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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.

35R10 Functional partial differential equations
45J05 Integro-ordinary differential equations
Full Text: DOI
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