Positive solutions for fractional differential equations with non-separated type nonlocal multi-point and multi-term integral boundary conditions
DOI:
https://doi.org/10.24193/subbmath.2021.4.08Keywords:
Fractional differential equations, Riemann-Liouville fractional derivative, multi-term fractional integral boundary condition, fixed point theorems.Abstract
In this paper, we investigate a class of nonlinear fractional differential equations that contain both the multi-term fractional integral boundary condition and the multi-point boundary condition. By the Krasnoselskii fixed point theorem we obtain the existence of at least one positive solution. Then, we obtain the existence of at least three positive solutions by the Legget-Williams fixed point theorem. Two examples are given to illustrate our main results.
Mathematics Subject Classification (2010): 34A08, 34B15, 34B18.
References
Agarwal, R.P., Alsaedi, A., Alsharif, A., Ahmad, B., On nonlinear fractional-order boundary value problems with nonlocal multi-point conditions involving Liouville-Caputo derivatives, Differ. Equ. Appl., 9(2017), no. 2, 147-160.
Bouteraa, N., Benaicha, S., Existence of solutions for three-point boundary value problem for nonlinear fractional differential equations, Bull. Transilv. Univ. Brasov Ser. III., 10(59)(2017), no. 1.
Bouteraa, N., Benaicha, S., Djourdem, H., Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions, Universal Journal of Mathematics and Applications, 1(2018), 39-45.
Cabada, A., Wang, G., Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389(2012), 403-411.
Cui, Y., Zou, Y., Existence of solutions for second-order integral boundary value problems, Nonlinear Anal. Model. Control., 21(2016), 828-838.
Delbosco, D., Fractional calculus and function spaces, J. Fract. Calc., 6(1996), 45-53.
Diethelm, L., Freed, K., On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity, in: Keil, F., Mackens, W., Voss, H., Werthers, J. (eds.), Scientific Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer, Heidelberg, 1999.
Djourdem, H., S. Benaicha, S., Existence of positive solutions for a nonlinear three-point boundary value problem with integral boundary conditions, Acta Math. Univ. Comenianae, 87(2018), no. 2, 167-177.
Glockle, W.G., Nonnenmacher, T.F., A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68(1995), 46-53.
Guo, D., Lakshmikantham, V., Nonlinear Problems in Abstract Cones, Math. Anal. Appl., (1988).
Guo, L., Liu, L., Wu, Y., Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions, Nonlinear Anal. Model. Control, 21(2016), no. 5, 635-650.
He, J., Some applications of nonlinear fractional differential equations and their approximations, Bull. Am. Soc. Inf. Sci. Technol., 15(1999), 86-90.
Henderson, J., Luca, R., Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems, NoDEA Nonlinear Differential Equations Appl., 20(2013), 1035-1054.
Henderson, J., Luca, R., Systems of Riemann’Liouville fractional equations with multi-point boundary conditions, Appl. Math. Comput., 309(2017), 303-323.
Henderson, J., Luca, R., Tudorache, A., On a system of fractionl differential equations with coupled integral boundary conditions, Fract. Calc. Appl. Anal., 18(2015), 361-386.
Infante, G., Positive solutions of nonlocal boundary value problems with singularities, Discrete Contin. Dyn. Syst., (2009), Supplement, 377-384.
Ji, Y.D., Guo, Y.P., Qiu, J.Q., Yang, L.Y., Existence of positive solutions for a boundary value problem of nonlinear fractional differential equations, Adv. Differ. Equ., 2015(2015), Art. 13.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
Leggett, R.W., Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28(1979), 673-688.
Liu, L., Jiang, J., Wu, Y., The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems, J. Nonlinear Sci. Appl., 9(5)(2016), 2943-2958.
Ma, D-X., Positive solutions of multi-point boundary value problem of fractional differential equation, Arab J. Math. Sci., 21(2015), no. 2, 225-236.
Mainardi, F., Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics, in: Carpinteri, C.A., Mainardi, F. (eds.), Fractal and Fractional Calculus in Continuum Mechanics, Springer, Vienna, 1997.
Metzler, F., Schick, W., Kilian, H.G., Nonnenmache, T.F., Relaxation in filled polymers: a fractional calculus approach, J. Chem. Phys., 103(1995), 7180-7186.
Nica, O., Precup, R., On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babes-Bolyai Math., 56(2011), no. 3, 113-125.
Nyamoradi, N., Existence of solutions for multi-point boundary value problems for fractional differential equations, Arab J. Math. Sci., 18(2012), 165-175.
Oldham, K.B., Spanier, J., The fractional calculus, Math. Anal. Appl., (1974). [27] Podlubny, I., Fractional differential equations, Math. Anal. Appl., (1999).
Pu, R., Zhang, X., Cui, Y., Li, P., Wang, W., Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions, Funct. Spaces, 2017, Art. ID 5892616, 2017, 8 pages.
Ross B., (ed.), The Fractional Calculus and its Applications, Lecture Notes in Math, Vol. 475, Springer-Verlag, Berlin, 1975.
Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993.
Shah, K., Zeb, S., Khan, R.A., Existence of triple positive solutions for boundary value problem of nonlinear fractional differential equations, Comput. Methods Differ. Equ., 5(2017), no. 2, 158-169.
Sun, Y., Zhao, M., Positive solutions for a class of fractional differential equations with integral boundary conditions, Appl. Math. Lett., 34(2014), 17-21.
Tariboon, J., Ntouyas, S.K., Sudsutad, W., Positive solutions for fractional differential equations with three-point multi-term fractional integral boundary conditions, Adv. Difference Equ., 2014(2014), 17 pages.
Wang, Y., Liang S., Wang, Q., Multiple positive solutions of fractional-order boundary value problem with integral boundary conditions, J. Nonlinear Sci. Appl., 10(2017), 6333-6343.
Webb, J.R.L., Nonlocal conjugate type boundary value problems of higher order, Nonlinear Anal., 71(2009), 1933-1940.
Zhou L., Jiang, W., Positive solutions for fractional differential equations with multi-point boundary value problems, Journal of Applied Mathematics and Physics, 2(2014), 108-114.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
