J. Nonl. Evol. Equ. Appl. 2020 (4), pp. 55-63, published on May 31, 2020:

A Cauchy problem for some fractional differential equation via deformable derivatives

Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, M.D. 21251, USA

Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, M.D. 21251, USA

Received on September 14, 2019
Accepted on January 9, 2020

Communicated by Maximilian F. Hasler

Abstract. Abstract. In this paper we investigate further properties of the new concept of deformable derivative and use the results to study the existence (and uniqueness) of mild solutions to the Cauchy problem for the nonlinear diﬀerential equation with nonlocal conditions D^αx(t) = f(t, x(t)), t ∈ [0, T ], x(0) + g(x) = x₀, where D^αx(t) is the deformable derivative of x, 0 < α < 1. We use the Krasnoselkii’s theorem to achieve our main result.
Keywords:	Deformable derivative, Krasnoselskii’s theorem, mild solution.
2010 Mathematics Subject Classification: 34G20, 34A08.