Existence of solutions of a second order equation defined on unbounded intervals with integral conditions on the boundary

Alberto Cabada; Rabah Khaldi

doi:10.26637/mjm0903/006

Abstract

In this paper we shall use the upper and lower solutions method to prove the existence of at least one solution for the second order equation defined on unbounded intervals with integral conditions on the boundary:
$$u^{\prime \prime}(t)-m^2 u(t)+f\left(t, e^{-m t} u(t), e^{-m t} u^{\prime}(t)\right)=0, \text{for all}\, t \in[0,+\infty),$$
$$
u(0)-\frac{1}{m} u^{\prime}(0)=\int_0^{+\infty} e^{-2 m s} u(s) d s, \lim _{t \rightarrow+\infty}\left\{e^{-m t} u(t)\right\}=B
$$
where $m>0,m\neq \frac{1}{6},B\in \mathbb{R}$ and $f:\left[ 0,+\infty \right) \times \mathbb{R}^{2}\rightarrow \mathbb{R} $ is a continuous function satisfying a suitable locally $L^1$ bounded condition and a kind of Nagumo's condition with respect to the first derivative.

Keywords:

Boundary value problems, Integral boundary conditions, Upper and lower solutions method, Existence of solution

Mathematics Subject Classification:

34B40, 34B15, 74H20

Authors Imprint References Funding Related Articles Downloads

Alberto Cabada Departamento de Estat´ ıstica, Análise Matemática e Optimización Facultade de Matemáticas, Universidade de Santiago de Compostela, Spain. https://orcid.org/0000-0003-1488-935X
Rabah Khaldi Laboratory of Advanced Materials, Department of Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria. https://orcid.org/0000-0002-6462-4424

Pages: 117-128
Date Published: 01-07-2021
Vol. 9 No. 03 (2021): Malaya Journal of Matematik (MJM)

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