Abstract
An impulsive boundary value problem with nonlinear boundary conditions for a second order ordinary differential equation is studied. In particular, sufficient conditions are provided so that a compression expansion cone theoretic fixed point theorem can be applied to imply the existence of positive solutions. The nonlinear forcing term is assumed to satisfy usual sublinear or superlinear growth as t → ∞ or t → 0+. The nonlinear impulse terms and the nonlinear boundary terms are assumed to satisfy the analogous asymptotic behavior.
Highlights
A well-known compression - expansion cone theoretic fixed point theorem due to Krasnosel’skii [18] and Guo [16] has been employed extensively to many types of boundary value problems
Applications appear in the case of partial differential equations, [3, 4, 6], for example; in a landmark paper [14], Erbe and Wang introduced the applications to ordinary differential equations
They showed that under the assumptions that the nonlinear term exhibits superlinear or sublinear growth, the fixed point theorem applies readily to boundary value problems whose solutions exhibit a natural type of concavity
Summary
A well-known compression - expansion cone theoretic fixed point theorem due to Krasnosel’skii [18] and Guo [16] has been employed extensively to many types of boundary value problems. Applications appear in the case of partial differential equations, [3, 4, 6], for example; in a landmark paper [14], Erbe and Wang introduced the applications to ordinary differential equations For their primary applications, they showed that under the assumptions that the nonlinear term exhibits superlinear or sublinear growth, the fixed point theorem applies readily to boundary value problems whose solutions exhibit a natural type of concavity. Shen and Wang [21] considered a second order impulsive problem with nonlinear boundary conditions and they employed the method of upper and lower solutions and the Schauder fixed point theorem. We do not pursue these variations or generalizations in this short article; we restrict ourselves to the posed problem
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