Global existence of mild solutions of second order nonlinear Volterra integrodifferential equations in Banach spaces

Abstract

In this paper, we study the existence and uniqueness of mild solutions for second order initial value problems, with nonlocal conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family.

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

“…Only a few authors have dealt with the second-order problem in a Banach space whose r.h.s. also depends on the first derivative (sometimes solving the second-order problem by reducing it to a first-order equation)-see, e.g., [22][23][24][25][26][27][28][29][30][31] and the references therein. In [22,23], the operator A (in Equation (1) below) was supposed to be bounded, and impulses were not present; however, boundary value problems were studied there instead of the Cauchy problem.…”

“…In [22,23], the operator A (in Equation (1) below) was supposed to be bounded, and impulses were not present; however, boundary value problems were studied there instead of the Cauchy problem. In [24,25], a mild solution to a not impulsive problem with a nonlocal boundary condition was obtained, and in the second paper, the nonlinearity was allowed to also depend on an integral term, but in both cases, the r.h.s. must satisfy the Lipschitz condition.…”

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

“…Moreover, g and h are assumed completely continuous and bounded or globally Lipschitz continuous. In [21], the nonlinear term depends also on the weighted average of the solution and g and h depends also on ẋ, but the nonlinear term, g and h are assumed globally Lipschitz continuous.…”

Global Existence for Volterra-Fredholm Type Functional Impulsive Integrodifferential Equations

“…Also in [3], the authors studied second order impulsive functional differential inclusions by using Schaefer's theorem combined with a selection of theorem of Bressan and Colombo for lower semicontinuous multivalued operators with decomposable values. For recent results on local and global existence for ordinary, functional or neutral integrodifferential equations see [8,16,18,19,21,[25][26][27][28][29][30][31][32][33][34].…”