Stability criteria for Volterra equations

Burton, T. A.; Mahfoud, W. E.

doi:10.1090/s0002-9947-1983-0704607-8

Cited by 47 publications

(28 citation statements)

References 8 publications

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“…If a > Z 1 0 k(s) ds; (1.2) it is proved in [7] that the following all hold. If a > Z 1 0 k(s) ds; (1.2) it is proved in [7] that the following all hold.…”

Section: Introductionmentioning

confidence: 99%

Subexponential solutions of linear integro-differential equations and transient renewal equations

Appleby

Reynolds

2002

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper studies the asymptotic behaviour of the solutions of the scalar integro-di® erential equationThe kernel k is assumed to be positive, continuous and integrable. Ifit is known that all solutions x are integrable and x(t) ! 0 as t ! 1 , but also that x = 0 cannot be exponentially asymptotically stable unless there is some ® > 0 such that Z 1 0 k(s)e ® s ds < 1 :Here, we restrict the kernel to be in a class of subexponential functions in which k(t) ! 0 as t ! 1 so slowly that the above condition is violated. It is proved here that the rate of convergence of x(t) ! 0 as t ! 1 is given byThe result is proved by determining the asymptotic behaviour of the solution of the transient renewal equationIf the kernel h is subexponential, then lim t! 1 r(t) h(t) = 1 (1 ¡ R 1 0 h(s) ds) 2 :

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“…If a > Z 1 0 k(s) ds; (1.2) it is proved in [7] that the following all hold. If a > Z 1 0 k(s) ds; (1.2) it is proved in [7] that the following all hold.…”

Section: Introductionmentioning

confidence: 99%

Subexponential solutions of linear integro-differential equations and transient renewal equations

Appleby

Reynolds

2002

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Brauer [6] showed that the solution could not be stable if ∞ 0 k(s)ds > a, Burton and Mahfoud [7] proved the zero solution is asymptotically stable if ∞ 0 k(s)ds < a, Kordonis and Philos [8] discussed the stability of the solution under condition ∞ 0 k(s)ds = a. Therefore, a necessary condition for lim t→∞ x(t) = 0 for all solutions is that ∞ 0 k(s)ds ≤ a.…”

Section: K(t − S)x(s)dsmentioning

confidence: 99%

Mean square exponential and non-exponential asymptotic stability of impulsive stochastic Volterra equations

Zhao

Han

2011

J Inequal Appl

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In this article, some inequalities on convolution equations are presented firstly. The mean square stability of the zero solution of the impulsive stochastic Volterra equation is studied by using obtained inequalities on Liapunov function, including mean square exponential and non-exponential asymptotic stability. Several sufficient conditions for the mean square stability are presented. Results in this article indicate that not only the impulse intensity but also the time of impulse can influence the stability of the systems. At last, an example is given to show application of some obtained results. Mathematics classification Primary (2000): 60H10, 60F15, 60J70, 34F05.

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“…Nonlinear integro-differential equations have been widely studied by researchers due to its wide applications in biology, ecology, medicine, physics, engineering and other scientific areas, (see [3], [5], [20] and reference therein). In recent years, qualitative properties of solutions such as stability, instability, boundedness, existence etc.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability of nonlinear integrodifferential equation

Jackreece

Aniaku

2018

IJAMR

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Stability criteria for Volterra equations

Abstract: Abstract. We consider a system of integro-differential equations of the form (1

Cited by 47 publications

References 8 publications

Subexponential solutions of linear integro-differential equations and transient renewal equations

Subexponential solutions of linear integro-differential equations and transient renewal equations

Mean square exponential and non-exponential asymptotic stability of impulsive stochastic Volterra equations

Asymptotic stability of nonlinear integrodifferential equation

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