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Existence of solutions to a second order partial differential equation with nonlocal conditions. (English) Zbl 1041.35045

The author considers the nonlocal Cauchy problem for the second order equation \[ \begin{aligned} u''(t)&= Au(t)+f(t,u(t),u'(t)), \quad t\in [0,a],\\ u(0)&= x_0 + q(u, u'),\\ u'(0)&= y_0 + p(u, u'),\end{aligned} \] where \(A\) is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators on a Banach space \(X\), \(f, q\) and \(p\) are appropriate continuous functions. Existence of mild and classsical solutions is proved.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
34G20 Nonlinear differential equations in abstract spaces
47D09 Operator sine and cosine functions and higher-order Cauchy problems
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