Articles

ULAM-HYERS-RASSIAS STABILITY AND EXISTENCE OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH MULTIPOINT SUMMATION BOUNDARY CONDITION

  • Syed Sabyel HAIDER ,
  • Mujeeb Ur REHMAN
Expand
  • School of Natural Sciences, National University of Sciences and Technology, H-12 Islamabad, 44000, Pakistan

Received date: 2018-10-26

  Revised date: 2019-04-25

  Online published: 2020-05-26

Abstract

The purpose of this study is to acquire some conditions that reveal existence and stability for solutions to a class of difference equations with non-integer order μ∈ (1, 2]. The required conditions are obtained by applying the technique of contraction principle for uniqueness and Schauder's fixed point theorem for existence. Also, we establish some conditions under which the solution of the considered class of difference equations is generalized Ulam-Hyers-Rassias stable. Example for the illustration of results is given.

Cite this article

Syed Sabyel HAIDER , Mujeeb Ur REHMAN . ULAM-HYERS-RASSIAS STABILITY AND EXISTENCE OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH MULTIPOINT SUMMATION BOUNDARY CONDITION[J]. Acta mathematica scientia, Series B, 2020 , 40(2) : 589 -602 . DOI: 10.1007/s10473-020-0219-1

References

[1] Atici F M, Eloe P W. Discrete fractional calculus with the nabla operator. Electron J Qual Theory Differ Equ, 2009, 3:1-12
[2] Atici F M, Eloe P W. Initial value problems in discrete fractional calculus. Proc Amer Math Soc, 2009, 137(3):981-989
[3] Anastassiou G A. Nabla discrete fractional calculus and nabla inequalities. Math Comput Modelling, 2010, 51(5):562-571
[4] Abdeljawad T. On riemann and caputo fractional differences. Comput Math Appl, 2011, 62(3):1602-1611
[5] Abdeljawad T, Baleanu D. Fractional differences and integration by parts. J Comput Anal Appl, 2011, 13(3):574-582
[6] Cheng J F, Chu Y M. On the fractional difference equations of order (2, q). Abstr Appl Anal, 2011, 2011:1-16
[7] Cheng J F. Solutions of fractional difference equations of order (k, q). Acta Math Appl Sin, 2011, 34(2):313-330
[8] Bohner M, Peterson A. Dynamic Equations on Time Scales an Introduction with Applications. New York:Birkhauser Boston, 2001
[9] Gray H, Zhang N. On a New Definition of the Fractional Difference. Math Comput, 1988, 50:513-529
[10] Atici F M, Eloe P W. A transform method in discrete fractional calculus. Int J Differ Equ, 2007, 2:165-176
[11] Rehman M U, Iqbal F, Seemab A. On existence of positive solutions for a class of discrete fractional boundary value problems. Positivity, 2017, 21:1173-1187
[12] Nefzi B, Brahim K, Fitouhi A. On the finite Mellin transform in quantum calculus and application. Acta Math Sin, 2018, 38(4):1393-1410
[13] Chen F, Zhou Y. Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem. Discrete Dyn Nat Soc, 2013, 2013:1-7
[14] Tian Y, Ma D, Ge W. Multiple positive solutions of four point boundary value problems for finite difference equations. J Differ Equ Appl, 2006, 12:57-68
[15] Atici F M, Eloe P W. Two-point boundary value problems for finite fractional difference equations. J Differ Equ Appl, 2011, 17:445-456
[16] Goodrich C S. On discrete sequential fractional boundary value problems. J Math Anal Appl, 2012, 385:111-124
[17] Pan Y, Han Z, Sun S, Zhao Y. The existence of solutions to a system of discrete fractional boundary value problems. Abstr Appl Anal, 2012, 2012:1-15
[18] Ferreira R A C. Positive solutions for a class of boundary value problems with fractional q-differences. Comput Math Appl, 2011, 61:367-373
[19] Goodrich C S. Existence of a positive solution to a system of discrete fractional boundary value problems. Appl Math Comput, 2011, 217:4740-4753
