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)
Nonlinear differential equations with Marchaud-Hadamard-type fractional derivative in the weighted space of summable functions. (English) Zbl 1132.26314

Summary: The paper is devoted to the study of a Cauchy-type problem for the nonlinear differential equation of fractional order 0<α<1,

(D 0+,μ α y)(x)=f(x,y(x)),(x μ 𝒥 0+,μ 1-α y)(0+)=b,b,

containing the Marchaud-Hadamard-type fractional derivative (D 0+,μ α y)(x), on the half-axis + =(0,+) in the space X c,0 p,α ( + ) defined for α>0 by

X c,0 p,α ( + )={yX c p ( + ):D 0+,μ α yX c,0 p ( + )}·

Here X c,0 p ( + ) is the subspace of X c p ( + ) of functions g with compact support on infinity: g(x)0 for large enough x>R. The equivalence of this problem and a nonlinear Volterra integral equation is established. The existence and uniqueness of the solution y(x) of the above Cauchy-type problem is proved by using the Banach fixed point theorem. The solution in closed form of the above problem for the linear differential equation with {f(x,y(x))=λy(x)+f(x)} is constructed. The corresponding assertions for the differential equations with the Marchaud-Hadamard fractional derivative (D 0+ α y)(x) are presented. Examples are given.

MSC:
26A33Fractional derivatives and integrals (real functions)
34K30Functional-differential equations in abstract spaces
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
45D05Volterra integral equations
47N20Applications of operator theory to differential and integral equations