New results on the stability, integrability and boundedness in Volterra integro-differential equations

The authors of this article deal with a first order non-linear Volterra integro-differential equation (NVIDE). To this end, the conditions are obtained which are sufficient for stability (S), boundedness (B), and for every solution x of (NVIDE) is integrable. For properties of solutions of (NVIDE) considered three new theorems on (S), (B) and integrability properties of solutions are proved. The methods of the proofs involve constructing of a suitable Lyapunov functional (LF) which gives meaningful results for the problems to be investigated. The conditions to be given involve nonlinear improvement and extensions of those conditions found in the literature. An example is provided to illustrate the effectiveness of the proposed results. The results obtained are new and complements that found in the literature.