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

Abstract

We study the existence of mild solutions for semilinear differential equations with nonlocal initial conditions in a separable Banach space X. We derive conditions in terms of the Hausdorff measure of noncompactness under which mild solutions exist in L p (0, b; X). For illustration, a partial integral differential system is worked out.

Measure of noncompactness and semilinear nonlocal functional differential equations in Banach spaces

“…In some previous papers the authors assumed that the space X is a separable Banach space and the semigroup T (t) is equicontinuous (see, e.g., [27,28]). We mention here that these assumptions are not necessary.…”

Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces

“…Benchohra and Ntouyas [6] discussed second order differential equations under compact conditions. For more details on the nonlocal problem, we refer to the papers of [3,[7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein.…”

“…In recent years, the technique of the measure of noncompactness was often applied to study the nonlocal Cauchy problem, such as [9][10][11][12][14][15][16][17][18][19][20][21][22]. Recently, Obukhovski and Zecca [21] used the measure of noncompactness with values in R 2 to discuss the controllability for classical initial problems.…”

“…Recently, Obukhovski and Zecca [21] used the measure of noncompactness with values in R 2 to discuss the controllability for classical initial problems. By making full use of the measure of noncompactness, Xue [15][16][17] established new existence theorems for semilinear and nonlinear nonlocal problem (1.1). In particular, in [15], Xue established the existence results without the assumptions of compactness.…”

On the existence and asymptotic behaviour of solutions of an evolution equation and an application to the Feynman–Kac theorem

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]). However, only in a few papers the problem (1)- (2) was considered on an unbounded interval [8,18].…”