Nonlinear differential equations with Marchaud-Hadamard-type fractional derivative in the weighted space of summable functions

The paper is devoted to the study of the Cauchy-type problem for the nonlinear differential equation of fractional order 0 < alpha < 1: [GRAPHICS] containing the Marchaud-Hadamard-type fractional derivative (D-0+(alpha),mu y)(x), on the half-axis R+ = (0, +infinity) in the space X-c,(0)p,(alpha) (R+) defined for alpha > 0 by [GRAPHICS] where X-c(p),(0)(R+) is the subspace of X-c(p)(R+) of functions g is an element of X-c(p)(R+) with compact support on infinity: g(x) equivalent to 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)] = lambda y(x) + f (x) is constructed. The corresponding assertions for the differential equations with the Marchaud-Hadamard fractional derivative (D-0+(alpha) y)(x) are presented. Examples are given.