Research paper
Razumikhin qualitative analyses of Volterra integro-fractional delay differential equation with caputo derivatives

https://doi.org/10.1016/j.cnsns.2021.106037Get rights and content

Highlights

  • A non-linear system of Volterra integro-fractional delay differential equations with Caputo fractional derivatives is considered.

  • New sufficient conditions for uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution of the unperturbed system, and the boundedness of all solutions of the perturbed system, are presented.

  • The technique of proof involves the Razumikhin method with an appropriate Lyapunov function.

  • For illustrative purposes, two examples are provided.

Abstract

A non-linear system of Volterra integro-fractional delay differential equations with Caputo fractional derivatives is considered. New sufficient conditions for uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution of the unperturbed system, and the boundedness of all solutions of the perturbed system, are presented. The technique of proof involves the Razumikhin method with an appropriate Lyapunov function. For illustrative purposes, two examples are provided.

Introduction

The theory of fractional calculus goes back to the seventeenth century where integrals and derivatives of non-integer order were first discussed. In recent years, qualitative properties of solutions of fractional differential and integral equations, both with and without delays, has been a highly active area of study. Stability theorems for various fractional differential and fractional delay differential equations have been established. Uniformly stability, asymptotic stability, and Mittag-Leffler stability of solutions are among the properties that have been investigated by many researchers due to their significant roles in applications in the applied sciences and engineering. But, to the best of our knowledge, there are no such results known for delay fractional integro-differential equations except for those in [1].

During the investigations of qualitative aspects of solutions to such equations, researchers sometimes encounter difficulties in finding suitable techniques that can lead to meaningful results. Often times these difficulties involve the calculations that are needed in finding estimates for the derivatives of the Lyapunov functionals being used. The Razumikhin method provides a direct and effective approach to overcome some such problems. The corresponding results are then sometimes referred to as stability theorems of the Razumikhin type (see [2], [3], [4], [5]).

Of significance here is not just the fact that our results appear to be among the first ones for this type of equation, but that we hope that other researchers will see the benefits of the Razumikhin approach and apply it to many other types of problems.

In this paper, we consider an initial value problem (IVP) for the system of non-linear fractional integro-differential equations with a delay and a Caputo fractional derivative of order q(0,1)Dt0Ctqx(t)=A(t)x(t)+tτtK(t,s)f(s,x(s))ds+Q(t,x(t),x(tτ)),t>t0,x(t0+θ)=ϕ(θ),x(0)=ϕ(0)=x0,θ[τ,0], where t0R+=[0,), the delay τ is a positive constant, xRn, ϕC([τ,0],Rn) is the initial function, A(t)C(R+,Rn×n) is an n×n matrix, K(t,s)C([τ,)×[τ,),Rn×n) is the kernel in Eq. (1a), the nonlinear function f(s,x)C([τ,)×Rn,Rn) satisfies f(s,0)=0, and QC(R+×Rn×Rn,Rn) is a perturbation term.

By way of background and motivation for the problem considered here, we mention, for example, the paper of Agarwal et al. [6] who considered the integro-differential system x(t)=A(t)x(t)+B(t)0tK(t,s)C(s)x(s)dsand proved a theorem that essentially characterizes any solution of this system as having exponential form.

Alahmadi et al. [7] studied the boundedness and stability of solutions of the nonlinear Volterra integro-differential equation y(t)=A(t)y(t)+f(y)+0tC(t,s)h(y(s))ds+p(t).using a Lyapunov functional coupled with Laplace transforms. In [8], Andreev and Peregudova examined the stability of a nonlinear Volterra integro-differential equations with unbounded delay. They studied the limiting properties of the solutions by using Lyapunov functionals with a semidefinite time derivative.

Berezansky and Domoshnitsky [9] obtained explicit tests for uniform exponential stability of solutions of the second-order linear scalar integro-differential equation x(t)+g(t)tG(t,s)x(s)ds+h(t)tH(t,s)x(s)ds=0.Chang and Ponce [10] examined the uniform exponential stability of solutions to the Volterra equation in a Banach space X: u(t)=A(t)u(t)+0ta(ts)Au(s)ds.

In the above mentioned paper by Hristova and Tunç [1], the authors considered the fractional integral–differential equation with a Caputo derivative of order 0<q<1toCDtqx(t)=a(t)f(x(t))+tτtB(t,s)g(s,x(s))ds+h(t,x(t)),x(tτ(t)).Using Lyapunov functions and a Razumikhin approach, they were able to obtain conditions for the stability and uniform stability of solutions to this equation.

