#### 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
##### Abstract

We study the existence of solutions of the second-order boundary value problem at resonance ${u}^{\prime \prime }=f\left(t,u,{u}^{\prime }\right)$ satisfying the boundary conditions $u\left(0\right)+u\left(1\right)=0$, ${u}^{\prime }\left(0\right)-{u}^{\prime }\left(1\right)=0$, or $u\left(0\right)-u\left(1\right)=0$, ${u}^{\prime }\left(0\right)+{u}^{\prime }\left(1\right)=0$. 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.

##### Keywords
boundary value problem, resonance, shift
Primary: 34B15
Secondary: 34B27
##### Milestones
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