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Existence of solutions for a class of abstract differential equations with nonlocal conditions. (English) Zbl 1221.47079

The authors consider the system

${y}^{\text{'}}\left(t\right)=Ay\left(t\right)+f\left(t,y\left(t\right)\right)\phantom{\rule{1.em}{0ex}}\left(0\le t\le a\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}y\left(0\right)=g\left(u\right)+{y}_{0}\phantom{\rule{2.em}{0ex}}\left(1\right)$

in a Banach space $E,$ where $A$ is the infinitesimal generator of an analytic semigroup. The nonlinearity $f$ maps $\left[0,a\right]×{E}_{\alpha }$ into $E,$ where ${E}_{\alpha }$ is the domain of the fractional power ${\left(-A\right)}^{\alpha }$ $\left(0<\alpha <1\right)·$ Finally, $g$ maps $C\left(I,{E}_{\alpha }\right)$ into $E,$ where $I\subset \left(0,a\right]·$ The results are on existence and uniqueness of solutions of (1), where “solution” is understood in various ways, two of them classical and mild (the latter means a solution of the integral equation version of (1)). The results are applied to partial differential equations, with nonlocal conditions involving partial derivatives or nonlinear expressions of the solution.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34K30 Functional-differential equations in abstract spaces 34B10 Nonlocal and multipoint boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations
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