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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 $f:\bbfR^+\to\bbfR$ 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 $\bbfR^+$. The fractional derivative of order $0<s<1$ of a continuous function $f:\bbfR^+\to\bbfR$ 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 $\bbfR^+$. The authors consider the fractional differential equation $$D^su= f(x,u),\tag1$$ where $0<s<1$ and $f:[0,a]\times \bbfR\to\bbfR$, $0<a\le+\infty$, is a given function, continuous in $(0,a)\times \bbfR$. 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 $u\in C(\bbfR^+)\cap L^1_{\text{loc}}(\bbfR^+)$ 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 $0\le\sigma<s<1$, $f:[0,1]\times \bbfR\to\bbfR$ is a continuous function in $(0,1]\times\bbfR$ and $t^\sigma f(t,y)$ is continuous on $[0,1]\times\bbfR$, then (1) has at least one continuous solution on $[0,\delta]$ for a suitable $\delta\le 1$. 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 $a\in\bbfR^+$ and $b\in\bbfR$.
Reviewer: D.M.Bors (Iaşi)

##### MSC:
 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 26A33 Fractional derivatives and integrals (real functions) 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions
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