On the stability of impulsive functional differential equations with infinite delays

On the stability of impulsive functional differential equations with infinite delays Xiaodi Li∗,a, T. Caraballo∗,b, R. Rakkiyappanc, X. Hana aSchool of Mathematical Sciences, Shandong Normal University, Ji’nan, 250014, P.R. China bDpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain cDepartment of Mathematics, Bharathiar University, Coimbatore - 641 046, Tamilnadu, India Abstract In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Key words: Infinite delays; Impulsive differential equations; Lyapunov functions; Stability 1. Introduction As a popular and important topic, the stability problem of impulsive functional differential equations has generated a considerable interest in recent years, and a number of papers dealing with the stability problem of impulsive functional differential equations have appeared, see [1–12] and the references cited therein. In particular, stability of impulsivefunctional differential equations with infinite delays has recently received significant attention, see [13–21]. For example, Luo and Shen [13–15] studied the uniform asymptotic stability of impulsive functional differential equations with infinite delays by using Lyapunov functionals or/and Lyapunov functions and Razumikhin technique. In [16], Zhang and Sun extended the technique developed in [17] to impulsive systems and derived some new results on uniform stability of impulsive functional differential equations with infinite delays. Quite recently, Faria et al.[18] studied the existence and global stability for a class of non-autonomous impulsive functional ∗Corresponding Author. Email address : car[email protected] (T. Caraballo) 1 Stability of IFDEs with infinite delays differential equations with infinite delays via some analysis techniques. Our research group [19-21] also studied the stability problem of impulsive functional differential equations with infinite delays from impulsive perturbation and impulsive control point of view, respectively. In the present paper, we will further investigate the stability problem of impulsive functional differential equations with infinite delays. By using Lyapunov functions and Razumikhin technique, some new results to guarantee uniform stability and the uniformly asymptotic stability are obtained. One of the most remarkable advantages of the results in this paper is that the Razumikhin condition is independent of impulses and enables one to deal with impulsive infinite delay differential equations with large impulsive perturbations. The methods developed in this paper extend and improve the results in [13–15,19,20]. Moreover, they can be applied to the some cases not covered by the results in [9,11,12,16,18]. The paper is organized as follows. In Section 2, we introduce some definitions and notations. In Section 3, we present some new theorems on uniform stability and uniform asymptotic stability for impulsive functional differential equations with infinite delays. Two examples are given to illustrate the advantages of the results in Section 4. In Section 5, we draw a conclusion. 2. Preliminaries Let Rdenote the set of real numbers,R+the set of positive real numbers and Rnthe ndimensional real space equipped with the Euclidean norm |•|. Let Z+denote the set of positive integers, i.e., Z+={1,2,...