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Existence of solutions for fractional multi-point boundary value problems at resonance with three-dimensional kernels. (English) Zbl 1445.34032
Summary: In this paper, by using Mawhin’s continuation theorem, we investigate the existence of solutions for a class of fractional differential equations with multi-point boundary value problems at resonance, and the dimension of the kernel for a fractional differential operator is three. An example is given to show our main result.
##### MSC:
 34A08 Fractional ordinary differential equations and fractional differential inclusions 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] Podlubny, I, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal., 5, 367-386, (2002) · Zbl 1042.26003 [2] Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) · Zbl 0789.26002 [3] Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003 [4] Zhou, Y, Wang, J, Zhang, L: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2016) · Zbl 1360.34003 [5] Podlubny, I: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1998) · Zbl 0924.34008 [6] Magin, RL, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59, 1586-1593, (2010) · Zbl 1189.92007 [7] El-Saka, HA, The fractional-order SIS epidemic model with variable population size, J. Egypt. Math. Soc., 22, 50-54, (2014) · Zbl 1336.92078 [8] Shabibi, M; Postolache, M; Rezapour, S; Vaezpour, SM, Investigation of a multi-singular pointwise defined fractional integro-differential equation, J. Math. Anal., 7, 61-77, (2016) · Zbl 1362.34018 [9] Shabibi, M; Rezapour, Sh; Vaezpour, S, A singular fractional integro-differential equation, UPB Sci. Bull., Ser. A, Appl. Math. Phys., 79, 109-118, (2017) [10] Khalil, H; Khan, RA; Baleanu, D; Rashidi, MM, Some new operational matrices and its application to fractional order Poisson equations with integral type boundary constrains, Comput. Appl. Math., (2016) [11] Baleanu, D; Uğurlu, E, Regular fractional dissipative boundary value problems, Adv. Differ. Equ., 2016, (2016) · Zbl 1419.34010 [12] Jiao, F; Zhou, Y, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62, 1181-1199, (2011) · Zbl 1235.34017 [13] Ge, B; Rădulescu, V; Zhang, J, Infinitely many positive solutions of fractional boundary value problems, Topol. Methods Nonlinear Anal., 49, 647-664, (2017) · Zbl 1375.35181 [14] Boucenna, A; Moussaoui, T, Existence results for a fractional boundary value problem via critical point theory, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math., 54, 47-64, (2015) · Zbl 1354.34013 [15] Rezapour, Sh; Shabibi, M, A singular fractional differential equation with Riemann-Liouville integral boundary condition, J. Adv. Math. Stud., 8, 80-88, (2015) · Zbl 1318.34009 [16] Baleanu, D; Hedayati, V; Rezapour, S, On two fractional differential inclusions, SpringerPlus, 5, (2016) [17] Agarwal, RP; O’Regan, D; Staněk, S, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371, 57-68, (2010) · Zbl 1206.34009 [18] Bai, Z; Lü, H, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505, (2005) · Zbl 1079.34048 [19] Liu, X; Jia, M; Ge, W, The method of lower and upper solutions for mixed fractional four-point boundary value problem with $$p$$-Laplacian operator, Appl. Math. Lett., 65, 56-62, (2017) · Zbl 1357.34018 [20] Jia, M; Liu, X, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions, Appl. Math. Comput., 232, 313-323, (2014) · Zbl 1410.34019 [21] Zhang, W; Liu, W; Chen, T, Solvability for a fractional $$p$$-Laplacian multipoint boundary value problem at resonance on infinite interval, Adv. Differ. Equ., 2016, (2016) · Zbl 1419.34047 [22] Chen, Y; Tang, X, Solvability of sequential fractional order multi-point boundary value problems at resonance, Appl. Math. Comput., 218, 7638-7648, (2012) · Zbl 1252.34004 [23] Bai, Z; Zhang, Y, The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl., 60, 2364-2372, (2010) · Zbl 1205.34018 [24] Jiang, W, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74, 1987-1994, (2011) · Zbl 1236.34006 [25] Jiang, W, Solvability of fractional differential equations with $$p$$-Laplacian at resonance, Appl. Math. Comput., 260, 48-56, (2015) · Zbl 1410.34020 [26] Baleanu, D; Mousalou, A; Rezapour, S, A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-fabrizio derivative, Adv. Differ. Equ., 2017, (2017) · Zbl 1377.45004 [27] Baleanu, D; Mousalou, A; Rezapour, S, On the existence of solutions for some infinite coefficient-symmetric Caputo-fabrizio fractional integro-differential equations, Bound. Value Probl., 2017, (2017) · Zbl 1377.45004 [28] Aydogan, S; Baleanu, D; Mousalou, A; Rezapour, S, On approximate solutions for two higher-order Caputo-fabrizio fractional integro-differential equations, Adv. Differ. Equ., 2017, (2017) · Zbl 1377.45004 [29] Sen, M; Hedayati, V; Gholizade Atani, Y; Rezapour, S, The existence and numerical solution for a $$k$$-dimensional system of multi-term fractional integro-differential equations, Nonlinear Anal., Model. Control, 22, 188-209, (2017) [30] Khalil, H; Khan, R; Baleanu, D; Saker, S, Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions, Adv. Differ. Equ., 2016, (2016) · Zbl 1419.34087 [31] Mawhin, J: Topological Degree and Boundary Value Problems for Nonlinear Differential Equations: Topological Methods for Ordinary Differential Equations. Springer, Berlin (1993) · Zbl 0798.34025 [32] Mawhin, J: Topological Degree Methods in Nonlinear Boundary Value Problems. NSF-CBMS Regional Conference Series in Mathematics, vol. 40. Am. Math. Soc., Providence (1979) · Zbl 0414.34025 [33] Lin, X; Zhao, B; Du, Z, A third-order multi-point boundary value problem at resonance with one three dimensional kernel space, Carpath. J. Math., 30, 93-100, (2014) · Zbl 1324.34035 [34] Li, S; Yin, J; Du, Z, Solutions to third-order multi-point boundary-value problems at resonance with three dimensional kernels, Electron. J. Differ. Equ., 2014, (2014) · Zbl 1290.34022 [35] Liang, J; Ren, L; Zhao, Z, Existence of solutions for a multi-point boundary value problems with three dimension kernel at resonance, Q. J. Math., 26, 138-143, (2011) · Zbl 1240.34060 [36] Zhao, Z; Liang, J; Ren, L, A multi-point boundary value problems with three dimension kernel at resonance, Adv. Math., 38, 345-358, (2009)
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