## 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, \begin{aligned} ^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], \tag{1} \end{aligned} satisfying the nonlocal conditions, $u(0) + g(u) = u_0, \tag{2}$ where $$0<q<1$$, $$T>0$$, \begin{aligned} \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), \end{aligned} 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 \begin{aligned} (\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, \end{aligned} 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).

### 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
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