}.For any interval J⊆R, any subset S⊆Rk(1 ≤k≤n),C(J,S)= {ϕ:J→Sis continuous}and PC(J,S)={ϕ:J→Sis continuous everywhere except at finite number of points t, at which ϕ(t+), ϕ(t−) exist and ϕ(t+)=ϕ(t)}.The impulse times tksatisfy 0 ≤t0<t1< . . . < tk< . . . , limk→+∞tk= +∞. Denote by αa constant satisfying −∞ ≤ α≤0.In the case when α=−∞, the interval [t+α, t] is understood to be replaced by (−∞,t]. Consider the impulsive functional differential equations of the form                      x′(t)=f(t,xt),t≥σ, t,tk, ∆x|t=tk=x(tk)−x(t− k)=Ik(tk,x(t− k)),k∈Z+, xσ=φ(s), α ≤s≤0, (1) 2 Stability of IFDEs with infinite delays where σ≥t0, φ ∈C,f∈C([tk−1,tk)×C,Rn),f(t,0) =0,Cis some open set in PC([α, 0],Rn). Given a function x(·) : [α, +∞)→S, for each t≥t0,we denote by xtis the element in C defined by xt(s)=x(t+s),s∈[α, 0].Define PCB ={ϕ∈C:ϕis bounded}and for ϕ∈PCB, the norm of ϕis defined by kϕk=supα≤θ≤0|ϕ(θ)|. Define PCBδ={ϕ∈PCB :kϕk ≤ δ}. Ik(t,x)∈C([0,∞)×Rn,Rn) and Ik(t,0) =0,k∈Z+.Moreover, for any given ρ > 0,there exists a ρ1∈(0, ρ) such that x∈S(ρ1) implies that x+Ik(tk,x)∈S(ρ),where S(ρ)={x:|x|< ρ, x∈Rn}. In this paper, we assume that fand Iksatisfy certain conditions such that the solution of (1) exists on [σ, +∞) and is unique, see [3, 20] for detailed information. We denote by x(t)= x(t, σ, φ) the solution of (1) with initial value (σ, φ). Since f(t,0) =0,Ik(t,0) =0,k∈Z+, then x(t)=0 is a solution of (1), which is called the trivial solution. We introduce some definitions (see [3]) as follows: Definition 2.1. The function V: [α, ∞)×C→R+is said to belong to class ν0if (i) Vis continuous on each of the sets [tk−1,tk)×Cand lim(t,ϕ)→(t− k,ψ)V(t, ϕ)=V(t− k, ψ) exists; (ii) V(t,x) is locally Lipschitzian in xand V(t,0) ≡0. Definition 2.2. Given a function V∈ν0, for any (t, ψ)∈[tk−1,tk)×C, the upper right-hand Dini derivative of Valong the solution of (1) is defined by D+V(t, ψ(0)) =limsup h→0+ {V(t+h, ψ(0) +hf(t, ψ)) −V(t, ψ(0))}/h. Definition 2.3. The trivial solution x=0 of (1) is said to be (P1) stable, if for any σ≥t0and ε > 0, there exists a δ=δ(ε, σ)>0 such that φ∈PCBδ implies |x(t, σ, φ)|< ε, t≥σ; (P2) uniformly stable, if the δin (P1) is independent on σ; (P3) uniformly asymptotically stable, if (P2) holds and there exists some δ > 0 such that for any ε > 0 there exists some T=T(ε, δ)>0 such that φ∈PCBδimplies |x(t, σ, φ)|< ε, t≥σ+T. In addition, we define the following classes of functions for later use: 3 Stability of IFDEs with infinite delays K1={a∈C(R+,R+)|a(0) =0 and a(s)>0 for s>0}; K2={a∈C(R+,R+)|a∈K1and ais non-decreasing in s}. 3. Stability results Now we can state our main stability result. Theorem 3.1. Assume that there exist some functions W1,W2∈K2,P,G∈K1,q∈C(R+,R+), g∈PC(R+,R+),V(t,x)∈ν0and constants βk≥0,k∈Z+such that (i) W1(|x|)≤V(t,x)≤W2(|x|),(t,x)∈[t0,∞)×S(ρ); (ii) For any (tk, ψ)∈R+×PC([α, 0],S(ρ1)), V(tk, ψ(0) +Ik(tk, ψ)) −V(t− k, ψ(0)) ≤βkV(t− k, ψ(0)), where P∞ k=1βk. =β < ∞; (iii)For anyσ≥t0andψ∈PC([α, 0],S(ρ)),if P(V(t, ψ(0))) >V(t+θ, ψ(θ)) formax{α, −q(V(t))} ≤ θ≤0,then D+V(t, ψ(0)) ≤ −g(t)G(V(t, ψ(0))),t∈[tk−1,tk),k∈Z+, where P(s)>s for s >0; (iv) For any given ε2> ε1>0,there exists a η=η(ε1, ε2)>0such that for any A >0 implies that ZA+η Ag(t)dt >(1 +β)W2(ε2) M, where M =inf0.5W1(ε1)≤s≤W2(ε2)G(s). Then the trivial solution of (1) is uniformly asymptotically stable. Proof. We first show that the trivial solution of (1) is uniformly stable. For any ε∈(0, ρ1),one may choose δ > 0 such that W2(δ)≤β⋆−1W1(ε), where β⋆. = Q∞ k=1(1+βk)+1. For any σ≥t0and φ∈PCBδ,let x(t)=x(t, σ, φ) be a solution of (1) through (σ, φ). Note that φ∈PCBδ,it is obvious that W1(|x|)≤V(t,x(t)) ≤W2(δ)≤β⋆−1W1(ε)<W1(ε), σ +α≤t≤σ, 4 Stability of IFDEs with infinite delays which implies that |x(t)|< ρ1,t∈[σ+α, σ]. Suppose that σ∈[tm−1,tm) for some m∈Z+,then we can prove for t∈[σ, tm) V(t,x(t)) ≤β⋆−1W1(ε).