Citation: | J. Vanterler da C. Sousa, Leandro S. Tavares, César E. Torres Ledesma. A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A ψ-HILFER FRACTIONAL OPERATOR[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1610-1630. doi: 10.11948/20200343 |
Boundary value problems driven by fractional operators has drawn the attention of several researchers in the last decades due to its applicability in several areas of Science and Technology. The suitable definition of the fractional derivative and its associated spaces is a natural problem that arise on the study of this kind of problem. A manner to avoid of such problem is to consider a general definition of fractional derivative. The purpose of this manuscript is to contribute, in the mentioned sense, by presenting the $\psi-$fractional spaces $\mathbb{H}_{p}^{\alpha, \beta; \psi}([0, T], \mathbb{R})$. As an application we study a problem, by using the Mountain Pass Theorem, which includes an wide class of equations.
[1] | R. P. Agarwal, M. B. Ghaemi and S. Saiedinezhad, The Nehari manifold for the degenerate $p$-Laplacian quasilinear elliptic equations, Adv. Math. Sci. Appl., 2010, 20(1), 37-50. |
[2] | R. Almeida, Further properties of Osler's generalized fractional integrals and derivatives with respect to another function, Rocky Mountain J. Math., 2019, 49(8), 2459-2493. |
[3] | A. Ambrosetti, Critical Points and Nonlinear Variational Problems, Bull. Soc. Math. France, 120, Memoire 49, 1992. |
[4] | A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Func. Anal., 1973, 14, 349-381. doi: 10.1016/0022-1236(73)90051-7 |
[5] | L. Bai, B. Dai and F. Li, Solvability of second-order Hamiltonian systems with impulses via variational method, Appl. Math. Comput., 2013, 219(14), 7542-7555. |
[6] | A. Benhassine, Existence of infinitely many solutions for a class of fractional Hamiltonian systems, J. Elliptic Parabolic Equ., 2019, 5(1), 105-123. doi: 10.1007/s41808-019-00034-z |
[7] | A. Boucenna and T. Moussaoui, Existence of a positive solution for a boundary value problem via a topological-variational theorem, J. Fract. Calc. Appl., 2014, 5(3S), 1-9. |
[8] | L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations, J. Math. Anal. Appl., 2013, 399(1), 239-251. doi: 10.1016/j.jmaa.2012.10.008 |
[9] | H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer Science & Business Media, 2010. |
[10] | J. M. Carcione and F. Mainardi, On the relation between sources and initial conditions for the wave and diffusion equations, Comput. Math. Appl., 2017, 73(6), 906-913. doi: 10.1016/j.camwa.2016.04.019 |
[11] | G. Chai and J. Chen, Existence of solutions for impulsive fractional boundary value problems via variational method, Boundary Value Probl., 2017, 2017(1), 1-120. doi: 10.1186/s13661-016-0733-1 |
[12] | G. Cruz, A. Mendez and C. E. Torres Ledesma, Multiplicity of solutions for fractional Hamiltonian systems with Liouville-Weyl fractional derivatives, Frac. Calc. Appl. Anal., 2015, 18(4), 875-890. |
[13] | M. Ferrara and A. Hadjian, Variational approach to fractional boundary value problems with two control parameters, Elect. J. Diff. Equ., 2015, 2015(138), 1-15. |
[14] | G. J. Fix and J. P. Roof, Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl., 2004, 48(7), 1017-1033. |
[15] | H. Hassani, J. A. Tenreiro Machado, Z. Avazzadeh and E. Naraghirad, Generalized shifted Chebyshev polynomials: Solving a general class of nonlinear variable order fractional PDE, Commun. Nonlinear Sci. Numer. Simul., 2020, 85, 105229. doi: 10.1016/j.cnsns.2020.105229 |
[16] | F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 2011, 62(3), 1181-1199. doi: 10.1016/j.camwa.2011.03.086 |
[17] | F. Jiao and Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Inter. J. Bifur. Chaos, 2012, 22(04), 1250086. doi: 10.1142/S0218127412500861 |
[18] | M. Karkulik, Variational formulation of time-fractional parabolic equations, Comput. Math. Appl., 2018, 75(11), 3929-3938. doi: 10.1016/j.camwa.2018.03.003 |
[19] | A. A. Kilbas, O. I. Marichev and S. G. Samko, Fractional integral and derivatives (theory and applications), 1993. |
[20] | Y. Li and B. Dai, Existence and multiplicity of nontrivial solutions for Liouville-Weyl fractional nonlinear Schrödinger equation, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas, 2018, 112(4), 957-967. doi: 10.1007/s13398-017-0405-8 |
[21] | W. Liu, M. Wang and T. Shen, Analysis of a class of nonlinear fractional differential models generated by impulsive effects, Boundary Value Probl., 2017, 2017(1), 175. doi: 10.1186/s13661-017-0909-3 |
