Existence Results for a Nonlinear Fourth Order Ordinary Differential Equation with Four-Point Boundary Value Conditions

Abstract

The aim of this paper is to study the more accurate existence results of positive solution for a nonlinear fourth order ordinary differential equation (for short NLFOODE) using four-point boundary value conditions (for short BVCs). The upper and lower solution method and Schauder’s fixed point theorem have been applied to demonstrate the obtained existence results. First, the Green’s function of the corresponding linear boundary value problem (for short BVP) has been constructed and then it is used to solve the considered BVP of this paper. An example has also been included at the end of this paper to support the analytic proof.

Keywords

• NLFOODE with four-point BVCs
• Upper and lower solution method
• Schauder’s fixed point theorem
• Existence of positive solution

Kaynakça

1. E. Dulácska, The structures, soil settlement effects on buildings, developments in geotechnical engineering, Elsevier, Amsterdam, 69, (1992).
2. S.P. Timoshenko, Theory of elastic stability, McGraw-Hill, New York, (1961).
3. W. Soedel, Vibrations of shells and plates, Dekker, New York, (1993).
4. D.G. Zill, M.R. Cullen, Differential equations with boundary-value problems, 5th edition, Brooks/Cole, (2001).
5. R.P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential and Integral Equations, 2, (1989) 91-110.
6. C.P. Gupta, Existence and uniqueness theorem for a bending of an elastic beam equation, Appl. Anal., 26, (1988) 289-304.
7. S. Chen, W. Ni, and C. Wang, Positive solution of fourth order ordinary differential equation with four-point boundary conditions, Appl. Math. Letters, 19, (2006) 161-168.
8. S. Benaicha, F. Haddouchi, Positive solutions of a nonlinear fourth-order integral boundary value problem, Annals of West University of Timisoara - Math. Comput. Sci., 54(1), (2016) 73-86.

9. X. Lv, L. Wang and M. Pei, Monotone positive solution of a fourth-order BVP with integral boundary conditions, Boundary Value Problems, 2015(72), (2015).
10. K. Wang and Z. Yang, Positive solutions for a fourth-order boundary value problem, J. Math., Hind. Publ. Corp., 2013, Article ID 316576, (2013), 8 pages.
11. Z. Hao, L. Liu, L. Debnath, A necessary and sufficiently condition for the existence of positive solution of fourth-order singular boundary value problems, Appl. Math. Letters, 16, (2003) 279-285.
12. J.R. Graef, C. Quian, B. Yang, A three-point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl., 287, (2003) 217-233.
13. R.P. Agarwal, O. O’Regan, P.J.Y. Wong, Positive solutions of differential, difference, and integral equations, Kluwer Academic, Dordrecht, (1998).
14. A. Cabada, The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl., 185, (1994) 302-320.
15. Z.X. Zhang, J.Wang, The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, J. Comput. Appl. Math., 147, (2003) 41-52.
16. M. Asaduzzaman and M. Zulfikar Ali, On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary conditions: A fixed point theory approach, J. Nonlinear Sci. Appl., 13, (2020) 364-377.
17. M. Asaduzzaman and 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 (1), (2019) 80-98.
18. M. Asaduzzaman and M. Zulfikar Ali, Existence of three positive solutions for nonlinear third order arbitrary two-point boundary value problems, Differential Equations and Control Processes, Russia, 2019(2), (2019) 83-100.
19. M. Tuz, The Existence of symmetric positive solutions of fourth-order elastic beam equations, Symmetry, 11(1)121, (2019).
20. J. Leray, J. Schauder, Topologie et equations fonctionels, Ann. Sci. École Norm. Sup., 51, (1934) 45-78.
21. Q.M. Al-Mdallal and M. A. Hajji, A convergent algorithm for solving higher-order nonlinear fractional boundary value problems. Frac. Calc. Appl. Anal., 18(6)1423, (2015).
22. Q.M. Al-Mdallal and M.I. Syam, The Chebyshev collocation-path following method for solving sixth-order Sturm–Liouville problems, Appl. Math. Comput., 232, (2014) 391-398.
23. M.A. Hajji, Q.M. Al-Mdallal, F.M. Allan, An efficient algorithm for solving higher-order fractional Sturm–Liouville eigenvalue problems, J. Comput. Phy., 272, (2014) 550-558.
24. J. He, M. Jia, X. Liu and H. Chen, Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity. Adv. Differ. Equ., 2018(1)49, (2018).
25. H. Khan, C. Tunc, A. Khan, Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations, Discrete & Continuous Dynamical Systems-S, 13(9), (2020).
26. X. Wang, Existence of solutions for nonlinear impulsive higher order fractional differential equations, Elec. J. Qual. Theo. Differ. Equ., 2011(80), (2011) 1-2.
27. X. Zhang, Positive solutions for singular higher-order fractional differential equations with nonlocal conditions. J. Appl. Math. Comp., 49(1-2), (2015) 69-89.
28. M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22, (1906) 1-74.
29. W. Rudin, Functional Analysis, McGraw-Hill, Springer-Verlag, New York, (1973).