(2) If this is not true, then there exists some t∈[σ, tm) such that V(t,x(t)) > β⋆−1W1(ε). Define t⋆=inf{t∈[σ, tm),V(t,x(t)) > β⋆−1W1(ε)}, then it is obvious that t⋆≥σ, V(t⋆,x(t⋆)) =β⋆−1W1(ε) and V(t,x(t)) ≤β⋆−1W1(ε),t∈[σ, t⋆]. Meanwhile, we know D+V|(1)(t⋆,x(t⋆)) ≥0.(3) In this case, it holds P(V(t⋆,x(t⋆))) >V(t⋆,x(t⋆)) =β⋆−1W1(ε)≥V(s,x(s)),t⋆+α≤s≤t⋆. By condition (iii),g∈PC(R+,R+),and G∈K1, we obtain D+V(t⋆,x(t⋆)) ≤ −g(t⋆)G(V(t⋆,x(t⋆))) =−g(t⋆)G(β⋆−1W1(ε)) <0, which is a contradiction with (3). Thus (2) holds. It implies that x(t− m)∈S(ρ1),x(tm)∈S(ρ). Then note that V(tm,x(tm)) ≤(1 +βm)V(t− m,x(t− m)) ≤β⋆−1(1 +βm)W1(ε). We next can prove that for t∈[tm,tm+1) V(t,x(t)) ≤β⋆−1(1 +βm)W1(ε). Suppose that this is not true, then we can define t∗=inf{t∈[tm,tm+1),V(t,x(t)) > β⋆−1(1 +βm)W1(ε)}. Thus, we can obtain a contradiction by the same arguments as the proof of (2), and we will therefore omit the details. By induction hypothesis, we may prove that for t∈[σ, tm)∪[tk,tk+1),k≥m, V(t,x(t)) ≤β⋆−1(1 +βm)(1 +βm+1)···(1 +βk)W1(ε), 5 Stability of IFDEs with infinite delays which yields W1(||x||)≤V(t,x(t)) ≤β⋆−1Y σ<tk<t (1 +βk)W1(ε)<W1(ε),t≥σ. Hence, |x(t)|< ε, t≥σ. In view of the choice of δ, the trivial solution of (1) is uniformly stable. Next we show the uniformly asymptotic stability. Since the trivial solution of (1) is uniformly stable, for any given ε2∈(0, ρ1), σ ≥t0,we can find a corresponding δ=δ(ε2)>0 such that for any φ∈PCBδimplies that |x(t)| ≤ ε2< ρ1,t≥σand V(t,x(t)) ≤W2(ε2),t≥σ. In the sequel, we assume without loss of generality that σ∈[tm1−1,tm1),m1∈Z+. For any ε∈(0, ε2),choose constants Mand aas follows: M=M(ε2, ε)=inf 0.5W1(ε)≤s≤W2(ε2)G(s), a=a(ε2, ε)=mininf 0.5W1(ε)≤s≤W2(ε2)[P(s)−s],0.5W1(ε). Then it is obvious that M>0,a>0. Also, from condition (iv), we know that there exists η=η(ε, ε2)>0 such that for any A>0 implies that ZA+η Ag(t)dt >(1 +β)W2(ε2) M.(4) Now we choose N∈Z+such that 0.5W1(ε)+(N−1)a≤W2(ε2)<0.5W1(ε)+Na. Since P∞ k=1βk<∞,there exists a large enough integer N0>m1such that ∞ X i=N0 βi<a 3W2(ε2),and βk<a 3NW1(ε),k≥N0.(5) Suppose that tN0=σ+λη, where λis a constant. Then we show that there exists T1>tN0such that V(T1,x(T1)) <0.5W1(ε)+(N−1)a.(6) Suppose on the contrary, then for all t>tN0 V(t,x(t)) ≥0.5W1(ε)+(N−1)a≥0.5W1(ε). 6 Stability of IFDEs with infinite delays In view of the definition of a, we have P(V(t,x(t))) ≥V(t,x(t)) +a ≥0.5W1(ε)+(N−1)a+a =0.5W1(ε)+Na >W2(ε2)≥V(s,x(s)),t+α≤s≤t,t>tN0. By assumption (iii), we obtain that the inequality D+V(t,x(t)) ≤ −g(t)G(V(t,x(t))) holds for all t>tN0,t,tk. Integrating above inequality from tN0to tN0+η, by (4) we get V(tN0+η, x(tN0+η)) ≤V(tN0,x(tN0)) −ZtN0+η tN0 g(s)G(V(s))ds +X tN0<t<tN0+η [V(tk)−V(t− k)] ≤V(tN0,x(tN0)) −MZtN0+η tN0 g(s)ds +X tN0<t<tN0+η βkV(t− k) ≤W2(ε2)−MZtN0+η tN0 g(s)ds +X tN0<t<tN0+η βkW2(ε2) ≤W2(ε2)(1 +β)−MZtN0+η tN0 g(s)ds <0, which is a contradiction. Thus (6) holds. One may choose T1=tN0+η=σ+(λ+1)η. We next show that for all t>T1 V(t,x(t)) <0.5W1(ε)+(N−1)a+a 2.(7) Suppose this is not true, then there exists τ2>T1such that V(τ2,x(τ2)) ≥0.5W1(ε)+(N−1)a+a 2(8) and V(t,x(t)) <0.5W1(ε)+(N−1)a+a 2for all T1≤t< τ2.