# Ψ-Stability of Nonlinear Volterra Integro-Differential Systems with Time Delay

Authors: Lianzhong Li; Maoan Han; Xin Xue; Yuanyuan Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We give some sufficient conditions forΨ-uniform stability of the trivial solutions of a nonlinear differential system and of nonlinear Volterra integro-differential systems with time delay.


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## Body

## 1. Introduction

Akinyele [1] introduced the notion of Ψ-stability of the degree k with respect to a function Ψ∈C(R+-R+), increasing and differentiable on R and such that Ψ(t)≥1 for t≥0 and limt→∞Ψ(t)=b, b∈[1,∞). Constantin [2] introduced the notions of degree of stability and degree of boundedness of solutions of an ordinary differential equation, with respect to a continuous positive and nondecreasing function Ψ:R+→R+; some criteria for these notions are proved there too.Morchało [3] introduced the notions of Ψ-stability, Ψ-uniform stability, and Ψ-asymptotic stability of trivial solution of the nonlinear system x′=f(t,x). Several new and sufficient conditions for the mentioned types of stability are proved for the linear system x′=A(t)x; in this paper Ψ is a scalar continuous function. In [4, 5], Diamandescu gives some sufficient conditions for Ψ-asymptotic stability and Ψ-(uniform) stability of the nonlinear Volterra integro-differential system x′=A(t)x+∫0t‍F(t,s,x(s))ds; in these papers Ψ is a matrix function. Furthermore, in [6], sufficient conditions are given for the uniform Lipschitz stability of the system x′=f(t,x)+g(t,x).In paper [7], for the nonlinear system
(1)y′=f(t,y)+g(t,y)
and the nonlinear Volterra integro-differential system
(2)z′=f(t,z)+∫0t‍F(t,s,z(s))ds,
by using the knowledge of fundamental matrix and nonlinear variation of constants, we give some sufficient conditions for Ψ-(uniform) stability of trivial solution for the system. The purpose of this paper is to provide sufficient conditions for Ψ-uniform stability of trivial solutions for the nonlinear delayed system
(3)x′(t)=f(t,x(t))+g(t,x(t-τ(t)))
and the nonlinear delayed Volterra integro-differential systems
(4)x′(t)=f(t,x(t))+g(t,x(t-τ(t)))+p(t,x(t))∫0t‍q(s,x(s-τ(s)))ds,(5)x′(t)=f(t,x(t))+g(t,x(t-τ(t)))+p(t,x(t-τ(t)))∫0t‍q(s,x(s))ds,
where f,g,p,q∈C(ℝ+×ℝn,ℝn), f(t,0)=g(t,0)=p(t,0)=q(t,0)=0 for t∈ℝ+, and τ∈C1(ℝ+,ℝ+) with τ(t)≤t on ℝ+. The systems studied in [7] do not include time delay, whereas all the systems studied in this paper have time delay.In this paper, we investigate conditions on the functionsf,g,p,q under which the trivial solutions of systems (3), (4), and (5) are Ψ-stability on ℝ+; the main tool used is the integral inequalities and the integral technique. Here Ψ is a matrix function whose introduction allows us to obtain a mixed behavior for the components of solutions.Letℝn denote the Euclidean n-space. For x=(x1,x2,x3,…,xn)T∈ℝn, let ∥x∥=max{|x1|,|x2|,…,|xn|} be the norm of x. For an n×n matrix A=(aij), we define the norm |A|=sup∥x∥≤1∥Ax∥. It is well known that
(6)|A|=max1≤i≤n∑j=1n‍|aij|.LetΨi:ℝ+→(0,∞), i=1,2,…,n, be continuous functions and Ψ=diag[Ψ1,Ψ2,…,Ψn].Now we give the definitions ofΨ-(uniform) stability that we will need in the sequel.Definition 1 (see [4, 8]).
The trivial solution of (3) ((4) or (5)) is said to be Ψ-stable on ℝ+ if for every ε>0 and any t0∈ℝ+, there exists δ=δ(ε,t0)>0 such that any solution x(t) of (3) ((4) or (5)), which satisfies the inequality ∥Ψ(t0)x(t0)∥<δ, exists and satisfies the inequality ∥Ψ(t)x(t)∥<ε for all t≥t0.Definition 2 (see [4, 8]).
The trivial solution of (3) ((4) or (5)) is said to be Ψ-uniformly stable on ℝ+ if it is Ψ-stable on ℝ+ and the previous δ is independent of t0.