[20] Henderson J, Ntouyas S, Purnaras I. Positive solutions for systems of nonlinear discrete boundary value problem. J Differ Equ Appl, 2009, 15:895-912
[21] Pan Y, Han Z, Sun S, Huang Z. The existence and uniqueness of solutions to boundary value problems of fractional difference equations. Math Sci, 2012, 6(1):1-7
[22] Ferreira R A C. Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one. J Differ Equ Appl, 2013, 19:712-718
[23] Goodrich C S. On a discrete fractional three-point boundary value problem. J Differ Equ Appl, 201218:397-415
[24] Chen Y, Tang X. The difference between a class of discrete fractional and integer order boundary value problems. Commun Nonlinear Sci Numer Simul, 2014, 19:4057-4067
[25] Chen H Q, Cui Y Q, Zhao X L. Multiple solutions to fractional difference boundary value problems. Abstr Appl Anal, 2014, Article ID 879380:1-6
[26] Chen H Q, Jin Z, Kang S G. Existence of positive solutions for Caputo fractional difference equation. Adv Differ Equ, 2015, 44:1-12
[27] Dong W, Xu J, Regan D O. Solutions for a fractional difference boundary value problem. Adv Differ Equ, 2013, 1:319
[28] Sitthiwirattham T. Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three-point fractional sum boundary conditions. Math Methods Appl Sci, 2015, 38:2809-2815
[29] Sitthiwirattham T. Boundary value problem for p-Laplacian Caputo fractional difference equations with fractional sum boundary conditions. Math Methods Appl Sci, 2016, 39:1522-1534
[30] Reunsumrit J, Sitthiwirattham T. On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations. Math Methods Appl Sci, 2016, 39:2737-2751
[31] Soontharanon J, Jasthitikulchai N, Sitthiwirattham T. Nonlocal fractional sum boundary value problems for mixed types of Riemann-Liouville and Caputo fractional difference equations. Dynam Systems Appl, 2016, 25:409-429
[32] Laoprasittichok S, Sitthiwirattham T. Existence and uniqueness results of nonlocal fractional sum-difference boundary value problems for fractional difference equations involving sequential fractional difference operators. J Comput Anal Appl, 2017, 23:1097-1111
[33] Kaewwisetkul B, Sitthiwirattham T. On nonlocal fractional sum-difference boundary value problems for Caputo fractional functional difference equations with delay. Adv Differ Equ, 2017, 1:219
[34] Reunsumrit J, Sitthiwirattham T. Positive solutions of three-point fractional sum boundary value problem for Caputo fractional difference equations via an argument with a shift. Positivity, 2016, 20(4):861-876
[35] Goodrich C S. On semipositone discrete fractional boundary value problems with non-local boundary conditions. J Differ Equ Appl, 2013, 19(11):1758-1780
[36] Costabile F A, Napoli A. A multipoint Birkhoff type boundary value problem. J Numer Math, 2015, 23:1-11
[37] Agarwal R P. On multipoint boundary value problems for discrete equations. J Math Anal Appl, 1983, 96:520-534
[38] Rassias T M. On the stability of linear mappings in Banach spaces. Proc Amer Math Soc, 1978, 72:297-300
[39] Jonnalagadda J M. Hyers-Ulam Stability of Fractional Nabla Difference Equations. Int J Anal, 2016:1-5
[40] Ulam S M. A collection of mathematical problems. New York:Interscience Publishers, 1961
[41] Hyers H D, Isac G, Rassias T. Stability of functional equations in several variables. 34. Springer Science and Business Media, 2012
[42] Jung S M. Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis. 48. Springer Science and Business Media, 2011
[43] Feng Q. Some new generalized Gronwall-Bellman type discrete fractional inequalities. Appl Math Comput, 2015, 259:403-411
[44] Goodrich C S, Peterson A. Discrete Fractional Calculus. New York:Springer, 2010
[45] Zeidler E. Nonlinear Functional Analysis and its Applications Part 1, Fixed Point Theorems. New York:Springer, 1986
Options
Outlines

/