Existence of mild solutions for semilinear evolution equations with non-local initial conditions

Existence of solutions for delay evolution equations with nonlocal conditions

“…The existence problem on the compact interval OE0; a, in the very simplest case when q D 0, i.e., when the delay is absent, was studied by Liu and Yuan [1]. In this case C.OE q; 0; X / is identified with X, f .t; u; u 0 / is identified with a function f from OE0; a X to X , and so in the paper [1] the following problem was studied: …”

“…On the basis of Schauder's fixed point theorem and approximation theory, the authors established some existence results of mild solutions for the problem (2) in [1]. As we can easily see, the general problem (1) contains as particular case (2).…”

“…In Section 3, we consider the existence of mild solutions for the problem (1.1) under the situation that g satisfies Lipschitz-type condition. By utilizing fixed point theorem and Kuratowski measure of noncompactness, we establish some existence results of mild solutions for the problem (1). In Section 4, we discuss the existence of mild solutions for the problem (1) under the case that g is completely continuous.…”

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].…”

“…However, only in a few papers the problem (1)- (2) was considered on an unbounded interval [8,18]. Additionally, in assumptions concerning the semigroup {G(t)} t≥0 or the function f (t, x), rather restrictive conditions have been imposed frequently that require the compactness of f (t, x) or {G(t)} t≥0 , or equicontinuity of the semigroup {G(t)} t≥0 [1][2][3][4][5][6][7][8][9][11][12][13][14][15][16][17][18].…”