼  ci ðtÞSi ðtÞ þ γ i ðtÞ; dt

i ¼ 1; 2; …; N:

ð2:4Þ

for a.e. t A ½0; T Þ. The function γ which satisfies (2.4) is called an output solution associated with the state x. With this definition, it turns out that the state z ¼ ðxT ; ST ÞT is a solution of (1.3) in the sense of Filippov since it satisfies N N X X dxi ðtÞ Dij ðtÞco½f j ðxj ðtÞÞ þ Dτij ðtÞco½f j ðxj ðt τij ðtÞÞÞ A  ai ðtÞxi ðtÞ þ dt j¼1 j¼1

þ

N X j¼1

Dkij ðtÞ

Z

þ1 0

co½f j ðxj ðt  sÞÞkij ðsÞ dsþ Bi ðtÞSi ðtÞ þ I i ðtÞ;

dSi ðtÞ A  ci ðtÞSi ðtÞ þ co½f i ðxi ðtÞÞ; dt

i ¼ 1; 2; …; N:

ð2:5Þ

Definition 2.3 (IVP). For any continuous function ϕ ¼ ðϕ1 ; ϕ2 ; …; ϕ2N ÞT : ð  1; 0-R2N and any measurable selection ψ ¼ ðψ 1 ; ψ 2 ; …; ψ N ÞT : ð  1; 0-RN , such that ψ j ðsÞ A co½f j ðϕj ðsÞÞ ðj ¼ 1; 2; …; NÞ for a.e. s A ð  1; 0 by an initial value problem associated to (1.3) with initial condition ½ϕ; ψ, we mean the following problem: find a couple of functions ðxT ; ST ÞT : ð  1; T Þ-R2N and γ : ð 1; T Þ-RN , such that ðxT ; ST ÞT is a solution of (1.3) on ð  1; T Þ for some T 4 0, γ is an output solution associated to x, and

dSi ðtÞ > > ¼ ci ðtÞSi ðtÞ þ γ i ðtÞ for a:e: t A ½0; T Þ; i ¼ 1; 2; …; N; > > > > dt > > > > > γ j ðtÞ A co½f j ðxj ðtÞÞ for a:e: t A ½0; T Þ; > > T T > T > > ðx ðsÞ; S ðsÞÞ ¼ ϕðsÞ; 8 s A ð  1; 0; > > : γðsÞ ¼ ψðsÞ for a:e: s A ð  1; 0:

Definition 2.4. A solution x(t) of the given IVP of system (1.3) on ½0; þ1Þ is a periodic solution with period ω if xðt þ ωÞ ¼ xðtÞ for all t Z 0. Next, let us introduce some basic concepts and facts from multi-valued analysis which will be used throughout this paper. The reader is referred to the books [46–50] and the papers [51–56] for a thorough treatment. Let X be a Banach space with norm ‖  ‖X and P(X) denotes the family of nonempty subsets of X. For the sake of convenience, we introduce the following notations:

Let A  P cl ðXÞ, then the distance from x to A is given by dist ðx; AÞ ¼ inf f J x  a J : a A Ag. Definition 2.5. A multi-valued map F with nonempty values is said to be upper semi-continuous (USC) at x0 A E, if βðFðxÞ; Fðx0 ÞÞ-0 as x-x0 , here βðFðxÞ; Fðx0 ÞÞ 9 supfdistðy; Fðx0 ÞÞ : yA FðxÞg. F(x) is said to have a closed (convex, compact) image if for each x A E, F(x) is closed (convex, compact). Definition 2.6. A nonempty closed subset P of X is called a cone if (1) αx þ βy A P for all x; y A P and all α; β Z 0; (2) x;  x A P implies x ¼ θ, where θ stands for a zero element of X. Definition 2.7. The Kuratowskii measure of noncompactness is the map α : P b ðXÞ-½0; þ 1 defined by

for a.e. t A ½0; T Þ. For an initial value problem (IVP) associated to the competitive neural networks (1.3), we follow the definition introduced by Forti et al. (see [20,21]).

8 N N X X > dxi ðtÞ > > ¼  ai ðtÞxi ðtÞ þ Dij ðtÞγ j ðtÞ þ Dτij ðtÞγ j ðt  τij ðtÞÞ > > > dt > j¼1 j¼1 > > > > Z þ1 > N X > > > Dkij ðtÞ γ j ðt  sÞkij ðsÞ dsþ Bi ðtÞSi ðtÞ þI i ðtÞ; þ > > > < 0 j¼1

Throughout this paper, the initial functions ϕ and ψ (as described in Definition 2.3) satisfy the following: ϕ is a bounded continuous function from ð  1; 0 to R2N and ψ is an essentially bounded measurable function from ð  1; 0 to RN .

P cl ðXÞ ¼ fY  X : Y closedg; P cv ðXÞ ¼ fY  X : Y convexg; P cp ðXÞ ¼ fY  X : Y compactg; P b ðXÞ ¼ fY  X : Y boundedg:

0

j¼1

779

ð2:6Þ

αðYÞ ¼ inf fϵ 40 : Y admits a finite cover by sets of diameter rϵg where Y A P b ðXÞ: Let Z be a nonempty subset of X and F : Z-PðXÞ (P(X) denotes the family of nonempty subsets of X). F is called a k-set contraction if αðFðYÞÞ r kαðYÞ for all bounded sets Y of Z and F(z) is bounded, here FðYÞ ¼ ⋃y A Y FðyÞ and k Z0 is a constant. For notational purposes, for ϱ 4 0 let Ωϱ ¼ fx A X : ‖x‖X o ϱg;

∂Ωϱ ¼ fx A X : ‖x‖X ¼ ϱg;

Ω ϱ ¼ fx A X : ‖x‖X r ϱg:

As a matter of convenience we recall the fixed point theorem for multivalued maps due to Agarwal and O'Regan [51, Theorems 2.3 and 2.7]. Lemma 2.1. Let X ¼ ðX; ‖  ‖X Þ be a Banach space and E  X a closed, convex, nonempty set with αu þ βv A E for all α Z 0; β Z 0 and u; v A E. And let r; R be positive constants with 0 o r oR. Suppose F : Ω R -P cp;cv ðEÞ is a USC, k-set contractive (here 0 rk o1) map and assume the following conditions hold: (1) x2 = λFðxÞ; for λ A ½0; 1Þ and x A ∂ΩR , (2) for any δ 4 0 and x A ∂Ωr , there exist a v A E⧹fθg such that x2 = FðxÞ þ δv. Then F : Ω R -P cp;cv ðEÞ has a fixed point in fx : x A E and r r ‖x‖X r Rg, i.e., there exists at least one fx : x A E and r r ‖x‖X r Rg, such that x ¼ FðxÞ.

780

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

Lemma 2.2. Let X ¼ ðX; ‖  ‖X Þ be a Banach space and E  X a closed, convex, nonempty set with αu þ βv A E for all α Z 0; β Z0 and u; v A E. And let r; R be positive constants with 0 o r o R. Suppose F : Ω R -P cp;cv ðEÞ is a USC, k-set contractive (here 0 r k o 1) map and assume that the following conditions hold:

þ

(1) x2 = λFðxÞ; for λ A ½0; 1Þ and x A ∂Ωr , (2) for any δ 4 0 and x A ∂ΩR , there exist a v A E⧹fθg such that x2 = FðxÞ þ δv.