In [11], Ngoc and Anh the authors investigated the stability of the zero solution of the nonlinear Volterra integro-differential equation x(t)=h(t,x(t))+0tq(t,s,x(s))dsby using spectral properties of Metzler matrices and the comparison principle. They obtained some explicit criteria for uniform asymptotic stability and exponential stability of solutions. Finally, Raffoul and Rai [12] used a modified version of the Lyapunov functionals approach to obtain criterion for the stability of the zero solution of the infinite delay nonlinear Volterra integro-differential equation x(t)=Px(t)+tC(t,s)g(x(s))ds.

As indicated above, we plan to investigate the uniformly stability, asymptotic stability, and Mittag-Leffler stability of the zero solution of (1) with Q0, and the boundedness of all solutions of (1) with Q0, by using the Razumikhin method (see Hale [2]). It should be noted that the Caputo derivative is applicable to continuously differentiable quadratic Lyapunov functions to study qualitative properties of solutions of fractional differential equations and fractional delay differential equations (see, for example, [1], [13], [14]).

It is known that the presence of the fractional derivatives in the system requires that we use appropriately defined fractional derivatives of Lyapunov functions. In the literature, four types of fractional derivatives are commonly applied to calculate the derivatives of Lyapunov functions; these are the Caputo fractional derivative, the Caputo fractional Dini derivative, the Riemann–Liouville fractional derivative, and the Grünwald–Letnikov fractional derivative [13], [14], [15]. Not all of these will be employed here. The results presented below are new contributions to the literature on delay fractional integro-differential equations with Caputo derivatives.

Section snippets

Preliminaries

We begin by considering the system of fractional delay differential equations with a Caputo derivative of order q(0,1)Dt0Ctqx(t)=F(t,xt),tJ=[t0τ,T),T+,0t0t,where xRn, F(t,ϕ)J×C([τ,0],Rn), F(t,0)=0, x(t0+s)=ϕ(s) for s[τ,0], x(t0+)=ϕ(0), ϕC([τ,0],Rn), and τ>0 is the constant delay. For ϕC([τ,0],Rn), we use the usual Euclidean norms and t0 defined by xt=supτs0|x(t+s)|andϕt0=supt0τst0ϕ(t),respectively.

Since the function F is continuous, for any initial data (t0,ϕ)

Razumikhin analysis of solutions

In this section we consider system (1) with Q0, i.e., we consider system (1) with (1a) replaced by Dt0Ctqx(t)=A(t)x(t)+tτtK(t,s)f(s,x(s))ds.We will make use of the following hypotheses in our main results.

  • (H1)

    The matrix A(t)C(R+,Rn×n) is positive definite and symmetric and its eigenvalues satisfy λmλi(A(t))λM for all tR+,where λm>0, λM>0, λm,λMR.

  • (H2)

    There exists a constant f0>0 such that f(s,0)=0,f(s,x)f(s,y)f0xyfor all s[τ,) and all xyRn.

  • (H3)

    There exists a constant K0>0 such that tτ

Boundedness of Solutions of System (1)

We now turn our attention to the perturbed system (see Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6).

Theorem 4.1

The solutions of system(1)are bounded if conditions (H1), (H2), and (H5) hold.

Proof

With the Lyapunov function defined in (5), we differentiate to obtain (see (7)) Dt0CtqV0(t,x(t))2λm21f0(τ+1)tτtK(t,s)dsx(t)2+xQ(t,x(t),x(tτ))2λm21f0(τ+1)tτtK(t,s)ds12|q(t)|x(t)20. Hence, V0(t,x(t))V0(t0,x(t0))=x2(t0),and so solutions are bounded. 

It should be clear from this theorem that

Conclusions

In this paper we considered systems of non-linear Volterra integro-fractional delay differential equations with Caputo derivatives. Some new results on the uniform, asymptotic, and Mittag-Leffler stability of the zero solution of the unperturbed equation and the boundedness of solutions of the perturbed equation were obtained. The method of proof is based on the application of the Razumikhin method. The application of the Razumikhin method for this kind of the problem appears to be new. This

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank the reviewers for carefully reading the paper and making many helpful suggestions for improvement.

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