# Stability Analysis of Solutions for a Kind of Integro-Differential Equations with a Delay

Authors: Jing Zhao; Fanwei Meng
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Stability of zero solution for second-order integro-differential equations with a delay is analyzed and some new results are presented. Through constructing Lyapunov functional, we give the corresponding sufficient conditions on stability of zero solution for two integro-differential equations. Moreover, an illustrative example is considered to support our new results.


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## Body

## 1. Introduction

Stability and the existence of solutions for nonlinear differential equations have been studied by many scholars [1–20] due to their many applications to problems in information theory, control theory, mechanics, chemistry, physics, and so on. In [5], the authors considered a second-order functional integro-differential equation with multiple delays(1)x′′t+atft,xt,x′tx′t+gt,xt,x′t+∑i=1nhixt-τi=∫0tCt,ξx′ξdξ,and they gave some new conditions on the continuability and boundedness of solutions. Li [6] studied Ψ-stability of the trivial solutions of the following three nonlinear systems with time delay,(2)x′t=ft,xt+gt,xt-τt,and two Volterra integro-differential equations(3)x′t=ft,xt+gt,xt-τt+pt,xt∫0tqs,xs-τsds,(4)x′t=ft,xt+gt,xt-τt+pt,xt-τt∫0tqs,xsds.In [1], Afuwape studied the asymptotic stability and the uniformly ultimate boundedness of the solutions for a kind of third-order delay differential equations and gave some sufficient conditions.This paper investigates second-order integro-differential equations with a delay and obtains some new results on the stability of zero solution. By constructing Lyapunov functional, the corresponding sufficient conditions are present on stability of zero solution for two integro-differential equations. Moreover, an illustrative example is considered to show that our results are effective.The rest of this paper is organized as follows. Section2 presents the main results of this paper. In Section 3, an illustrative example is given to support our new results.


