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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)
34K37Functional-differential equations with fractional derivatives
34K30Functional-differential equations in abstract spaces
34K10Boundary value problems for functional-differential equations
47N20Applications of operator theory to differential and integral equations
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