zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 theorems for nonlinear fractional differential equations. (English) Zbl 0964.34004
Summary: The authors study the following Cauchy-type problem for the nonlinear differential equation of fractional order $\alpha\in \bbfC$, $\text{Re}(\alpha)> 0$, $$(D^\alpha_{a+}y)(x)= f[x, y(x)],\quad n-1< \text{Re}(\alpha)\le n,\quad n= -[-\text{Re}(\alpha)],$$ $$(D^{\alpha- k}_{a+} y)(a+)= b_k,\quad b_k\in \bbfC,\quad k= 1,2,\dots, n,$$ containing the Riemann-Liouville fractional derivative $D^\alpha_{a+}y$, on a finite interval $[a,b]$ of the real axis $\bbfR= (-\infty, \infty)$ in the space of summable functions $L(a,b)$. An equivalence of this problem and a nonlinear Volterra integral equation are established. The existence and uniqueness of the solution $y(x)$ to the above Cauchy-type problem are proved by using the method of successive approximations. Corresponding assertions for the ordinary differential equations are presented. Examples are given.

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