Existence results for differential evolution equations with nonlocal conditions in Banach space

Abstract

Our aim in this paper is to study the existence and uniqueness of a mild solution to an initial value problem (IVP for short) for a class of nonlinear differential evolution equations with nonlocal initial conditions in a Banach space. We assume that the linear part is not necessarily densely defined and generates an evolution family. We give two results, the first one is based on a Krasnosel'skii fixed point Theorem, and in the second approach we make use Mönch fixed point Theorem combined with the measure of noncompactness and condensing. KeywordsNonlocal initial value problem, evolution family, measure of noncompactness, condensing map, nondensely defined operators, mild solution, Mönch fixed point Theorem.

On solutions of a semilinear measure‐driven evolution equation with nonlocal conditions on infinite interval

“…The theory of MDEs in ℝ 𝑛 space has been developed recently, see [10,13,14,25,33], for example. In particular, in [10] the authors investigated the practical stability of Equation (1) with initial condition 𝑥(𝑡 0 ) = 𝑥 0 . In addition, many authors discussed nonlinear retarded measure-driven equations with finite or infinite delay formulated as 𝑑𝑥 = 𝑓(𝑡, 𝑥 𝑡 )𝑑𝑔, 𝑡 ∈ [0, 𝑎],…”