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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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