## 2.Ψ-Stability of the Systems

To prove our theorems, we need the following lemmas.Lemma 3.
Leth,k,p,q∈C(ℝ+×ℝ+,ℝ+) with (t,s)↦∂th(t,s), ∂tk(t,s), ∂tp(t,s), ∂tq(t,s)∈C(ℝ+×ℝ+,ℝ+). Assume, in addition, that b∈C(ℝ+,ℝ+) and α∈C1(ℝ+,ℝ+) are nondecreasing functions and α(t)≤t for t≥0. If u∈C(ℝ+,ℝ+) satisfies
(7)u(t)≤b(t)+∫0t‍h(t,s)u(s)ds+∫0α(t)‍k(t,s)u(s)ds+∫0t‍p(t,s)u(s)(∫0α(s)‍q(s,v)u(v)dv)ds,
for t≥0, and b(t)∫0t‍R(s)Q(s)ds<1, then
(8)u(t)≤b(t)Q(t)1-b(t)∫0t‍R(s)Q(s)ds,t≥0,
where Q(t)=exp(∫0t‍h(t,s)ds+∫0α(t)‍k(t,s)ds), R(t)=(d/dt)∫0t‍p(t,s)(∫0α(s)‍q(s,v)dv)ds.Proof.
LetT≥0 be fixed and denote
(9)x(t)=∫0t‍h(t,s)u(s)ds+∫0α(t)‍k(t,s)u(s)ds+∫0t‍p(t,s)u(s)(∫0α(s)‍q(s,v)u(v)dv)ds,t≥0,
then u(t)≤b(t)+x(t),  and x is nondecreasing on ℝ+. For t∈[0,T], by calculations we get the following:
(10)x′(t)=[h(t,t)u(t)+∫0t‍∂th(t,s)u(s)ds]+[k(t,α(t))u(α(t))α′(t)+∫0α(t)‍∂tk(t,s)u(s)ds]+[p(t,t)u(t)∫0α(t)‍q(t,v)u(v)dv+∫0t‍∂tp(t,s)u(s)(∫0α(s)‍q(s,v)u(v)dv)ds]x′(t)≤[b(T)+x(t)][ddt(∫0t‍h(t,s)ds+∫0α(t)‍k(t,s)ds)]+[b(T)+x(t)]2ddt∫0t‍p(t,s)(∫0α(s)‍q(s,v)dv)ds,
Suppose that b(0)>0 (if b(0)=0, carry out the following arguments with b(t)+ε instead of b(t), where ε>0 is an arbitrary small constant, and subsequently pass to the limit as ε→0 to complete the proof), then we get
(11)x′(t)[b(T)+x(t)]2-1b(T)+x(t)ddt(∫0t‍h(t,s)ds+∫0α(t)‍k(t,s)ds)≤ddt∫0t‍p(t,s)(∫0α(s)‍q(s,v)dv)ds.
Let
(12)z(t)=1b(T)+x(t),q(t)=∫0t‍h(t,s)ds+∫0α(t)‍k(t,s)ds,Q(t)=exp(q(t))Q(t)=exp(∫0t‍h(t,s)ds+∫0α(t)‍k(t,s)ds),R(t)=ddt∫0t‍p(t,s)(∫0α(s)‍q(s,v)dv)ds,
then, we have
(13)z′(t)+z(t)(ddtq(t))≥-R(t),
Multiplying the above inequality by eq(t)=Q(t), we get
(14)ddt(z(t)Q(t))≥-Q(t)R(t).
Consider now the integral on the interval [0,t] to obtain
(15)z(t)Q(t)≥z(0)-∫0t‍Q(s)R(s)ds,0≤t≤T,
so
(16)z(t)=1b(T)+x(t)≥[1b(T)-∫0t‍Q(s)R(s)ds]1Q(t)=1-b(T)∫0t‍Q(s)R(s)dsb(T)Q(t)
for 0≤t≤T. Let t=T, since b(T)∫0T‍Q(s)R(s)ds<1, then we have
(17)b(T)+x(T)≤b(T)Q(T)1-b(T)∫0T‍Q(s)R(s)ds.
Since T≥0 was arbitrarily chosen, considering u(t)≤b(t)+x(t), we get (8).Lemma 4.
Leth,k,p,q,b,α be as in Lemma 3. If u∈C(ℝ+,ℝ+) satisfies
(18)u(t)≤b(t)+∫0t‍h(t,s)u(s)ds+∫0α(t)‍k(t,s)u(s)ds+∫0α(t)‍p(t,s)u(s)(∫0s‍q(s,v)u(v)dv)ds,
for t≥0, and b(t)∫0t‍R(s)Q(s)ds<1, then
(19)u(t)≤b(t)Q(t)1-b(t)∫0t‍R(s)Q(s)ds,t≥0,
where Q(t)=exp(∫0t‍h(t,s)ds+∫0α(t)‍k(t,s)ds), R(t)=(d/dt)∫0α(t)‍p(t,s)(∫0s‍q(s,v)dv)ds.The proof is similar to the proof of Lemma3, we omit the details.Theorem 5.
If there exist functionsa(t,s),b(t,s)∈C(ℝ+×ℝ+,ℝ+) with (t,s)↦∂ta(t,s), ∂tb(t,s)∈C(ℝ+×ℝ+,ℝ+) such that
(20)∥Ψ(t)f(s,x)∥≤a(t,s)∥Ψ(s)x∥,∥Ψ(t)g(s,x)∥≤b(t,s)∥Ψ(s)x∥,
for 0≤s≤t and for all x∈ℝn. Moreover,
(21)limsupt→∞∫0t‍(a(t,s)+b(t,s))ds=L1,|Ψ(t)Ψ-1(s)|≤L2for0≤s≤t,
and |Ψ(t)x(α(t))|≤|Ψ(α(t))x(α(t))|, where L1, L2 are nonnegative constants. If α(t)=t-τ(t) is an increasing diffeomorphism of ℝ+. Then, the trivial solution of system (3) is Ψ-uniformly stable on ℝ+.Proof.
