Well ordered monotone iterative technique for nonlinear second order four point Dirichlet BVPs
Abstract
In this article, we develop a monotone iterative technique (MI-technique) with lower and upper (L-U) solutions for a class of four-point Dirichlet nonlinear boundary value problems (NLBVPs), defined as, 
where 
, the non linear term 
is continuous function in x, one sided Lipschitz in ψ and Lipschitz in 
. To show the existence result, we construct Green’s function and iterative sequences for the corresponding linear problem. We use quasilinearization to construct these iterative schemes. We prove maximum principle and establish monotonicity of sequences of lower solution 
and upper solution 
such that 
Then under certain sufficient conditions we prove that these sequences converge uniformly to the solution ψ(x) in a specific region where 
Keyword : Green’s function, monotone iterative technique, maximum principle
How to Cite
Verma, A., & Urus, N. (2022). Well ordered monotone iterative technique for nonlinear second order four point Dirichlet BVPs. Mathematical Modelling and Analysis, 27(1), 59–77. https://doi.org/10.3846/mma.2022.14198
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Zh. Bai and Z. Du. Positive solutions for some second-order four-point boundary value problems. Journal of mathematical analysis and applications, 330(1):34– 50, 2007. https://doi.org/10.1016/j.jmaa.2006.07.044
Zh. Bai, W. Ge and Y. Wang. Multiplicity results for some second-order fourpoint boundary-value problems. Nonlinear Analysis: Theory, Methods & Applications, 60(3):491–500, 2005. https://doi.org/10.1016/j.na.2004.08.039
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A. Cabada. The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. Journal of Mathematical Analysis and Applications, 185(2):302–320, 1994. ISSN 0022-247X. https://doi.org/10.1006/jmaa.1994.1250
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M. Cherpion, C. De Coster and P. Habets. A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions. Applied Mathematics and Computation, 123(1):75–91, 2001. https://doi.org/10.1016/S0096-3003(00)00058-8
C. De Coster and P. Habets. Two-Point Boundary Value Problems: Lower and Upper Solutions, volume 205. Elsevier, 2006.
P. Drábek, G. Holubová, A. Matas and P. Nečesal. Nonlinear models of suspension bridges: discussion of the results. Applications of Mathematics, 48(6):497– 514, 2003. https://doi.org/10.1023/B:APOM.0000024489.96314.7f
C.P. Gupta. A Dirichlet type multi-point boundary value problem for second order ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications, 26(5):925–931, 1996. https://doi.org/10.1016/0362-546X(94)00338-X
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V.A. Ilyin and E.I. Moiseev. Nonlocal boundary value problem of the second kind for a Sturm Liouville operator on its differential and finite difference aspects. Differential Equations, 23(8):979–987, 1987.
G. Infante. Nonlocal boundary value problems with two nonlinear boundary conditions. Communications in Applied Analysis, 12(3):279, 2008.
G. Infante, F.M. Minhos and P. Pietramala. Non-negative solutions of systems of ODEs with coupled boundary conditions. Communications in Nonlinear Science and Numerical Simulation, 17(12):4952–4960, 2012. https://doi.org/10.1016/j.cnsns.2012.05.025
A.C. Lazer and P.J. McKenna. Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis. SIAM Review, 32(4):537–578, 1990. https://doi.org/10.1137/1032120
B. Liu. Positive solutions of a nonlinear four-point boundary value problems. Applied Mathematics and Computation, 155(1):179–203, 2004. https://doi.org/10.1016/S0096-3003(03)00770-7
B. Liu. Positive solutions of a nonlinear four-point boundary value problems in Banach spaces. Journal of Mathematical Analysis and Applications, 305(1):253– 276, 2005. https://doi.org/10.1016/j.jmaa.2004.11.037
R. Ma and L. Ren. Positive solutions for nonlinear m-point boundary value problems of Dirichlet type via fixed-point index theory. Applied mathematics letters, 16(6):863–869, 2003. https://doi.org/10.1016/S0893-9659(03)90009-7
E. Picard. Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. Journal de Mathématiques pures et appliquées, 6:145–210, 1890.
