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Global existence for second order semilinear ordinary and delay integrodifferential equations with nonlocal conditions. (English) Zbl 0906.35110
The following initial value problem with nonlocal initial conditions is studied: $x''(t)= Ax(t)+f \bigl(t,x(t), x'(t)\bigr), \quad a.a.\;t\in [0,b], \quad x(0) +g(x)= x_0,\;x'(0)= \eta$ where $$A$$ is a linear infinitesimal generator of a strongly continuous cosine family $$C(t)$$ and $$f$$ is continuous and locally bounded in $$x,x'$$. Using fixed point theory and the Leray-Schauder alternative, it is shown that this system has a unique mild solution on some interval $$[0,b]$$, provided $$f$$ is sublinear in $$x,x',C(t)$$ is compact for $$t>0$$ and $$g$$ is uniformly bounded. A similar result is given in the functional differential case.

MSC:
 35R10 Functional partial differential equations 45J05 Integro-ordinary differential equations
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References:
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