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. Google Scholar

  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. Google Scholar

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

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

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

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

  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. Google Scholar

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

  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. Google Scholar

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

  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). Google Scholar

  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). Google Scholar

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

  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. Google Scholar


COPYRIGHT INFORMATION

Copyright © 2021 IJAA, unless otherwise stated.