A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A <i>ψ</i>-HILFER FRACTIONAL OPERATOR

J. Vanterler da C. Sousa; Leandro S. Tavares; César E. Torres Ledesma

doi:10.11948/20200343

2021 Volume 11 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(3): 1610-1630

Next Article Previous Article

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

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

A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A ψ-HILFER FRACTIONAL OPERATOR

1.
Department of Applied Mathematics, State University of Campinas, Imecc, 13083-859, Campinas, SP, Brazil
2.
Centro de Cincias e Tecnologia, Universidade Federal do Cariri, Juazeiro do Norte, CE, CEP: 63048-080, Brazil and Departamento de Matemática, UnBUniversidade de Brasília, Brasília, DF, CEP: 70910-900, Brazil
3.
Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo Ⅱ s/n. Trujillo-Perú

Corresponding author: Email: vanterlermatematico@hotmail.com(J. V. Sousa)

Abstract

Abstract

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.
- $\psi$-fractional derivative space /
- variational structure /
- fractional differential equations /
- boundary value problem /
- mountain pass theorem
MSC: 26A33, 34B15, 35J20, 58E05

FullText(HTML)

References

[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. Google Scholar
[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. Google Scholar
[3]	A. Ambrosetti, Critical Points and Nonlinear Variational Problems, Bull. Soc. Math. France, 120, Memoire 49, 1992. Google Scholar
[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 CrossRef Google Scholar
[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. Google Scholar
[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 CrossRef Google Scholar
[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. Google Scholar
[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 CrossRef Google Scholar
[9]	H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer Science & Business Media, 2010. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[19]	A. A. Kilbas, O. I. Marichev and S. G. Samko, Fractional integral and derivatives (theory and applications), 1993. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[22]	F. Mainardi, The two forms of fractional relaxation of distributed order, J. Vibr. Control, 2007, 13(9), 1249-1268. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[25]	P. Rabinowitz, Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., 65, 1986. Google Scholar
[26]	R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publicacions Matematiques, 2014, 58(1), 133-154. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[33]	C. E. Torres Ledesma, Boundary value problem with fractional $p$-Laplacian operator, Adv. Nonlinear Anal., 2016, 5(2), 133-146. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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. Google Scholar
[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 CrossRef Google Scholar
[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. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[41]	C. E. Torres Ledesma, Mountain pass solution for a fractional boundary value problem, J. Frac. Cal. Appl., 2012, 5(1), 1-10. Google Scholar
[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 CrossRef Google Scholar
[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 CrossRef Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar