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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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[1] Bochenek J., Ann. Pol. Math. 54 pp 155– (1991)
[2] DOI: 10.1016/0022-247X(91)90164-U · Zbl 0748.34040 · doi:10.1016/0022-247X(91)90164-U
[3] Dugundji J., Monographie Matematyczne,PNW Warasawa 1 (1982)
[4] DOI: 10.1006/jmaa.1997.5425 · Zbl 0884.34069 · doi:10.1006/jmaa.1997.5425
[5] DOI: 10.1080/00036819708840525 · Zbl 0874.35126 · doi:10.1080/00036819708840525
[6] Webb G., Acta Math. 32 pp 75– (1978)
[7] DOI: 10.1090/S0002-9947-1974-0382808-3 · doi:10.1090/S0002-9947-1974-0382808-3
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