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Solvability of fractional multi-point boundary value problems with nonlinear growth at resonance. (English) Zbl 07269815

J. Contemp. Math. Anal., Armen. Acad. Sci. 55, No. 2, 126-142 (2020) and Izv. Nats. Akad. Nauk Armen., Mat. 55, No. 2, 46-64 (2020).
Summary: This work is concerned with the solvability of multi-point boundary value problems for fractional differential equations with nonlinear growth at the resonance. Existence results are obtained with the use of the coincidence degree theory. As an application, we discuss an example to illustrate the obtained results.

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Abbas, S.; Benchohra, M.; Graef, J. R.; Henderson, J., Implicit Fractional Differential and Integral Equations: Existence and Stability (2018), Berlin: Walter de Gruyter, Berlin · Zbl 1390.34002
[2] Abbas, S.; Benchohra, M.; N’Guérékata, G. M., Topics in Fractional Differential Equations (2012), New York: Springer, New York · Zbl 1273.35001
[3] Ahmad, B.; Eloe, P., A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices, Commun. Appl. Nonlinear Anal., 17, 69-80 (2010) · Zbl 1275.34005
[4] Agarwal, R. P.; O’Regan, D.; Stanek, S., Positive solutions for Dirichlet problems of singular non linear fractional differential equations, J. Math. Anal. Appl., 371, 57-68 (2010) · Zbl 1206.34009 · doi:10.1016/j.jmaa.2010.04.034
[5] Bai, Z., On solutions of some fractional m-point boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equations, 2010, 37 (2010) · Zbl 1210.34004
[6] Bai, Z., Solvability for a class of fractional m-point boundary value problem at resonance, Comput. Math. Appl., 62, 1292-1302 (2011) · Zbl 1235.34006 · doi:10.1016/j.camwa.2011.03.003
[7] 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 · doi:10.1016/j.camwa.2010.08.030
[8] Bai, Z.; Zhang, Y., Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput., 218, 1719-1725 (2011) · Zbl 1235.34007
[9] Benchohra, M.; Hamani, S.; Ntouyas, S. K., “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Anal.: Theory, Methods Appl., 71, 2391-2396 (2009) · Zbl 1198.26007
[10] 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
[11] Djebali, S.; Guedda, L., A third order boundary value problem with nonlinear growth at resonance on the half-axis, Math. Methods Appl. Sci., 40, 2859-2871 (2017) · Zbl 1384.34028 · doi:10.1002/mma.4202
[12] Hu, Z.; Liu, W.; Chen, T., Existence of solutions for a coupled system of fractional differential equations at resonance, Boundary Value Probl., 2012, 98 (2012) · Zbl 1281.34009 · doi:10.1186/1687-2770-2012-98
[13] Hu, Z.; Liu, W.; Chen, T., Two-point boundary value problems for fractional differential equations at resonance, Bull. Malays. Math. Sci. Soc., 36, 747-755 (2013) · Zbl 1274.34007
[14] Hu, L.; Zhang, S.; Shi, A., Existence of solutions for two-point boundary value problem of fractional differential equations at resonance, Int. J. Differ. Equations, 2014, 632434 (2014) · Zbl 1306.34008 · doi:10.1155/2014/632434
[15] Jafari, H.; Gejji, V. D., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput., 180, 700-706 (2006) · Zbl 1102.65136
[16] 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
[17] Jiang, W., “The existence of solutions to boundary value problems of fractional differential equations at resonance,” Nonlinear Anal.: Theory, Methods Appl., 74, 1987-1994 (2011) · Zbl 1236.34006
[18] Jiang, W., Solvability for a coupled system of fractional differential equations at resonance, Nonlinear Anal.: Real World Appl., 13, 2285-2292 (2012) · Zbl 1257.34005 · doi:10.1016/j.nonrwa.2012.01.023
