Vol. 12, No. 1, 2019

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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

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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boundary value problem, resonance, shift
Mathematical Subject Classification 2010
Primary: 34B15
Secondary: 34B27
Received: 24 January 2018
Revised: 13 February 2018
Accepted: 14 February 2018
Published: 31 May 2018

Communicated by Johnny Henderson
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