# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $f:{ℝ}^{+}\to ℝ$ is given by

${I}^{s}f\left(x\right)={\left({\Gamma }\left(s\right)\right)}^{-1}{\int }_{0}^{x}{\left(x-t\right)}^{s-1}f\left(t\right)dt$

provided the right-side is pointwise defined on ${ℝ}^{+}$. The fractional derivative of order $0 of a continuous function $f:{ℝ}^{+}\to ℝ$ is given by

${D}^{s}f\left(x\right)={\left({\Gamma }\left(1-s\right)\right)}^{-1}·\frac{d}{dx}{\int }_{0}^{x}{\left(x-t\right)}^{-s}f\left(t\right)dt$

provided the right-side is pointwise defined on ${ℝ}^{+}$.

The authors consider the fractional differential equation

${D}^{s}u=f\left(x,u\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $0 and $f:\left[0,a\right]×ℝ\to ℝ$, $0, is a given function, continuous in $\left(0,a\right)×ℝ$. Under some assumptions, equation (1) is equivalent to the integral equation $u\left(x\right)={I}^{s}f\left(x,u\left(x\right)\right)$, reduction used systematically in this paper.

A real-valued function $u\in C\left(0,a\right)\cap {L}^{1}\left(0,a\right)$, or $u\in C\left({ℝ}^{+}\right)\cap {L}_{\text{loc}}^{1}\left({ℝ}^{+}\right)$ in the case $a=+\infty$, with fractional derivative ${D}^{s}u$ on $\left(0,a\right)$, is a solution of (1) if ${D}^{s}u\left(x\right)=f\left(x,u\left(x\right)\right)$ for all $x\in \left(0,a\right)$.

The authors prove that if $0\le \sigma , $f:\left[0,1\right]×ℝ\to ℝ$ is a continuous function in $\left(0,1\right]×ℝ$ and ${t}^{\sigma }f\left(t,y\right)$ is continuous on $\left[0,1\right]×ℝ$, then (1) has at least one continuous solution on $\left[0,\delta \right]$ 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 $u\left(a\right)=b$ with $a\in {ℝ}^{+}$ and $b\in ℝ$.

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