Existence theory for nabla fractional three-point boundary value problems via continuation methods for contractive maps
DOI:
https://doi.org/10.12775/TMNA.2022.043Keywords
Nabla fractional calculus, boundary value problem, Green's function, continuation method for contractive maps, fixed point, existence, uniquenessAbstract
In this article, we analyse an $\alpha$-th order, $1 < \alpha \leq 2$, nabla fractional three-point boundary value problem (BVP). We construct the Green's function associated to this problem and derive a few of its important properties. We then establish sufficient conditions on existence and uniqueness of solutions for the corresponding nonlinear BVP using the modern ideas of continuation methods for contractive maps. Our results extend recent results on nabla fractional BVPs. Finally, we provide an example to illustrate the applicability of main results.References
T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl. 62 (2011), no. 3, 1602–1611.
T. Abdeljawad and F.M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal. (2012), Art. ID 406757, 13 pp.
R. P. Agarwal and D. O’Regan, Fixed point theory for generalized contractions on spaces with two metrics, J. Math. Anal. Appl. 248 (2000), no. 2, 402–414.
R.P. Agarwal, D. O’Regan and R. Precup, Fixed point theory and generalized Leray–Schauder alternatives for approximable maps in topological vector spaces, Topol. Methods Nonlinear Anal. 22 (2003), no. 1, 193–202.
S.S. Almuthaybiri and C.C. Tisdell, Global existence theory for fractional differential equations: new advances via continuation methods for contractive maps, Analysis 39 (2019), no. 4, 117–128.
D.R. Anderson, Solutions to second-order three-point problems on time scales, J. Difference Equ. Appl. 8 (2002), no. 8, 673–688.
D. Ariza-Ruiz and A. Jimenez-Melado, A continuation method for weakly Kannan maps, Fixed Point Theory Appl. (2010), Art. ID 321594, 12 pp.
F.M. Atici and P.W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. (2009), Special Edition I, no. 3, 12 pp.
M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Boston, MA, 2001.
C. Chen, M. Bohner and B. Jia, Existence and uniqueness of solutions for nonlinear Caputo fractional difference equations, Turk. J. Math. 44 (2020), 857–869.
W.S. Cheung and J. Ren, Positive solution for discrete three-point boundary value problems, Aust. J. Math. Anal. Appl. 1 (2004), no. 2, Art. 9, 7 pp.
W.S. Cheung, J. Ren, P.J.Y. Wong and D. Zhao, Multiple positive solutions for discrete nonlocal boundary value problems, J. Math. Anal. Appl. 330 (2007), no. 2, 900–915.
C.S. Goodrich, On a discrete fractional three-point boundary value problem, J. Difference Equ. Appl. 18 (2012), no. 3, 397–415.
C.S. Goodrich, Coercivity of linear functionals on finite dimensional spaces and its application to discrete BVPs, J. Difference Equ. Appl. 22 (2016), no. 5, 623–636.
C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015.
A. Granas and J. Dugundji, Fixed Point Theory, Springer–Verlag, New York, 2003.
J. Hein, Z. McCarthy, N. Gaswick, B. McKain and K. Speer, Laplace transforms for the nabla-difference operator, PanAmer. Math. J. 21 (2011), no. 3, 79–97.
J. Henderson, S.K. Ntouyas and I.K. Purnaras, Positive solutions for systems of nonlinear discrete boundary value problems, J. Difference Equ. Appl. 15 (2009), no. 10, 895–912.
A. Ikram, Lyapunov inequalities for nabla Caputo boundary value problems, J. Difference Equ. Appl. 25 (2019), no. 6, 757–775.
J.M. Jonnalagadda, On a nabla fractional boundary value problem with general boundary conditions, AIMS Mathematics 5 (2019), no. 1, 204–215.
J.M. Jonnalagadda, On two-point Riemann–Liouville type nabla fractional boundary value problems, A dv. Dyn. Syst. Appl. 13 (2018), no. 2, 141–166.
W.G. Kelley and A.C. Peterson, Difference Equations. An Introduction with Applications, Second edition, Harcourt/Academic Press, San Diego, CA, 2001.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006.
W.A. Kirk and N. Shahzad, Continuation methods in certain metric and geodesic spaces, Bull. Math. Sci. 6 (2016), no. 2, 311–323.
X. Liu, B. Jia, S. Gensler, L. Erbe and A. Peterson, Convergence of approximate solutions to nonlinear Caputo nabla fractional difference equations with boundary conditions, Electron. J. Differential Equations (2020), no. 4, 19 pp.
R. Ma and Y.N. Raffoul, Positive solutions of three-point nonlinear discrete second order boundary value problem, J. Difference Equ. Appl. 10 (2004), no. 2, 129–138.
D. O’Regan, Fixed point theorems for nonlinear operators, J. Math. Anal. Appl. 202 (1996), no. 2, 413–432.
D. O’Regan and R. Precup, Theorems of Leray–Schauder Type and Applications, Gordon and Breach Science Publishers, Amsterdam, 2001.
D. O’Regan and R. Precup, Compression-expansion fixed point theorem in two norms and applications, J. Math. Anal. Appl. 309 (2005), no. 2, 383–391.
I. Podlubny, 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, CA, 1999.
R. Precup, Discrete continuation method for boundary value problems on bounded sets in Banach spaces. Fixed point theory with applications in nonlinear analysis, J. Comput. Appl. Math. 113 (2000), no. 1–2, 267–281.
R. Precup, Existence theorems for nonlinear problems by continuation methods, Nonlinear Anal. 30 (1997), no. 6, 3313–3322.
R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht, 2002.
R. Precup, On the continuation method and the nonlinear alternative for Caristi-type non-self-mappings, J. Fixed Point Theory Appl. 16 (2014), no. 1–2, 3–10.
J.R.L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc. (2) 74 (2006), no. 3, 673–693.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Jagan Mohan Jonnalagadda
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
