# Complete Controllability of Impulsive Stochastic Integrodifferential Systems in Hilbert Space

Authors: Xisheng Dai; Feng Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper concerns the complete controllability of the impulsive stochastic integrodifferential systems in Hilbert space. Based on the semigroup theory and Burkholder-Davis-Gundy's inequality, sufficient conditions of the complete controllability for impulsive stochastic integro-differential systems are established by using the Banach fixed point theorem. An example for the stochastic wave equation with impulsive effects is presented to illustrate the utility of the proposed result.


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## Body

## 1. Introduction

It is well known that controllability is one of the fundamental concepts and plays an important role in control theory and engineering. The problem which is about controllability of linear and nonlinear stochastic systems represented by SODE (stochastic ordinary differential equation) in finite dimensional space has been extensively studied (e.g., [1–4] and references therein). The controllability for infinite dimensional stochastic systems represented by SPDE (stochastic partial differential equation) is natural generalization of stochastic systems in finite dimensional space [5]. According to the literature, at least three types of infinite dimensional stochastic systems have been studied, that is, approximate, complete, and S-controllability [6], so the controllability research of the infinite dimensional stochastic systems is usually more complicated than that of the finite dimensional. For linear stochastic system, the controllability problem has been studied by some authors [6, 7], which is shown as the following SPDE:
(1)dx(t)=[Ax(t)+Bu(t)]dt+Σ(t)dW(t),t∈[0,T],x(0)=x0∈ℒ2(Ω,H),
where x0 is ℱ0-measurable, H is separable Hilbert space, A is the infinitesimal generator of a strongly continuous semigroup S(t) on H, B∈ℒ(U,H), u(t) is feedback control, W(t) is Q-Wiener process, andΣ∈ℒ2(Q1/2E,H). For nonlinear stochastic systems in infinite dimensional space, there are also many results on the controllability theory (see [8–13]).On the other hand, the impulsive effects exist widely in many evolution processes in which the states are changed abruptly at certain moments of time, involving fields such as finance, economics, mechanics, electronics, and telecommunications (see [14] and references of therein). Impulsive differential systems have emerged as an important area investigation in applied sciences, and many papers have been published about the controllability of impulsive differential systems both in finite and infinite dimensional space. Sakthivel et al. [15] established the sufficient conditions for approximate controllability of nonlinear impulsive differential systems by Schauder’s fixed point theorem; Li et al. [16] investigated the complete controllability of the first-order impulsive functional differential systems in Banach space using Schaefer’s fixed point theorem; Chang [17] studied the complete controllability of impulsive functional differential systems with infinite delay; Sakthivel et al. [18] discussed complete controllability of second-order nonlinear impulsive differential systems. However, the complete controllability problem of impulsive stochastic integro-differential systems has not been investigated in infinite dimensional space yet, to the best of our knowledge, although [19–22], respectively, investigated the controllability of impulsive stochastic control systems in finite dimensional space by using contraction mapping principle; and Subalakshmi and Balachandran [23] studied the approximate controllability of nonlinear stochastic impulsive systems in Hilbert spaces by using Nussbaum’s fixed point theorem. Based on Banach fixed point theorem, the proposed work in this paper on the complete controllability of the integro-differential stochastic systems with impulsive effects in Hilbert spaces is new in the literature.In this paper, our main purpose is to show the complete controllability of following impulsive stochastic integro-differential systems in Hilbert space,(2)dx(t)=[(t,x(t),∫0t‍f(t,s,x(s))ds)Ax(t)+Bu(t)0+F(t,x(t),∫0t‍f(t,s,x(s))ds)]dt0+G(t,x(t),∫0t‍g(t,s,x(s))ds)dw(t),00000000000000000000000t≠tk,t⩾0,Δx(tk)=Ik(xtk-),t=tk,k=1,2,…,m,x(0)=x0∈H,
where F:[0,T]×H×H→H, G:[0,T]×H×H→ℒ2(Q1/2E,H), f,g:[0,T×[0,T]×H→H are measurable mappings. Ik(xtk-)=x(tk+)-x(tk-),  t=tk,  k=1,2,…,ρ, where x(tk+) and x(tk-) denote the right and left limits of x(t) at t=tk, respectively. Also Δx(tk)=x(tk+)-x(tk-) represents the jump in the state x at time tk with Ik determining the size of the jump. For systems (2), if Ik=0, the controllability problem was studied by Subalakshmi et al. [11]. If Ik≠0 and G=0,  f=0, [15] discussed the approximate controllability problem. When A, B are matrices with appropriate dimensions, F, G are vectors (in fact, matrix is aspecial form of operator), and f=g=0, Karthikeyan et al. [19] obtained the controllability results, so system (2) is of the more general form and has great diversity.The outline of this paper is as follows: Section2 contains basic notations, lemmas, and preliminary facts. The controllability results are given in Section 3 by fixed point methods. In Section 4, we provide an example to demonstrate the effectiveness of our method. Finally, conclusions are given in Section 5.