N. Urus, A.K. Verma and M. Singh. Some new existence results for a class of four point nonlinear boundary value problems. JNPG-The Journal of Revelations, 3:7–13, 2019.
A.K. Verma, B. Pandit, L. Verma and R.P. Agarwal. A review on a class of second order nonlinear singular BVPs. Mathematics, 8(7):1045, 2020. https://doi.org/10.3390/math8071045
A.K. Verma and M. Singh. A note on existence results for a class of three-point nonlinear BVPs. Mathematical Modelling and Analysis, 20(4):457–470, 2015. https://doi.org/10.3846/13926292.2015.1065293
A.K. Verma, N. Urus and M. Singh. Monotone iterative technique for a class of four point BVPs with reversed ordered upper and lower solutions. International Journal of Computational Methods, 17(9), 2020. https://doi.org/10.1142/S021987621950066X
L. Yang, C. Shen and Y. Liang. Existence, multiplicity of positive solutions for four-point boundary value problem with dependence on the first order derivative. Fixed Point Theory, 11(1):147–159, 2010.
Y. Zhang. Positive solutions of singular sublinear Dirichlet boundary value problems. SIAM journal on mathematical analysis, 26(2):329–339, 1995. https://doi.org/10.1137/S0036141093246087
Md. Asaduzzaman. Existence results for a nonlinear fourth order ordinary differential equation with four-point boundary value conditions. Advances in the Theory of Nonlinear Analysis and its Application, 4(4):233–241, 2020. https://doi.org/10.31197/atnaa.774794
Md. Asaduzzaman, Md. Ali et al. On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with fourpoint boundary value conditions: a fixed point theory approach. Journal of Nonlinear Sciences & Applications (JNSA), 13(6):364–377, 2020. https://doi.org/10.22436/jnsa.013.06.06
Zh. Bai and Z. Du. Positive solutions for some second-order four-point boundary value problems. Journal of mathematical analysis and applications, 330(1):34– 50, 2007. https://doi.org/10.1016/j.jmaa.2006.07.044
Zh. Bai, W. Ge and Y. Wang. Multiplicity results for some second-order fourpoint boundary-value problems. Nonlinear Analysis: Theory, Methods & Applications, 60(3):491–500, 2005. https://doi.org/10.1016/j.na.2004.08.039
Zh. Bai, W. Li and W. Ge. Existence and multiplicity of solutions for four-point boundary value problems at resonance. Nonlinear Analysis: Theory, Methods & Applications, 60(6):1151–1162, 2005. https://doi.org/10.1016/j.na.2004.10.013
P. Bailey, L.F. Shampine and P. Waltman. Nonlinear second order boundary value problems: Existence and regions of uniqueness. Journal of Mathematical Analysis and Applications, 14(3):433–444, 1966. https://doi.org/10.1016/0022-247X(66)90004-7
P. Bailey, L.F. Shampine, P. Waltman et al. Existence and uniqueness of solutions of the second order boundary value problem. Bulletin of the American Mathematical Society, 72(1, Part 1):96–98, 1966. https://doi.org/10.1090/S0002-9904-1966-11434-9
I.V. Barteneva, A. Cabada and A.O. Ignatyev. Maximum and antimaximum principles for the general operator of second order with variable coefficients. Applied Mathematics and Computation, 134(1):173–184, 2003. https://doi.org/10.1016/S0096-3003(01)00280-6
A. Cabada. The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. Journal of Mathematical Analysis and Applications, 185(2):302–320, 1994. ISSN 0022-247X. https://doi.org/10.1006/jmaa.1994.1250
A. Cabada. An overview of the lower and upper solutions method with nonlinear boundary value conditions. Boundary value problems, 2011(1):893753, 2011. https://doi.org/10.1155/2011/893753
A. Cabada, P. Habets and S. Lois. Monotone method for the Neumann problem with lower and upper solutions in the reverse order. Applied Mathematics and Computation, 117(1):1–14, 2001. https://doi.org/10.1016/S0096-3003(99)00149-6
M. Cherpion, C. De Coster and P. Habets. A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions. Applied Mathematics and Computation, 123(1):75–91, 2001. https://doi.org/10.1016/S0096-3003(00)00058-8