þ

The following lemmas will be used in the proof. Lemma 2.3 (Aubin and Frankowska [48], Lasota and Opial [53]). If dimðXÞ o 1 and F : ½0; ω  X-PðXÞ is L1-Carathéodory, then set F(x) is nonempty for each fixed x A X. Lemma 2.4 (Hong and Wang [55]). Let J be a compact real interval, F : J  E-P b;cl;cv ðEÞ a L1-Carathéodory multivalued map, SF;x (there SF;x ¼ ff x A L1 ðJ; EÞ : f x ðtÞ A Fðt; xÞ for a:e: t A Jg) be nonempty for each fixed x A E and let Γ be a linear continuous mapping from L1 ðJ; EÞ to CðJ; EÞ. Then the map Γ1SF : CðJ; EÞ-P b;cl;cv ðCðJ; EÞÞ is a closed graph map in CðJ; EÞ  CðJ; EÞ. 3. Periodicity and multiperiodicity Under some assumptions, the periodicity and multiperiodicity of IVP for the competitive neural network system (1.3) with discontinuous activation functions were studied. The applied methods are based on the application of fixed point theorem for multivalued maps due to Agarwal and O'Regan [51] and the functional differential inclusions theory. Lemma 3.1. Vector function zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT is an ω-periodic solution to the system (1.3) in the sense of Filippov, if and only if zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT is an ω-periodic solution of the following integral inclusions: xi ðtÞ A

þ

t þω

Gi ðt; vÞ

t

:

Dij ðvÞco½f j ðxj ðvÞÞ þ

j¼1

N X

Z Dkij ðvÞ

Z

tþω t

þ1 0

j¼1

Si ðtÞ A

8 N
N X

9 = co½f j ðxj ðv  sÞÞkij ðsÞ ds þ Bi ðvÞSi ðvÞ þI i ðvÞ dv; ;

GN þ i ðt; vÞfco½f i ðxi ðvÞÞg dv;

i ¼ 1; 2; …; N for t A ½0; ω;

Dkij ðtÞ

Gi ðt; vÞ ¼ R t þ ω ai ðsÞ ds e t 1

and

Rv c ðsÞ ds e t i

GN þ i ðt; vÞ ¼ R t þ ω ; ci ðsÞ ds e t 1

i ¼ 1; 2; …; N:

Proof. Suppose that zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT is an ω-periodic solution to the system (1.3) in the sense of Filippov. According to the definition of Filippov Solution (Definition 2.2), one can obtain from (2.4) that N N X X dxi ðtÞ Dij ðtÞco½f j ðxj ðtÞÞ þ Dτij ðtÞco½f j ðxj ðt  τij ðtÞÞÞ A  ai ðtÞxi ðtÞ þ dt j¼1 j¼1

Z Dkij ðtÞ

Z xi ðtÞ A

þ

tþω

t

0

dSi ðtÞ A  ci ðtÞSi ðtÞ þ co½f i ðxi ðtÞÞ; dt

co½f j ðxj ðt  sÞÞkij ðsÞ ds þ Bi ðtÞSi ðtÞ þ I i ðtÞ ; ;

0

ð3:3Þ

8
N X

Z Dkij ðvÞ

Z Si ðtÞ A

t þω

t

j¼1

þ1

co½f j ðxj ðv  sÞÞkij ðsÞ ds þBi ðvÞSi ðvÞ þ I i ðvÞ

0

j¼1

GN þ i ðt; vÞfco½f i ðxi ðvÞÞg dv;

;

dv;

i ¼ 1; 2; …; N for t A ½0; ω:

t þω

Si ðtÞ ¼ t

GN þ i ðt; vÞγ i ðvÞ dv;

i ¼ 1; 2; …; N:

ð3:4Þ

Deviating the two sides of (3.4) about t, one can obtain that 8
8 N
Dij ðtÞγ j ðtÞ þ

j¼1

þ

N X j¼1

þ1

γ j ðt  sÞkij ðsÞ ds

0

 þ Bi ðtÞSi ðtÞ þ Ii ðtÞ þ

Z t

N X

Dkij ðvÞ

Dτij ðtÞγ j ðt  τij ðtÞÞ

j¼1

Z Dkij ðtÞ

N X

Z

þ1 0

tþω

8 N N X ∂Gi ðt; vÞ< X Dij ðvÞγ j ðvÞ þ Dτij ðvÞγ j ðv  τij ðvÞÞ ∂t :j ¼ 1 j¼1

9 = γ j ðv  sÞkij ðsÞ ds þ Bi ðvÞSi ðvÞ þI i ðvÞ dv; ;

dSi ðtÞ ¼ GN þ i ðt; t þ ωÞγ i ðt þ ωÞ  GN þ i ðt; tÞγ i ðtÞ þ dt

ð3:2Þ

9 =

That is, z(t) is an ω-periodic solution of the integral inclusions (3.1). On the other hand, suppose that z(t) is an ω-periodic solution of integral inclusions (3.1). By the integral representation theorem [47], there exist a measurable function γ ¼ ðγ 1 ; γ 2 ; …; γ N ÞT : ½0; þ1Þ-RN with γ i ðtÞ A co½f i ðxi ðtÞÞ for a.e. t A ½0; þ1Þ and i ¼ 1; 2; …; N, such that 8 Z t þω N N
j¼1

co½f j ðxj ðt  sÞÞkij ðsÞ dsþ Bi ðtÞSi ðtÞ þ I i ðtÞ;

9 =

þ1

for a.e. t A ½0; þ1Þ and i ¼ 1; 2; …; N. Note that the periodicity of z(t), by integrating both sides of functional differential inclusions (3.3) over the interval ½t; t þ ω ð0 r t r ωÞ, one can obtain the following integral inclusions:

þ

þ1

Z

Rt Rt c ðsÞ ds 0 c ðsÞ ds  Ae 0 i fco½f i ðxi ðtÞÞg; ½Si ðtÞe 0 j

j¼1

Rv a ðsÞ ds e t i

j¼1

Dτij ðtÞco½f j ðxj ðt τij ðtÞÞÞ

j¼1

j¼1

where

þ

N X

Z Dτij ðvÞco½f j ðxj ðv  τij ðvÞÞÞ

ð3:1Þ

N X

N X j¼1

Then F : Ω R -P cp;cv ðEÞ has a fixed point in fx : x A E and r r ‖x‖X r Rg, i.e., there exists at least one fx : x A E and r r ‖x‖X r Rg, such that x ¼ FðxÞ.

Z

for a.e. t A ½0; þ1Þ and i ¼ 1; 2; …; N. Thus 8 Rt Rt N
for a.e. t A ½0; þ1Þ and i ¼ 1; 2; …; N.

Z t

tþω

∂GN þ i ðt; vÞ γ i ðvÞ dv; ∂t

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

here

From the periodicity of z(t), one has N X

N X

dxi ðtÞ ¼  ai ðtÞxi ðtÞ þ Dij ðtÞγ j ðtÞ þ Dτij ðtÞγ j ðt τij ðtÞÞ dt j¼1 j¼1 þ

Z

N X

Dkij ðtÞ

Γ i ðvÞ ¼

γ j ðt sÞkij ðsÞ ds þ Bi ðtÞSi ðtÞ þ I i ðtÞ;

dSi ðtÞ ¼  ci ðtÞSi ðtÞ þ γ i ðtÞ; dt

for a.e. t A ½0; þ 1Þ and i ¼ 1; 2; …; N. According to the definition of Filippov solution (Definition 2.2), zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT is an ω-periodic solution to the system (1.3) in the sense of Filippov. The proof of Lemma 3.1 is completed.□

Z ¼ fzðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT ¼ ðx1 ðtÞ; x2 ðtÞ; …; xN ðtÞ; S1 ðtÞ; S2 ðtÞ; …;

1rirN

jxi j1 ¼ max j xi ðtÞj ; t A ½0;ω

1rirN

jSi j1 ¼ max j Si ðtÞj ;

i ¼ 1; 2; …; n:

t A ½0;ω

g L ¼ min gðsÞ;

