Volume 10, Issue 5, pp 2366--2383
Publication Date: 2017-05-24
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.
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.
Fractional ordinary differential equation, boundary value problem, p-Laplacian operator, existence.
 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.
 R. P. Agarwal, Y. Zhou, Y.-Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095–1100.
 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.
 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.
 M. Bergounioux, A. Leaci, G. Nardi, F. Tomarelli, Fractional Sobolev spaces and functions of bounded variation, ArXiv, 2016 (2016), 19 pages.
 G. Bonanno, G. Riccobono, Multiplicity results for Sturm-Liouville boundary value problems, Appl. Math. Comput., 210 (2009), 294–297.
 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.
 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.
 G. Cerami, An existence criterion for the critical points on unbounded manifolds, (Italian) Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332–336.
 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.
 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.
 I. Ekeland, Convexity methods in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Springer-Verlag, Berlin, (1990).
 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.
 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).
 D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. Funct. Spaces Appl., 2013 (2013), 15 pages.
 W.-H. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74 (2011), 1987–1994.
 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.
 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.
 H. Jin, W.-B. Liu, Eigenvalue problem for fractional differential operator containing left and right fractional derivatives, Adv. Difference Equ., 2016 (2016), 12 pages.
 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).
 N. Kosmatov, A boundary value problem of fractional order at resonance, Electron. J. Differential Equations, 2010 (2010), 10 pages.
 V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69 (2008), 3337–3343.
 K. Q. Lan, W. Lin, Positive solutions of systems of Caputo fractional differential equations, Commun. Appl. Anal., 17 (2013), 61–85.
 J. Leszczynski, T. Blaszczyk, Modeling the transition between stable and unstable operation while emptying a silo, Granul. Matter, 13 (2011), 429–438.
 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.
 F. Mainardi, Fractional diffusive waves in viscoelastic solids, J. L. Wegner, F. R. Norwood (Eds.), IUTAM Symposium– Nonlinear Waves in Solids, ASMEAMR, Fairfield, NJ, (1995), 93–97.
 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.
 J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, Springer- Verlag, New York, (1989).
 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.
 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).
 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).
 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).
 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.
 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.
 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).
 X. H. Tang, L. Xiao, Homoclinic solutions for ordinary p-Laplacian systems with a coercive potential, Nonlinear Anal., 71 (2009), 1124–1132.
 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.
 C. Torres, Mountain pass solution for a fractional boundary value problem, J. Fract. Calc. Appl., 5 (2014), 1–10.
 Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal., 71 (2009), 3249–3256.