## 2. Preliminaries

Let(Ω,ℱ,ℙ) be a complete probability space with a filtration {ℱt}t⩾0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all ℙ-null sets). We consider three Hilbert spaces E,  H, and U, and a Q-Wiener process on (Ω,ℱ,ℙ) with the covariance operator Q∈ℒ(E) such that trQ<∞. Let 〈·〉 and ∥·∥ denote inner product and norm of H, respectively. ℒ(X,Y) is the space of all linear bounded operator from a Hilbert space X to a Hilbert space Y. We also employ the same notation ∥·∥ for the norm of ℒ(X,Y). We assume that there exists a complete orthonormal {ek} in E, a bounded sequence of nonnegative real numbers λk such that Qek=λkek,  k=1,2,…, and a sequence {βk} of independent Brownian motions such that
(3)〈w(t),e〉=∑k=1∞λk〈ek,e〉βk(t),e∈E,t∈[0,T],
and ℱt=ℱtβ, where ℱtβ is the σ-algebra generated by {β(s):0⩽s⩽t}. Let ℒ20=ℒ2(Q1/2E,H) be the space of all Hilbert-Schmidt operator from Q1/2E to H with the inner product 〈Ψ,Φ〉ℒ20=tr[ΨQΦ*] and the norm ∥·∥ℒ20. L2(ℱT,H) is the Hilbert space of all ℱT-measurable square integrable random variables with values in Hilbert space H. L2ℱT([0,T],H) is the Hilbert space of square integrable and ℱT-adapted processes with values in H.LetPC([0,T],L2(Ω,ℱ,ℙ;H))={ϕ:ϕ is a function from [0,T] into L2(Ω,ℱ,ℙ;H) such that ϕ(t) is continuous at t≠tk, left continuous at t=tk, and the right limit ϕ(tk+) exists for k=1,2,…,ρ}. ℋ2(U2) is the closed subspace of PC([0,T],L2(Ω,ℱ,ℙ;H)) consisting of measurable and ℱt-adapted H-valued(U-valued) process ϕ(·)∈PC([0,T],L2(Ω,ℱ,ℙ;H))(ϕ(·)∈PC([0,T],L2(Ω,ℱ,ℙ;U))) endowed with the norm ∥ϕ∥ℋ22=sup0⩽t⩽T𝔼∥ϕ(t)∥2.By a solution of system (2), we mean a mild solution of the following nonlinear integral equation:
(4)x(t)=S(t)x0+∫0tS(t-s)Bu(s)ds+∫0t‍S(t-s)F(s,x(s),∫0sf(s,τ,x(τ))dτ)ds+∫0t‍S(t-s)G(s,x(s),∫0sg(s,τ,x(τ))dτ)dw(s)+∑k=1ρS(t-tk)Ik(x(tk-)),
where u∈Uad∶=U2, S(t)t⩾0 denotes the strongly continuous semigroup generated by the operator A.Now let us introduce the controllability operatorΠsT associated with (4) (see[8]),
(5)ΠsT{·}=∫sT‍S(T-t)BB*S*(T-t)dt,
which belongs to ℒ(H,H); B* is the adjoint operator of B.Definition 1.
System (2) is completely controllable on [0,T] if
(6)ℛT(x0)=L2ℱT([0,T],H).
That is, all the points in L2ℱT([0,T],H) can be reached from the point x0 at time T, where ℛt(x0)={x(t;x0,u):u∈L2ℱ([0,T],H)}.Lemma 2 (Burkholder-Davis-Gundy’s inequality [23]).
For anyr⩾1 and for arbitrary ℒ20-valued predictable process Ψ(t),  t∈[0,T], one has
(7)𝔼(sup0⩽t⩽T|∫0t‍Ψ(s)dw(s)|2r)⩽Cr𝔼(∫0t‍∥Ψ(s)∥ℒ202ds)r,
where
(8)Cr=(r(2r-1))r(2r2r-1)2r2.Lemma 3 (Mahmudov [6]).
The following linear system(9)dx(t)=[Ax(t)+Bu(t)]dt+D(t)dw(t),x(0)=x0
is completely controllable if and only if Π0T⩾γI, where γ is constant and I is unit operator.Lemma 4.
Assume that the operatorΠsT is invertible. Then for arbitrary target xT∈L2(ℱT,H), the control
(10)u(t)=B*S*(T-t)𝔼×{(Π0T)-1[∑k=1ρ‍xT-S(T)x0-∫0T‍S(T-s)F-(s)ds00000000000-∫0T‍S(T-s)G-(s)dw(s)00000000000-∑k=1ρ‍S(T-tk)Ik(x(tk-))]∣ℱt}
transfers the systems (4) from x0 to xT at time T, where F-(s)=F(s,x(s),∫0s‍f(s,τ,x(τ))dτ), G-(s)=G(s,x(s),∫0s‍g(s,τ,x(τ))dτ)).Proof.
Substituting (10) into (4), we can obtain that
(11)x(t)=S(t)x0+∫0t‍S(t-s)×BB*S*(T-s)𝔼×{(Π0T)-1[∑k=1ρ‍xT-S(T)x0-∫0T‍S(T-s)F-(s)ds00000000000-∫0T‍S(T-s)G-(s)dw(s)00000000000-∑k=1ρ‍S(T-tk)Ik(x(tk-))]∣ℱt}+∫0t‍S(T-s)F-(s)ds+∫0t‍S(T-s)G-(s)dw(s)+∑k=1ρ‍S(T-tk)Ik(x(tk-))=S(t)x0+Π0t{∑k=1ρ‍S*(T-t)(Π0T)-1000000000000×[xT-S(T)x0-∑k=1ρ‍S(T-tk)Ik(x(tk-))}-∫0T‍S(T-s)F-(s)ds000000000000000-∫0t‍S(t-s)G-(s)dw(s)000000000000000-∑k=1ρ‍S(T-tk)Ik(x(tk-))]}+∫0t‍S(T-s)F-(s)ds+∫0t‍S(T-s)G-(s)dw(s)+∑k=1ρ‍S(T-tk)Ik(x(tk-)).
The proof is completed by letting t=T in (11).