Dkij ðvÞ

j¼1

γðvÞ ¼ ðγ 1 ðvÞ; γ 2 ðvÞ; …; γ N ðvÞÞT A ðco½f 1 ðx1 ðvÞÞ; co½f 2 ðx2 ðvÞÞ; …; co½f N ðxN ðvÞÞÞT ¼ co½f ðxðvÞÞ:

(H2) There exist two positive constants R0 ; R1 with 0 o R0 o R1 , such that ΦðR0 Þ ¼ max Φi ðR0 Þ r R0

and

Ψ ðR1 Þ

¼ min Ψ i ðR1 Þ ZR1 : 1 r i r 2N

Gi ðt; vÞ

t

Dkij ðvÞ

j¼1

φN þ i ðzÞ ¼

Z

Z

:j ¼ 1

þ1 0

tþω t

8 N
Dij ðvÞco½f j ðxj ðvÞÞ þ

N X

Dτij ðvÞco½f j ðxj ðv  τij ðvÞÞÞ

j¼1

9 = co½f j ðxj ðv  sÞÞkij ðsÞ ds þ Bi ðvÞSi ðvÞ þI i ðvÞ dv; ;

t A ½0;ωz A ∂Ωr ⋂P γðvÞ A co½f ðxðvÞÞ

inf

1 r i r 2N

constants

R0 ; R1 ; R2

with

Ψ ðR0 Þ ¼ min Ψ i ðR0 Þ ZR0 1 r i r 2N

Ψ ðR2 Þ ¼ min Ψ i ðR2 Þ ZR2 : 1 r i r 2N

(H4) There exist 2m þ 1 positive constants R0 ; R1 ; …; R2m with 0 oR0 oR1 o ⋯ o R2m , such that ΦðR2k þ 1 Þ ¼ max Φi ðR2k þ 1 Þ r R2k þ 1 1 r i r 2N

¼ min Ψ i ðR2k Þ Z R2k ;

and

Ψ ðR2k Þ

k ¼ 0; 1; …; m  1:

(H5) For i ¼ 1; 2; …; N and zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT A Ω R1 \ P, 8 n n
i ¼ 1; 2; …; N:

ΦN þ i ðrÞ ¼ max

sup

Ψ N þ i ðrÞ ¼ min

inf

inf

γ i ðtÞ A co½f i ðxi ðtÞÞ

fγ i ðtÞg 4 0:

(H5n) For i ¼ 1; 2; …; N and zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT A Ω R2 \ P, 8 N N
γ i ðtÞ A co½f i ðxi ðtÞÞ

fγ i ðtÞg 4 0:

t

Z

t þω

inf

t A ½0;ωz A ∂Ωr ⋂P γðvÞ A co½f ðxðvÞÞ

t

Z

Gi ðt; vÞΓ i ðvÞ dv;

tþω

sup

t A ½0;ωz A ∂Ωr ⋂P γðvÞ A co½f ðxðvÞÞ

t

Z

t þω

inf

t A ½0;ωz A ∂Ωr ⋂P γðvÞ A co½f ðxðvÞÞ

i ¼ 1; 2; …; N;

Ψ ðR0 Þ ¼ min Ψ i ðR0 ÞZ R0 :

j¼1

GN þ i ðt; vÞfco½f i ðxi ðvÞÞg dv;

T It follows from Lemma 3.1 that, z~ ðtÞ ¼ ðx~ T ðtÞ; S~ ðtÞÞT A Z is a fixed point of the multi-valued map φðzÞ, if and only if T z~ ðtÞ ¼ ðx~ T ðtÞ; S~ ðtÞÞT is a positive ω-periodic solution of the system (1.3). In the following discussion, we will solve the fixed point problem by virtue of Lemmas 2.1 and 2.2. Before doing so, let us further present additional assumptions and notations which are useful for the subsequent proof. Denote Z tþω Φi ðrÞ ¼ max sup sup Gi ðt; vÞΓ i ðvÞ dv;

Ψ i ðrÞ ¼ min

and

(H3) There exist three positive 0 oR0 oR1 o R2 , such that

1 r i r 2N

φðzÞ ¼ ðφ1 ðzÞ; φ2 ðzÞ; …; φ2N ðzÞ; ÞT ;

t þω

ΦðR1 Þ ¼ max Φi ðR1 Þ r R1

and

P ¼ fzðtÞ A Z : xi ðtÞ Z κi jxi j1 and Si ðtÞ Z κ N þ i jSi j1 ; i ¼ 1; 2; …; Ng; L b b Gi b i ¼ M1 , Gi ¼ eL ai , G b N þ i ¼ M1 , , κ N þ i ¼ GGNN þþ ii and G where κ i ¼ a a c cL Gi e i 1 e i 1 e i 1 ei GN þ i ¼ c L . e i 1 And define the multi-valued map φ : Z-PðZÞ by

where

(H2n) There exist two positive constants R0 ; R1 with 0 o R0 o R1 , such that

1 r i r 2N

s A ½0;ω

where g(t) is an ω-periodic function. Define a cone P in Z by

þ

N X

γ j ðv  sÞkij ðsÞ dsþ Bi ðvÞSi ðvÞ þ I i ðvÞ;

ΦðR1 Þ ¼ max Φi ðR1 Þ r R1 ;

g M ¼ max gðsÞ;

s A ½0;ω

N X

j¼1

1 r i r 2N

Then Z is a Banach space with the above norm ‖  ‖Z . For convenience, we shall introduce the notations

φi ðzÞ ¼

Dτij ðvÞγ j ðv τij ðvÞÞ þ

2N

SN ðtÞÞ A CðR; R Þ : zðt þωÞ ¼ zðtÞg;   ‖z‖Z ¼ max max jxi j1 ; max jSi j1 ;

Z

N X

1 r i r 2N

Let us define T

Dij ðvÞγ j ðvÞ þ

þ1 0

ð3:5Þ

N X j¼1

Z

þ1 0

j¼1

781

t

GN þ i ðt; vÞγ i ðvÞ dv; GN þ i ðt; vÞγ i ðvÞ dv;

(H5nn) For i ¼ 1; 2; …; N and zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT A Ω R2m \ P, 8 N N
782

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

fγ i ðtÞg 4 0:

inf

z00 ¼ y00 þ μ00 ηn . Note that ηn ¼ ðηn1 ; ηn2 ; …; ηn2N ÞT A P⧹fθg, there exists at least one i0 A f1; 2; …; 2Ng, such that ηni0 a 0.

γ i ðtÞ A co½f i ðxi ðtÞÞ

Case 1. If i0 A f1; 2; …; Ng, then 00 n x00 i0 ¼ yi0 þ μ00 ηi0 :

3.1. Periodicity

Theorem 3.1. If the assumptions (H1), (H2) and (H5) hold, then there exists at least one positive ω-periodic solution of system (1.3). Proof. To prove that the result of Theorem 3.1 is true, it is enough to show that φ has at least one fixed point z in fz : z A P and R0 r J z J Z r R1 g. The discussion will be divided into three steps. Step 1: The set-valued map φ : Ω R1 \ P-P cp;cv ðPÞ is a k-set contractive map with k ¼0. For the proof of Step 1, see Lemma A.1 in the Appendix. Step 2: The set-valued map φ : Ω R1 \ P-P cp;cv ðPÞ is an upper semi-continuous (USC) map. For the proof of Step 2, see Lemma A.2 in the Appendix. Step 3: In this step, we prove that the conditions (1) and (2) of Lemma 2.1 hold. First, for any zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT A P with R0 r J z J Z r R1 , and yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; …; y2N ðtÞÞT A φðzÞ. There exists a measurable function γ ¼ ðγ 1 ; γ 2 ; …; γ N ÞT : ½0; þ 1Þ-RN such that γ i ðtÞ A co½f i ðxi ðtÞÞ for a.e. t A ½0; þ 1Þ, and Z t þω Gi ðt; vÞΓ i ðvÞ dv; yi ðtÞ ¼ t

yN þ i ðtÞ ¼

Z

t þω

t

GN þ i ðt; vÞγ i ðvÞ dv;

i ¼ 1; 2; …; N;

