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Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator

  • Wengui Yang
Published/Copyright: January 13, 2020
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Abstract

This paper is concerned with the existence and nonexistence of positive solutions for the eigenvalue problems of nonlinear Hadamard fractional differential equations with p-Laplacian operator. By applying the properties of the Green function and Guo-Krasnosel’skii fixed point theorem on cones, some existence and nonexistence results of positive solutions are obtained based on different eigenvalue intervals. Finally, some examples are presented to demonstrate the feasibility of our main results.


This work was supported by the Scientific Research Foundation of Graduate School of Southeast University under Grant No. YBJJ1824, and the Key Scientific Research Programme of Higher Education of Henan Province-Guidance Plan under Grant No. 16B110011.


  1. Communicated by Michal Fečkan

References

[1] Ahmad, B.—Luca, R.: Existence of solutions for a sequential fractional integro-differential system with coupled integral boundary conditions, Chaos Solitons Fractals 104 (2017), 378–388.10.1016/j.chaos.2017.08.035Search in Google Scholar

[2] Ahmad, B.—Alsaedi, A.—Aljoudi, S.—Ntouyas, S. K.: A six-point nonlocal boundary value problem of nonlinear coupled sequential fractional integro-differential equations and coupled integral boundary conditions, J. Appl. Math. Comput. 56 (2018), 367–389.10.1007/s12190-016-1078-8Search in Google Scholar

[3] Ahmad, B.—Ntouyas, S. K.—Alsaedi, A.: New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions, Bound. Value Probl. 2013 (2013), #275.10.1186/1687-2770-2013-275Search in Google Scholar

[4] Ahmad, B.—Ntouyas, S. K.: Boundary value problems of Hadamard-type fractional differential equations and inclusions with nonlocal conditions, Vietnam J. Math. 45 (2017), 409–423.10.1007/s10013-016-0213-zSearch in Google Scholar

[5] Ahmad, B.—Ntouyas, S. K.: On Hadamard fractional integro-differential boundary value problems, J. Appl. Math. Comput. 47 (2015), 119–131.10.1007/s12190-014-0765-6Search in Google Scholar

[6] Ahmad, B.—Ntouyas, S. K.: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal. 17 (2014), 348–360.10.2478/s13540-014-0173-5Search in Google Scholar

[7] Alesemi, M.: Solvability for a class of nonlinear Hadamard fractional differential equations with parameters, Bound. Value Probl. 2019 (2019), #101.10.1186/s13661-019-1213-1Search in Google Scholar

[8] Baleanu, D.—Wu, G.—Zeng, S.: Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Solitons Fractals 102 (2017), 99–105.10.1016/j.chaos.2017.02.007Search in Google Scholar

[9] Baleanu, D.—Fernandez, A.: On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simul. 59 (2018), 444–462.10.1016/j.cnsns.2017.12.003Search in Google Scholar

[10] Benchohra, M.—Bouriah, S.—Graef, J. R.: Boundary value problems for nonlinear implicit Caputo-Hadamard-type fractional differential equations with impulses, Mediterr. J. Math. 14 (2017), #206.10.1007/s00009-017-1012-9Search in Google Scholar

[11] Cabada, A.—Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (2012), 403–411.10.1016/j.jmaa.2011.11.065Search in Google Scholar

[12] Cabada, A.—Dimitrijevic, S.—Tomovic, T.—Aleksic, S.: The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions, Math. Methods Appl. Sci. 40 (2017), 1880–1891.10.1002/mma.4105Search in Google Scholar

[13] Diethelm, K.: The Analysis of Fractional Differential Equations. Lectures Notes in Math., Springer, 2010.10.1007/978-3-642-14574-2Search in Google Scholar

[14] Guo, D. J.—Lakshmikantham, V.: Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988.Search in Google Scholar

[15] Han, Z.—Lu, H.—Zhang, C.: Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian, Appl. Math. Comput. 257 (2015), 526–536.10.1016/j.amc.2015.01.013Search in Google Scholar

[16] Hao, X.—Wang, H.—Liu, L.—Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator, Bound. Value Prob. 2017 (2017), #182.10.1186/s13661-017-0915-5Search in Google Scholar

[17] Henderson, J.—Luca, R.: Systems of Riemann-Liouville fractional equations with multi-point boundary conditions, Appl. Math. Comput. 309 (2017), 303–323.10.1016/j.amc.2017.03.044Search in Google Scholar

[18] Kilbas, A. A.—Srivastava, H. M.—Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, Elsevier B. V., Amsterdam, 2006.Search in Google Scholar

[19] Krasnosel ’skii, M. A.: Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.Search in Google Scholar

[20] Li, X.—Liu, X.—Jia, M.—Li, Y.—Zhang, S.: Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval, Math. Methods Appl. Sci. 40 (2017), 1892–1904.10.1002/mma.4106Search in Google Scholar