## 3. Main Results

In this section, by using contraction mapping principle in Banach space we discuss the complete controllability criteria of semilinear impulsive stochastic systems (2). For the proof of the main result we impose the following assumptions on data of the problem.Assumption A.
The functionsF, G, and I are continuous and satisfy the usual linear growth condition; that is, there exist positive real constants L1, αk for arbitrary x∈H, and 0⩽t⩽T such that
(12)∥F(t,x,y)∥2+∥G(t,x,y)∥ℒ202⩽L1(1+∥x∥2+∥y∥2),∥Ik(x)∥2⩽αk(1+∥x∥2),k=1,2,…,ρ,∥∫0t‍f(t,s,x(s))ds∥2+∥∫0t‍g(t,s,x(s))ds∥2⩽k1∥x∥2.Assumption B.
The functionsF, G, and I satisfy the following Lipschitz condition and for every t⩾0 and x,y∈H there exist positive real constants L2, βk, k2 such that
(13)∥F(t,x1,y1)-F(t,x2,y2)∥20000+∥G(t,x1,y1)-G(t,x2,y2)∥ℒ20200⩽L2(∥x1-x2∥2+∥y1-y2∥2),∥Ik(x)-Ik(y)∥⩽βk∥x-y∥,k=1,2,…,ρ,∫0t‍{∫0t‍∥f(t,s,x(s))-f(t,s,y(s))∥20000+∫0t‍∥g(t,s,x(s))-g(t,s,y(s))∥2}ds00⩽k2∥x-y∥2.Assumption C.
The linear system (9) is completely controllable. By Lemma 3, for some γ>0, 𝔼〈Π0Tz,z〉⩾γ𝔼∥z∥2, for all z∈L2(ℱT,H). Consequently, ∥(Π0T)-1∥⩽1/γ=l2.Assumption D.
Letp=[6Tl1L2(Ml1l2+1)(T+4)(1+k2T)+6l1ρ(Ml2+1)∑k=1ρ‍βk] be such that 0⩽p<1.Now for convenience, let us introduce the following notations:(14)l1=max0⩽t⩽T∥S(t)∥2,M=maxs∈[0,T]∥ΠsT∥2.Theorem 5.
Suppose that assumptions A, B, C, and D are satisfied. Then system (4) is completely controllable on [0,T].Proof.
For arbitrary initial datax0∈ℋ2, we can define a nonlinear operator Φ from H2 to H2 as the following:
(15)(Φx)(t)=S(t)x0+∫0t‍S(t-s)Bu(s)ds+∫0t‍S(t-s)F-(s,x)ds+∫0t‍S(t-s)G-(s,x)dw(s)+∑k=1ρ‍S(t-tk)Ik(x(tk-)),
where u(t) is defined by (10).
