On Simultaneous Approximation and Combinations of Lupas Type Operators
Download PDF
Authors: T. A. K. SINHA, K. K. SINGH AND A. K. SHARMA
DOI: 10.46793/KgJMat2105.761B
Abstract:
This work deals with the BVP multi-point existence of solutions of a nonlinear fractional differential equations at resonance, where the kernel’s dimension of the fractional differential operator is equal to three. Our results are based on Mawhin’s theory of coincidence. As application, we give an example to illustrate our results.
Keywords:
Fractional differential equations, fractional Caputo derivative, multi-point boundary value problem, resonance, three dimensional kernels, coincidence degree theory.
References:
[1] B. Ahmad and P. Eloe, A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices, Comm. Appl. Nonlinear Anal. 17 (2010), 69–80.
[2] R. P. Agarwal, D. O’Regan and S. Stanek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl. 371 (2010), 57–68.
[3] Z. Bai, On solutions of some fractional m-point boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equ. 37 (2010), 15 pages, Paper ID MR2676127.
[4] Z. Bai, Solvability for a class of fractional m-point boundary value problem at resonance, Comput. Math. Appl. 62 (2011), 1292–1302.
[5] Z. Bai and Y. Zhang, The existence of solutions for a fractional multi-point boundary value problem, Comput. Math. Appl. 60(8) (2010), 2364–2372.
[6] Z. Bai and Y. Zhang, Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput. 218 (2011), 1719–1725.
[7] M. Benchohra, S. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009), 2391–2396.
[8] S. Djebali and L. Guedda, A third order boundary value problem with nonlinear growth at resonance on the half-axis, Math. Meth. Appl. Sci. (2016).
[9] Z. Hu, W. Liu and T. Chen, Existence of solutions for a coupled system of fractional differential equations at resonance, Bound. Value Probl. 2012(98) (2012), 13 pages.
[10] Z. Hu, W. Liu and T. Chen, Two-point boundary value problems for fractional differential equations at resonance, Bull. Malays. Math. Sci. Soc. 36(3) (2013), 747–755.
[11] L. Hu, S. Zhang and A. Shi, Existence of solutions for two-point boundary value problem of fractional differential equations at resonance, Hindawi Publishing Corporation International Journal of Differential Equations (2014), Article ID 632434, 7 pages.
[12] H. Jafari and V. D. Gejji, Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput. 180 (2006) 700–706.
[13] M. Jia and X. Liu, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions, Appl. Math. Comput. 232 (2014), 313–323.
[14] W. Jiang and N. Kosmatov, Solvability of a third-order differential equation with functional boundary conditions at resonance, Bound. Value Probl. 81 (2017), DOI 10.1186/s13661-017-0811-z.
[15] W. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal. TMA 74 (2011), 1987–1994.
[16] W. Jiang, Solvability for a coupled system of fractional differential equations at resonance, Nonlinear Anal. Real World Appl. 13 (2012), 2285–2292.
[17] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam, 2006.
[18] N. Kosmatov, Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear Anal. 68 (2010), 2158–2171.
[19] N. Kosmatov, A boundary value problem of fractional order at resonance, Electron. J. Differ. Equ. 135 (2010), 10 pages.
[20] N. Kosmatov and W. Jiang, Second-order functional problems with a resonance of dimension one, Differ. Equ. Appl. 3 (2016), 349–365.
[21] N. Kosmatov and W. Jiang, Resonant functional problems of fractional order, Chaos Solitons Fractals 91 (2016), 573–579.
[22] S. Li, J. Yin and Z. Du, Solutions to third-order multi-point boundary-value problems at resonance with three dimensional kernels, Electron. J. Differ. Equ. (2014), Article ID 61.
[23] S. Liang and J. Zhang, Existence and uniqueness of positive solutions to m-point boundary value problem for nonlinear fractional differential equation, J. Appl. Math. Comput. 38 (2012), 225–241.
[24] X. Lin, B. Zhao and Z. Du, Athird-order multi-point boundary value problem at resonance with one three dimensional kernel space, Carpathian J. Math. 30(1) (2014), 93–100.
[25] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, NSF-CBMS Regional Conference Series in Mathematics 40, American Mathematical Society, Providence, RI, 1979.
[26] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley Sons, New York, NY, USA, 1993.
[27] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[28] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), 367–386.
[29] M. Rehman and P. Eloe, Existence and uniqueness of solutions for impulsive fractional differential equations, Appl. Math. Comput. 224 (2013), 422–431.
[30] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
[31] N. Xu, W. Liu, and L. Xiao, The existence of solutions for nonlinear fractional multipoint boundary value problems at resonance, Bound. Value Probl. 2012 (2012), Article ID 65, 10 pages.
[32] Y. Zhang and Z. Bai, Existence of solution for nonlinear fractional three-point boundary value problems at resonances, J. Appl. Math. Comput. 36 (2011), 417–440.
[33] X. Zhang, L. Wang and Q. Sun, Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Appl. Math. Comput. 226 (2014), 708–718.
Login:
