# 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 of mild solutions to impulsive fractional differential equations. (English) Zbl 1187.34108

Summary: Our aim in this paper is to study the existence and the uniqueness of the solution for the fractional semilinear differential equation:

$\left\{\begin{array}{c}{D}_{t}^{\alpha }x\left(t\right)=Ax\left(t\right)+f\left(t,x\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in I=\left[0,T\right],\phantom{\rule{4pt}{0ex}}t\ne {t}_{k},\hfill \\ x\left(0\right)={x}_{0}\in X,\hfill \\ {{\Delta }x|}_{t={t}_{k}}={l}_{k}\left(x\left({t}_{k}^{-}\right)\right),\phantom{\rule{1.em}{0ex}}k=1,\cdots ,m,\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $0<\alpha <1$, the operator $A:D\left(A\right)\subset X\to X$ is a generator of ${𝒞}_{0}$-semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on a Banach space $𝕏$, ${D}_{t}^{\alpha }$ is the Caputo fractional derivative, $f:I×𝕏\to 𝕏$ is a given continuous function ${I}_{k}:𝕏\to 𝕏$, $0={t}_{0}<{t}_{1}<\cdots <{t}_{m}<{t}_{m+1}=T$. ${{\Delta }x|}_{t={t}_{k}}=x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)$, $x\left({t}_{k}^{+}\right)={lim}_{h\to {0}^{+}}x\left({t}_{k}+h\right)$ and $x\left({t}_{k}^{-}\right)={lim}_{h\to 0}-x\left({t}_{k}+h\right)$ represent respectively the right and left limits of $x\left(t\right)$ at $t={t}_{k}$.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34K05 General theory of functional-differential equations 34K37 Functional-differential equations with fractional derivatives 34K45 Functional-differential equations with impulses