## 2. Main Results

Consider the following integro-differential equation with a delay:(5)x′′+ft,x,x′x′+gt,x,x′+hxt-τ=pt,xt∫0tqs,x′sds,where R+=[0,+∞), τ>0 is a constant, f, g:R+×R2→R and h:R+→R are continuous with h(0)=0, and p,q:R+×R→R with p(t,0)=0 and q(t,0)=0.We can rewrite (5) as the system(6)x′=y,y′=-ft,x,yy-gt,x,y-hx+∫t-τth′xsysds+pt,x∫0tqs,ysds.Theorem 1.
Consider system (6). There exist nonnegative constants K∗, R, b, M, and N, a positive number l, and functions P(t),Q(t):R+→R+, such that the following conditions hold:(C1
)h(x)x-1≥l when x≠0, h′x≤K∗.(C2
)g(t,x,0)=0, g(t,x,y)y-1≥b when y≠0.(C3
)pt,x≤P(t)≤M, qt,y≤Qty, Q(t)≤N.(C4
)N∫0∞P(s)ds+M∫0∞Q(s)ds+K∗(τ+1)≤2f(t,x,y)+2b.
Then the zero solution of system (6) is stable.Proof.
Define a Lyapunov functionalV(t)=V(t,x,y) as(7)Vt=12y2+∫0xhηdη+λ∫-τ0ds∫t+sty2θdθ+12∫0tds∫t∞pθ,xθQsy2sdθ,where λ is a positive constant to be determined. Using condition (C1), we have(8)∫0xhηdη=∫0xhηηηdη≥∫0xlηdη=l2x2>0,and V(t)≥l/2x2+1/2y2.
Suppose that(x(t),y(t)) is a solution of (6). Then(9)dVdt=yy′+hxy+λ∫-τ0y2t-y2t+sds+12Qty2t∫t∞pθ,xθdθ-12pt,xt∫0tQsy2sds=-ft,x,yy2-gt,x,yy+λτy2+yt∫t-τth′xsysds+ytpt,xt∫0tqs,ysds-λ∫t-τty2sds+12Qty2t∫t∞pθ,xθdθ-12pt,xt∫0tQsy2sds.
By conditions (C1) and (C3) and 2ab≤a2+b2, we have(10)yt∫t-τth′xsysds≤yt∫t-τth′xsysds≤K∗2∫t-τty2t+y2sds=K∗2τy2+K∗2∫t-τty2sds,(11)ytpt,xt∫0tqs,ysds≤ytpt,xt∫0tqs,ysds≤pt,xtyt∫0tQsysds≤12pt,xt∫0tQsy2t+y2sds≤My22∫0∞Qsds+12pt,xt∫0tQsy2sds.Then,(12)dVdt≤-ft,x,y-b+λτ+K∗2τy2+K∗2-λ∫t-τty2sds+N∫0∞Psds+M∫0∞Qsds2y2.
Choosingλ=K∗/2, we have(13)dVdt≤-2ft,x,y-2b+K∗τ+1+N∫0∞Psds+M∫0∞Qsds2y2≤0.Therefore, the zero solution of system (6) is stable and the proof is completed.Theorem1 can be generalized to the form with a variable delay τ(t). It only needs a new condition. Then, the following result is obtained.Corollary 2.
Consider system (6) with a variable delay τ(t). Conditions (1)-(3) of Theorem 1 are satisfied. Moreover,(C4′
)N∫0∞P(s)ds+M∫0∞Q(s)ds+2-β/1-βK∗τ≤2f(t,x,y)+2b,(C5
) there areτ>0 and 0<β<1, such that 0≤τ(t)≤τ and τ′(t)≤1.
Then the zero solution of system (6) with a variable delay τ(t) is stable.Proof.
The proof is similar to that of Theorem1. But(14)dVdt≤-ft,x,y-b+λτt+K∗2τty2+K∗2-λ1-β∫t-τtty2sds+N∫0∞Psds+M∫0∞Qsds2y2.Thus, choosing λ=K∗/2(1-β), we have(15)dVdt≤-2ft,x,y-2b+2-β/1-βK∗τ+N∫0∞Psds+M∫0∞Qsds2y2≤0.Therefore, the zero solution of system (6) with a variable delay τ(t) is stable and the proof is completed.Next, we consider another second-order integro-differential system with time delay(16)x′′+ft,x,x′x′+gt,x,x′+hxt=pxt-τ∫0tqs,x′sds,System (16) can be rewritten as follows:(17)x′=yy′=-ft,x,yy-gt,x,y-hxt-px∫0tqs,ysds+∫t-τtp′xsysds∫0tqs,ysds,Theorem 3.
Consider system (17). There are nonnegative constants K, R, L1, L2, and N, positive numbers l, M, and M1, and functions P(t) and Q(t), such that the conditions hold:(D1
)h(x)x-1≥l when x≠0, p′x≤K.(D2
)f(t,x,0)=0, g(t,x,0)=0, f(t,x,y)y-2≥L1, g(t,x,y)y-1≥L2 when y≠0.(D3
)px≤P(t), M1≤P(t)≤M, qt,y≤Q(t)y, Q(t)≤N.(D4
)M∫0∞Q(s)ds+(M+Kτ)N/M1∫0∞P(s)ds≤2L2.(D5
)Kτ∫0∞Q(s)ds≤2L1.
Then the zero solution of system (17) is stable.Proof.
Define a Lyapunov functionalV(t)=V(t,x,y) as(18)Vt=12y2+∫0xhηdη+λ∫-τ0ds∫t+sty4θdθ+μ∫0tds∫t∞pxθQsy2sdθ,where λ and μ are two positive constants to be determined. Using condition (D1), we have(19)∫0xhηdη=∫0xhηηηdη≥∫0xlηdη=l2x2,and then V(t)≥l/2x2+l/2y2.
Let(x(t),y(t)) be a solution of (17). Then,(20)dVdt=yy′+hxy+λ∫-τ0y4t-y4t+sds+μQty2t∫t∞pxθdθ-μpxt∫0tQsy2sds=-ft,x,yy2-gt,x,yy-ypx∫0tqs,ysds+y∫t-τtp′xsysds∫0tqs,ysds+λτy4-λ∫t-τty4sds+μQty2∫t∞pxθdθ-μpxt∫0tQsy2sds.
Applying conditions (D1) and (D3) and 2ab≤a2+b2, we have(21)y∫t-τtp′xsysds∫0tqs,ysds≤y∫t-τtp′xsysds∫0tqs,ysds≤K2∫t-τty2t+y2sds∫0tQθyθdθ≤Kτ2y2t∫0tQθyθdθ+K2∫t-τty2s∫0tQθyθdθds≤Kτ4∫0tQθy4t+y2θdθ+K4∫t-τt∫0tQθy4s+y2θdθds≤Kτ4y4∫0∞Qsds+Kτ2∫0tQsy2sds+K4∫0∞Qsds∫t-τty4sds,(22)-ytpxt∫0tqs,ysds≤ytpt,xt∫0tqs,ysds≤pxtyt∫0tQsysds≤12pxt∫0tQsy2t+y2sds≤My22∫0∞Qsds+M2∫0tQsy2sds.
By condition (D2), we have(23)dVdt≤-L1-Kτ4∫0∞Qsds-λτy4-L2-M2∫0∞Qsds-μN∫0∞Psdsy2+K4∫0∞Qsds-λ∫t-τty4sds+M+Kτ2-μM1∫0tQsy2sds.
Thus, we chooseλ=K/4∫0∞Q(s)ds and μ=M+Kτ/2M1. Using conditions (D4) and (D5), we have(24)dVdt≤-L1-Kτ2∫0∞Qsdsy4-L2-M2∫0∞Qsds-M+KτN2M1∫0∞Psdsy2≤0.
Therefore, the zero solution of system (17) is stable and the proof is completed.Corollary 4.
Consider system (17) with a variable delay τ(t). Conditions (1)-(3) of Theorem 3 are satisfied. Moreover,(D5′
)(2-β)Kτ∫0∞Q(s)ds≤4(1-β)L1,(D6
) there areτ>0 and 0<β<1, such that 0≤τ(t)≤τ and τ′(t)≤1.
Then the zero solution of system (17) with a variable delay τ(t) is stable.Proof.
The proof is similar to that of Theorem3. But(25)dVdt≤-L1-Kτt4∫0∞Qsds-λτty4-L2-M2∫0∞Qsds-μN∫0∞Psdsy2+K4∫0∞Qsds-λ1-τ′t∫t-τtty4sds+M+Kτt2-μPt∫0tQsy2sds≤-L1-Kτ4∫0∞Qsds-λτy4-L2-M2∫0∞Qsds-μN∫0∞Psdsy2+K4∫0∞Qsds-λ1-β∫t-τtty4sds+M+Kτ2-μM1∫0tQsy2sds.Thus, choosing λ=K/4(1-β)∫0∞Q(s)ds and μ=M+Kτ/2M1, we have(26)dVdt≤-L1-2-βKτ41-β∫0∞Qsdsy4-L2-M2∫0∞Qsds-M+KτN2M1∫0∞Psdsy2≤0.Then, the zero solution of system (6) with a variable delay τ(t) is stable and the proof is completed.