(9) 7 Stability of IFDEs with infinite delays Suppose that T1∈[tm,tm+1),m≥N0,m∈Z+,then we claim that τ2≥tm+1. Otherwise, then τ2∈[T1,tm+1).Since (6) holds, it is clear that there exists τ1∈[T1, τ2) such that V(τ1,x(τ1)) =0.5W1(ε)+(N−1)a and V(τ1,x(τ1)) ≤V(t,x(t)) ≤V(τ2,x(τ2)), τ1≤t≤τ2. Then we have for t∈[τ1, τ2] P(V(t,x(t))) ≥V(t,x(t)) +a ≥0.5W1(ε)+Na >W2(ε2)≥V(s,x(s)),t+α≤s≤t. Using condition (iii), we have D+V(t,x(t)) ≤ −g(t)G(V(t)) ≤0, τ1≤t≤τ2, which implies V(τ2,x(τ2)) ≤V(τ1,x(τ1)). This is a contradiction in view of (8). Then we have proven that τ2≥tm+1.Without loss of generality, we may suppose that τ2∈[tm+q,tm+q+1),q≥1.Next we shall claim that there exists τ′ 1∈(T1, τ2) such that 0.5W1(ε)+(N−1)a<V(τ′ 1,x(τ′ 1)) <0.5W1(ε)+(N−1)a+a 2.(10) By virtue of (9), we only need to prove the left-hand inequality of (10). Suppose this inequality does not hold, then for all t∈(T1, τ2), V(t,x(t)) ≤0.5W1(ε)+(N−1)a. Then by (8), we know that there must be τ2=tm+q. It follows that V(tm+q,x(tm+q)) ≥0.5W1(ε)+(N−1)a+a 2,V(t− m+q,x(t− m+q)) ≤0.5W1(ε)+(N−1)a, which together with condition (ii) yields a 2≤βm+qV(t− m+q,x(t− m+q)) ≤βm+qW2(ε2). 8 Stability of IFDEs with infinite delays Thus, it leads to βm+q≥a 2W2(ε2). This contradicts the first inequality of (5) and thus (10) holds. Defining now eτ1=sup{t∈[T1, τ2],V(t,x(t)) <0.5W1(ε)+(N−1)a}, then V(eτ1−,x(eτ1−)) ≤0.5W1(ε)+(N−1)a, V(eτ1,x(eτ1)) =V(eτ1+,x(eτ1+)) ≥0.5W1(ε)+(N−1)a (11) and 0.5W1(ε)+(N−1)a≤V(t,x(t)) ≤0.5W1(ε)+(N−1)a+a 2,t∈[eτ1, τ2].(12) By virtue of (10), we know that eτ1< τ2.Note that τ2∈[tm+q,tm+q+1), we further show that eτ1<tm+q.Suppose on the contrary that eτ1∈[tm+q, τ2), then there is no impulse point tk between eτ1and τ2. From (12), we have P(V(t,x(t))) ≥V(t,x(t)) +a =0.5W1(ε)+Na >W2(ε2)≥V(s,x(s)),t+α≤s≤t,eτ1≤t≤τ2. By assumption (iii), we obtain D+V(t,x(t)) ≤ −g(t)G(V(t)) ≤0,eτ1≤t≤τ2, which implies that V(τ2,x(τ2)) ≤V(eτ1,x(eτ1)). This is a contradiction with the definition of eτ1. Consequently, we have that eτ1<tm+q. Suppose that eτ1∈[tm+k,tm+k+1),1≤k<q,then wenowconsiderthe followingtwopossible cases: Case 1: If eτ1>tm+k,i.e., eτ1∈(tm+k,tm+k+1),then considering the definition of eτ1, we have V(eτ1,x(eτ1)) =0.5W1(ε)+(N−1)a. 9 Stability of IFDEs with infinite delays It is easy to check that the condition in Property 4.2 is satisfied. Hence, the trivial solution of (11) with above parameters is uniformly asymptotically stable. Remark 4.3. Note that that function |b(t)|can be unbounded and the impulse constant M= Q(1 +βk) can be large enough. Thus our results have wider range and can be applied to some cases not covered by the results in [13–15,18-21]. 5. Conclusion In this paper, we further investigated the stability problem of impulsive functional differential equations with infinite delays. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability were obtained. Our results are milder and more general than several previously known results. But the results in this paper were only given from the impulsiveperturbation point of view. How to obtain the different results, from impulsive control point of view, would be a difficult problem and need further consideration in the future. Acknowledgement This work was jointly supported by the Project of Shandong Province Higher Educational Science and Technology Program (J12LI04), Research Fund for Excellent Young and Middleaged Scientists of Shandong Province (BS2012DX039) and National Natural Science Foundation of China (11301308). The research of T. Caraballo has been partially supported by FEDER (European Community) and the Spanish Ministerio de Economia y Competitividadproject MTM2011-22411, and the Consejeria deInnovaci´on, Ciencia yEmpresa (Juntade Andalucia) undergrant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492. References [1] A. Anokhin, On linear impulsive systems for functional differential equations, Soviet Mathematics Doklady 33 (1986) 220–223. 