On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary value conditions: a fixed point theory approach
Volume 13, Issue 6, pp 364--377
http://dx.doi.org/10.22436/jnsa.013.06.06
Publication Date: April 24, 2020
Submission Date: October 20, 2018
Revision Date: February 07, 2020
Accteptance Date: March 03, 2020
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Authors
Md. Asaduzzaman
- Department of Mathematics, Islamic University, Kushtia-7003, Bangladesh.
Md. Zulfikar Ali
- Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
Abstract
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
\[
\begin{cases}
u(t) = u(1-t),\,\, u^{\prime\prime\prime}(0)-u^{\prime\prime\prime}(1)=u^{\prime\prime}(t_{1})+u^{\prime\prime}(t_{2}),\\
v(t) = v(1-t),\,\, v^{\prime\prime\prime}(0)-v^{\prime\prime\prime}(1)=v^{\prime\prime}(t_{1})+v^{\prime\prime}(t_{2}), \,\,\,0<t_{1}<t_{2}<1.
\end{cases}
\]
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.
Share and Cite
ISRP Style
Md. Asaduzzaman, Md. Zulfikar Ali, On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary value conditions: a fixed point theory approach, Journal of Nonlinear Sciences and Applications, 13 (2020), no. 6, 364--377
AMA Style
Asaduzzaman Md., Ali Md. Zulfikar, On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary value conditions: a fixed point theory approach. J. Nonlinear Sci. Appl. (2020); 13(6):364--377
Chicago/Turabian Style
Asaduzzaman, Md., Ali, Md. Zulfikar. "On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary value conditions: a fixed point theory approach." Journal of Nonlinear Sciences and Applications, 13, no. 6 (2020): 364--377
Keywords
- Fourth order four-point boundary value problem
- existence of symmetric positive solution
- Krasnoselskii's fixed point theorem
- Green's function
MSC
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