## 3. An Illustrative Example

In this section, we give an illustrative example to the effectiveness of results obtained in this paper. Moreover, we list a graph of solutions of an integral-differential equation with a delay to verify the correctness of the conclusion.Example 1.
Consider the following example:(27)x′′t+2et+x2t+x′2tx′t+14x′t+12xt-2=e-2t∫0te-sx′sds.In the example,f(t,x,y)=2et+x2+y2, g(t,x,y)=1/4y, h(x)=1/2x, p(t,x)=e-2t, q(t,y)=∫0te-sy(s)ds, and a delay τ=2. Obvious, this system is the same form as (5). Moreover,(C1
)h(x)x-1≥l=1/2, h′x=1/2≤K∗=1/2;(C2
)g(t,x,0)=0, g(t,x,y)y-1≥b=1/4;(C3
)pt,x=P(t)≤M=1, qt,y≤Qty, Q(t)=e-t≤1;(C4
)N∫0∞P(s)ds+M∫0∞Q(s)ds+K∗(τ+1) = ∫0∞e-2tdt+∫0∞e-tdt+1/2×2=7/2, 2f(t,x,y)+2b≥4+2×1/4≥7/2.Thus, all conditions of Theorem1 are satisfied and the zero solution of system (27) is stable by the obtained result. To show the effectiveness of the result, we carry out a simulation result with the following choices. Initial Condition: x(t)=x′(t)=0 for t∈[-2,0). The simulation result is shown in Figure 1, which is the stability of solutions for system (27).Figure 1
Stability of solutions for system (27).



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