Suppose thatx(t,t0,x0):=x(t) is the unique solution of system (3) which satisfies x(t0)=x0, since
(22)x(t)=x0+∫t0t‍f(s,x(s))ds+∫t0t‍g(s,x(α(s)))ds=x0+∫t0t‍f(s,x(s))ds+∫α(t0)α(t)‍g(α-1(r),x(r))α′(α-1(r))dr,
after performing the change of variables r=α(s) in the second integral, and α-1 is the inverse of the diffeomorphism α then, it follows that
(23)∥Ψ(t)x(t)∥≤∥Ψ(t)Ψ-1(t0)Ψ(t0)x0∥+∫t0t‍∥Ψ(t)f(s,x(s))∥ds+∫α(t0)α(t)‍∥Ψ(t)g(α-1(r),x(r))α′(α-1(r))∥ds≤L2∥Ψ(t0)x0∥+∫t0t‍a(t,s)∥Ψ(s)x(s)∥ds+∫α(t0)α(t)‍b(t,α-1(r))α′(α-1(r))∥Ψ(r)x(r)∥dr,
this implies by Lemma 3 that
(24)∥Ψ(t)x(t)∥≤L2∥Ψ(t0)x0∥exp×(∫t0t‍a(t,s)ds+∫α(t0)α(t)‍b(t,α-1(r))α′(α-1(r))dr)=L2∥Ψ(t0)x0∥exp(∫t0t‍(a(t,s)+b(t,s))ds)≤L2eL1∥Ψ(t0)x0∥,
so for every ε>0, choose δ=ε/(L2eL1), then
(25)∥Ψ(t)x(t)∥≤L2eL1∥Ψ(t0)x0∥<ε
for ∥Ψ(t0)x0∥<δ and for all 0≤t0≤t<∞. Hence, the conclusion of the theorem follows.Theorem 6.
Let all the conditions in Theorem5 hold. Suppose further that there exist functions m(t,s),n(t,s)∈C(ℝ+×ℝ+,ℝ+) with (t,s)↦∂tm(t,s), ∂tn(t,s)∈C(ℝ+×ℝ+,ℝ+) such that
(26)∥Ψ(t)p(s,x)Ψ-1(s)∥≤m(t,s)∥Ψ(s)x∥,∥Ψ(t)q(s,x)∥≤n(t,s)∥Ψ(s)x∥,
for 0≤s≤t and for all x∈ℝn, moreover,
(27)limsupt→∞∫0t‍m(t,s)(∫0s‍n(s,u)du)ds=L3,
where L3 is a nonnegative constant. Then, the trivial solutions of systems (4) and (5) are Ψ-uniformly stable on ℝ+.Proof.
For that system (4), suppose x(t,t0,x0):=x(t) is the unique solution of system (4) which satisfies x(t0)=x0, since
(28)x(t)=x0+∫t0t‍f(s,x(s))ds+∫t0t‍g(s,x(α(s)))ds+∫t0t‍p(s,x(s))∫0s‍q(u,x(α(u)))duds,0≤t0≤t,
it follows that
(29)∥Ψ(t)x(t)∥≤∥Ψ(t)Ψ-1(t0)Ψ(t0)x0∥+∫t0t‍∥Ψ(t)f(s,x(s))∥ds+∫α(t0)α(t)‍∥Ψ(t)g(α-1(r),x(r))∥α′(α-1(r))dr+∫t0t‍∥Ψ(t)p(s,x(s))Ψ-1(s)∥×(∫0α(s)‍∥Ψ(s)q(α-1(r),x(r))∥α′(α-1(r))dr)ds≤L2∥Ψ(t0)x0∥+∫t0t‍a(t,s)∥Ψ(s)x(s)∥ds+∫α(t0)α(t)‍b(t,α-1(r))α′(α-1(r))∥Ψ(r)x(r)∥dr+∫t0t‍m(t,s)∥Ψ(s)x(s)∥×(∫0α(s)‍n(s,α-1(r))∥Ψ(r)x(r)∥α′(α-1(r))dr)ds
after performing the change of variables r=α(s)  (or r=α(u)) at some intermediate step, and α-1 is the inverse of the diffeomorphism α. Denote
(30)Q(t)=exp(∫t0t‍a(t,s)ds+∫α(t0)α(t)‍b(t,α-1(r))α′(α-1(r))dr)=exp(∫t0t‍(a(t,s)+b(t,s))ds),R(t)=ddt[∫t0t‍m(t,s)(∫0α(s)‍n(s,α-1(r))α′(α-1(r))dr)ds]=ddt[∫t0t‍m(t,s)(∫0s‍n(s,u)du)ds].
This implies by Lemma 3 that
(31)∥Ψ(t)x(t)∥≤L2∥Ψ(t0)x0∥Q(t)1-L2∥Ψ(t0)x0∥∫0t‍Q(v)R(v)dv≤∥Ψ(t0)x0∥L2eL11-L2∥Ψ(t0)x0∥eL1∫0t‍R(v)dv=∥Ψ(t0)x0∥×L2eL11-L2∥Ψ(t0)x0∥eL1∫t0t‍m(t,s)(∫0s‍n(s,u)du)ds≤∥Ψ(t0)x0∥L2eL11-L2L3∥Ψ(t0)x0∥eL1
for L2L3∥Ψ(t0)x0∥eL1<1 and 0≤t0≤t. So, for every ε>0 and t0≥0, let 0<q<1/L2L3eL1 be a constant and choose δ=min{q,((1-qL2L3eL1)ε)/L2eL1}, then
(32)∥Ψ(t)x(t)∥<(1-qL2L3eL1)εL2eL1×L2eL11-qL2L3eL1=ε
for ∥Ψ(t0)x0∥<δ and for all 0≤t0≤t<∞. This proves that the trivial solution of system (4) is Ψ-uniformly stable on ℝ+.
Using Lemma4, the proof of system (5) is similar to that of system (4) and the details are left to the readers.Remark 7.
ForΨi=1, i=1,2,…,n, we obtain the theorems of classical stability and uniform stability.


