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

Delbosco, Domenico; Rodino, Luigi

J. Math. Anal. Appl. 204, No. 2, 609-625 (1996).

This paper deals with fractional calculus. The fractional primitive of order \(s>0\) of a function \(f:\mathbb{R}^+\to\mathbb{R}\) is given by \[ I^sf(x)= (\Gamma(s))^{-1} \int^x_0(x- t)^{s-1}f(t)dt \] provided the right-side is pointwise defined on \(\mathbb{R}^+\). The fractional derivative of order \(0<s<1\) of a continuous function \(f:\mathbb{R}^+\to\mathbb{R}\) is given by \[ D^sf(x)= (\Gamma(1-s))^{-1}\cdot{d\over dx} \int^x_0(x-t)^{- s}f(t)dt \] provided the right-side is pointwise defined on \(\mathbb{R}^+\).
The authors consider the fractional differential equation \[ D^su= f(x,u),\tag{1} \] where \(0<s<1\) and \(f:[0,a]\times \mathbb{R}\to\mathbb{R}\), \(0<a\leq+\infty\), is a given function, continuous in \((0,a)\times \mathbb{R}\). Under some assumptions, equation (1) is equivalent to the integral equation \(u(x)= I^sf(x,u(x))\), reduction used systematically in this paper.
A real-valued function \(u\in C(0,a)\cap L^1(0,a)\), or \(u\in C(\mathbb{R}^+)\cap L^1_{\text{loc}}(\mathbb{R}^+)\) in the case \(a=+\infty\), with fractional derivative \(D^su\) on \((0,a)\), is a solution of (1) if \(D^su(x)= f(x,u(x))\) for all \(x\in(0,a)\).
The authors prove that if \(0\leq\sigma<s<1\), \(f:[0,1]\times \mathbb{R}\to\mathbb{R}\) is a continuous function in \((0,1]\times\mathbb{R}\) and \(t^\sigma f(t,y)\) is continuous on \([0,1]\times\mathbb{R}\), then (1) has at least one continuous solution on \([0,\delta]\) for a suitable \(\delta\leq 1\). 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 \(a\in\mathbb{R}^+\) and \(b\in\mathbb{R}\).

Reviewer: D.M.Bors (Iaşi)

MSC:

34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
26A33	Fractional derivatives and integrals
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Keywords:

fractional calculus; fractional differential equation; initial value problems