Existence and uniqueness results to a fractional q-difference coupled system with integral boundary conditions via topological degree theory

Document Type : Research Paper


1 Laboratory of Mathematics And Applied Sciences, University of Ghardaia, 47000, Algeria

2 Faculty of Sciences, Saad Dahlab University, Blida 1, Algeria


This paper aims to highlight the existence and uniqueness results for a coupled system of nonlinear fractional $q$-difference subject to nonlinear more general four-point boundary conditions are treated. Our analysis relies on two approaches, the topological degree for condensing maps via a priori estimate method and the Banach contraction principle fixed point theorem. Finally, Two examples illustrating the effectiveness of the theoretical results are presented.


[1] S. Abbas, M. Benchohra, B. Samet and Y. Zhou, Coupled implicit Caputo fractional q-difference systems, Adv.
Difference Equ. 2019 (2019) 527.
[2] C.R. Adams, On the linear ordinary q-difference equation, Ann. Math. 30 (1928) 195–205.
[3] R.P. Agarwal, Certain fractional q-integrals and q-derivatives, Proc. Camb. Philos. Soc. 66 (1969) 365–370.
[4] R.P. Agarwal and D. O’Regan, Toplogical degree theory and its applications, Tylor and Francis, 2006.
[5] G. Adomian and G.E. Adomian, Cellular systems and aging models, Comput. Math. Appl. 11 (1985) 283–291.
[6] B. Ahmad, A. Alsaedi and A. Assolami, Caputo type fractional differential equations with nonlocal RiemannLiouville integral boundary conditions, J. Appl. Math. Comput. 41 (2013) 339–350.
[7] B. Ahmad, S.K. Ntouyas and I.K. Purnaras, Existence results for nonlocal boundary value problems of nonlinear
fractional q–difference equations, Adv. Differnce Equ. 2012 (2012) 140.
[8] A. Ali, M. Sarwar, M. B. Zada and K. Shah, Degree theory and existence of positive solutions to coupled system
involving proportional delay with fractional integral boundary conditions, Math. Methods Appl. Sci. (2020). DOI:
[9] W.A. Al-Salam, Some fractional q-integrals and q-derivatives, Proc. Edinb. Math. Soc. 15 (1969) 135–140.
[10] M.H. Annaby and Z.S. Mansour, q–Fractional Calculus and Equations, Lecture Notes in Mathematics, vol. 2056,
Springer, Heidelberg, 2012.
[11] A. Ali, B. Samet, K. Shah and R.A. Khan, Existence and stability of solution to a toppled systems of differential
equations of non-integer order, Bound. Value Prob. 2017(1) (2017) 1–13.
[12] N. Ali, K. Shah, D. Baleanu, M. Arif and R.A. Khan, Study of a class of arbitrary order differential equations by
a coincidence degree method, Bound. Value Probl. 2017(1) (2017) 1–14.
[13] P.W. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear Dyn. Evol. Equ. Fields
Inst. Commun. 48 (2006) 13–52.
[14] M. Ba˘ca, P. Jeyanthi, N. T. Muthuraja, P. N. Selvagopal and A. Semani˘cov´a-Feˇnovˇc´ıkov´a, Ladders and fan graphs
are cycle-antimagic, Hacettepe J. Math. Statist. 2020 (2020) 1–14.
[15] Z. Baitiche, C. Derbazi and M. Benchohra, ψ-Caputo fractional differential equations with multi-point boundary
conditions by topological degree theory, Results Nonlinear Anal. 3(4) (2020) 166–178.
[16] A. Boutiara, K. Guerbati and M. Benbachir, Caputo-Hadamard fractional differential equation with three-point
boundary conditions in Banach spaces, AIMS Math. 5(1) (2020) 259–272.
[17] A. Boutiara, M. Benbachir, K. Guerbati, Measure Of Noncompactness for Nonlinear Hilfer Fractional Differential
Equation in Banach Spaces, Ikonion Journal of Mathematics, 1(2)(2019).
[18] A. Boutiara, M. Benbachir and K. Guerbati, Caputo type fractional differential equation with nonlocal Erd´elyiKober type integral boundary conditions in Banach spaces, Surv. Math. Appl. 15 (2020) 399–418.
[19] R.D. Carmichael, The general theory of linear q–difference equations, Am. J. Math. 34 (1912) 147–168.
[20] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
[21] N.R. Deepak, T. Ray and R.R. Boyce, Evolutionary algorithm shape optimization of a hypersonic flight experiment