[21] Li, Y.—Qi, A.: Positive solutions formulti-point boundary value problems of fractional differential equations with p-Laplacian, Math. Methods Appl. Sci. 39 (2016), 1425–1434.10.1002/mma.3579Search in Google Scholar

[22] Li, Y.—Lin, S.: Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation with p-Laplacian, J. Func. Spac. Appl. 2013 (2013), Art. ID: 951643.10.1155/2013/951643Search in Google Scholar

[23] Liu, X.—Jia, M.—Ge, W.: The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator, Appl. Math. Lett. 65 (2017), 56–62.10.1016/j.aml.2016.10.001Search in Google Scholar

[24] Ntouyas, S. K.—Tariboon, J.: Fractional integral problems for Hadamard-Caputo fractional Langevin differential inclusions, J. Appl. Math. Comput. 51 (2016), 13–33.10.1007/s12190-015-0888-4Search in Google Scholar

[25] Pei, K.—Wang, G.—Sun, Y.: Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain, Appl. Math. Comput. 312 (2017), 158–168.10.1016/j.amc.2017.05.056Search in Google Scholar

[26] Prasad, K. R.—Sreedhar, N.—Wesen, L. T.: Existence of positive solutions for higher order p-Laplacian boundary value problems, Mediterr. J. Math. 15 (2018), #19.10.1007/s00009-017-1064-xSearch in Google Scholar

[27] Rao, S. N.: Multiplicity of positive solutions for coupled system of fractional differential equation with p-Laplacian two-point BVPs, J. Appl. Math. Comput. 55 (2017), 41–58.10.1007/s12190-016-1024-9Search in Google Scholar

[28] Shah, K.—Khan, E. A.: Iterative scheme for a coupled system offractional-order differential equations with three-point boundary conditions, Math. Methods Appl. Sci. 41 (2018), 1047–1053.10.1002/mma.4122Search in Google Scholar

[29] Su, X.—Jia, M.—Fu, X.: On positive solutions of eigenvalue problems for a class of p-Laplacian fractional differential equations, J. Appl. Anal. Comput. 8 (2018), 152–171.Search in Google Scholar

[30] Wang, Y.—Jiang, J.: Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian, Adv. Differ. Equ. 2017 (2017), #337.10.1186/s13662-017-1385-xSearch in Google Scholar

[31] Wang, G.—Wang, T.: On a nonlinear Hadamard type fractional differential equation with p-Laplacian operator and strip condition, J. Nonlinear Sci. Appl. 9 (2016), 5073–5081.10.22436/jnsa.009.07.10Search in Google Scholar

[32] Wang, T.—Wang, G.—Yang, X.: On a Hadamard-type fractional turbulent flow model with deviating arguments in a porous medium, Nonlinear Anal. Model. Contr. 22 (2017), 765–784.10.15388/NA.2017.6.3Search in Google Scholar

[33] Yang, W.: Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions, J. Appl. Math. Comput. 44 (2014), 39–59.10.1007/s12190-013-0679-8Search in Google Scholar

[34] Yang, W.: Positive solution for fractional q-difference boundary value problems with ϕ-Laplacian operator, Bull. Malays. Math. Sci. Soc. 36 (2013), 1195–1203.Search in Google Scholar

[35] Yang, W.: Monotone iterative technique for a coupled system of nonlinear Hadamard fractional differential equations, J. Appl. Math. Comput. 59 (2019), 585–596.10.1007/s12190-018-1192-xSearch in Google Scholar

[36] Yang, W.: Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions, J. Appl. Math. Comput. 49 (2015), 357–381.10.1007/s12190-014-0843-9Search in Google Scholar

[37] Yang, W.: Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations, J. Nonlinear Sci. Appl. 8 (2015), 110–129.10.22436/jnsa.008.02.04Search in Google Scholar

[38] Yang, W.—Qin, Y.: Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions, ScienceAsia 43 (2017), 201–206.10.2306/scienceasia1513-1874.2017.43.201Search in Google Scholar

[39] Yuan Q.—Yang, W.: Positive solution for q-fractional four-point boundary value problems with p-Laplacian operator, J. Inequal. Appl. 2014 (2014), #481.10.1186/1029-242X-2014-481Search in Google Scholar

[40] Yukunthorn, W.—Suantai, S.—Ntouyas S. K.—Tariboon J.: Boundary value problems for impulsive multi-order Hadamard fractional differential equations, Bound. Value Probl. 2015 (2015), #148.10.1186/s13661-015-0414-5Search in Google Scholar

[41] Zhi, E.—Liu, X.—Li, F.: Nonlocal boundary value problem for fractional differential equations with p-Laplacian, Math. Methods Appl. Sci. 37 (2014), 2651–2662.10.1002/mma.3005Search in Google Scholar

[42] Zhou, Y.: Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.10.1142/9069Search in Google Scholar

Received: 2019-02-02
Accepted: 2019-07-12
Published Online: 2020-01-13
Published in Print: 2020-02-25

© 2020 Mathematical Institute Slovak Academy of Sciences

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