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Existence of solution for a systems of coupled fractional boundary value problem

Year 2020, Volume: 2 Issue: 1, 48 - 59, 30.06.2020

Abstract

This paper deals with the existence and uniqueness of solutions for a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. The existence results are obtained by using Leray-Shauder nonlinear alternative and Banach contraction principle. An illustrative example is presented at the end of the paper to illustrate the validity of our results.

References

  • [1] C. R. Adamas, On the linear ordinary q-difference equation, Ann. Math., 30 (1928), 195-205.
  • [2] O. Agrawal, Some generalized fractional calculus operators and their applications in integralequations, Fract. Cal. Appl. Anal., 15 (2012), 700-711.
  • [3] R. P. Agarwal, D. Baleanu, V. Hedayati, Sh. Rezapour, Two fractional derivative inclusion problems via integral boundary condition, Appl. Math. Comput., 257 (2015), 205-212.
  • [4] N. Bouteraa and S. Benaicha, Existence of solutions for three-point boundary value problem for nonlinear fractional equations, Analele Universitatii Oradia Fasc. Mathematica, Tom XXIV (2017), Issue No. 2, 109-119.
  • [5] N. Bouteraa, S. Benaicha and H. Djourdem, Positive solutions for nonlinear fractional equation with nonlocal boundary conditions, Universal journal of Mathematics and Applications. 1 (1) (2018), 39-45.
  • [6] N. Bouteraa and S. Benaicha, The uniqueness of positive solution for higher-order nonlinear fractional differential equation with nonlocal boundary conditions, Advances in the Theory of Nonlinear and it Application, 2(2018) No 2, 74-84.
  • [7] N. Bouteraa and S. Benaicha, The uniqueness of positive solution for nonlinear fractional differential equation with nonlocal boundary conditions, Analele Universitatii Oradia Fasc. Mathematica, Tom XXIV (2018), Issue No. 2, 53-65. 19
  • [8] N. Bouteraa, S. Benaicha, Existence of solution for nonlocal boundary value problem for Caputo nonlinear fractional inclusion, Journal of Mathematical Sciences and Modeling. vol.1 (1) (2018), 45-55.
  • [9] N. Bouteraa, S. Benaicha, Existence results for fractional differential inclusion with nonlocal boundary conditions, Riv. Mat. Univ. Parma, To appear
  • [10] S. Benaicha and N. Boutetaa, Positive solutions for systems of fourth-order two-point boundary value problems with parameter, Journal of Mathematical Sciences and Modelling, 2 (1) (2019), 30-38.
  • [11] H. Djourdem and S. Benaicha, Triple positive solutions for a fractional boundary value problem, Maltepe Journal of Mathematics. Volume I, Issue 2 (2019), pages 96-109.
  • [12] Y. Ding, Z. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Technol. 20 (2012), 763-769.
  • [13] N. J. Ford, M. L Morgado, Fractional boundary value problems: analysis and numerical methods, Frac. Calc. Appl. Anal. 14(2011), 554-567.
  • [14] A. Granas and J. Dugundji, Fixed Point Theory. Springer, NewYork (2005).
  • [15] J. Henderson, R. Luca and A. Tudorache, On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 18 (2015), 361-386.
  • [16] G. Infante and P. Pietramala, Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions. Math. Methods Appl. Sci. 37(14) (2014), 2080-2090.
  • [17] A. A. Kilbas, H. M. Srivastava, J. J. Trijullo, Theory and applications of fractional differential equations, Elsevier Science b. V, Amsterdam, (2006). 25
  • [18] S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc.; New York, (1993).
  • [19] D. Mozyrska, Z. Bartosiewicz, On Observability of Nonlinear Discrete-Time Fractional Order Control Systems New Trends in Nanotechnology and Fractional calculus Applications (2010), 305-312.
  • [20] N. Pongarm, S. Asawasamrit, J. Tariboon, sequential derivatives of nonlinear q-difference equations with three-point q-integrale boundary conditions, J. Appl. Math., 2013 (2013), 9 pages.
  • [21] P. M. Rajkovic, S. D. Marancovic, M. S. Stankovic, On q-analogues of Caputo derivative and Mettag-Leffter function, Frac. Calc. Appl. Anal. 10(2007), 359-373.
  • [22] A. Ral, V. Gupta, R. P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, (2013).
  • [23] D. R. Smart, Fixed Point Theorems, Compbridge University Press, London-New York, (1974). 1.4
  • [24] B. Senol, C. Yeroglu, Frequency boundary of fractional order systems with nonlinear uncertainties. J. Franklin Inst. 350 (2013), 1908-1925.
  • [25] I. M. Sokolov, J. Klafter and A. Blumen, Fractional kinetics. Phys.Today 55 (2002), 48-54.
  • [26] J. Tariboon, S. K. Ntouyas, W. Sudsutad, Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions. J. Nonlinear Sci. Appl. 9 (2016), 295-308.
  • [27] C. Zhai and L. Xu, Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul. 19 (2014),2820-2827.
  • [28] X. G. Zhang, S. L. Liu, Y. H. Wu, Theuniqueness of positive solutions for a fractional order mode lof turbulent flow in a porousme dium. Appl. Math. Lett. 37(2014), 26-33.
  • [29] X. G. Zhang, C. L. Mao, Y. H. Wu, H. Su, Positive solutions of a singular nonlocal fractional order differential system via Schauder’s fixed point theorem. Abstr. Appl. Anal. 2014.
Year 2020, Volume: 2 Issue: 1, 48 - 59, 30.06.2020

