Existence of solutions for conformable fractional problems with nonlinear functional boundary conditions

Bouharket Bendouma; Alberto Cabada; Ahmed Hammoudi

doi:10.26637/MJM0704/0013

Abstract

In this article, we study the existence of solutions for nonlinear conformable fractional differential equations with nonlinear functional boundary conditions. We obtain the exact expression of the fractional Green’s function related to the linear problem. Moreover, the method of upper and lower solutions together with Schauder’s fixed point theorem is developed for the nonlinear conformable fractional problems with nonlinear functional boundary conditions.

Keywords:

Conformable fractional derivative, nonlinear boundary conditions, Green’s function, upper and lower solutions method, , existence theorems, maximum principles

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Bouharket Bendouma Ibn Khaldoun, Tiaret University PB78, 14000 zaaroura, Tiaret, Algeria
Alberto Cabada University of Sidi Bel Abbes, P.O. Box 89, Sidi Bel Abb ` es, 22000, Algeria https://orcid.org/0000-0003-1488-935X
Ahmed Hammoudi Laboratory of Mathematics, Ain Temouchent University P.O. Box 89, 46000 Ain T ´ emouchent, Algeria.

Pages: 700-708
Date Published: 01-10-2019
Vol. 7 No. 04 (2019): Malaya Journal of Matematik (MJM)

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015), 57-66.

T. Abdeljawad, M. AlHorani and R. Khalil, Conformable fractional semigroups of operators, J. Semig. Theory. Appl. 2015 (2015), 9 pages.

D.R. Anderson and R.I. Avery, Fractional-order boundary value problem with Stum-Liouville boundary conditions, Electr. J. Differ. Equ. 2015 (2015), no. 29, 10 pages.

J. Appell, P. P. Zabrejko, Nonlinear superposition operators. Cambridge Tracts in Mathematics, 95. Cambridge University Press, Cambridge, 1990. viii+311 pp.

H. Batarfi, J. Losada, J.J. Nieto and W. Shammakh, ThreePoint Boundary Value Problems for Conformable Fractional Differential Equations, J. Funct. Spaces. 2015 (2015), 6 pages.

B. Bayour and D.F.M. Torres, Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math. 312(2016), 127-133.

M.Benchohra, A. Cabada and D. Seba, An existence result for nonlinear fractional differential equations on Banach spaces, Bound. Value. Probl. 2009, 11 pages.

P. L. Butzer and U. Westphal, An access to fractional differentiation via fractional difference quotients, in Fractional calculus and its applications (Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974), 116-145. Lecture Notes in Math., 457, Springer, Berlin, 1975.

A. Cabada, The monotone method for first-order problems with linear and nonlinear boundary conditions, Appl. Math. Comput. 63 (1994), 163-188.

A. Cabada, Green's Functions in the Theory of Ordinary Differential Equations, Springer Briefs Math., Springer, New York, 2014.

A. Cabada and Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions, Appl. Math. Comput. 228 (2014), 251-257.

A. Cabada and Z. Hamdi, Multiplicity results for integral boundary value problems of fractional order with parametric dependence, Fract. Calc. Appl. Anal. 18 (2015), no. 1, 223-237.

A. Cabada and Z. Hamdi, Existence results for nonlinear fractional Dirichlet problems on the right side of the first eigenvalue, Georgian Math. J. 24(2017), Issue 1, 41-53.

A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389(2012), Issue 1, 403-411.

W. S. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math. $290(2015), 150-158$.

A. Gökdoğan, E. Ünal and E. Çelik, Existence and Uniqueness Theorems for Sequential Linear Conformable Fractional Differential Equations, 2015.

H.L. Gray and N.F. Zhang, On a new definition of the fractional difference, Math. Comp. 50 (1988), 182: 513529.

S. Heikkilä and A. Cabada, On first order discontinuous differential equations with nonlinear boundary conditions, Nonlinear World 3 (1996), 487-503.

O.S. Iyiola and E.R. Nwaeze, Some new results on the new conformable fractional calculuswith application using D'Alambert approach, Progr. Fract. Dier. Appl. $2(2016),(2), 115-122$.

T. Jankowski, Boundary problems for fractional differential equations, Appl. Math. Lett. 28 (2014), 14-19.

U.N. Katugampola, A new fractional derivative with classical properties, preprint, 2014.

R. Khaldi and A. Guezane-Lakoud, Lyapunov inequality for a boundary value problem involving conformable derivative, Prog. Fract. Differ. Appl. 2017.

R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. $264(2014), 65-70$.

A. Kilbas, M.H. Srivastava and J.J. Trujillo, Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies 204, 2006.

R.L. Magin, Fractional calculus in Bioengineering, CR in Biomedical Engineering 32 (2004), no. 1, 1-104.

K.S. Miller and B. Ross, Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, (1989), 139-152.

I. Podlubny, Fractional Differential Equations, Academic Press: San Diego CA, (1999).

S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, ,Switzerland, 1993.

K. Shugui, C. Huiqing, Y. Yaqing and G. Ying, Existence and uniqueness of the solutions for the fractional initial value problem, Electr. J. Shanghai Normal University (Natural Sciences), 45(20016), no. 3, 313-319.

A. Shi and S. Zhang, Upper and lower solutions method and a fractional differential equation boundary value problem, Electr. J. Qual. Theory. Differ. Equ. 30 (2009), 13 pages.

Y. Wang, J. Zhou and Y. Li, Fractional Sobolev's Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales, Adv. Math. Phys. (2016), 1-21.

$mathrm{Z}$. Wei, Q. Li, and J. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 367 (2009), no. 1, 260-272.

X.J. Yang, D. Baleanu and J.A.T. Machado, Application of the local fractional Fourier series to fractal signals, in Discontinuity and complexity in nonlinear physical systems, Nonlinear Syst. Complex, Springer, Cham, 2014, $63-89$.

X.J. Yang, Advanced local fractional calculus and its applications, World Science Publisher, New York, 2012.

$mathrm{S}$. Zhang and $mathrm{X}$. Su, The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order, Comput. Math. Appl. 62 (2011), no. 3 , 1269-1274.