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Solvability of various boundary value problems for the equation $$x''=f(t,x,x',x'')-y$$. (English) Zbl 0585.34020
Summary: Some of the solvability results of Granas, Guenther and Lee for various homogeneous boundary value problems for the equation $$x''=f(t,x,x')$$ are extended in an essentially constructive way to the equation (*): $$x''=f(t,x,x',x'')$$ where f is assumed to satisfy the growth condition: $$| f(t,x,r,q)| \leq A(t,x)r^ 2+B| q| +C(t,x)$$ for r, q in R with A and C bounded functions on each compact subset of [0,T]$$\times R$$ and B in (0,1) and some further conditions stated below. Our proofs are based on the author’s continuation theorem for semilinear A-proper maps and the approach used by Granas, Guenther and Lee in obtaining the a priori bounds for the solutions of equation (*).

##### MSC:
 34B99 Boundary value problems for ordinary differential equations
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