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Existence of multiple solutions for some functional boundary value problems. (English) Zbl 0782.34074
The problem (1) \(x'''= q(t,x,x',x'')\), \(t\in [0,1]\), \(\alpha(x)=\beta(x')=0\), \(x''(0)= x''(1)\) with \(\alpha\), \(\beta\) continuous, increasing functionals, \(\alpha(0)=\beta(0)=0\), is investigated. Using Schauder’s fixed point theorem sufficient conditions for the existence of (a) at least one solution of (1) with \(x''(t)\geq 0\) on \([0,1]\), (b) at least one solution of (1) with \(x''(t)\leq 0\) on \([0,1]\), (c) at least two different solutions \(x_ 1\), \(x_ 2\) of (1) with \(x_ 1{''}(t)\leq 0\leq x_ 2{''}(t)\) on \([0,1]\) are obtained. The proofs are based on a priori estimates, degree theory and lower and upper solutions.

MSC:
34K10 Boundary value problems for functional-differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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