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
References
-
[1]
R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2 (1989), 91--110
-
[2]
M. Asaduzzaman, M. Zulfikar Ali, Existence of three positive solutions for nonlinear third order arbitrary two-point boundary value problems, Differ. Uravn. Protsessy Upr., 2019 (2019), 83--100
-
[3]
M. Asaduzzaman, M. Zulfikar Ali, Fixed Point Theorem Based Solvability of Fourth Order Nonlinear Differential Equation with Four-point Boundary Value Conditions, Adv. Fixed Point Theory, 9 (2019), 260--272
-
[4]
M. Asaduzzaman, M. Zulfikar Ali, The unique symmetric positive solutions for nonlinear fourth order arbitrary two-point boundary value problems: A fixed point theory approach, Adv. Fixed Point Theory, 9 (2019), 80--98
-
[5]
E. Cetin, F. S. Topal, Symmetric positive solutions of fourth order boundary value problems for an increasing homeomorphism and homomorphism on time-scales, Comput. Math. Appl., 63 (2012), 669--678
-
[6]
S. H. Chen, W. Ni, C. P. Wang, Positive solution of fourth order ordinary differential equation with four-point boundary conditions, Appl. Math. Lett., 19 (2006), 161--168
-
[7]
E. Dulacska, Soil settlement effects on buildings, in: Developments in Geotechnical Engineering, Amsterdam (1992)
-
[8]
H. Y. Feng, D. L. Bai, M. Q. Feng, Multiple symmetric positive solutions to four-point boundary-value problems of differential systems with P-Laplacian, Electron. J. Differential Equations, 2012 (2012), 11 pages
-
[9]
M. Q. Feng, P. Li, S. J. Sun, Symmetric positive solutions for fourth-order $n$-dimensional $m$-Laplace systems, Bound. Value Probl., 2018 (2018), 20 pages
-
[10]
D. J. Guo, Nonlinear Functional Analysis, Shangdong Sci. Tech. Press, Jinan (1985)
-
[11]
C. P. Gupta, Existence and uniqueness theorem for a bending of an elastic beam equation, Appl. Anal., 26 (1988), 289--304
-
[12]
J. Henderson, H. B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc., 128 (2000), 2373--2379
-
[13]
F. Y. Li, Y. Zhang, Multiple symmetric nonnegative solutions of second-order ordinary differential equations, Appl. Math. Lett., 17 (2004), 261--267
-
[14]
Y. H. Li, X. Y. Zhang, Positive Solutions for Systems of Nonlinear Higher Order Differential Equations with Integral Boundary Conditions, Abstr. Appl. Anal., 2014 (2014), 7 pages
-
[15]
X. Z. Lv, L. B. Wang, M. H. Pei, Monotone positive solution of a fourth-order BVP with integral boundary conditions, Bound. Value Probl., 2015 (2015), 12 pages
-
[16]
P .K. Palamides, A. P. Palamides, Fourth-Order Four-Point Boundary Value Problem: A Solutions Funnel Approach, Int. J. Math. Math. Sci., 2012 (2012), 18 pages
-
[17]
H. D. Qu, The Symmetric Positive Solutions of Four-Point Problems for Nonlinear Boundary Value Second-Order Differential Equations, Int. J. Math. Anal. (Ruse), 3 (2009), 1969--1979
-
[18]
H. D. Qu, The symmetric positive solutions of three-point boundary value problems for nonlinear second-order differential equations, Bull. Inst. Math. Acad. Sin. (N.S.), 7 (2012), 405--416
-
[19]
W. Soedel, Vibrations of Shells and Plates, Dekker, New York (1993)
-
[20]
Y. P. Sun, Existence and multiplicity of symmetric positive solutions for three-point boundary value problem, J. Math. Anal. Appl., 329 (2007), 998--1009
-
[21]
S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York (1961)
-
[22]
M. Tuz, The Existence of Symmetric Positive Solutions of Fourth-Order Elastic Beam Equations, Symmetry, 11 (2019), 14 pages
-
[23]
A. K. Verma, N. Urus, Region of existence of multiple solutions for a class of 4-point BVPs, arXiv, 2019 (2019), 21 pages
-
[24]
J. R. L. Webb, G. Infante, D. Franco, Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 427--446
-
[25]
Q. L. Yao, Existence and iteration of n symmetric positive solutions for a singular two-point boundary value problem, Comp. Math. Appl., 47 (2004), 1195--1200
-
[26]
C. Zhai, R. Song, Q. Han, The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem, Comput. Math. Appl., 62 (2011), 2639--2647
-
[27]
X. Y. Zhang, Fixed point theorems for a class of nonlinear operators in Banach spaces and applications, Nonlinear Anal., 69 (2008), 536--543
-
[28]
X. M. Zhang, M. Q. Feng, W. G. Ge, Symmetric positive solutions for $p$-Laplacian fourth-order differential equations with integral boundary conditions, J. Comput. Appl. Math., 222 (2008), 561--573
-
[29]
D. G. Zill, M. R. Cullen, Differential Equations with Boundary-Value Problems, Brooks/Cole, Boston (2001)
