Abstract
In this paper, we consider certain non-linear scalar Volterra integro-differential equations and Volterra integro-differential systems of first order. We investigate the boundedness, stability, uniformly asymptotic stability, integrability and square integrability of solutions to the scalar equations and the system considered. The technique used to prove the results of the paper is based on the second method of Lyapunov. From the obtained results, we extend and improve some related results that can be found in the literature.
Highlights
The theory of linear and non-linear Volterra integrodifferential equations and systems provides important mathematical models for many real-world phenomena in science and engineering. It is very important during investigations which are related to sciences and engineering to have information about the qualitative properties of solutions of linear and non-linear Volterra integro-differential equations and systems without solving them
The main theories, techniques and methods in the literature used to investigate the qualitative behaviour of paths of solutions of Volterra integro-differential equations and systems, without needing to find their analytical solutions, include the second method of Lyapunov, continuation methods, coincidence degree theory, perturbation theory, fixed point method or theory, iterative techniques and the variation of constants formula
When we look at the literature mentioned above, it can be seen that in the most of the papers, sufficient conditions are obtained for stability, boundedness, uniform asymptotic stability and integrability of solutions, instead of necessary and sufficient conditions
Summary
The theory of linear and non-linear Volterra integrodifferential equations and systems provides important mathematical models for many real-world phenomena in science and engineering. To summarize the above information, it follows that, motivated by the papers, books and monographs mentioned above, and by the paper of Burton and Mahfoud (1983) in particular, the aim of this paper is to prove new five theorems on the stability, boundedness, uniform asymptotic stability and integrability of solutions of scalar Volterra integro-differential Equations (3) and (4) and uniform asymptotic stability of the zero solution and square integrability of solutions of the Volterra integro-differential system (6). In what follows, we write x instead of xðtÞ: 2. Qualitative criteria for solutions Let qðt; xÞ 0
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