# IJPAM: Volume 113, No. 2 (2017)

# Title

APPROXIMATING SOLUTION OF AN INITIAL ANDA PERIODIC BOUNDARY VALUE PROBLEM FOR

FIRST ORDER QUADRATIC FUNCTIONAL

DIFFERENTIAL EQUATIONS

# Authors

Dnyaneshwar V. Mule, Bhimrao R. AhirraoDepartment of Mathematics

North Maharashtra University

Jalgaon 425001, INDIA

Department of Mathematics

Mahatma Phule College

Kingaon, Ahmedpur, 413515, INDIA

Department of Mathematics

Z.B. Patil College

Dhule, 424002, INDIA

# Abstract

In this paper we prove the algorithms for the existence and approximation of the solutions for an initial and a periodic boundary value problem of nonlinear first order ordinary hybrid quadratic functional differential equations via iteration method embodied in a recent hybrid fixed point principle of Dhage (2014) in a partially ordered normed linear space. A numerical example is also provided to illustrate the abstract theory developed in the paper.# History

**Received: **October 30, 2016
**Revised: **December 22, 2016
**Published: **March 19, 2017

# AMS Classification, Key Words

**AMS Subject Classification: **34A12, 34H34, 47H07, 47H10
**Key Words and Phrases: **Hybrid differential equation, Hybrid fixed point theorem, Approximation theorem.

# Download Section

**Download paper from here.**

You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

## Bibliography

- 1
- B.C. Dhage, Periodic boundary value problemsof first order Carathéodory and discontinuous differential equations,
*Nonlinear Funct. Anal. & Appl.***13**(2) (2008), 323-352. - 2
- B.C. Dhage, Quadratic perturbationsof periodic boundary value problems of second order ordinarydifferential 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, Nonlinear -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, S. B. Dhage, D. V. Mule, Local attractivity and stability results for hybrid functional nonlinearfractional integral equations, Nonlinear Funct. Anal. Appl.
**19**(2014), 415-433. - 8
- Dnyaneshwar V. Mule, Bhimrao R. Ahirrao ,
*Approximating positive solutions of nonlinear first orderordinary quadratic differential equations with maxima*, Advances in Inequalities and Applications, Vol 2016 (2016), Article ID 11. - 9
- B.C. Dhage,
*Some generalizations of a hybrid fixed point theorem in a partially ordered metric space and nonlinear functional integral equations*, Differ. Equ Appl.**8**(2016), 77-97. - 10
- Dnyaneshwar V. Mule, Bhimrao R. Ahirrao ,
*Approximating positive solutions of quadratic functional integral equations*, Adv.Fixed Point Theory, 6 (2016), No. 3, 295-307. - 11
- B.C. Dhage, V. Lakshmikantham, Basic results onhybrid differential equations,
*Nonlinear Analysis: Hybrid Systems***4**(2010), 414-424. - 12
- A. Granas, J. Dugundji,
*Fixed Point Theory*, Springer Verlag, 2003. - 13
- J.K. Hale,Theory of Functional Differential Equations, Springer-Verlag, New York-Berlin, 1977.
- 14
- S. Heikkilä, V. Lakshmikantham,
*Monotone Iterative Techniques forDiscontinuous Nonlinear Differential Equations*,Marcel Dekker inc., New York 1994. - 15
- J. Hale and S.M. Verduyn Lunel,
*Introduction to Functional Differential Equa-tions*, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. - 16
- V. Kolmanovskii, and A. Myshkis, em Introduction to the Theory and Applications ofFunctional-Differential Equations , Mathematics and its Applications, 463, KluwerAcademic Publishers, Dordrecht, 1999.
- 17
- J.J. Nieto, Basic theory for nonresonanceimpulsive periodic problems of first order,
*J. Math. Anal. Appl.***205**(1997), 423-433. - 18
- S. B. Dhage, B. C. Dhage,
*Dhage iteration method for approximating positive solutions of nonlinear first order ordinary quadratic differential equations with maxima*, Nonlinear Anal. Forum**16**(1) (2016), 87-100. - 19
- E. Zeidler,
*Nonlinear Functional Analysis and Its Applications : Part.I*, Springer-Verlag, New York (1985). - 20
- J.J. Nieto, R. Rodriguez-Lopez, Existence and application of solution for nonlinear differential equation with peridic boundry conditions,Compt.Math.Appl.
*Order***40**(2000),435-442.223-239. - 21
- G. Gasper, M. Rahman,
*Basic Hypergeometric Series*, Cambridge University Press, Cambridge (1990). - 22
- M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, In:
*Operator Theory: Advances and Applications*, Birkhäuser, Basel (1994), 369-396. - 23
- D.S. Moak, The -analogue of the Laguerre polynomials,
*J. Math. Anal. Appl.*,**81**(1981), 20-47.

# How to Cite?

**DOI: 10.12732/ijpam.v113i2.6**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2017

**Volume:**113

**Issue:**2

**Pages:**251 -

Google Scholar; DOI (International DOI Foundation); WorldCAT.

**This work is licensed under the Creative Commons Attribution International License (CC BY).**