## 3. Examples

Example 8.
Consider the nonlinear differential system(33)x1′(t)=x1(t)+x1(t2)sint,x2′(t)=-x2(t)+x2(t2)cost.
In (33), f(t,x(t))=(x1(t),-x2(t))T, g(t,x(t/2))=(x1(t/2)sint,x2(t/2)cost)T. Let Ψ(t)=(e-t00e-t), then a(t,s)=b(t,s)=e-(t-s) for 0≤s≤t≤∞, it is easy to verify that L1=2, L2=1, and all the assumptions in Theorem 5 satisfied, so the trivial solution of system (33) is ψ-uniformly stable on ℝ+.Example 9.
Consider the nonlinear Volterra integro-differential system as follows:(34)x1′(t)=x1(t)+x1(t)e-t∫0t‍x1(s2)cossds,x2′(t)=-x2(t)+x2(t)e-t∫0t‍x2(s2)sinsds.
In (34), f(t,x(t))=(x1(t),-x2(t))T, g≡0, p(t,x(t))=(x1(t)e-t,x2(t)e-t)T, q(s,x(s/2))=(x1(s/2)coss,x2(s/2)sins)T. Choose the same matrix function Ψ(t), then a(t,s)=n(t,s)=e-(t-s), b(t,s)≡0, m(t,s)=e-2(t-s) for 0≤s≤t≤∞, it is easy to verify that L1=L2=1, L3=1/2, and all the assumptions in Theorem 6 are satisfied, so the trivial solution of system (34) is ψ-uniformly stable on ℝ+.



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