On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary value conditions: a fixed point theory approach

Asaduzzaman, Md.¹ ; Ali, Md. Zulfikar ²

¹ Department of Mathematics, Islamic University, Kushtia-7003, Bangladesh
² Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh

Journal of nonlinear sciences and its applications, Tome 13 (2020) no. 6, p. 364-377.

Résumé

The purpose of this paper is to investigate the existence of symmetric positive solutions of the following nonlinear fourth order system of ordinary differential equations

$ \begin{cases} -u^{(4)}(t) = f(t,\, v),\\ -v^{(4)}(t) = g(t,\, u) , \,\,\,t\in[0,\,1], \end{cases} $

with the four-point boundary value conditions

$ \begin{cases} u(t) = u(1-t),\,\, u^{\prime\prime\prime}(0)-u^{\prime\prime\prime}(1)=u^{\prime\prime}(t_{1})+u^{\prime\prime}(t_{2}),\\ v(t) = v(1-t),\,\, v^{\prime\prime\prime}(0)-v^{\prime\prime\prime}(1)=v^{\prime\prime}(t_{1})+v^{\prime\prime}(t_{2}), \,\,\,0$

By applying Krasnoselskii's fixed point theorem and under suitable conditions, we establish the existence of at least one or at least two symmetric positive solutions of the above mentioned fourth order four-point boundary value problem in cone. Some particular examples are provided to support the analytic proof.

Détail

DOI : 10.22436/jnsa.013.06.06

Classification : 34B10, 34B15, 34B18
Keywords: Fourth order four-point boundary value problem, existence of symmetric positive solution, Krasnoselskii's fixed point theorem, Green's function

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