ð3:6Þ

t A ½0;ω

N X j¼1

N X j¼1

Dki0 j ðvÞ

Z

Dτi0 j ðvÞco½f j ðx00 j ðv  τ i0 j ðvÞÞÞ

þ1

0

co½f j ðx00 j ðv  sÞÞki0 j ðsÞ ds þ Bi0 ðvÞSi0 ðvÞ þ I i0 ðvÞ

9 = ;

dv;

which, together with z00 A ∂ΩR0 \ P, implies that 8 Z tþω
N X j¼1

Z Dki0 j ðvÞ

þ1

0

γ 00 j ðv  sÞki0 j ðsÞds þ Bi0 ðvÞSi0 ðvÞ þ I i0 ðvÞ

9 = ;

dv

ZΨ i0 ðR0 Þ ZΨ ðR0 Þ ZR0 ; 00 where γ 00 j ðtÞ A co½f j ðxj ðtÞÞ ðj ¼ 1; 2; …; NÞ. Hence

which is a contradiction. That is, if i0 A f1; 2; …; Ng, then the condition (2) of Lemma 2.1 holds. Case 2. If i0 A fN þ 1; N þ 2; …; 2Ng, then 00 n S00 i0  N ¼ yi0 þ μ00 ηi0 :

The rest proof is similar as the proof of Case 1. Hence, we omit it here.

t

r Φi ðR1 Þ r ΦðR1 Þ r R1 ;  Z t þ ω   j yN þ i ðtÞj 1 ¼ max  GN þ i ðt; vÞγ i ðvÞ dv t A ½0;ω

Z

t

tþω

¼ max

t

 GN þ i ðt; vÞγ i ðvÞ dv

r ΦN þ i ðR1 Þ r ΦN þ i ðR1 Þ r R1 ;

i ¼ 1; 2; …; N:

T T

Hence, for any z ¼ ðxT ; S Þ A ∂ΩR1 \ P, J y J Z ¼ max j yi ðtÞj 1 r max Φi ðR1 Þ ¼ ΦðR1 Þ r R1 : 1 r i r 2N

1 r i r 2N

Thus, we claim that the condition (1) in Lemma 2.1 is true. Otherwise, there exist z0 A ΩR1 \ P and some constant λ0 A ½0; 1Þ such that z0 A λ0 φðz0 Þ

þ

00 n n R0 Z j z00 i0 j 1 ¼ j yi0 þ μ00 ηi0 j 1 Z R0 þ μ00 ηi0 4 R0 ;

where Γ i ðvÞ is defined as the above. Then, we have  Z t þ ω   j yi ðtÞj 1 ¼ max  Gi ðt; vÞΓ i ðvÞ dv t A ½0;ω t Z t þ ω  ¼ max Gi ðt; vÞΓ i ðvÞ dv

t A ½0;ω

00 00 T Note that x00 ¼ ðx00 1 ; x2 ; …; xn ÞÞ A ∂ΩR0 \ P, 00 00 then we have δi0 j x00 j r x 1 i0 i0 ðtÞ r j xi0 j 1 . In addition 8 Z tþω N
for z0 A ∂ΩR1 \ P:

Then there exists y0 A φðz0 Þ with z0 ¼ λ0 y0 . Therefore R1 ¼ J z0 J Z ¼ J λ0 y0 J Z ¼ j λ0 j  J y0 J Z o J y0 J Z r J y J Z r R1 ; which is a contradiction. That is to say, the condition (1) of Lemma 2.1 is satisfied. Next, we prove that the condition (2) of Lemma 2.1 holds. We should prove that for any z ¼ ðxT ; ST ÞT A ∂ΩR0 \ P and any μ4 0, there exists a η ¼ ðη1 ; η2 ; ⋯; η2N ÞT A P⧹fθg such that z2 = φðzÞ þ μη. Otherwise, for any ηn ¼ ðηn1 ; ηn2 ; ⋯; ηn2N ÞT A P⧹fθg, there exist z00 A ∂ΩR0 \ P and some μ00 4 0, such that z00 A φðz00 Þ þ μ00 ηn : 00 00 T 00 Correspondingly, there exists y00 ¼ ðy00 1 ; y2 ; …; y2N Þ A φðz Þ with

By now, we have proved that all the requirements of Lemma 2.1 are satisfied. Based on Lemma 2.1, the set-valued map φ has at T least one fixed point z~ ¼ ðx~ T ; S~ ÞT A P with R0 o J z~ J Z o R1 , such that z~ ðtÞ A φðz~ ÞðtÞ. As a consequence, the system (1.3) has at least one positive ω-periodic solution. The proof is completed.□ Applying Lemma 2.2 and similar to the proof of Theorem 3.1, we also have the following Theorem 3.2. Theorem 3.2. If the assumptions (H1), (H2n) and (H5) hold, then there exists at least one positive ω-periodic solution of system (1.3). Remark 1. It is well known that a very basic and important problem in the study of neural networks with discontinuous neuron activations is the global stability of periodic solution (or equilibrium point). Much progress has been made in this direction [20–24,27– 31,44]. However, by comparison we find that Theorems 3.1 and 3.2 obtained in this section on the existence of periodic solution for the neural network dynamic system (1.2) with discontinuous neuron activations make the following improvements: (1) It is worthy to point out that the used method is also valid for many kinds of neural network system with discontinuous neuron activations, such as Hopfield neural network system, Cohen– Grossberg neural network system, BAM neural network system. (2) In the earlier papers [20–24,27–31,44], many results on the existence of periodic solution or equilibrium point for neural networks with discontinuous neuron activations are conducted under the assumptions such as the discontinuous activation function f i ðÞ is bounded or satisfies a growth condition (g.c.). It

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

is worthy to point out that these assumptions are not required in Theorems 3.1 and 3.2. In fact, in Theorems 3.1 and 3.2, the discontinuous neuron activation functions may be unbounded, may be super nonlinear growth condition, even to be super exponential growth. Moreover, the restriction condition þ  f i ðρik Þ 4 f i ðρik Þ (where fi is discontinuous at ρik) in the earlier literature [20–24,27–31,44] is not required in Theorems 3.1 and 3.2. Therefore, the discontinuous activation functions of this paper are more general and more practical. (3) It is well known that most of the existing results concerning the delayed neural network dynamical systems with discontinuous right-hand sides have not consider the time-varying and distributed delays situation [20–24,27–31,44].Moreover, many results on the existence of periodic solution or equilibrium point for continuous and discontinuous neural networks with timevarying delays are conducted under the following assumption:  Time-varying delay τij ðtÞ is a continuously differentiable function and satisfies τ0ij ðtÞ o1. It is noted that this assumption is not required in Theorems 3.1 and 3.2. In fact, the derivative of the time-varying delay τij ðtÞ is allowed to be very large or does not exist in Theorems 3.1 and 3.2. That is, there are less and weaker hypotheses in our paper than those in papers mentioned above. Therefore, Theorems 3.1 and 3.2 are much more general and practical.

