Title: The Dhage Iteration Principle for Coupled PBVPs of Nonlinear Second Order Differential Equations
Author(s): Bapurao C. Dhage
Pages: 53-62
Cite as:
Bapurao C. Dhage, The Dhage Iteration Principle for Coupled PBVPs of Nonlinear Second Order Differential Equations, Int. J. Anal. Appl., 8 (1) (2015), 53-62.

Abstract


The present paper proposes a new monotone iteration principle for the existence as well as approximations of the coupled solutions for a coupled periodic boundary value problem of second order ordinary nonlinear differential equations. An algorithm for the coupled solutions is developed and it is shown that the sequences of successive approximations defined in a certain way converge monotonically to the coupled 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. We claim that the method as well as the results of this paper are new to literature on nonlinear analysis and applications.

Full Text: PDF

 

References


  1. B.C. Dhage, Periodic boundary value problems of first order Carath´eodory and discontinuous differential equations, Nonlinear Funct. Anal. & Appl. 13(2) (2008), 323-352.

  2. B.C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Differ. Equ. Appl. 5 (2013), 155-184.

  3. B.C. Dhage, Partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math. 45 (4) (2014), 397-426.

  4. B.C. Dhage, Nonlinear D-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, Malaya J. Mat. 3(1)(2015), 62-85.

  5. B.C. Dhage, Approximating coupled solutions of coupled PBVPs of nonlinear first order ordinary differential equations, Taiwnese J. Math. (Submitted)

  6. B.C. Dhage, S.B. Dhage, Approximating solutions of nonlinear first order ordinary differential equations, Global Jour. Math. Sci. 3 (2014), (In press).

  7. B.C. Dhage, S.B. Dhage, Coupled hybrid fixed point theorems in partially ordered metric spaces with application, Nonlinear Studies 21(4)(2014), 675-686.

  8. B.C. Dhage, S.B. Dhage, Approximating solutions of nonlinear pbvps of hybrid differential equations via hybrid fixed point theory, Indian J. Math. 57(1) (2015), 103-119.

  9. B.C. Dhage, S.B. Dhage, Approximating positive solutions of pbvps of nonlinear first order ordinary quadratic differential equations, Appl. Math. Lett. 46 (2015), 133-142.

  10. B.C. Dhage, S.B. Dhage, S.K. Ntouyas, Approximating solutions of nonlinear second order ordinary differential equations, Malaya J. Mat. 3(3) (2015), 00-00.

  11. T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis: TMA 65 (2006), 1379-1393.

  12. D. Guo, V, Lakshmikantham, Coupled fixed point of nonlinear operators with applicatons, Nonlinear Anal. 11 (1987), 623-632.

  13. S. Heikkil¨a, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker inc., New York 1994.