By Lemma4, the control (10) transfers system (4) from the initial state x0 to the final state xT provided that the operators Φ has a fixed point in ℋ2. So, if the operator Φ has a fixed point then system (2) is completely controllable. As mentioned before, to prove the complete controllability of the system (2), it is enough to show that Φ has a fixed point in ℋ2. To do this, we can employ the contraction mapping principle. In the following, we will divide the proof into two steps.
Firstly, we show thatℋ2 maps ℋ2 into itself. From (15) we have
(16)sup0⩽t⩽T𝔼∥(Φx)(t)∥2=sup0⩽t⩽T𝔼∥∑k=1ρ‍S(t)x0+∫0t‍S(t-s)Bu(s)ds00000000000000000+∫0t‍S(t-s)F-(s,x)ds00000000000000000+∫0t‍S(t-s)G-(s,x)dw(s)00000000000000000+∑k=1ρ‍S(t-tk)Ik(x(tk-))∥200000000000000⩽5sup0⩽t⩽T𝔼∥S(t)x0∥200000000000000000+5sup0⩽t⩽T𝔼∥∫0t‍S(t-s)Bu(s)ds∥200000000000000000+5sup0⩽t⩽T𝔼∥∫0t‍S(t-s)F-(s,x)ds∥200000000000000000+5sup0⩽t⩽T𝔼∥∫0t‍S(t-s)G-(s,x)dw(s)∥200000000000000000+5sup0⩽t⩽T𝔼∥∑k=1ρ‍S(t-tk)Ik(x(tk-))∥200000000000000≜∑i=15‍Ai.
Using Holder inequality, B-D-G inequality (hereC1=4), and Assumption C, we have the following estimates:
(17)A1⩽5l1𝔼∥x0∥2,A3⩽5Tsup0⩽t⩽T𝔼∫0t‍∥S(t-s)F-(s,x)∥2ds⩽5Tl1L1∫0T‍(1+(1+k1)sup0⩽s⩽T𝔼∥x(s)∥2)ds,A4⩽20l1L1∫0T‍(1+(1+k1)sup0⩽s⩽T𝔼∥x(s)∥2)ds,A5⩽5ρsup0⩽t⩽T𝔼∑k=1ρ‍∥S(t-tk)Ik(x(tk-))∥2⩽5ρl1∑i-1ρ‍αk(1+sup0⩽t⩽T𝔼∥x(s)∥2).
Meanwhile by control function (15), we have
(18)A2=5sup0⩽t⩽T𝔼×∥∑k=1ρ‍S(T-tk)Ik(x(tk-))]}∫0t‍S(t-r)BB*S*(T-r)𝔼000×{(Π0T)-1×[∑k=1ρ‍x(T)-S(T)x00000000000000000-∫0T‍S(T-s)F-(s,x)ds0000000000000000-∫0T‍S(T-s)G-(s,x)dw(s)0000000000000000-∑k=1ρS(T-tk)Ik(x(tk-))]}dr∥2=5sup0⩽t⩽T𝔼∥∑k=1ρ‍S(T-tk)Ik(x(tk-)))∫0t‍S(t-r)BB*S*(t-r)S*(T-t)(Π0T)-10000000×[∑k=1ρ‍x(T)-S(T)x0-∫0T‍S(T-s)F-(s,x)ds0000000000-∫0T‍S(T-s)G-(s,x)dw(s)0000000000-∑k=1ρ‍S(T-tk)Ik(x(tk-))]dr∥2=5sup0⩽t⩽T𝔼∥Π0tS*(T-t)(Π0T)-1∑k=1ρ‍S(T-tk)Ik(x(tk-))0000000000×[xT-S(T)x0∑k=1ρ‍S(T-tk)Ik(x(tk-))]-∫0T‍S(T-s)F-(s)ds0000000000000-∫0t‍S(t-s)G-(s,x)dw(s)0000000000000-∑k=1ρ‍S(T-tk)Ik(x(tk-))]∥2.
So similar as in (17), we get
(19)A2⩽25Ml1l2(∑i=1ρ‍𝔼∥xT∥2+l1∥x0∥20000000000000+Tl1L1∫0T‍(1+(1+k1)sup0⩽s⩽T∥x(s)∥2)ds0000000000000+4l1L1∫0T‍(1+(1+k1)sup0⩽s⩽T∥x(s)∥2)ds0000000000000+ρ∑i=1ρ‍αk(1+sup0⩽t⩽T∥x(t)∥2)).
