Tripled Periodic Boundary Value Problems of Nonlinear Second Order Differential Equations

Gupta, Animesh; Manro, Saurabh

Tripled Periodic Boundary Value Problems of Nonlinear Second Order Differential Equations

Authors

Animesh Gupta ¹
Saurabh Manro ²

¹ Department of Mathematics & Computer Science, R.D.V.V. Jabalpur (M.P.) India.

² School of Mathematics and Computer Applications, Thapar University, Patiala, Punjab, India.

Abstract

The present paper proposes a new monotone iteration principle for the existence as well as approximations
of the tripled solutions for a tripled periodic boundary value problem of second order ordinary nonlinear
differential equations. An algorithm for the tripled solutions is developed and it is shown that the sequences
of successive approximations defined in a certain way converge monotonically to the tripled solutions of the
related differential equations under some suitable hybrid conditions. A numerical example is also indicated
to illustrate the abstract theory developed in the paper.

Keywords

References

[1] H. Aydi, E. Karapinar, M. Postolache, Tripled coincidence point theorems for weak φ−contractions in partially
ordered metric spaces, Fixed Point Theory Appl., 2012 (2012), 12 pages.

[2] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fundamenta
Mathematicae, 3 (1922), 133–181.

[3] V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric
spaces, Nonlinear Anal., 74 (2011), 4889–4897.

[4] A. Gupta, An application of Meir-Keeler type tripled fixed point theorem, Int. J. Adv. Appl. Math. Mech., 2
(2014), 21–38.

[5] S. Heikkilä, V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations,
Marcel Dekker inc., New York (1994).

[6] A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28 (1969), 326–329.

[7] J. J. Nieto, R. R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to
ordinary differential equations, Acta Math. Sinica Engl. Ser., 23 (2007), 2205–2212.

[8] A. C. M. Ran, M. C. B Reurings, A fixed point theorem in partially ordered sets and some applications to matrix
equations, Proc. Am. Math. Soc., 132 (2004), 1435–1443.

Communications in Nonlinear Analysis