# IJPAM: Volume 117, No. 3 (2017)

# Title

MULTIPLE POSITIVE SOLUTIONS TO NONLINEARBOUNDARY VALUE PROBLEMS OF A SYSTEM FOR

GENERALIZED -LAPLACIAN FRACTIONAL

DIFFERENTIAL EQUATIONS

# Authors

A. Kameswara RaoDepartment of Mathematics

Gayatri Vidya Parishad College of Engineering for Women

Madhurawada, Visakhapatnam, 530 048, INDIA

# Abstract

In this article we study the existence of positive solutions for a coupled system of generalized -Laplacian fractional order boundary value problems. We prove that the boundary value problem has at least three positive solutions by apply the five functionals fixed-point theorem. An example demonstrates the main results.# History

**Received: **2017-09-23
**Revised: **2017-10-27
**Published: **January 16, 2018

# AMS Classification, Key Words

**AMS Subject Classification: **34B15, 34B18
**Key Words and Phrases: **fractional boundary value problems, positive solutions, generalized -laplacian operator, fixed point, cone

# 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
- K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
- 2
- I. Podlubny, Fractional Differential Equations, Academic Press, New York/London/Toronto, 1999.
- 3
- R. P. Agarwal, Formulation of Euler-Larange equations for fractional variational problems, J. Math. Anal. Appl. 272 (2002) 368-379.
- 4
- H. Weitzner, G.M. Zaslavsky, Some applications of fractional equations, Commun. Nonlinear Sci. Numer. Simul. 8 (2003) 273-281.
- 5
- F. C. Meral, T.J. Royston, R. Magin, Fractional calculus in viscoelasticity, an experimental study, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 939-945.
- 6
- J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1140-1153.
- 7
- A. A. Kilbas, H. H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006.
- 8
- S. G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives (Theorey and Applications), Gordon and Breach, Switzerland, 1993.
- 9
- S. Sun, Q. Li, Y. Li, Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations, Comput. Math. Appl. 64 (2012) 3310-3320.
- 10
- Z. Bai, H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005) 495-505.
- 11
- G. Wang, L. Zhang, Sotiris K. Ntouyas, Existence of multiple positive solutions of a nonlinear arbitrary order boundary value problem with advanced arguments, Electron. J. Qual. Theory Differ. Equ. 15 (2012) 1-13.
- 12
- Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear Anal. 71 (2009) 2724-2733.
- 13
- S. Sun, Y. Zhao, Z. Han, Y. Li, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 4961-4967.
- 14
- S. Sun, Y. Zhao, Z. Han, M. Xu, Uniqueness of positive solutions for boundary value problems of singular fractional differential equations, Inverse Probl. Sci. Eng. 20 (2012) 299-309.
- 15
- Y. Zhao, S. Sun, Z. Han, M. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput. 217 (2011) 6950-6958.
- 16
- W. Feng, S. Sun, Z. Han, Y. Zhao, Existence of solutions for a singular system of nonlinear fractional differential equations, Comput. Math. Appl. 62 (2011) 1370-1378.
- 17
- Y. Zhao, S. Sun, Z. Han, Q. Li, The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2086-2097.
- 18
- X. Yang, Z. Wei, W. Dong, Existence of positive solutions for the boundary value problem of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 85-92.
- 19
- X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal. 71 (2009) 4676-4688.
- 20
- T. Chen, W. Liu, An anti-periodic boundary value problem for fractional differential equation with -Laplacian operator, Appl. Math. Lett. 25 (2012) 1671-1675.
- 21
- G. Chai, Positive solutions for boundary value problem of fractional differential equation with -Laplacian operator, Bound. Value Probl. 2012 (2012) 1-18.
- 22
- H. Lu, Z. Han, S. Sun, J. Liu, Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with p-Laplacian, Adv. Differ. Equ. 30 (2013) 1-16.
- 23
- M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional differential equation, Abs. Appl. Anal. 2007 (2007) 1-8. Article ID 10368.
- 24
- H. Lu, Z. Han, S. Sun, Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p- Laplacian, Bound. Value Probl. 26 (2014) 1-17.
- 25
- J. Wang, H. Xiang, Upper and lower solutions method for a class of singular fractional boundary value problems with -Laplacian operator, Abs. Appl. Anal. 2010 (2010) 1-12. Article ID 971824.
- 26
- S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Anal. 71 (2009) 5545-5550.
- 27
- J. G. Dix, G.L. Karakostas, A fixed-point theorem for S-type operators on Banach spaces and its applications to boundary-value problems, Nonlinear Anal. 71 (2009) 3872-3880.
- 28
- J. Xu, Z. Wei, W. Dong, Uniqueness of positive solutions for a class of fractional boundary value problems, Appl. Math. Lett. 25 (2012) 590-593.
- 29
- T. Chen, W. Liu, Z. Hu, A boundary value problem for fractional differential equation with -Laplacian operator at resonance, Nonlinear Anal. 75 (2012) 3210-3217.
- 30
- X. Han, H. Gao, Existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations, Adv. Differ. Equ. 2012 (66) (2012) 1-8.
- 31
- X. Zhang, L. Liu, B. Wiwatanapataphee, Y. Wu, Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives, Abs. Appl. Anal. 2012 (2012) 1-16. Article ID 512127.
- 32
- D. X. Zhao, H. Z. Wang, W. G. Ge; Existence of triple positive solutions to a class of p-Laplacian boundary value problems, J. Math. Anal. Appl. 328 (2007), 972-983.
- 33
- R. W. Leggett, L.R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979) 673-688.
- 34
- H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal. Appl. 281 (2003) 287-306.
- 35
- G. Isac, Leray-Schauder Type Alternatives, Complementarity Problems and Variational Inequalities, US, Springer, 2006.
- 36
- R.I. Avery, A generalization of the Leggett-Williams fixed point theorem, Math. Sci. Res. Hot-Line 3 (1999) 9-14.

# How to Cite?

**DOI: 10.12732/ijpam.v117i3.16**

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:**117

**Issue:**3

**Pages:**547 - 562

-LAPLACIAN FRACTIONAL DIFFERENTIAL EQUATIONS%22&as_occt=any&as_epq=&as_oq=&as_eq=&as_publication=&as_ylo=&as_yhi=&as_sdtAAP=1&as_sdtp=1" title="Click to search Google Scholar for this entry" rel="nofollow">Google Scholar; DOI (International DOI Foundation); WorldCAT.

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