Dhage Iteration Method for Nonlinear First Order Hybrid Differential Equations with a Linear Perturbation of Second Type

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B.C. Dhage

Abstract

In this paper the authors prove algorithms for the existence and approximation of the solutions for an initial and a periodic boundary value problem of nonlinear first order ordinary hybrid differential equations with a linear perturbation of second type via Dhage iteration method. Examples are furnished to illustrate the hypotheses and main abstract results of this paper.

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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, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Differ. Equ. Appl. 2 (2010), 465-486.
  3. 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.
  4. B.C. Dhage, Partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, Tamkang J. Math. 45 (2014), 397-426.
  5. B.C. Dhage, Approximation methods in the theory of hybrid differential equations with linear perturbations of second type, Tamkang J. Math. 45 (2014), 39-61.
  6. B.C. Dhage, Global existence and convergence analysis for hybrid differential equations, Comm. Appl. Anal. 19 (2015), 439-452.
  7. 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.
  8. B.C. Dhage, Operator theoretic techniques in the theory of nonlinear hybrid differential equations, Nonlinear Anal. Forum 20 (2015), 15-31.
  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, GJMS Vol. 2, No. 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 nonlinear first order ordinary quadratic differential equations, Cogent Mathematics (2015), 2: 1023671.
  12. 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.
  13. B.C. Dhage and N. S. Jadhav, Basic results on hybrid differential equations with linear perturbation of second type, Tamkang J. Math. 44 (2) (2013), 171-186.
  14. B.C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Analysis: Hybrid Systems 4 (2010), 414-424.
  15. A. Granas, J. Dugundji, Fixed Point Theory, Springer Verlag, 2003.
  16. S. Heikkilä, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equa- tions, Marcel Dekker inc., New York 1994.
  17. J.J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. 205 (1997), 423-433.
  18. E. Zeidler, Nonlinear Functional Analysis and Its Applications : Part.I, Springer-Verlag, New York (1985).