From (17)–(19), we have
(20)sup0⩽t⩽T𝔼∥Φx(t)∥200⩽C[1+∫0T‍(1+(1+k1)sup0⩽s⩽T𝔼∥x(s)∥2)ds]00⩽C[1+(1+(1+k1)Tsup0⩽s⩽T𝔼∥x(s)∥2)]
for all t∈[0,T], where C is constant. This implies that Φ maps ℋ2 into itself.
Secondly, we prove thatΦ is a contraction mapping on ℋ2, for any x,y∈ℋ2,
(21)sup0⩽t⩽T𝔼∥(Φx)(t)-(Φy)(t)∥200=sup0⩽t⩽T𝔼∥∑k=1ρ‍Π0tS*(T-t)(Π0T)-100000000000×[∑k=1ρ‍∫0T‍S(T-s)(F-(s,y)-F-(s,x))ds00000000000000+∫0T‍S(T-s)(G-(s,y)-G-(s,x))dw(s)00000000000000+∑k=1ρ‍S(T-tk)(Ik(y(tk-))-Ik(x(tk-)))]00000000000+∫0t‍S(t-s)(F-(s,x)-F-(s,y))ds00000000000+∫0t‍S(t-s)(G-(s,x)-G-(s,y))dw(s)00000000000+∑k=1ρ‍S(t-tk)(Ik(x(tk-))-Ik(y(tk-)))∥200⩽6Ml1l2{∥∑k=1ρ‍S(T-tk)(Ik(x(tk-))-Ik(y(tk-)))∥2∥∫0T‍S(T-s)(F-(s,x)-F-(s,y))ds∥200000000000000+∥∫0T‍S(T-s)(G-(s,x)-G-(s,y))ds∥200000000000000+∥∑k=1ρ‍S(T-tk)(Ik(x(tk-))-Ik(y(tk-)))∥2}0000+6sup0⩽t⩽T𝔼∥∫0t‍S(t-s)(F-(s,x)-F-(s,y))ds∥20000+6sup0⩽t⩽T𝔼∥∫0t‍S(t-s)(G-(s,x)-G-(s,y))dw(s)∥20000+6sup0⩽t⩽T𝔼∥∑k=1ρ‍S(t-tk)(Ik(x(tk-))-Ik(y(tk-)))∥200≜6Ml1l2∑i=13‍Bi+∑i=46‍Bi.
Using Lipschitz condition, similiar to A1–A5, we have the following estimates:
(22)B1⩽Tl1L2(1+k2T)∫0T‍sup0⩽s⩽T𝔼∥x(s)-y(s)∥2ds,(23)B2⩽4l1L2(1+k2T)∫0T‍sup0⩽s⩽T𝔼∥x(s)-y(s)∥2ds,(24)B3⩽l1ρ∑k=1ρ‍βksup0⩽s⩽T𝔼∥x(s)-y(s)∥2,(25)B4⩽6Tl1L2(1+k2T)∫0T‍sup0⩽s⩽T𝔼∥x(s)-y(s)∥2ds,(26)B5⩽24l1L2(1+k2T)∫0T‍sup0⩽s⩽T𝔼∥x(s)-y(s)∥2ds,(27)B6⩽6l1ρ∑k=1ρ‍βksup0⩽s⩽T𝔼∥x(s)-y(s)∥2
together with inequalities (22)–(27):
(28)sup0⩽t⩽T𝔼∥(Φx)(t)-(Φy)(t)∥200⩽6TMl12l2L2(1+k2T)∫0T‍sup0⩽s⩽T𝔼∥x(s)-y(s)∥2ds0000+24Ml12l2L2(1+k2T)∫0T‍sup0⩽s⩽T𝔼∥x(s)-y(s)∥2ds0000+(6Ml1l2ρ∑k=1ρ‍βk+6l1ρ∑k=1ρ‍βk)sup0⩽t⩽T𝔼∥x(t)-y(t)∥200⩽[∑k=1ρ‍6Tl1L2(Ml1l2+1)(T+4)(1+k2T)00000+6l1ρ(Ml2+1)∑k=1ρ‍βk]sup0⩽t⩽T𝔼∥x(t)-y(t)∥2.
Theorefore,Φ is a contraction mapping from ℋ2 to ℋ2, and hence Φ has a unique fixed point. Thus the system (4) is completely controllability on [0,T].