[19] W. Jiang, and N. Kosmatov, ‘‘Solvability of a third-order differential equation with functional boundary conditions at resonance,’’ Boundary Value Probl. 2017, art. ID, 81 (2017). · Zbl 1366.34034
[20] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (Elsevier, Amsterdam, 2006). · Zbl 1092.45003
[21] Kosmatov, N., “Multi-point boundary value problems on an unbounded domain at resonance,” Nonlinear Anal.: Theory, Methods Appl., 68, 2158-2171 (2010) · Zbl 1138.34006
[22] Kosmatov, N., A boundary value problem of fractional order at resonance, Electron. J. Differ. Equations, 2010, 135 (2010) · Zbl 1210.34007
[23] Kosmatov, N.; Jiang, W., Resonant functional problems of fractional order, Chaos, Solitons Fractals, 91, 573-579 (2016) · Zbl 1372.34048 · doi:10.1016/j.chaos.2016.08.003
[24] Liang, S.; Zhang, J., Existence and uniqueness of positive solutions to m-point boundary value problem for nonlinear fractional differential equation, J. Appl. Math. Comput., 38, 225-241 (2012) · Zbl 1296.34021 · doi:10.1007/s12190-011-0475-2
[25] Mawhin, J., “Topological degree methods in nonlinear boundary value problems,” NSF-CBMS Regional Conference Series in Mathematics (1979), Providence, RI: Am. Math. Soc., Providence, RI · Zbl 0414.34025
[26] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York: Wiley, New York · Zbl 0789.26002
[27] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), New York: Academic, New York · Zbl 0428.26004
[28] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus Appl. Anal., 5, 367-386 (2002) · Zbl 1042.26003
[29] Rehman, M.; Eloe, P., Existence and uniqueness of solutions for impulsive fractional differential equations, Appl. Math. Comput., 224, 422-431 (2013) · Zbl 1334.34019
[30] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Yverdon: Gordon and Breach, Yverdon · Zbl 0818.26003
[31] Xu, N.; Liu, W.; Xiao, L., The existence of solutions for nonlinear fractional multipoint boundary value problems at resonance, Boundary Value Probl., 2012, 65 (2012) · Zbl 1278.34005 · doi:10.1186/1687-2770-2012-65
[32] Zhang, Y.; Bai, Z., Existence of solution for nonlinear fractional three-point boundary value problems at resonances, J. Appl. Math. Comput., 36, 417-440 (2011) · Zbl 1225.34013 · doi:10.1007/s12190-010-0411-x
[33] Zhang, X.; Wang, L.; Sun, Q., Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Appl. Math. Comput., 226, 708-718 (2014) · Zbl 1354.34049
[34] Zhou, Y., Basic Theory of Fractional Differential Equations (2014), Singapore: World Scientific, Singapore · Zbl 1336.34001
[35] Zhou, Y., Attractivity for fractional differential equations in Banach space, Appl. Math. Lett., 75, 1-6 (2018) · Zbl 1380.34025 · doi:10.1016/j.aml.2017.06.008
[36] Zhou, Y., Attractivity for fractional evolution equations with almost sectorial operators, Fract. Calculus Appl. Anal., 21, 786-800 (2018) · Zbl 1405.34012 · doi:10.1515/fca-2018-0041
[37] Zhou, Y.; Shangerganesh, L.; Manimaran, J.; Debbouche, A., A class of time-fractional reaction-diffusion equation with nonlocal boundary condition, Math. Methods Appl. Sci., 41, 2987-2999 (2018) · Zbl 1391.35404 · doi:10.1002/mma.4796
[38] Zhou, Y.; Peng, L.; Huang, Y. Q., Duhamel’s formula for time-fractional Schrödinger equations, Math. Methods Appl. Sci., 41, 8345-8349 (2018) · Zbl 1405.35254 · doi:10.1002/mma.5222
[39] Zhou, Y.; Peng, L.; Huang, Y. Q., Existence and Hölder continuity of solutions for time-fractional Navier-Stokes equations, Math. Methods Appl. Sci., 41, 7830-7838 (2018) · Zbl 1404.35482 · doi:10.1002/mma.5245
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