Abstract
In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.
Highlights
We consider nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators: Z t α,γD0+ u(t) + Au(t) = g t, u(t), k(t, s) f (s, u(s))ds, t ∈ (0, T ] = J, (1) (1−α)(1−γ) I0+Received: 31 December 2020Accepted: 4 March 2021Published: 8 March 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in [u(t)]|t=0 + h(u(t)) = u0, α,γ where D0+ is the Hilfer fractional derivative of order α ∈ (0, 1) and type γ ∈ [0, 1]
Our main results are proved in relation to the following hypotheses: Hypothesis 1 (H1)
We prove the existence of a mild solution to the problem in Equations (1) and (2) when the associated semigroup is compact (Theorem 5) and noncompact (Theorem 6)
Summary
We consider nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators:. Is the Riemann–Liouville integral of order 1 − q, 0 < q < 1, A is an almost sectorial operator on a complex Banach space, and f is a given function. Motivated by these results, here, we extend the previous available results of the literature to a class of Hilfer fractional integro-differential equations in which the closed operator is almost sectorial.
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