Solvability of fractional p-Laplacian boundary value problems with controlled parameters

Volume 10, Issue 5, pp 2366--2383 http://dx.doi.org/10.22436/jnsa.010.05.09

Publication Date: May 24, 2017 Submission Date: December 19, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2199 Downloads
• 3871 Views

Authors

Tengfei Shen - School of Mathematics, China University of Mining and Technology, Xuzhou 221116, P. R. China. Wenbin Liu - School of Mathematics, China University of Mining and Technology, Xuzhou 221116, P. R. China.

Abstract

This paper aims to investigate existence of solutions of several boundary value problems for fractional one-dimensional p-Laplacian equation under controlled parameters. By employing fixed point theory and critical point theory, some new results are obtained, which enrich and generalize the previous results.

Share and Cite

Keywords

• Fractional ordinary differential equation
• boundary value problem
• p-Laplacian operator
• existence.

MSC

•  26A33
•  34B15
•  34G20

References

• [1] R. P. Agarwal, D. O’Regan, S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371 (2010), 57–68.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] R. P. Agarwal, Y. Zhou, Y.-Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095–1100.
  • View Article
  • Google Scholar


• [3] C.-Z. Bai, Existence of positive solutions for boundary value problems of fractional functional differential equations , Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 14 pages.
  • MathSciNet
  • Google Scholar


• [4] M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340–1350.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] M. Bergounioux, A. Leaci, G. Nardi, F. Tomarelli, Fractional Sobolev spaces and functions of bounded variation, ArXiv, 2016 (2016), 19 pages.
  • Google Scholar


• [6] G. Bonanno, G. Riccobono, Multiplicity results for Sturm-Liouville boundary value problems, Appl. Math. Comput., 210 (2009), 294–297.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] G. Bonanno, R. Rodríguez-López, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 717–744.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] D. Bonheure, P. Habets, F. Obersnel, P. Omari, Classical and non-classical solutions of a prescribed curvature equation, J. Differential Equations, 243 (2007), 208–237.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] G. Cerami, An existence criterion for the critical points on unbounded manifolds, (Italian) Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332–336.
  • MathSciNet
  • Google Scholar


• [10] T.-Y. Chen, W.-B. Liu, Solvability of fractional boundary value problem with p-Laplacian via critical point theory, Bound. Value Probl., 2016 (2016), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] B. Du, X.-P. Hu, W.-G. Ge, Positive solutions to a type of multi-point boundary value problem with delay and onedimensional p-Laplacian, Appl. Math. Comput., 208 (2009), 501–510.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] I. Ekeland, Convexity methods in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Springer-Verlag, Berlin (1990)
  • View Article
  • MathSciNet
  • Google Scholar


• [13] M. Fečkan, Y. Zhou, J.-R. Wang, On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3050–3060.
  • View Article
  • MathSciNet
  • Google Scholar


• [14] D. J. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA (1988)
  • MathSciNet


• [15] D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. Funct. Spaces Appl., 2013 (2013), 15 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] W.-H. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74 (2011), 1987–1994.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181–1199.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 17 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] H. Jin, W.-B. Liu, Eigenvalue problem for fractional differential operator containing left and right fractional derivatives, Adv. Difference Equ., 2016 (2016), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [20] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
  • MathSciNet


• [21] N. Kosmatov, A boundary value problem of fractional order at resonance, Electron. J. Differential Equations, 2010 (2010), 10 pages.
  • MathSciNet
  • Google Scholar


• [22] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69 (2008), 3337–3343.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] K. Q. Lan, W. Lin, Positive solutions of systems of Caputo fractional differential equations, Commun. Appl. Anal., 17 (2013), 61–85.
  • MathSciNet
  • Google Scholar


• [24] J. Leszczynski, T. Blaszczyk, Modeling the transition between stable and unstable operation while emptying a silo, Granul. Matter, 13 (2011), 429–438.
  • View Article
  • Google Scholar


• [25] Y.-J. Liu, W.-G. Ge, Multiple positive solutions to a three-point boundary value problem with p-Laplacian, J. Math. Anal. Appl., 277 (2003), 293–302.
  • View Article
  • MathSciNet
  • Google Scholar


• [26] F. Mainardi, Fractional diffusive waves in viscoelastic solids, J. L. Wegner, F. R. Norwood (Eds.), IUTAM Symposium– Nonlinear Waves in Solids, ASME/AMR, Fairfield, NJ, (1995), 93–97.
  • Google Scholar


• [27] J. Mawhin, Some boundary value problems for Hartman-type perturbations of the ordinary vector p-Laplacian, Lakshmikantham’s legacy: a tribute on his 75th birthday, Nonlinear Anal., 40 (2000), 497–503.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer- Verlag, New York (1989)
  • Google Scholar


• [29] M. L. Morgado, N. J. Ford, P. M. Lima, Analysis and numerical methods for fractional differential equations with delay, J. Comput. Appl. Math., 252 (2013), 159–168.
  • View Article
  • MathSciNet
  • Google Scholar


• [30] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA (1999)
  • Google Scholar


• [31] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1986)
  • View Article
  • MathSciNet
  • Google Scholar


• [32] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolski˘ı, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
  • Google Scholar


• [33] T.-F. Shen,W.-B. Liu, X.-H. Shen, Existence and uniqueness of solutions for several BVPs of fractional differential equations with p-Laplacian operator, Mediterr. J. Math., 13 (2016), 4623–4637.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] J. Simon, Régularité de la solution d’un probléme aux limites non linéaires, (French) [[Regularity of the solution of a nonlinear boundary problem]] Ann. Fac. Sci. Toulouse Math., 3 (1981), 247–274.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] E. Szymanek, The application of fractional order differential calculus for the description of temperature profiles in a granular layer, Adv. Theory Appl. Non-integer Order Syst., Springer Inter. Publ., Switzerland (2013)
  • View Article
  • Google Scholar


• [36] X. H. Tang, L. Xiao, Homoclinic solutions for ordinary p-Laplacian systems with a coercive potential, Nonlinear Anal., 71 (2009), 1124–1132.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] Y. Tian, W.-G. Ge, Second-order Sturm-Liouville boundary value problem involving the one-dimensional p-Laplacian, Rocky Mountain J. Math., 38 (2008), 309–327.
  • View Article
  • MathSciNet
  • MATH


• [38] C. Torres, Mountain pass solution for a fractional boundary value problem, J. Fract. Calc. Appl., 5 (2014), 1–10.
  • MathSciNet
  • Google Scholar


• [39] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal., 71 (2009), 3249–3256.
  • View Article
  • MathSciNet
  • Google Scholar