3.2. Multi-periodicity Next, we discuss the multiplicity of positive ω-periodic solutions for the competitive neural networks (1.3) with discontinuous activation functions. Theorem 3.3. If the assumptions (H1), (H3) and (H5n) hold, then there exists at least two positive ω-periodic solutions of system (1.3). Proof. It follows from Theorem 3.1 that φ has at least one fixed point in fzð1Þ : zð1Þ A P and R0 r J zð1Þ J Z r R1 g. Meanwhile, Theorem 3.2 implies that φ has at least one fixed point in fzð2Þ : zð2Þ A P and R1 r J zð2Þ J Z r R2 g. That is, φ has at least two fixed points in zð1Þ ; zð2Þ ðzð1Þ ; zð2Þ A PÞ with R0 r J zð1Þ J X r R1 r J zð2Þ J X r R2 . Thus the system (1.3) has at least two positive ω-periodic solutions. From the proof of Theorem 3.1–3.3, it is easy to obtain the following result. Theorem 3.4. If the assumptions (H1), (H4) and (H5nn) hold, then there exists at least 2m positive ω-periodic solutions of system (1.3).

Remark 2. In the earlier papers, a series of results on the multiperiodicity for neural networks with continuous neuron activation functions were obtained [10,17–19]. However, for neural networks with discontinuous neuron activation functions, there are few papers consider the multiperiodicity of them. In [32,33], Huang et al. studied the multiperiodicity of neuron network system with r-level discontinuous activation functions. Note that, for neural networks with discontinuous neuron activation functions, the neural networks with r-level discontinuous activation functions are only a very special case of them. Moreover, for neural networks with more general discontinuous neuron activation functions, the methods used in [32,33] will be invalid. Thus, we have to search another ‘new way’ to study the multiperiodicity of neural networks with more general discontinuous neuron activation functions. Fortunately, we overcome this difficulty due to Agarwal and O'Regan [51]. Based on retarded differential inclusions theory and fixed point theorem of set-valued maps due to Agarwal and O'Regan [51], we obtain the existence of multiple positive periodic solutions for the neural networks with more general discontinuous neuron activation functions. Thus, Theorems 3.3 and 3.4 are completely new. 4. Numerical examples In this section, we consider two numerical examples, with which the time-varying and distributed delayed neural network systems have different discontinuous neuron activation functions, to show the effectiveness of the theoretical results given in the previous sections. Example 1. Consider the following generalized competitive neural networks: dx1 ðtÞ ¼  1:2x1 ðtÞ þ ð0:3 þ 0:2 sin 2tÞf 1 ðx1 ðtÞÞ þ 0:3f 2 ðx2 ðtÞÞ dt þ 0:2f 1 ðx1 ðt  τðtÞÞÞ þ0:2f 2 ðx2 ðt  τðtÞÞÞ þ ð0:3 þ 0:1 sin 2tÞS1 ðtÞ þ 0:8 þ0:4 cos 2t; dx2 ðtÞ ¼  1:2x2 ðtÞ þ 0:3f 1 ðx1 ðtÞÞ þ ð0:3  0:2 cos 2tÞf 2 ðx2 ðtÞÞ dt þ 0:2f 1 ðx1 ðt  τðtÞÞÞ þð0:3  0:1 sin 2tÞf 2 ðx2 ðt  τðtÞÞÞ þ ð0:3  0:1 cos 2tÞS2 ðtÞ þ 0:8  0:4 sin 2t; dS1 ðtÞ ¼  ð0:9 þ 0:05j sin tj ÞS1 ðtÞ þ f 1 ðx1 ðtÞÞ; dt dS2 ðtÞ ¼  ð0:9  0:05j cos tj ÞS2 ðtÞ þf 2 ðx2 ðtÞÞ; dt

1

ð4:1Þ

1.4

0.9

1.2

0.8

1

0.7

x2

0.6

x1

783

0.5

0.8 0.6

0.4 0.3

0.4

0.2 0.2

0.1 0

0

20

40

60 time t

80

100

0

0

20

40

60 time t

Fig. 1. Time-domain behavior of the state variable x1 and x2 for the system (4.1), respectively.

80

100

784

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

Example 2. Consider the following generalized competitive neural networks:

where

8 s > < 0:1e ; 2 ; 0:4 0:1s f 1 ðsÞ ¼ f 2 ðsÞ ¼ > : 0:01s3 ;

s r 1;  1 o s r 1;

dx1 ðtÞ ¼  1:8x1 ðtÞ þ ð1:3 þ 0:1 sin 2tÞf 1 ðx1 ðtÞÞ þ 0:01f 2 ðx2 ðtÞÞ dt þ 0:1f 1 ðx1 ðt  τðtÞÞÞ þ0:1f 2 ðx2 ðt  τðtÞÞÞ

s 4 1;

and τðtÞ  0:5. It is easy to see that the assumptions (H1) and (H5) hold. Take R0 ¼ 0.001 and R1 ¼ 3, one can easily check, Φi ð3Þ o 3, Φ2 þ i ð3Þ o 3, Ψ i ð0:001Þ 4 0:001, Ψ 2 þ i ð0:001Þ 40:001, i¼1,2. Hence, Φð3Þ o 1 and Ψ ð0:001Þ 4 0:001, i.e., the assumption (H2n) holds. Hence, all the assumptions of Theorem 3.2 hold. Therefore, it follows from Theorem 3.2 that the non-autonomous system (4.1) has at least one π-periodic solution. Consider the IVP of the system (4.1) with initial conditions ϕðsÞ ¼ ð0; 0; 0; 0ÞT for s A ½  0:5; 0. As shown in Figs. 1–3, numerical simulations also confirm that the system (4.1) has at least one positive π-periodic solution by MATLAB.

þ 0:01S1 ðtÞ þ 0:5 þ 0:3 cos 2t; dx2 ðtÞ ¼  1:8x2 ðtÞ þ 0:01f 1 ðx1 ðtÞÞ þ ð1:3  0:1 cos 2tÞf 2 ðx2 ðtÞÞ dt þ 0:1f 1 ðx1 ðt  τðtÞÞÞ þð0:1  0:1 sin 2tÞf 2 ðx2 ðt  τðtÞðÞ þ 0:01S2 ðtÞ þ 0:5  0:3 sin 2t; dS1 ðtÞ ¼  ð1  0:05j sin tj ÞS1 ðtÞ þ f 1 ðx1 ðtÞÞ; dt dS2 ðtÞ ¼  ð0:9 þ 0:05j cos tj ÞS2 ðtÞ þf 2 ðx2 ðtÞÞ; dt

Remark 3. The discontinuous activation functions of the system (4.1) are described by 8 s s r 1; > < 0:1e ; f 1 ðsÞ ¼ f 2 ðsÞ ¼ 0:4 0:1s2 ;  1 o s r 1; > : 0:01s3 ; s 4 1:

where 8  3s þ 0:4; > :  3s þ 3; e 8  3s þ 0:4; > :  3s þ 3; e

It is easy to see that the activation functions f 1 ðÞ and f 2 ðÞ are discontinuous, unbounded and satisfy exponential growth condition. That is, the following restriction conditions in the earlier papers [20–24,27–31] have been eliminated successfully, such as (i) f i ðxi Þ ði ¼ 1; 2; …; nÞ are unbounded; (ii) f i ðxi Þ ði ¼ 1; 2; …; nÞ satisfy the growth condition: there exist nonnegative constants ki ; hi , such that J co½f i ðxi Þ J ¼

sup ζ A co½f i ðxi Þ

j ζj r ki j xi j þ hi ;

i ¼ 1; 2; …; n;

sup J γ J r cð1 þ J x J α Þ;

γ A co½f ðxÞ

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.15

0.1

0.1

0.05

0.05 0

20

40

60

time t

80

100

s r1; 1 o s o 2; s Z 2;

0.2

0.15

0

s Z 2;

Remark 4. The discontinuous activation functions of the system (4.2) are described by 8  3s þ 0:4; s r 1; > :  3s þ 3; s Z 2; e

S2

S1

where co½f ðxÞ ¼ ðco½f 1 ðx1 Þ; co½f 2 ðxn Þ; …; co½f n ðxn ÞÞT . Moreover, the activation functions f i ðÞ ði ¼ 1; 2Þ are discontinuous at s¼1 and  s ¼  1. By easy calculation, one has f i ð  1Þ ¼ 0:1e  1 o þ  þ 0:3 ¼ f i ð  1Þ ði ¼ 1; 2Þ, f i ð þ1Þ ¼ 0:3 4 0:01 ¼ f i ð þ1Þ ði ¼ 1; 2Þ.  i þ i Hence, the restriction condition f i ðρk Þ of i ðρk Þ in the literature [20–24,27–31]does not require in this paper. Therefore, to discuss the existence of periodic solution to the system (4.1), the methods used in the papers will be invalid. That is to say, the assumptions in this paper (Theorems 3.1 and 3.2) are much less conservative than that in mentioned above.

