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Existence and uniqueness for a nonlinear fractional differential equation. (English) Zbl 0881.34005

This paper deals with fractional calculus. The fractional primitive of order s>0 of a function f: + is given by

I s f(x)=(Γ(s)) -1 0 x (x-t) s-1 f(t)dt

provided the right-side is pointwise defined on + . The fractional derivative of order 0<s<1 of a continuous function f: + is given by

D s f(x)=(Γ(1-s)) -1 ·d dx 0 x (x-t) -s f(t)dt

provided the right-side is pointwise defined on + .

The authors consider the fractional differential equation

D s u=f(x,u),(1)

where 0<s<1 and f:[0,a]×, 0<a+, is a given function, continuous in (0,a)×. Under some assumptions, equation (1) is equivalent to the integral equation u(x)=I s f(x,u(x)), reduction used systematically in this paper.

A real-valued function uC(0,a)L 1 (0,a), or uC( + )L loc 1 ( + ) in the case a=+, with fractional derivative D s u on (0,a), is a solution of (1) if D s u(x)=f(x,u(x)) for all x(0,a).

The authors prove that if 0σ<s<1, f:[0,1]× is a continuous function in (0,1]× and t σ f(t,y) is continuous on [0,1]×, then (1) has at least one continuous solution on [0,δ] for a suitable δ1. Then, the authors show that uniqueness and global existence of solutions of (1) can be obtained a uniform Lipschitz-type assumption.

The last section of the paper concerns initial value problems of the type (1) and u(a)=b with a + and b.

Reviewer: D.M.Bors (Iaşi)

MSC:
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
26A33Fractional derivatives and integrals (real functions)
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions