Cui, Yujun; Zou, Yumei Existence and uniqueness of solutions for fourth-order boundary-value problems in Banach spaces. (English) Zbl 1191.34079 Electron. J. Differ. Equ. 2009, Paper No. 33, 8 p. (2009). Using Mönch’s fixed point theorem, an existence result is proved for the solutions of the following boundary value problem \[ x^{(4)}= f(t,x,x'') \quad (t \in (0,1)), \quad x(0) = x(1) = x''(0) = x''(1) = 0, \]where \(f : [0,1] \times E \times E \to E\) is continuous and \(E\) is a Banach space. The function \(f\) has to satisfy some linear growth conditions with sufficiently small coefficients, as well as some restrictions with respect to a measure of compactness. The uniqueness is proved under stronger conditions. Reviewer: Jean Mawhin (Louvain-La-Neuve) Cited in 1 Document MSC: 34G20 Nonlinear differential equations in abstract spaces 47H10 Fixed-point theorems 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:boundary-value problems in Banach spaces; fixed point theorems PDFBibTeX XMLCite \textit{Y. Cui} and \textit{Y. Zou}, Electron. J. Differ. Equ. 2009, Paper No. 33, 8 p. (2009; Zbl 1191.34079) Full Text: EuDML EMIS