Vol. 122, No. 1, 1986

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ISSN: 0030-8730
Solvability of various boundary value problems for the equation x′′ = f(t,x,x,x′′) y

Walter Volodymyr Petryshyn

Vol. 122 (1986), No. 1, 169–195

In this paper 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)| ≤ A (t,x)r2 + B|q|+ C(t,x )

for r, q in R with A and C bounded functions on each compact subset of [0,T] ×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 ().

Mathematical Subject Classification 2000
Primary: 34B15
Secondary: 34A45, 34C25
Received: 20 August 1984
Revised: 25 February 1985
Published: 1 March 1986
Walter Volodymyr Petryshyn