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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 $$ y'(t) = Ay(t) + f(t, y(t)) \quad (0 \le t \le a) \, , \quad y(0) = g(u) + y_0 \tag1$$ in a Banach space $E,$ where $A$ is the infinitesimal generator of an analytic semigroup. The nonlinearity $f$ maps $[0, a] \times E_\alpha$ into $E,$ where $E_\alpha$ is the domain of the fractional power $(-A)^\alpha$ $(0 < \alpha < 1).$ Finally, $g$ maps $C(I, E_\alpha)$ into $E,$ where $I \subset (0, a].$ 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.

47D06One-parameter semigroups and linear evolution equations
34K30Functional-differential equations in abstract spaces
34B10Nonlocal and multipoint boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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