Existence of a mild solution to fractional differential equations with $\psi$-Caputo derivative, and its $\psi$-Hölder continuity

Year 2021, Volume: 5 Issue: 3, 337 - 350, 30.09.2021

Bui Nghia

https://doi.org/10.31197/atnaa.932760

Abstract

This paper is devoted to the study existence of locally/globally mild solutions for fractional differential equations with $\psi$-Caputo derivative with a nonlocal initial condition. We firstly establish the local existence by making use usual fixed point arguments, where computations and estimates are essentially based on continuous and bounded properties of the Mittag-Leffler functions. Secondly, we establish the called $\psi$-H\"older continuity of solutions, which shows how $|u(t')-u(t)|$ tends to zero with respect to a small difference $|\psi(t')-\psi(t)|^{\beta}$, $\beta\in(0,1)$. Finally, by using contradiction arguments, we discuss on the existence of a global solution or maximal mild solution with blowup at finite time.

Keywords

Fractional calculus, Fractional differential equations, $\psi$-Caputo derivative, Fixed point theorem, Maximal mild solutions, $\psi$-H\"older continuity.

Supporting Institution

Nong Lam University, Ho Chi Minh City, Vietnam

Project Number

CS-CB21-KH-01

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