0.2

s r 1; 1 o s o2;

and τðtÞ  0:5. It is not difficult to verify that the coefficients of the system (4.2) satisfy all the conditions in Theorem 3.3. Therefore, it follows from Theorem 3.3 that the non-autonomous system (4.2) has at least two positive π-periodic solutions. Consider the IVP of the system (4.2) with 6 random initial conditions ϕðsÞ ¼ ð0; 0; 0; 0ÞT , ð1; 1; 1; 1ÞT , ð2; 2; 2; 2ÞT , ð3; 3; 3; 3ÞT , ð4; 4; 4; 4ÞT and ð5; 5; 5; 5ÞT for s A ½  0:5; 0. As shown in Figs. 4–6, numerical simulations also confirm that system (4.2) has at least two positive π-periodic solutions by MATLAB. Moreover, it is easy to see that there are two positive π-periodic solutions which is locally stable. For the issue of multistability of multiple periodic solutions, which is very interesting and we leave it in future work.

(iii) f i ðxi Þ ði ¼ 1; 2; …; nÞ satisfy the nonlinear growth condition: there exist nonnegative constant c and 0 o α o1, such that J co½f ðxÞ J ¼

ð4:2Þ

0

0

20

40

60

time t

Fig. 2. Time-domain behavior of the state variable S1 and S2 for the system (4.1), respectively.

80

100

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

785

0.4

1.5

0.3

S2

x2

1

0.5

0.2 0.1 0 0.4

0 1 0.5

x1

0

40

20

0

60

time

80

100

0.3 0.2

S1

t

0.1 0

0

20

40

60 time

80

100

t

5

5

4.5

4.5

4

4

3.5

3.5

3

3

S1

S2

Fig. 3. Three-dimensional trajectory of the state variables x1 and x2, S1 and S2 for the system (4.1), respectively.

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

20

40

60

80

0

100

0

20

40

time t

60

80

100

time t

5

5

4.5

4.5

4

4

3.5

3.5

3

3

x2

x1

Fig. 4. Time-domain behavior of the state variable x1 and x2 for the system (4.2) with 6 random initial conditions, respectively.

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

20

40

60

80

100

0

0

20

time t

40

60

80

100

time t

Fig. 5. Time-domain behavior of the state variable S1 and S2 for the system (4.2) with 6 random initial conditions, respectively.

and 8  3s þ 0:4; > :  3s þ 3; e

s r 1;

general discontinuous neuron activations functions, the methods in this paper are very effective and thus the results of this paper (Theorems 3.3 and 3.4) are essentially new.

1 o s o 2; s Z 2: 5. Conclusion

It is easy to see that the activation functions f 1 ðxÞ and f 2 ðxÞ are discontinuous, unbounded, and satisfy exponential growth condition. Therefore, the methods used in the papers [10,17–19] cannot be applied to discuss the existence of multiple periodic solutions to the system (4.2). To deal with the existence of multiple periodic solutions (or equilibrium) of neural networks systems with

In this paper, a class of general competitive neural networks with discontinuous right-hand sides, time-varying and distributed delays has been investigated. Under the framework of the theory of Filippov differential inclusions, and by applying fixed point theorem of set-valued analysis due to Agarwal and O'Regan, the

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

5

5

4

4

3

3

S2

x2

786

2 1

2 1

0 5

0 5

4

3

x1

2

1

0

0

20

40

60

80

4

100

3

2

S1

time t

1

0

40

20

0

60

80

100

time t

Fig. 6. Three-dimensional trajectory of the state variables x1 and x2, S1 and S2 for the system (4.2) with 6 random initial conditions, respectively.

existence of one and multiple periodic solutions for the neural network systems have been obtained. It is worthy to point out that, without assuming the boundedness or under linear growth condition of the discontinuous neuron activation functions, our results on the existence of one and multiple positive periodic solutions will also be valid. Finally, we give some numerical examples to show the applicability and effectiveness of our main results. We think it would be interesting to investigate the possibility of extending the results to more complex discontinuous neural network systems with time-varying and distributed delays, such as multistability of multiple periodic solutions, uncertain network systems and stochastic neural network systems [37,57]. These issues will be very interesting and thus they will be the topic of our future research.

Appendix A

bN þi yN þ i ðtÞ Z G

Proof. First, we will show that the set-valued map φ : Ω R1 \ P-P. In fact, for any zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT A Ω R1 \ P and yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; …; y2N ðtÞÞT A φðzÞ. There exists a measurable function γ ¼ ðγ 1 ; γ 2 ; …; γ N ÞT : ½0; þ 1Þ-RN such that γ i ðtÞ A co½f i ðxi ðtÞÞ for a.e. t A ½0; þ 1Þ and Z t þω yi ðtÞ ¼ Gi ðt; vÞΓ i ðvÞ dv; t

yN þ i ðtÞ ¼

Z

t þω

t

GN þ i ðt; vÞγ i ðvÞ dv;

i ¼ 1; 2; …; N:

ðA:1Þ

Thus, for t rv rt þ ω and z A Ω R1 \ P, we have Z tþω Γ i ðvÞ dv; 0 o yi ðtÞ r Gi t

Z

0 o yN þ i ðtÞ r GN þ i which implies Z j yi ðtÞj 1 r Gi

tþω t

tþω t

¼ κ N þ i j yN þ i ðtÞj 1 ;

γ i ðvÞ dv Z

bN þi G jy ðtÞj 1 GN þ i N þ i

i ¼ 1; 2; …; N:

ðA:4Þ

Therefore, for any z A Ω R1 \ P and y A φðzÞ, we have y A P. That is, φðzÞ A P for every fixed z A P, i.e., φ : Ω R1 \ P-P. Next, we will show that φðzÞ is convex for each z A Ω R1 \ P. In fact, for any z ¼ ðxT ; ST ÞT , if y ¼ ðy1 ; y2 ; …; y2N ÞT ; y~ ¼ ðy~ 1 ; y~ 2 ; …; y~ 2N ÞT A φðzÞ, then there exist γ ¼ ðγ 1 ; γ 2 ; …; γ N ÞT : ½0; þ 1Þ-RN and T γ~ ¼ ð~γ 1 ; γ~ 2 ; …; γ~ N Þ : ½0; þ 1Þ-RN with γ i ðtÞ; γ~ i ðtÞ A co½f i ðxi ðtÞÞ for a. e. t A ½0; þ 1Þ, such that for each t A ½0; þ 1Þ we have 8 Z tþω N N
Lemma A.1. If the conditions (H1)–(H2) be satisfied, then the setvalued map φ : Ω R1 \ P-P cp;cv ðPÞ is a k-set contractive map with k ¼0.

