Nonlinear differential equations with marchaud‐hadamard‐type fractional derivative in the weighted space of summable functions

Abstract

The paper is devoted to the study of the Cauchy‐type problem for the nonlinear differential equation of fractional order 0 < α < 1:

containing the Marchaud‐Hadamard‐type fractional derivative (D^α ₀₊, _μ y)(x), on the half‐axis R₊ = (0, +oo) in the space X^p,α _c,0 (R₊) defined for α > 0 by

where X^p _{c, 0} (R₊) is the subspace of X^p _c (R₊) of functions g X^p _c (R ₊ ) with compact support on infinity: g(x) = 0 for large enough x > R. The equivalence of this problem and of the nonlinear Volterra integral equation is established. The existence and uniqueness of the solution y(x) of the above Cauchy‐type problem is proved by using the Banach fixed point theorem. Solution in closed form of the above problem for the linear differential equation with f[x, y(x)] = λy(x) + f(x) is constructed. The corresponding assertions for the differential equations with the Marchaud‐Hadamard fractional derivative (D^α ₀₊ y)(x) are presented. Examples are given.

First Published Online: 14 Oct 2010

Keyword : differential equation of fractional order, Hadamard‐type fractional derivative, existence and uniqueness theorem, Mittag‐Leffler function