Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

6 Issues per year


Be aware of AN ONGOING FRAUD ATTEMPT to this journal, see details at  http://www.math.bas.bg/complan/fcaa/

IMPACT FACTOR 2016: 2.034
5-year IMPACT FACTOR: 2.359

CiteScore 2016: 2.18

SCImago Journal Rank (SJR) 2016: 1.372
Source Normalized Impact per Paper (SNIP) 2016: 1.492

Mathematical Citation Quotient (MCQ) 2016: 0.61

Online
ISSN
1314-2224
See all formats and pricing
More options …

Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions

Mustafa Günendi / İsmail Yaslan
Published Online: 2016-08-29 | DOI: https://doi.org/10.1515/fca-2016-0054

Abstract

In this article, we consider a nonlinear higher-order m-point fractional boundary-value problem with integral boundary conditions. We establish criteria for the existence of at least one, two and three positive solutions for higher order nonlinear m-point fractional boundary-value problems with integral boundary conditions by using the four functionals fixed point theorem, the Avery-Henderson fixed point theorem and the Legget-Williams fixed point theorem, respectively.

MSC 2010: Primary 26A33; Secondary 34B18, 34B27

Key Words and Phrases: boundary value problems; cone; fixed point theorems; positive solutions; Riemann-Liouville fractional derivative; integral boundary conditions

References

  • [1]

    D.R. Anderson, I.Y. Karaca, Higher-order three-point boundary value problem on time scales. Comput. Math. Appl. 56 (2008), 2429–2443.Google Scholar

  • [2]

    R.I. Avery, J. Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces. Comm. Appl. Nonlinear Anal. 8 (2001), 27–36.Google Scholar

  • [3]

    R.I. Avery, J. Henderson, D. O’Regan, Four functionals fixed point theorem. Math. Comput. Modelling 48 (2008), 1081–1089.Google Scholar

  • [4]

    M. Benchohra, J.R. Graef, S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations. Applicable Anal. 87, No 7 (2008), 851–863.Google Scholar

  • [5]

    A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389 (2012), 403–411.Google Scholar

  • [6]

    A. Dogan, J.R. Graef, L. Kong, Higher order semipositone multi-point boundary value problems on time scales. Comput. Math. Appl. 60 (2010), 23–35.Google Scholar

  • [7]

    J.R. Graef, L. Kong, B. Yang, Positive solutions for a semipositone fractional boundary value problem with a forcing term. Fract. Calc. Appl. Anal. 15, No 1 (2012), 8–24; ; http://www.degruyter.com/view/j/fca.2012.15.issue-1/issue-files/fca.2012.15.issue-1.xml.Crossref

  • [8]

    J.R. Graef, L. Kong, Q. Kong, M. Wang, Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions. Fract. Calc. Appl. Anal. 15, No 3 (2012), 509–528; ; http://www.degruyter.com/view/j/fca.2012.15.issue-3/issue-files/fca.2012.15.issue-3.xml.Crossref

  • [9]

    J.R. Graef, L. Kong, Existence of positive solutions to a higher order singular boundary value problem with fractional Q-derivatives. Fract. Calc. Appl. Anal. 16, No 3 (2013), 695–708; ; http://www.degruyter.com/view/j/fca.2013.16.issue-3/issue-files/fca.2013.16.issue-3.xml.Crossref

  • [10]

    W. Han, Y. Kao, Existence and uniqueness of nontrivial solutions for nonlinear higher-order three-point eigenvalue problems on time scales. Electron. J. Differential Equations 2008 (2008), Article ID 58, 1–15.Google Scholar

  • [11]

    V.A. Il’in, E.I. Moiseev, Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential Equations 23 (1987), 803–810.Google Scholar

  • [12]

    V.A. Il’in, E.I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equations 23 (1987), 979–987.Google Scholar

  • [13]

    M. Jia, X. Liu, X. Gu, Uniqueness and asymptotic behavior of positive solutions for a fractional-order integral boundary value problem. Abstr. Appl. Anal. 2012 (2012), Article ID 294694, 1–21.Google Scholar

  • [14]

    J. Jiang, L. Liu, Positive solutions for nonlinear second-order m-point boundary-value problems. Electron. J. Differential Equations 110 (2009), 1–12.Google Scholar

  • [15]

    J. Jin, X. Liu, M. Jia, Existence of positive solutions for singular fractional differential equations with integral boundary conditions. Electron. J. Differential Equations 2012 (2012), Article ID 63, 1–14.Google Scholar

  • [16]

    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Google Scholar

  • [17]

    R.W. Legget, L.R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach space. Indiana Univ. Math. J. 28 (1979), 673–688.Google Scholar

  • [18]

    S. Liang, Y. Song, Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions. Lithuanian Math. J. 52 (2012), 62–76.Google Scholar

  • [19]

    K.B. Oldham, J. Spanier, Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974).Google Scholar

  • [20]

    J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado (Eds.), Advances in Fractional Calculus. Springer (2007).Google Scholar

  • [21]

    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993).Google Scholar

  • [22]

    W. Sun, Y. Wang, Multiple positive solutions of nonlinear fractional differential equations with integral boundary value conditions. Fract. Calc. Appl. Anal. 17, No 3 (2014), 605–616; ; http://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml.Crossref

  • [23]

    Y. Wang, W. Ge, Existence of multiple positive solutions for even order multi-point boundary value problems. Georgian Mathematical J. 14 (2007), 775–792.Google Scholar

  • [24]

    Y. Wang, Y. Tang, M. Zhao, Multiple positive solutions for a nonlinear 2n-th order m-point boundary value problems. Electron. J. Qualitative Theo. Differential Equations 39 (2009), 1–13.Google Scholar

  • [25]

    L. Wang, X. Zhang, Existence of positive solutions for a class of higher-order nonlinear fractional differential equations with integral boundary conditions and a parameter. J. Appl. Math. Comput. 44 (2014), 293–316.Google Scholar

  • [26]

    S. Wong, Positive solutions of singular fractional differential equations with integral boundary conditions. Math. Comput. Modelling 57 (2013), 1053–1059.Google Scholar

  • [27]

    W. Yang, Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions. J. Appl. Math. Comput. 44 (2014), 39–59.Google Scholar

  • [28]

    İ. Yaslan, Higher order m-point boundary value problem on time scales. Comput. Math. Appl. 63 (2012), 739–750.Google Scholar

  • [29]

    K. Zhang, J. Xu, Unique positive solution for a fractional boundary value problem. Fract. Calc. Appl. Anal. 16, No 4 (2013), 937–948; ; http://www.degruyter.com/view/j/fca.2013.16.issue-4/issue-files/fca.2013.16.issue-4.xml.Crossref

  • [30]

    C. Zhao, Existence and uniqueness of positive solutions to higher-order nonlinear fractional differential equation with integral boundary conditions. Electron. J. Differential Equations 2012 (2012), Article ID 234, 1–11.Google Scholar

About the article

Received: 2015-03-06

Revised: 2016-05-01

Published Online: 2016-08-29

Published in Print: 2016-08-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 4, Pages 989–1009, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0054.

Export Citation

© 2016 Diogenes Co., Sofia. Copyright Clearance Center

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Ping Li and Meiqiang Feng
Advances in Difference Equations, 2018, Volume 2018, Number 1

Comments (0)

Please log in or register to comment.
Log in