Z

yN þ i ðtÞ ¼

tþω t

GN þ i ðt; vÞγ i ðvÞ dv;

and y~ i ðtÞ ¼

Z

tþω

Gi ðt; vÞ

t

þ

N X

Dkij ðvÞ

j¼1

y~ N þ i ðtÞ ¼

Z

tþω t

8 N
Z

i ¼ 1; 2; …; N:

Dij ðvÞ~γ j ðvÞ þ

j¼1

þ1

0

N X

ðA:5Þ

Dτij ðvÞ~γ j ðv  τij ðvÞÞ

j¼1

9 = γ~ j ðv  sÞkij ðsÞ ds þ Bi ðvÞSi ðvÞ þI i ðvÞ dv; ;

GN þ i ðt; vÞ~γ i ðvÞ dv;

i ¼ 1; 2; …; N:

ðA:6Þ

Let 0 rα r 1. In view of the properties of Filippov set-valued map, it is easy to see that αγ i ðtÞ þ ð1  αÞ~γ i ðtÞ A co½f i ðxi ðtÞÞ ði ¼ 1; 2; …; NÞ for a.e. t A ½0; þ 1Þ. Then for each t A ½0; ω we have 8 Z t þω N
γ i ðvÞ dv;

i ¼ 1; 2; …; N:

ðA:2Þ

þ

N X

Dτij ðvÞ½αγ j ðv τij ðvÞÞ þ ð1  αÞ~γ j ðv  τij ðvÞÞ

j¼1 t þω

t

j yN þ i ðtÞj 1 r GN þ i

þ

Γ i ðvÞ dv;

Z

t þω t

γ i ðvÞ dv;

N X

Dkij ðvÞ

j¼1

i ¼ 1; 2; …; N:

ðA:3Þ

For t r v r t þ ω and z A Ω R1 \ P, it follows from (A.1) and (A.3) that Z t þω b G bi Γ i ðvÞ dv Z i j yi ðtÞj 1 ¼ κi j yi ðtÞj 1 ; yi ðtÞ Z G G i t

Z

þ1 0

½αγ j ðv  sÞ þ ð1  αÞ~γ j ðv  sÞkij ðsÞ ds

) þ Bi ðvÞSi ðvÞ þ I i ðvÞ Z A

tþω t

dv

8 N
D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

þ

N X

Dτij ðvÞco½f j ðxj ðv  τij ðvÞÞÞ

r ωGi

j¼1

þ

N X

Z Dkij ðvÞ

j¼1

þ1

)

þ Bi ðvÞSi ðvÞ þI i ðvÞ

1rirN

and Z tþω GN þ i ðt; vÞ½αγ j ðvÞ þ ð1  αÞ~γ j ðvÞ dv ½αyN þ i ðtÞ þð1 αÞy~ N þ i ðtÞ ¼ t Z tþω GN þ i ðt; vÞco½f j ðxj ðvÞÞ dv; i ¼ 1; 2; …; N; A t

that is

;

b i; 9M

8 z A Ω R1 \ P:

Thus, φðΩ R1 \ PÞ is an uniformly bounded set for all z A Ω R1 \ P. For any zðtÞ ¼ ðxT ðtÞ; ST ðtÞÞT A Ω R1 \ P and yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; …; y2N ðtÞÞT A φðzÞ. There exists a measurable function γ ¼ ðγ 1 ; γ 2 ; …; γ N ÞT : ½0; þ 1Þ-RN such that γ i ðtÞ A co½f i ðxi ðtÞÞ for a.e. t A ½0; þ 1Þ, and Z tþω Gi ðt; vÞΓ i ðvÞ dv; yi ðtÞ ¼ t Z tþω yN þ i ðtÞ ¼ GN þ i ðt; vÞγ i ðvÞ dv; i ¼ 1; 2; …; N: t

½αyi ðtÞ þð1 αÞy~ i ðtÞ A φi ðzÞðtÞ and A φN þ i ðzÞðtÞ; i ¼ 1; 2; …; N:

½αyN þ i ðtÞ þ ð1  αÞy~ N þ i ðtÞ

Hence, ~ ½αyðtÞ þ ð1  αÞyðtÞ A φðzÞ; which implies that φðzÞ is convex for each z A Ω R1 \ P. Finally, we will show that the set-valued map φ : Ω R1 \ P-P is a k-set contractive map with k¼ 0. In fact, it is enough to show that φ : Ω R1 \ P-P is a compact map. According to the Ascoli–Arzela Theorem, it suffices to show that φðΩ R1 \ PÞ is an uniformly bounded and equi-continuous set. Note that fi has a finite number of discontinuous points on any compact interval of R. In particular, fi has a finite number of discontinuous points on the interval ½  R1 ; R1 . Without loss of generality, let fi discontinuous at points fρik : k ¼ 1; 2; …; li g on the interval ½  R1 ; R1 , and assume that R1 o ρi1 oρi2 o⋯ o ρili o R1 . Let us consider a series of continuous functions: ( f i ðsÞ if s A ½  R1 ; ρi1 Þ; 0 f i ðsÞ ¼  i f i ðρ1 Þ if s ¼ ρi1 ; 8 þ i f ðρ Þ > > < i k k f f i ðsÞ ¼ i ðsÞ > > : f  ðρi Þ i kþ1

l

9 =

J ψ J 1 ¼ esssups A ð  1;0 j ψ ðsÞj , which

and

b i; M b N þ i g; J y J Z r max fM dv;

i ¼ 1; 2; …; N;

f ii ðsÞ ¼

:j ¼ 1

τ M k M M M ½DM ij þ ðDij Þ þ ðDij Þ ðmaxfj M j j ; j mj j g þ J ψ J 1 Þ þ Bi R1 þ I i

where i ¼ 1; 2; …; N yields

co½f j ðxj ðv  sÞÞkij ðsÞds

0

8
787

if s ¼ ρik ; if s A ðρik ; ρik þ 1 Þ; k ¼ 1; 2; …; li 1: if s ¼ ρik þ 1 ;

8 þ < f i ðρil Þ

if s

: f i ðsÞ

if s A ðρili ; R1 :

i

Let t n ; t nn A ½0; ω, then for any z A Ω R1 \ P and i ¼ 1; 2; …; N, we have Z n  Z t nn þ ω  t þω    j yi ðt n Þ  yi ðt nn Þj ¼  Gi ðt n ; vÞΓ i ðvÞ dv  Gi ðt nn ; vÞΓ i ðvÞ dv nn  tn  t Z n  Z n  t þω  t þω   r Gi ðt n ; vÞΓ i ðvÞ dv  Gi ðt nn ; vÞΓ i ðvÞ dv n  tn  t Z n  Z tnn þ ω  t þω    nn nn Gi ðt ; vÞΓ i ðvÞ dv Gi ðt ; vÞΓ i ðvÞ dv þ  tn  t nn Z n   t þω    ½Gi ðt n ; vÞ  Gi ðt nn ; vÞΓ i ðvÞ dv ¼  tn  Z nn  Z nn   t   t þω      Gi ðt nn ; vÞΓ i ðvÞ dv þ  Gi ðt nn ; vÞΓ i ðvÞ dv; þ  tn   tn þ ω  and

Z n   t þω    n nn ½GN þ i ðt ; vÞ GN þ i ðt ; vÞγ i ðvÞ dv j yN þ i ðt Þ yN þ i ðt Þj r   tn  Z nn  Z nn   t   t þω      GN þ i ðt nn ; vÞγ i ðvÞ dv þ  GN þ i ðt nn ; vÞγ i ðvÞ dv: þ  tn   tn þ ω  n

nn

By applying the mean value theorem of derivations, we obtain j Gi ðt n ; vÞ  Gi ðt nn ; vÞj ¼ j Gi ðt n þ θðt nn t n Þ; vÞai ðt n þ θðt nn t n ÞÞ

