Abstract
This paper discusses a class of semilinear fractional evolution equations with infinite delay and almost sectorial operator on infinite interval in Banach space. By using the properties of analytic semigroups and Schauder’s fixed-point theorem, this paper obtains the existence of mild solutions of the fractional evolution equation. Moreover, this paper also discusses the existence of mild solution when the analytic semigroup lacks compactness by Kuratowski measures of noncompactness and Darbo–Sadovskii fixed-point theorem.
Highlights
Fractional differential models play a very important role in describing many complex phenomena such as chaotic system [1], fluid flow [2, 3], anomalous diffusion [4,5,6,7], and so on
The research on integer-order evolution equations has been relatively perfect [25, 26], but the research on fractional-order evolution equations is still in the preliminary stage. e existence of solutions for fractional evolution equations is the basis of the following study. e mild solution of integer-order evolution equations is defined by the constant variation method, which cannot be directly extended to fractional-order evolution equations
In order to obtain the existence of global mild solution of problem (3), we transform it into a fixed-point problem
Summary
Fractional differential models play a very important role in describing many complex phenomena such as chaotic system [1], fluid flow [2, 3], anomalous diffusion [4,5,6,7], and so on. Baliki et al [22] discussed a second-order evolution equation with infinite delay and obtained the existence and attractivity of mild solutions by Schauder’s fixed point as follows:. E existence of mild solutions for fractional evolution equations and evolution equations with infinite delay has been discussed in several papers (see [20, 21]). No paper is devoted to the existence of mild solutions with infinite delay and almost sectorial operator on infinite interval on Banach space. For the case that Q(t) is not compact, we expand the result of eorem 1.2.4 in Guo et al [27] from any compact interval to infinite interval (see Lemma 10) and obtain the existence of global mild solution by applying Kuratowski measures of noncompactness theory and Darbo–Sadovskii fixed-point theorem
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