Second Order Semilinear Volterra-Type Integro-Differential Equations with Non-Instantaneous Impulses

Abstract

We consider a non-instantaneous system represented by a second order nonlinear differential equation in a Banach space E. We use the family of linear bounded operators introduced by Kozak, Darbo fixed point method and Kuratowski measure of noncompactness. A new set of sufficient conditions is formulated which guarantees the existence of the solution of the non-instantaneous system. An example is also discussed to illustrate the efficiency of the obtained results.

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

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Nonlocal semilinear second-order differential inclusions in abstract spaces without compactness

“…Differential equations and inclusions in Banach spaces have been attracting quite big attention (see, e.g., [1,2,5,13,23,24]). In particular, as pointed out by Byszewski and Lakshmikantham in [11], the study of nonlocal conditions is of significance due to their applicability in many physical and engineering problems and also in other areas of applied mathematics.…”

Existence of solutions for non-autonomous second-order stochastic inclusions with Clarke’s subdifferential and non instantaneous impulses

“…Henríquez et al [29] considered NASO differential structure with nonlocal initial data and developed the existence of solutions by applying the principle of the Hausdorff measure of non-compactness. Benchohra et al [37] used a fixed point theorem developed by Darbo with the Kuratowski measure of non-compactness to build certain adequate conditions that guarantee the presence of a solution for a NASO non-instantaneous integro-differential system.…”