Hernández M., Eduardo Existence of solutions to a second order partial differential equation with nonlocal conditions. (English) Zbl 1041.35045 Electron. J. Differ. Equ. 2003, Paper No. 51, 10 p. (2003). 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. Reviewer: Michael I. Gil’ (Beer-Sheva) Cited in 15 Documents 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 Keywords:cosine function; differential equations in abstract spaces; nonlocal condition; existence of mild and classsical solutions PDFBibTeX XMLCite \textit{E. Hernández M.}, Electron. J. Differ. Equ. 2003, Paper No. 51, 10 p. (2003; Zbl 1041.35045) Full Text: EuDML EMIS