nose cone, J. Spacecraft Rockets 45(3) (2008) 428–437.
[22] M. El-Shahed and H.A. Hassan, Positive solutions of q-difference equation, Proc. Am. Math. Soc. 138 (2010)
[23] T. Ernst, A Comprehensive Treatment of q-Calculus, Birkh¨auser, Basel, 2012.[24] S. Etemad, S.K. Ntouyas and B. Ahmad, Existence theory for a fractional q–integro-difference equation with
q–integral boundary conditions of different orders, Math. 7 (2019) Article ID 659.
[25] RAC. Ferreira, Nontrivial solutions for fractional q-difference boundary value problems, Electron. J. Qual. Theory
Differ. Equ. 2010 (2010) 70.
[26] S. Harikrishnan, K. Shah, D. Baleanu and K. Kanagarajan, Note on the solution of random differential equations
via ψ-Hilfer fractional derivative, Adv Differ Equ 2018, 224 (2018).
[27] F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian. 75 (2006) 233–240.
[28] F.H. Jackson, q-difference equations, Am. J. Math. 32 (1910) 305–314.
[29] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.
[30] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations,
North-Holland Mathematics Sudies, Elsevier Science, 2006.
[31] K.S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, wiley, New
York, 1993.
[32] K.B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41 (2010) 9–12.
[33] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1993.
[34] P.M. Rajkovic, S.D. Marinkovic and M.S. Stankovic, Fractional integrals and derivatives in q–calculus, Appl.
Anal. Discrete Math. 1 (2007) 311–323.
[35] P.M. Rajkovic, S.D. Marinkovic and M.S. Stankovic, On q-analogues of Caputo derivative and Mittag-Leffler
function, Fract. Calc. Appl. Anal. 10 (2007) 359–373.
[36] K. Shah, A. Ali and R. A. Khan, Degree theory and existence of positive solutions to coupled systems of multi-point
boundary value problems, Bound. Value Probl. 2016 (2016) 43.
[37] K. Shah and R. A. Khan, Existence and uniqueness results to a coupled system of fractional order boundary value
problems by topological degree theory, Numer. Funct. Anal. Optim. 37 (2016) 887–899.
[38] K. Shah and W. Hussain, Investigating a class of nonlinear fractional differential equations and its Hyers-Ulam
stability by means of topological degree theory, Numer. Funct. Anal. Optim. 40 (2019) 1355–1372.
[39] K. Shah and M. Akram, Numerical treatment of non-integer order partial differential equations by omitting
discretization of data, Comp. Appl. Math. 37 (2018) 6700—6718.
[40] K. Shah and W. Hussain, Investigating a class of nonlinear fractional differential equations and Its Hyers-Ulam
stability by means of topological degree theory, Numer. Funct. Anal. Optim. 40(12) (2019) 1355–1372.
[41] M. Sher, K. Shah, M. Feˇckan and R.A. Khan, Qualitative analysis of multi-terms fractional order delay differential
equations via the topological degree theory, Math. 8(2) (2020) 218.
[42] M. Sher, K. Shah, J. Rassias, On qualitative theory of fractional order delay evolution equation via the prior
estimate method. Mathematical Methods in the Applied Sciences, 43(10)(2020), 6464-6475.
[43] M. Shoaib, K. Shah and R. Ali Khan, Existence and uniqueness of solutions for coupled system of fractional
differential equation by means of topological degree method, J. Nonlinear Anal. Appl. 2018 (2018) 124–135.
[44] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and
Media, Springer, Heidelberg & Higher Education Press, Beijing, (2010).
[45] J. Wang, Y. Zhou and W. Wei, Study in fractional differential equations by means of topological degree methods,
Numer. Funct. Anal. Optim. 33 (2012) 216–238.
[46] M. B. Zada, K. Shah and R. A. Khan, Existence theory to a coupled system of higher order fractional hybrid
differential equations by topological degree theory, Int. J. Appl. Comput. Math. 4 (2018) 102.
Volume 13, Issue 1
March 2022
Pages 3197-3211
  • Receive Date: 27 November 2020
  • Revise Date: 03 February 2021
  • Accept Date: 04 March 2021
  • First Publish Date: 15 January 2022