16 Stability of IFDEs with infinite delays [2] K. Gopalsamy, B. Zhang, On delay differential equations with impulses, Journal of Mathematical Analysis and Applications 139 (1989) 110–122. [3] X. Fu, B. Yan, Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, 2005. [4] I. Stamova, G. Stamov, Impulsivecontrol on global asymptotic stability for a class of impulsive bidirectional associative memory neural networks with distributed delays, Mathematical and Computer Modelling 53 (2011) 824–831. [5] X. Liu, G. Ballinger, Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Analysis 53 (2003) 1041–1062. [6] A. Ignatyev, On the stability of invariant sets of systems with impulse effect, Nonlinear Analysis 69 (2008) 53–72. [7] I. Stamova, G. Stamov, Stability analysis of impulsive functional systems of fractional order, Communications in Nonlinear Science and Numerical Simulation 19 (2014) 702–709. [8] J. Shen, J. Yan, Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear Analysis 33 (1998) 519–531. [9] F. Chen, X. Wen, Asymptotic stability for impulsive functional differential equation, Journal of Mathematical Analysis and Applications 336 (2007) 1149–1160. [10] S. Zavalishchin, A. Sesekin, Dynamic Impulse Systems Theory and Applications, Kluwer Academic Publishers Group, Dordrecht, 1997. [11] Q. Wang, X. Liu, Exponential stability for impulsive delay differential equations by Razumikhin method, Journal of Mathematical Analysis and Applications 309 (2005) 462–473. [12] B. Liu, X. Liu, K. Teo, Q. Wang, Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA Journal of Applied Mathematics 71 (2006) 47–61. [13] Z. Luo, J. Shen, Stability of impulsivefunctionaldifferential equations via the Liapunovfunctional, Applied Mathematics Letters 22 (2009) 163–169. [14] Z. Luo, J. Shen, Stability results for impulsive functional differential equations with infinite delays, Journal of Computational and Applied Mathematics 131 (2001) 55–64. [15] Z. Luo, J. Shen, Stability and boundedness for impulsive functional differential equations with infinite delays, Nonlinear Analysis 46 (2001) 475–493. [16] Y. Zhang, J. Sun, Stability of impulsive infinite delay differential equations, Applied Mathematics Letters 19 (2006) 1100–1106. [17] S. Zhang, A new technique in stability of infinite delay differential equations, Computers Mathematics with Applications 44 (2002) 1275–1287. [18] T. Faria, M. Gadotti, J. Oliveira, Stability results for impulsive functional differential equations with infinite delay, Nonlinear Analysis: Theory, Methods and Applications 75 (2012) 6570–6587. [19] X. Fu, X. Li, Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems, Journal of Computational and Applied Mathematics 224 (2009) 1–10. [20] X. Li, New results on global exponential stabilization of impulsive functional differential equations with 17 Stability of IFDEs with infinite delays infinite delays or finite delays, Nonlinear Analysis Series B: Real World Applications 11 (2010) 4194–4201. [21] X. Li, H. Akca, X. Fu, Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions, Applied Mathematics and Computation 219 (2013) 7329– 7337. 18