## 4. Example

Consider the impulsive stochastic integrao-differential wave equation with controlu(t,z)∈L2[0,1],
(29)∂2z∂t2=a∂2z∂ξ2+bu(t,z)+F(t,z(t),∫0t‍f(t,s,z(s))ds)+G(t,z(t),∫0t‍g(t,s,z(s))ds)∂w∂t,0000000000000000t≠tk,0⩽ξ⩽1,
where ∂w/∂t is white noise and initial and boundary conditions are
(30)z(t,0)=z(t,1)=0,t≠tk,Δz(tk)(ξ)=Ik1(ztk),Δz′(tk)(ξ)=Ik2(ztk),z(0,ξ)=z0(ξ),∂z(0,ξ)∂t=z1(ξ).
Let H=L2[0,1]; then A:H→H, Az=z′′. Domain of operator A is
(31)D(A)={z∈H∣z,z′iscontinuous,z′′∈H,0z(0)=z(1)=0z′′}.
Let
(32)Z=[z∂z∂t],Z(0)=[z0z1],B=[0b],(33)F-=[0F(t,z(t),∫0tf(t,s,z(s))ds)],G-=[0G(t,z(t),∫0tg(t,s,z(s))ds)].Then the system (29) is
(34)dZ=[𝒜Z+Bu+F-(t,Z)]dt+G-(t,Z)dW,t≠tk,ΔZk=[Ik1Ik2],t=tk,k=1,2,…,m,Z(0)=[z0z1],
where
(35)𝒜=[0I-A0],
and 𝒜 is the infinitesimal generator of a strongly continuous semigroup h(t),t⩾0, on X=D(A1/2)⊕H, for x×y∈X:
(36)S(t)[xy]=∑n=1∞‍[cos(nπt)(nπ)-1sin(nπt)-(nπ)sin(nπt)cos(nπt)][xnyn]en,en(θ)=2sin(nπθ),θ∈[0,1]. Write h(t) is
(37)h(t)=[C(t)S(t)AS(t)C(t)],
where
(38)S(t)x∶=∫0t‍C(s)xds,x∈H.
The mild solution of system (29) is
(39)z(t)=C(t)z0+S(t)z1+∫0t‍S(t-s)[Bu(s)+f(s,z)]ds+∫0t‍S(t-s)g(s,z)dws+∑0⩽tk⩽t‍C(t-tk)Ik1(ztk)+∑0⩽tk⩽t‍S(t-tk)Ik2(ztk)
by [7]; the stochastic linear system of (29) is complete controllable. Then from Theorem 5 one can easily prove system (29) is completely controllable, if the functions F, G, f, g, Ik1, Ik2 satisfy Lipschitz condition and linear growth condition.


## 5. Conclusions

The complete controllability of impulsive stochastic integro-differential systems in Hilbert space has been investigated in this paper. Sufficient conditions of complete controllability for impulsive stochastic integro-differential systems are established by using the Banach fixed point theorem. An example illustrates the efficiency of proposed results.



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*Source: 783098-2013-07-25.xml*