C. De Coster and P. Habets. Two-Point Boundary Value Problems: Lower and Upper Solutions, volume 205. Elsevier, 2006.
P. Drábek, G. Holubová, A. Matas and P. Nečesal. Nonlinear models of suspension bridges: discussion of the results. Applications of Mathematics, 48(6):497– 514, 2003. https://doi.org/10.1023/B:APOM.0000024489.96314.7f
C.P. Gupta. A Dirichlet type multi-point boundary value problem for second order ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications, 26(5):925–931, 1996. https://doi.org/10.1016/0362-546X(94)00338-X
J. Henderson, S.K. Ntouyas and I.K. Purnaras. Positive solutions for systems of nonlinear discrete boundary value problems. Journal of Difference Equations and Applications, 15(10):895–912, 2009. https://doi.org/10.1080/10236190802350649
V.A. Ilyin and E.I. Moiseev. Nonlocal boundary value problem of the second kind for a Sturm Liouville operator on its differential and finite difference aspects. Differential Equations, 23(8):979–987, 1987.
G. Infante. Nonlocal boundary value problems with two nonlinear boundary conditions. Communications in Applied Analysis, 12(3):279, 2008.
G. Infante, F.M. Minhos and P. Pietramala. Non-negative solutions of systems of ODEs with coupled boundary conditions. Communications in Nonlinear Science and Numerical Simulation, 17(12):4952–4960, 2012. https://doi.org/10.1016/j.cnsns.2012.05.025
A.C. Lazer and P.J. McKenna. Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis. SIAM Review, 32(4):537–578, 1990. https://doi.org/10.1137/1032120
B. Liu. Positive solutions of a nonlinear four-point boundary value problems. Applied Mathematics and Computation, 155(1):179–203, 2004. https://doi.org/10.1016/S0096-3003(03)00770-7
B. Liu. Positive solutions of a nonlinear four-point boundary value problems in Banach spaces. Journal of Mathematical Analysis and Applications, 305(1):253– 276, 2005. https://doi.org/10.1016/j.jmaa.2004.11.037
R. Ma and L. Ren. Positive solutions for nonlinear m-point boundary value problems of Dirichlet type via fixed-point index theory. Applied mathematics letters, 16(6):863–869, 2003. https://doi.org/10.1016/S0893-9659(03)90009-7
E. Picard. Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. Journal de Mathématiques pures et appliquées, 6:145–210, 1890.
N. Urus, A.K. Verma and M. Singh. Some new existence results for a class of four point nonlinear boundary value problems. JNPG-The Journal of Revelations, 3:7–13, 2019.
A.K. Verma, B. Pandit, L. Verma and R.P. Agarwal. A review on a class of second order nonlinear singular BVPs. Mathematics, 8(7):1045, 2020. https://doi.org/10.3390/math8071045
A.K. Verma and M. Singh. A note on existence results for a class of three-point nonlinear BVPs. Mathematical Modelling and Analysis, 20(4):457–470, 2015. https://doi.org/10.3846/13926292.2015.1065293
A.K. Verma, N. Urus and M. Singh. Monotone iterative technique for a class of four point BVPs with reversed ordered upper and lower solutions. International Journal of Computational Methods, 17(9), 2020. https://doi.org/10.1142/S021987621950066X
L. Yang, C. Shen and Y. Liang. Existence, multiplicity of positive solutions for four-point boundary value problem with dependence on the first order derivative. Fixed Point Theory, 11(1):147–159, 2010.
Y. Zhang. Positive solutions of singular sublinear Dirichlet boundary value problems. SIAM journal on mathematical analysis, 26(2):329–339, 1995. https://doi.org/10.1137/S0036141093246087