Title: Dhage Iteration Method for Approximating Positive Solutions of PBVPs of Nonlinear Quadratic Differential Equations with Maxima
Author(s): Shyam B. Dhage, Bapurao C. Dhage
Pages: 101-111
Cite as:
Shyam B. Dhage, Bapurao C. Dhage, Dhage Iteration Method for Approximating Positive Solutions of PBVPs of Nonlinear Quadratic Differential Equations with Maxima, Int. J. Anal. Appl., 10 (2) (2016), 101-111.

Abstract


In this paper authors prove the existence as well as approximation of the positive solutions for a periodic boundary value problem of first order ordinary nonlinear quadratic differential equations with maxima. An algorithm for the solutions is developed and it is shown that certain sequence of successive approximations converges monotonically to the positive solution of considered quadratic differential equations under some suitable mixed hybrid conditions. Our results rely on the Dhage iteration principle embodied in a recent hybrid fixed point theorem of Dhage (2014). A numerical example is also provided to illustrate the hypotheses and abstract theory developed in this paper.

Full Text: PDF

 

References


  1. D.D. Bainov, S. Hristova, Differential equations with maxima, Chapman & Hall/CRC Pure and Applied Mathematics, 2011.

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

  3. B.C. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Differ. Equ. Appl. 2 (2010), 465–486.

  4. 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.

  5. 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.

  6. 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.

  7. B.C. Dhage, Operator theoretic techniques in the theory of nonlinear hybrid differential equations, Nonlinear Anal. Forum 20 (2015), 15-31.

  8. B.C. Dhage, A new monotone iteration principle in the theory of nonlinear first order integro-differential equations, Nonlinear Studies 22 (3) (2015), 397-417.

  9. B.C. Dhage, S.B. Dhage, Approximating solutions of nonlinear first order ordinary differential equations, GJMS Special issue for Recent Advances in Mathematical Sciences and Applications-13, Global Journal of Mathematical Sciences, 2 (2014), 25-35.

  10. 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.

  11. 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.

  12. S.B. Dhage, B.C. Dhage, D. Octrocol, Dhage iteration method for approximating positive solutions of nonlinear first order ordinary quadratic differential equations with maxima, Fixed Point Theory, in press.

  13. B.C. Dhage, J. Henderson, S.K. Ntouyas, Periodic boundary value problems of first order differential equations in Banach algebras, J. Nonlinear Funct. Anal. & Diff. Equ. 1 (2007), 103-120.

  14. B.C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Analysis: Hybrid Systems 4 (2010), 414-424.

  15. D. Otrocol, I.A. Rus, Functional-differential equations with maxima of mixed type argument, Fixed Point Theory, 9(2008), no. 1, pp. 207–220.

  16. S. Heikkilä, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker inc., New York 1994.

  17. A.R. Magomedov, On some questions about differential equations with maxima, Izv. Akad. NaukAzerbaidzhan. SSRSer. Fiz.-Tekhn. Mat. Nauk, 1 (1977), 104–108 (in Russian).

  18. A.R. Magomedov, Theorem of existence and uniqueness of solutions of linear differential equations with maxima, Izv. Akad. NaukAzerbaidzhan. SSR Ser. Fiz. Tekhn. Mat. Nauk, 5 (1979), 116-118 (in Russian).

  19. A.D. Myshkis, on some problems of the theory of differential equations with deviating argument, Russian Math. Surveys 32 (1977), 181-210.

  20. J.J. Nieto, R. Rodriguez-Lopez, Existence and approximation of solution for nonlinear differential equations with periodic bounday conditions, Compt. Math. Appl. 40 (2000), 435-442.