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

Authors : Bui NGHİA

Pages : 337-350

Doi:10.31197/atnaa.932760

View : 11 | Download : 20

Publication Date : 2021-09-30

Article Type : Research Paper

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 $|uinsert ignore into journalissuearticles values(t`);-uinsert ignore into journalissuearticles values(t);|$ tends to zero with respect to a small difference $|\psiinsert ignore into journalissuearticles values(t`);-\psiinsert ignore into journalissuearticles values(t);|^{\beta}$, $\beta\ininsert ignore into journalissuearticles values(0,1);$. 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