Existence results and asymptotic behavior for nonlocal abstract Cauchy problems

Abstract

The purpose of this paper is to study the existence and asymptotic behavior of solutions for Cauchy problems with nonlocal initial datum generated by accretive operators in Banach spaces.

Existence results for semilinear fractional differential equations via Kuratowski measure of noncompactness

“…Garcia-Falset [18] studied the existence and asymptotic bahavior of solutions for Cauchy problems with nonlocal initial datum generated by accretive operators in Banach spaces. Xue [34] studied the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions under non-compactness conditions.…”

Existence of fixed points for the sum of two operators

“…Since then, several authors have investigated this type of problems, we refer the reader to [8,16,21,43] and references therein.…”

“…where A is the generator of a C 0 -semigroup, g : C([0, T ]; X) → X and f : [0, T ]×X → X are known functions (see [8,16,21,43]). …”

Lp theory for semilinear nonlocal problems with measure of noncompactness in separable Banach spaces

“…The maps f (t, ·) and g are neither compact (or Lipschitz) nor sums of compact and Lipschitz functions. Hence these cases are out of the scope of [12] and [29].…”

“…In [10] Cichoń and Majcher study the case when f is condensing with respect to a measure of noncompactness and g is a continuous function. In [12] García-Falset discusses the semilinear nonlocal problem when f is compact and g is a set-contraction with respect to the Kuratowski measure of noncompactness. Recently, Zhu and Li ( [29]) study the NIVP (1.1) in the case where g is completely continuous and f is multi-valued contracting and upper semicontinuous in general Banach spaces.…”