Infinite interval problems for Hilfer fractional evolution equations with almost sectorial operators. (English) Zbl 1519.34071

Zhou, Mian; Liang, Yong; Zhou, Yong

Rocky Mt. J. Math. 52, No. 6, 2257-2272 (2022).

In this article, there is considered the following Cauchy problem of fractional evolution equations on an infinite interval \[ \begin{cases} & ^H D^{\nu, \mu}_{0+} x(t)= A x(t) + f(t,x(t)), \ \ t \in (0,\infty), \\ & I^{(1-\nu)(1-\mu)}_{0+} x(0)=x_0, \end{cases} \tag{1} \] where \( ^H D^{\nu, \mu}_{0+}\) is the Hilfer fractional derivative of order \(0 < \mu < 1\) and type \(0 \leq \nu \leq 1\) ; \( I^{(1-\nu)(1-\mu)}_{0+} \) is the Riemann-Liouville integral of order \((1-\nu)(1-\mu)\); \(A\) is an almost sectorial operator in a Banach space \(X\) and \(f : [0,\infty) \times X \to X\).
The authors prove theorems for existence of at least one mild solution for the considered Cauchy problem (1) in the cases when the semigroup associated with the almost sectorial operator is compact as well as noncompact. The methods used are based on a generalized Ascoli-Arzela theorem, Schauder’s fixed point theorem and Kuratowski’s measure of noncompactness. It may be noted, that the authors do not assume that \( f (t, \cdot)\) satisfies a Lipschitz condition. Sufficient conditions for the attractiveness of the solutions in each of the considered cases are provided too.
Two examples are provided to illustrate the obtained results.

Reviewer: Hristo S. Kiskinov (Plovdiv)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34A08	Fractional ordinary differential equations
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26A33	Fractional derivatives and integrals
47N20	Applications of operator theory to differential and integral equations

Keywords:

fractional evolution equations; Hilfer derivative; almost sectorial operator; infinite interval

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