¼ ρili ;

nn n J t nn  t n j r Gi aM i j t t j ;

j GN þ i ðt n ; vÞ GN þ i ðt nn ; vÞj ¼ j GN þ i ðt n þ θðt nn  t n Þ; vÞci

Denote

8 9 ( ) < = 0 k li M i ¼ max max ff i ðsÞg; max max ff i ðsÞg ; max ff i ðsÞg i i i i :s A ½  R1 ;ρ1  ; 1 r k r li  1 s A ½ρ ;ρ  s A ½ρl ;R1  k kþ1 i

and

8 9 ( ) < = 0 k li mi ¼ min min ff i ðsÞg; min min ff i ðsÞg ; min ff i ðsÞg : :s A ½  R1 ;ρi1  ; 1 r k r li  1 s A ½ρi ;ρi  s A ½ρil ;R1  k kþ1

nn n ðt n þθðt nn t n ÞÞ J t nn t n j r GN þ i cM i j t t j ;

where 0 oθ o 1. And thus Z n   t þω    n nn b nn n ½Gi ðt ; vÞ  Gi ðt ; vÞΓ i ðvÞ dv r aM  i Mij t t j ;  tn  Z n   t þω    n nn nn n b ½GN þ i ðt ; vÞ  GN þ i ðt ; vÞγ i ðvÞ dv r cM  i MNþij t t j :  tn 

i

Obviously, we have sup z A Ω R1 \ P

j co½f i ðxi ðtÞÞj r maxfj M i j ; j mi j g;

i ¼ 1; 2; …; N:

Hence b N þ i; j yN þ i ðtÞj 1 r ωGN þ i maxfj M i j ; j mi j g 9 M Z t þω Γ i ðvÞ dv j yi ðtÞj 1 r Gi t

ðA:7Þ

Moreover, we have Z nn   t  M   b nn Gi ðt ; vÞΓ i ðvÞ dv r i j t nn t n j ;   tn  ω Z nn   t þω  M   b Gi ðt nn ; vÞΓ i ðvÞ dv r i j t nn  t n j ;   tn þ ω  ω Z nn   t  M b   GN þ i ðt nn ; vÞγ i ðvÞ dv r N þ i j t nn  t n j ;   tn  ω

788

D. Wang, D. Luo / Neurocomputing 168 (2015) 777–789

Z nn   t þω  M   b nn GN þ i ðt ; vÞγ i ðvÞ dv r N þ i j t nn  t n j :   tn þ ω  ω

That is J j F ðt; zÞ J j ¼ supfj uj : u A F ðt; zÞg r hq ðtÞ

Hence, for i ¼ 1; 2; …; N, we obtain   2 b nn n j yi ðt n Þ  yi ðt nn Þj r aM Mij t t j ; i þ ω   2 b M N þ i j t nn  t n j : þ j yN þ i ðt n Þ  yN þ i ðt nn Þj r cM i ω

for all J z J Z r q and a.e. t A ½0; ω. Hence, F ðt; zÞ is a L1-Carathéodory map. In view of Lemma 2.3, the set F(z) is nonempty for each fixed z A Ω R1 . Consider the linear continuous operator L : L1 ð½0; ω; Rn ÞCð½0; ω; R2N Þ, Z t þ ω Z tþω G1 ðt; vÞu1 ðvÞ dv; G2 ðt; vÞu2 ðvÞ dv; …; LuðtÞ ¼

As a result we have that     2 b Mi ; aM J yðt n Þ  y2 ðt nn Þ J Z r max max i þ ω 1rirN    2 b MNþi cM j t nn  t n j -0 as t nn -t n : max i þ ω 1rirN

Lemma A.2. Assume that the conditions (H1)–(H2) and (H4) hold. Then the multi-valued map φ : Ω R1 \ P-P cp;cv ðPÞ is an upper semicontinuous (USC) map.

N X

F i ðt; zÞ ¼

Dij ðtÞco½f j ðxj ðtÞÞ þ

j¼1

þ

N X

Dkij ðtÞ

N X

graph.

Set

Dτij ðtÞco½f j ðxj ðt  τij ðtÞÞÞ

j¼1

Z

þ1 0

j¼1

co½f j ðxj ðt  sÞÞkij ðsÞds þ Bi ðtÞSi ðtÞ þ I i ðtÞ;

F N þ i ðt; zÞ ¼ co½f i ðxi ðtÞÞ;

i ¼ 1; 2; …; N:

Let J j F ðt; zÞ J j ¼ supfj uj : u A F ðt; zÞg and L1 ð½0; ω; R2N Þ be the Banach space of all functions u ¼ ðu1 ; u2 ; …; u2N ÞT : ½0; ω-R2N which are Lebesgue integrable. Define the multi-valued operator F ¼ ðF 1 ; F 2 ; …; F 2N ÞT : Ω R0 -L1 ð½0; ω; R2N Þ by letting F i ðzÞ ¼ fui A L1 ð½0; ω; RÞ : ui ðtÞ A F i ðt; zðtÞÞ for a:e: t A ½0; ωg; i ¼ 1; 2; …; 2N: Next, we will show that F ðt; zÞ is a L1-Carathéodory map. In fact, for every z A Ω R1 , F ðt; zÞ is measurable with respect to t since each co½f i ðxi Þ ði ¼ 1; 2; …; nÞ has closed values for every z A Ω R1 and all the coefficients of F ðt; zÞ are continuous. Similar to the step 2 of Appendix in [25], it is easy to show that F ðt; zÞ is USC with respect to z for a.e. t A ½0; ω. Denote hq ðtÞ ¼

N X N X

τ M k M f½DM ij þ ðDij Þ þ ðDij Þ ðmaxfj M j j ; j mj j g þ J ψ J 1 Þ

i¼1j¼1 M þ BM i R1 þ I i þ maxfj M i j ; j mi j gg:

Then, for each q 4 0ðq r R1 Þ, there exists hq ðtÞ A L1 ð½0; ω; ½0; þ 1ÞÞ defined as the above and such that J j F i ðt; zÞ J j ¼ supfj ui j : ui A F i ðt; zÞg r

N X

τ M k M M M f½DM ij þðDij Þ þ ðDij Þ ðmaxfj M j j ; j mj j g þ J ψ J 1 Þ þ Bi R1 þ I i

j¼1

o hq ðtÞ; J j F N þ i ðt; zÞ J j ¼ supfj uN þ i j : uN þ i A F N þ i ðt; zÞg ¼ supfj uN þ i j : uN þ i A co½f i ðxi ðtÞÞg r maxfj M i j ; j mi j g o hq ðtÞ; i ¼ 1; 2; …; N:

t þω t

Thus, φðΩ R1 \ PÞ is an equi-continuous set in Z. Hence, the set-valued map φ : Ω R1 \ P-P is a compact map. Therefore, the multi-valued map φ : Ω R1 \ P-P cp;cv ðPÞ is a k-setcontractive map with k ¼0.□

Proof. We will show that φ has closed F ðt; zÞ ¼ ðF 1 ðt; zÞ; F 2 ðt; zÞ; …; F 2N ðt; zÞÞT , where

t

Z

T G2N ðt; vÞu2N ðvÞ dv ;

t

t A ½0; ω:

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Dongshu Wang was born in Anhui, China, in 1981. He received the B.S. degree in mathematics in 2003 from Huaibei Normal University, Huaibei, China, and the M.S. degree in applied mathematics from Huaqiao University, Quanzhou, China, in 2006. Presently he is working at Huaqiao University. He is currently pursuing the Ph. D. degree in applied mathematics at the College of Mathematics and Econometrics, Hunan University, Changsha, China. His current research interests include neural networks, mathematical biology, qualitative theory of differential equations and difference equations.

Daozhong Luo was born in Hubei, China, in 1976. He received the B.S. degree in mathematics in 2001 from China Three Gorges University, Yichang, China, and the M.S. degree in applied mathematics from Xiamen University, Xiamen, China, in 2005. Presently he is working at Huaqiao University. His current research interests include functional analysis, mathematical statistics, qualitative theory of differential equations and difference equations.