[22] | F. Mainardi, The two forms of fractional relaxation of distributed order, J. Vibr. Control, 2007, 13(9), 1249-1268. |
[23] | N. Nyamoradi and S. Tersian, Existence of solutions for nonlinear fractional order $p$-Laplacian differential equations via critical point theory, Fract. Calc. Appl. Anal., 2019, 22(4), 945-967. doi: 10.1515/fca-2019-0051 |
[24] | M. D. Ortigueira and J. Tenreiro Machado, On the properties of some operators under the perspective of fractional system theory, Commun. Nonlinear Sci. Numer. Simul., 2020, 82, 105022. doi: 10.1016/j.cnsns.2019.105022 |
[25] | P. Rabinowitz, Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., 65, 1986. |
[26] | R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 2014, 58(1), 133-154. |
[27] | C. J. Silva and Delfim F. M. Torres, Stability of a fractional HIV/AIDS model, Math. Comput. Simul., 2019, 164, 180-190. doi: 10.1016/j.matcom.2019.03.016 |
[28] | J. V. Sousa and E. Capelas de Oliveira, Leibniz type rule: $\psi$-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul. 2019, 77, 305-311. doi: 10.1016/j.cnsns.2019.05.003 |
[29] | J. V. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 2018, 60, 72-91. doi: 10.1016/j.cnsns.2018.01.005 |
[30] | J. V. Sousa, E. Capelas de Oliveira and L. A. Magna, Fractional calculus and the ESR test, AIMS Math., 2017, 2(4), 692-705. doi: 10.3934/Math.2017.4.692 |
[31] | J. V. Sousa, M. N. N. dos Santos, L. A. Magna and E. Capelas de Oliveira, Validation of a fractional model for erythrocyte sedimentation rate, Comput. Appl. Math., 2018, 37(5), 6903-6919. doi: 10.1007/s40314-018-0717-0 |
[32] | H. Sun and Q. Zhang, Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique, Comput. Math. Appl., 2012, 64(10), 3436-3443. doi: 10.1016/j.camwa.2012.02.023 |
[33] | C. E. Torres Ledesma, Boundary value problem with fractional $p$-Laplacian operator, Adv. Nonlinear Anal., 2016, 5(2), 133-146. |
[34] | C. E. Torres Ledesma, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Commun. Pure Appl. Anal., 2016, 15, 535-547. doi: 10.3934/cpaa.2016.15.535 |
[35] | C. E. Torres Ledesma, Existence of solution for a general fractional advection-dispersion equation, Anal. Math. Phys., 2019, 9(3), 1303-1318. doi: 10.1007/s13324-018-0234-8 |
[36] | C. E. Torres Ledesma, Existence of solutions for fractional Hamiltonian systems with nonlinear derivative dependence in $\mathbb{R}$, J. Fract. Calc. Appl., 2016, 7(2), 74-87. |
[37] | C. E. Torres Ledesma, Existence of a solution for the fractional forced pendulum, J. Appl. Math. Comput. Mechanics, 2014, 13(1), 125-142. doi: 10.17512/jamcm.2014.1.13 |
[38] | C. E. Torres Ledesma, Existence and symmetric result for Liouville-Weyl fractional nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul., 2015, 27(1), 314-327. |
[39] | C. E. Torres Ledesma and N. Nyamoradi, Impulsive fractional boundary value problem with $p$-Laplace operator, J. Appl. Math. Comput., 2017, 55(1), 257-278. doi: 10.1007/s12190-016-1035-6 |
[40] | C. E. Torres Ledesma and O. Pichardo, Multiplicity of Solutions for a Class of Perturbed Fractional Hamiltonian Systems, Bull. Malaysian Math. Sci. Soc., 2020, 43, 3897-3922. doi: 10.1007/s40840-020-00898-8 |
[41] | C. E. Torres Ledesma, Mountain pass solution for a fractional boundary value problem, J. Frac. Cal. Appl., 2012, 5(1), 1-10. |
[42] | Y. Wang, L. Liu and Y. Wu, Positive solutions for a class of fractional boundary value problem with changing sign non-linearity, Nonlinear Anal., Theory, Meth. Appl., 2011, 74(17), 6434-6441. doi: 10.1016/j.na.2011.06.026 |
[43] | D. Wu, C. Li and P. Yuan, Multiplicity Solutions for a Class of Fractional Hamiltonian Systems With Concave-Convex Potentials, Mediterr. J. Math., 2018, 15(2), 35. doi: 10.1007/s00009-018-1079-y |
[44] | Z. Xie, Y. Jin and C. Hou, Multiple solutions for a fractional difference boundary value problem via variational approach, Abst. Appl. Anal., 2012, 2012. |
[45] | J. Xu, D. O'Regan and K. Zhang, Multiple solutions for a class of fractional Hamiltonian systems, Frac. Calc. Appl. Anal., 2015, 18(1), 48-63. |
[46] | X. Zhang, L. Liu and Y. Wu, Variational structure and multiple solutions for a fractional advection-dispersion equation, Comput. Math. Appl., 2014, 68(1), 1794-1805. |
[47] | Y. Zhao, H. Chen and B. Qin, Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods, Appl. Math. Comput., 2015, 257, 417-427. |
[48] | Z. Zhang, and C. E. Torres Ledesma, Solutions for a class of fractional Hamiltonian systems with a parameter, J. Appl. Math. Comput., 2017, 54(1), 451-468. |