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 $$f:\mathbb{R}^+\to\mathbb{R}$$ 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
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