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

2021 Advances in the Theory of Nonlinear Analysis and its Applications  
This paper is devoted to the study existence of locally/globally mild solutions for fractional differential equations with ψ-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 ψ-Hölder continuity of solutions, which shows how |u(t ) − u(t)| tends to zero
more » ... with respect to a small difference |ψ(t ) − ψ(t)| β , β ∈ (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.
doi:10.31197/atnaa.932760 fatcat:ow4njsn6n5efnhynzylxug2adm