# 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 solution for fractional integro-differential equations of neutral type with nonlocal conditions. (English) Zbl 1313.34239
The existence of mild solutions are studied for the Caputo fractional neutral integro-differential equation, \align ^cD^q\Bigl (u(t)-G(t,\psi (t))\Bigr)=&\, A\Bigl (u(t)-G\bigl (t,\psi (t)\bigr)\Bigr) + F\Bigl (t,\phi (t)\Bigr)\\ &+ \int \limits _0^t k\Biggl (t,s,u(s),\int \limits _0^s\rho \bigl (s,\tau ,u(\tau)\bigr) {\operatorname {d}}\tau \Biggr) {\operatorname {d}}s, \ t \in [0,T], \tag1 \endalign satisfying the nonlocal conditions, $$u(0) + g(u) = u_0, \tag2$$ where $0<q<1$, $T>0$, \align \Bigl (t,\psi (t)\Bigr) &= \Bigl (t,u(t),u()\nu _1(t),\ldots ,u\bigl (\nu _m(t)\bigr)\Bigr),\\ \Bigl (t,\phi (t)\Bigr) &= \Bigl (t,u(t),u()\sigma _1(t),\ldots ,u\bigl (\sigma _n(t)\bigr)\Bigr), \endalign and where $A$ generates a compact semigroup of uniformly bounded linear operators $S(\cdot)$ on a Banach space $X$, $F\:[0,T] \times X_\alpha ^{n+1}\to X$, $G\:[0,T] \times X_\alpha ^{m+1}\to X_\alpha$, $g\:X_\alpha \to X_\alpha$, and $k\:\Delta \times X_\alpha \times X_\alpha \to X$ and $\rho \:\Delta \times X_\alpha \to X_\alpha$ are continuous, $X_\alpha = {\operatorname {Domain}} (A), 0<\alpha <1$, $\Delta =\bigl \{(t,s) \mid 0 \leq s \leq t \leq T\bigr \}$, $\nu _i(t) \leq t$ and $\sigma _j(t) \leq t$ are continuous and scalar valued. Under a list of five hypotheses, the author applies the Leray-Schauder nonlinear alternative to obtain a fixed point in $C\bigl ([0,T],X_\alpha \bigr)$, for the operator \align (\Psi u) (t) :=& \int \limits _0^\infty \xi _q(\theta) S(t^q \theta)\, {\operatorname {d}}\theta \Bigl (u_0 -g(u) -G\bigl (0,\psi (0)\bigr)\Bigr) + G\bigl (t,\psi (t)\bigr)\\ & + \int \limits _0^t q \int \limits _0^\infty \theta (t-s)^{q-1} \xi _q(\theta) S\bigl ((t-s)^q \theta \bigr) {\operatorname {d}}\theta \Bigl [F\bigl (s,\phi (s)\bigr)+ H(u)(s)\Bigr]\, {\operatorname {d}}s, \endalign where $\xi _q$ is a probability density function defined on $(0,\infty)$. The fixed point of $\Psi$ is a mild solution of (1), (2). Following that, under a listing of a new set of six hypotheses, the author applies the Contraction Mapping Principle to obtain a unique fixed point of $\Psi$ (although the author has renamed the operator $\tilde {\Psi}$), in $C\bigl ([0,T],X_\alpha \bigr)$. This unique fixed point corresponds to a unique solution of (1), (2).
Reviewer: Johnny Henderson (Waco)
##### MSC:
 34K37 Functional-differential equations with fractional derivatives 34K30 Functional-differential equations in abstract spaces 34K10 Boundary value problems for functional-differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text: