Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

The existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space is investigated in this manuscript. No assumptions about the compactness of a function or the Lipschitz continuity of a nonlinear function are imposed on the nonlocal item and the nonlinear function, respectively. However, we assumed that the nonlocal item is continuous, the nonlinear term is continuous and satisfies some specified assumptions, and the associated semigroup is compact. Our theorems are proved by means of approximate techniques, semigroup methods, and fixed point theorem. These methods are useful for fixing the noncompactness of operators caused by some specified given assumptions on this paper. The results obtained here improve some known results. Finlay, two examples are presented for illustration of our main results.

Automated summary

The manuscript investigates the existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space. No assumptions are made about the compactness of a function or the Lipschitz continuity of a nonlinear function for the nonlocal item and the nonlinear function, respectively. However, continuity of the nonlocal item, continuity of the nonlinear term, and satisfaction of specified assumptions for the nonlinear term are assumed, as well as compactness of the associated semigroup. The theorems are proved using approximate techniques, semigroup methods, and fixed point theorem, which address the noncompactness of operators caused by specified assumptions in the paper. The results obtained in this manuscript improve upon existing knowledge. Finally, two examples are presented to illustrate the main results.