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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, \[ \begin{aligned} ^cD^q\Bigl (u(t)-G(t,\psi (t))\Bigr)=&\, A\Bigl (u(t)-G\bigl (t,\psi (t)\bigr)\Bigr) + F\Bigl (t,\phi (t)\Bigr)\\ &+ \int \limits _0^t k\Biggl (t,s,u(s),\int \limits _0^s\rho \bigl (s,\tau ,u(\tau)\bigr) {\operatorname {d}}\tau \Biggr) {\operatorname {d}}s, \;t \in [0,T], \tag{1} \end{aligned} \] 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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