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Fractional Calculus and Applied Analysis

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Volume 18, Issue 1

Issues

Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence

Alberto Cabada
  • Departamento de Análise Matemática Facultade de Matemáticas Universidade de Santiago de Compostela 15782, Santiago de Compostela, SPAIN
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/ Zakaria Hamdi
Published Online: 2015-02-10 | DOI: https://doi.org/10.1515/fca-2015-0015

Abstract

This paper is devoted to the study of nonlinear fractional differential equation with parameter dependence and integral boundary value conditions. In the paper various existence and multiplicity results for positive solutions are derived depending of different values of the parameter. Some illustrative examples are also discussed.

MSC 2010: Primary 34A08; Secondary 26A33; 34K37

Key Words and Phrases: fractional differential equations; integral boundary conditions; parameter dependence; positive solution; Green's function; Krasnoselskii's fixed point theorem

References

  • [1] B. Ahmad, A. Alsaedi, B. Alghamdi, Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal. RWA. 9 (2008), 1727-1740.Google Scholar

  • [2] B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal.: Hybrid Syst. 4 (2010), 134-141.Web of ScienceGoogle Scholar

  • [3] G. A. Anastassiou, Fractional Differentiation Inequalities. Springer (2009).Google Scholar

  • [4] R. I. Avery, A. C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, No 3-5 (2001), 313-322.Web of ScienceGoogle Scholar

  • [5] M. Benchohra, J. J. Nieto, A. Ouahab, Second-order boundary value problem with integral boundary conditions. Bound. Value Probl. 2011 (2011), Art. No 260309.Google Scholar

  • [6] M. Benchohra, F. Ouaar, Existence results for nonlinear fractional differential equations with integral boundary conditions. Bull. Math. Anal. Appl. 2, No 4 (2010), 7-15.Google Scholar

  • [7] A. Cabada, J. A. Cid, Existence and multiplicity of solutions for periodic Hill's with parametric dependence and singularities. Abstr. Appl. Anal. 2011 (2011), Art. No 545264.Web of ScienceGoogle Scholar

  • [8] A. Cabada, D. Dimitrov, Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities. J. Math. Anal. Appl. 371, No 2 (2010), 518-533.Web of ScienceGoogle Scholar

  • [9] A. Cabada, Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions. Applied Math. Comput. 228 (2014), 251-257.Google Scholar

  • [10] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, No 1 (2012), 403-411.Google Scholar

  • [11] M. Feng, X. Zhang, W. Ge, New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2011 (2011), Art. No 720702.Google Scholar

  • [12] J. R. Graef, L. Kong. Q. Kong, M. Wang, Uniqueness of positive solutions of fractional boundary value problems with nonhomogeneous integral boundary conditions. Fract. Calc. Appl. Anal. 15, No 3 (2012), 509-528; DOI: 10.2478/s13540-012-0036-x; http://link.springer.com/article/10.2478/s13540-012-0036-x.CrossrefGoogle Scholar

  • [13] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones. Academic Press, New York (1988).Google Scholar

  • [14] T. Jankowski, Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions. Nonlinear Anal. TMA. 73 (2010), 1289-1299.Google Scholar

  • [15] J. Q. Jiang, L. S. Liu Y. H. Wu, Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. Appl. Math. Comput. 215 (2009), 1573-1582.Google Scholar

  • [16] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204, Elsevier Science BV, Amsterdam (2006).Google Scholar

  • [17] V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009).Google Scholar

  • [18] P. D. Phung and L. X. Truong, On a fractional differential inclusion with integral boundary conditions in banach space. Fract. Calc. Appl. Anal. 16, No 3 (2013), 538-558; DOI: 10.2478/s13540-013-0035-6; http://link.springer.com/article/10.2478/s13540-013-0035-6.CrossrefGoogle Scholar

  • [19] I. Podlubny, Fractional Differential Equations. Ser. Mathematics in Science and Engineering, Academic Press, New York (1999).Google Scholar

  • [20] H. A.H. Salem, Fractional order boundary value problem with integral boundary conditions involving Pettis integral. Acta Math. Sci. Ser. B Engl. Ed. 31, No 2 (2011), 661-672.Web of ScienceCrossrefGoogle Scholar

  • [21] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals 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; DOI: 10.2478/s13540-014- 0188-y; http://link.springer.com/article/10.2478/s13540-014-0188-y.CrossrefGoogle Scholar

  • [23] X. M. Zhang, M. Q. Feng, W. G. Ge, Existence result of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl. 353 (2009), 311-319.Google Scholar

About the article

Received: 2014-07-30

Published Online: 2015-02-10


Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 1, Pages 223–237, ISSN (Online) 1314-2224, DOI: https://doi.org/10.1515/fca-2015-0015.

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