Solutions of boundary value problems at resonance with periodic and antiperiodic boundary conditions

Garcia, Aldo E.; Neugebauer, Jeffrey T.

doi:10.2140/involve.2019.12.171

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Solutions of boundary value problems at resonance with periodic and antiperiodic boundary conditions

Aldo E. Garcia and Jeffrey T. Neugebauer

Vol. 12 (2019), No. 1, 171–180

DOI: 10.2140/involve.2019.12.171

Abstract

We study the existence of solutions of the second-order boundary value problem at resonance $u^{''} = f (t, u, u^{'})$ satisfying the boundary conditions $u (0) + u (1) = 0$ , $u^{'} (0) - u^{'} (1) = 0$ , or $u (0) - u (1) = 0$ , $u^{'} (0) + u^{'} (1) = 0$ . We employ a shift method, making a substitution for the nonlinear term in the differential equation so that these problems are no longer at resonance. Existence of solutions of equivalent boundary value problems is obtained, and these solutions give the existence of solutions of the original boundary value problems.

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Keywords

boundary value problem, resonance, shift

Mathematical Subject Classification 2010

Primary: 34B15

Secondary: 34B27

Milestones

Received: 24 January 2018

Revised: 13 February 2018

Accepted: 14 February 2018

Published: 31 May 2018

Communicated by Johnny Henderson

Authors

Aldo E. Garcia
Department of Mathematics and Statistics Eastern Kentucky University Richmond, KY United States

Jeffrey T. Neugebauer
Department of Mathematics and Statistics Eastern Kentucky University Richmond, KY United States

Vol. 12, No. 1, 2019