# Article

 Title: Existence of multiple solutions for some functional boundary value problems (English) Author: Staněk, Svatoslav Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 28 Issue: 1 Year: 1992 Pages: 57-65 Summary lang: English . Category: math . Summary: Let $X$ be the Banach space of $C^0$-functions on $\langle 0,1\rangle$ with the sup norm and $\alpha ,\beta \in X \rightarrow {R}$ be continuous increasing functionals, $\alpha (0)= \beta (0)=0$. This paper deals with the functional differential equation (1) $x^{\prime \prime \prime } (t) = Q [ x, x^\prime , x^{\prime \prime }(t)] (t)$, where $Q:{X}^2 \times {R} \rightarrow {X}$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $\alpha (x)=0=\beta (x^\prime )$, $x^{\prime \prime }(1)-x^{\prime \prime }(0)=0$ are given. The method of proof makes use of Schauder linearizatin technique and the Schauder fixed point theorem. The results are modified for 2nd order functional differential equations. (English) Keyword: Schauder linearization technique Keyword: Schauder differential equation Keyword: functional boundary conditions Keyword: boundary value problem MSC: 34B10 MSC: 34B15 idZBL: Zbl 0782.34074 idMR: MR1201866 . Date available: 2008-06-06T21:04:48Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107436 . Reference: [1] Castro A., and Shivaji R.: Nonnegative solutions for a class of radially symmetric non- positone problems.Proc. Amer. Math. Soc., in press. Reference: [2] Chiappinelli R., Mawhin J. and Nugari R.: Generalized Ambrosetti - Prodi conditions for nonlinear two-point boundary value problems.J. Differential Equations 69 (1987), 422-434. MR 0903395 Reference: [3] Dancer E.N. and Schmitt K.: On positive solutions of semilinear elliptic equations.Proc. Amer. 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