Nonlocal conditions for fractional differential equations
DOI:
https://doi.org/10.24193/subbmath.2024.4.08Keywords:
Fractional differential equations, nonlocal conditions, upper and lower solutions, iterative technique, fixed pointAbstract
In this work we use the method of lower and upper solutions to develop an iterative technique, which is not necessarily monotone, and combined with a fixed point theorem to prove the existence of at least one solution of nonlinear fractional differential equations with nonlocal boundary conditions of integral type.References
Agarwal, R. P., Benchohra, M., Hamani, S., Boundary value problems for differential
inclusions with fractional order, Adv. Stud. Contemp. Math.,16(2008), no. 2, 181-196.
Al-Refai, M., On the fractional derivatives at extreme points, Electron. J. Qual. Theory
Differ. Equ., (2012), no. 55, 1-5.
Balachandran, K., Uchiyama, K., Existence of solutions of nonlinear integrodi erential
equations of Sobolev type with nonlocal conditions in Banach spaces, Proc. Indian Acad.
Sci. Math. Sci., 110(2000), 225-232.
Benchohra, M., Hamani, S., Nonlinear boundary value problems for di erential inclu-
sions with Caputo fractional derivative, Topol. Methods Nonlinear Anal., 32(2008), 115-
Bitsadze, A.V., Samarskii, A.A., Some elementary generalizations of linear elliptic
boundary value problems, Dokl. Akad. Nauk SSSR, 185(1969), no. 4, 739-740.
Boucherif, A., Nonlocal conditions for two-endpoint problems, Int. J. Difference Eq.,
(2020), 321-334.
Boucherif, A., Bouguima, S. M., Benbouziane, Z., Al-Malki, N., Third order problems
with nonlocal conditions of integral type, Bound. Value Probl., (2014), no. 137.
Boucherif, A., Ntouyas, S.K., Nonlocal initial value problems for rst order fractional
differential equations, Dynam. Systems Appl., 20(2011), 247-260.
Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear
evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162(1991), 494-505.
Cabada, A., The method of lower and upper solutions for second, third, fourth and higher
order boundary value problems, J. Math. Anal. Appl., 185(1994), no. 2, 302-320.
Chang, Y.-K. and Nieto, J.J., Some new existence results for fractional differential in-
clusions with boundary conditions, Math. Comput. Model., 49(2009), 605-609.
Diethelm, K., The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin,
Ding, Y., Wei, Z., On the extremal solution for a nonlinear boundary value problems of
fractional p-Laplacian differential equation, Filomat, 30(2016), no. 14, 3771-3778.
Furati, K.M., Tatar, N.-E., An existence result for nonlocal fractional differential prob-
lem, J. Fract. Calc., 26(2004), 43-51.
Henderson, J., Thompson, H.B., Existence of multiple solutions for second order bound-
ary value problems, J. Di erential Equations, 166(2000), no. 2, 443-454.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional
Differential Equations, Elsevier, Amsterdam, 2006.
Lakshmikanthan, V., Leela, S., Devi, J.V., Theory of Fractional Dynamic Systems, Cam-
bridge Scienti c Publishers, 2009.
Lin, L., Liu, X., Fang, H., Method of upper and lower solutions for fractional differential
equations, Electron. J. Di erential Equations, (2012), no. 100, 1-13.
Ma, R., A survey on nonlocal boundary value problems, Appl. Math. E-Notes, 7(2007),
-279.
Mawhin, J., Topological degree methods in nonlinear boundary value problems, in: NS-
FCBMS Regional Conference Series in Math., Amer. Math. Soc. Colloq. Publ., 1979.
Mawhin, J., Szymanska-Debowska, K., Convexity, topology and nonlinear differential
systems with nonlocal boundary conditions: A survey, Rend. Istit. Mat. Univ. Trieste,
(2019), 125-166.
McRae, F.A., Monotone iterative technique for periodic boundary value problems of Ca-
puto fractional differential equations, Commun. Appl. Anal., 14(2010), 73-80.
Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Di erential Equa-
tions, John Wiley, 1993.
Pawar, G.U., Salunke, J.N., Upper and Lower Solution Method for Nonlinear Fractional
Di erential Equations Boundary Value Problem, J. Comput. Math., 9(2018), 164-173.
Podlubny, I., Fractional Differential Equations, Academic Press, 1999.
Rachankov, I., Upper and lower solutions and topological degree, J. Math. Anal. Appl.,
(1999), no. 1, 311-327.
Smart, D.R., Fixed Point Theorems, Cambridge University Press 1974.
Wang, Y., Liang, S.,Wang, Q., Existence results for fractional di erential equations with
integral and multi-point boundary conditions, Bound. Value Probl., (2018), no. 4.
Wang, X., Wang, L., Zeng, Q., Fractional di erential equations with integral boundary
conditions, J. Nonlinear Sci. Appl., 8(2015), 309-314.
Xie, W., Xiao, J., Luo, Z., Existence of extremal solutions for nonlinear fractional dif-
ferential equation with nonlinear boundary conditionals, Appl. Math. Lett., 41(2015),
-51.
Zhang, S., Positive solutions for boundary value problems of nonlinear fractional differ-
ential, Electron. J. Di erential Equations, (2006), no. 36, 1-12.
Zhou, Y., Basic Theory of Fractional Differential Equations,World Scienti c, Singapore,
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Transfer of copyright agreement: When the article is accepted for publication, the authors and the representative of the coauthors, hereby agree to transfer to Studia Universitatis Babeș-Bolyai Mathematica all rights, including those pertaining to electronic forms and transmissions, under existing copyright laws, except for the following, which the authors specifically retain: the authors can use the material however they want as long as it fits the NC ND terms of the license. The authors have all rights for reuse according to the license.