Abstract

References

  • [1] C. R. Adamas, On the linear ordinary q-difference equation, Ann. Math., 30 (1928), 195-205.
  • [2] O. Agrawal, Some generalized fractional calculus operators and their applications in integralequations, Fract. Cal. Appl. Anal., 15 (2012), 700-711.
  • [3] R. P. Agarwal, D. Baleanu, V. Hedayati, Sh. Rezapour, Two fractional derivative inclusion problems via integral boundary condition, Appl. Math. Comput., 257 (2015), 205-212.
  • [4] N. Bouteraa and S. Benaicha, Existence of solutions for three-point boundary value problem for nonlinear fractional equations, Analele Universitatii Oradia Fasc. Mathematica, Tom XXIV (2017), Issue No. 2, 109-119.
  • [5] N. Bouteraa, S. Benaicha and H. Djourdem, Positive solutions for nonlinear fractional equation with nonlocal boundary conditions, Universal journal of Mathematics and Applications. 1 (1) (2018), 39-45.
  • [6] N. Bouteraa and S. Benaicha, The uniqueness of positive solution for higher-order nonlinear fractional differential equation with nonlocal boundary conditions, Advances in the Theory of Nonlinear and it Application, 2(2018) No 2, 74-84.
  • [7] N. Bouteraa and S. Benaicha, The uniqueness of positive solution for nonlinear fractional differential equation with nonlocal boundary conditions, Analele Universitatii Oradia Fasc. Mathematica, Tom XXIV (2018), Issue No. 2, 53-65. 19
  • [8] N. Bouteraa, S. Benaicha, Existence of solution for nonlocal boundary value problem for Caputo nonlinear fractional inclusion, Journal of Mathematical Sciences and Modeling. vol.1 (1) (2018), 45-55.
  • [9] N. Bouteraa, S. Benaicha, Existence results for fractional differential inclusion with nonlocal boundary conditions, Riv. Mat. Univ. Parma, To appear
  • [10] S. Benaicha and N. Boutetaa, Positive solutions for systems of fourth-order two-point boundary value problems with parameter, Journal of Mathematical Sciences and Modelling, 2 (1) (2019), 30-38.
  • [11] H. Djourdem and S. Benaicha, Triple positive solutions for a fractional boundary value problem, Maltepe Journal of Mathematics. Volume I, Issue 2 (2019), pages 96-109.
  • [12] Y. Ding, Z. Wang and H. Ye, Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Technol. 20 (2012), 763-769.
  • [13] N. J. Ford, M. L Morgado, Fractional boundary value problems: analysis and numerical methods, Frac. Calc. Appl. Anal. 14(2011), 554-567.
  • [14] A. Granas and J. Dugundji, Fixed Point Theory. Springer, NewYork (2005).
  • [15] J. Henderson, R. Luca and A. Tudorache, On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 18 (2015), 361-386.
  • [16] G. Infante and P. Pietramala, Multiple nonnegative solutions of systems with coupled nonlinear boundary conditions. Math. Methods Appl. Sci. 37(14) (2014), 2080-2090.
  • [17] A. A. Kilbas, H. M. Srivastava, J. J. Trijullo, Theory and applications of fractional differential equations, Elsevier Science b. V, Amsterdam, (2006). 25
  • [18] S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc.; New York, (1993).
  • [19] D. Mozyrska, Z. Bartosiewicz, On Observability of Nonlinear Discrete-Time Fractional Order Control Systems New Trends in Nanotechnology and Fractional calculus Applications (2010), 305-312.
  • [20] N. Pongarm, S. Asawasamrit, J. Tariboon, sequential derivatives of nonlinear q-difference equations with three-point q-integrale boundary conditions, J. Appl. Math., 2013 (2013), 9 pages.
  • [21] P. M. Rajkovic, S. D. Marancovic, M. S. Stankovic, On q-analogues of Caputo derivative and Mettag-Leffter function, Frac. Calc. Appl. Anal. 10(2007), 359-373.
  • [22] A. Ral, V. Gupta, R. P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, (2013).
  • [23] D. R. Smart, Fixed Point Theorems, Compbridge University Press, London-New York, (1974). 1.4
  • [24] B. Senol, C. Yeroglu, Frequency boundary of fractional order systems with nonlinear uncertainties. J. Franklin Inst. 350 (2013), 1908-1925.
  • [25] I. M. Sokolov, J. Klafter and A. Blumen, Fractional kinetics. Phys.Today 55 (2002), 48-54.
  • [26] J. Tariboon, S. K. Ntouyas, W. Sudsutad, Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions. J. Nonlinear Sci. Appl. 9 (2016), 295-308.
  • [27] C. Zhai and L. Xu, Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul. 19 (2014),2820-2827.
  • [28] X. G. Zhang, S. L. Liu, Y. H. Wu, Theuniqueness of positive solutions for a fractional order mode lof turbulent flow in a porousme dium. Appl. Math. Lett. 37(2014), 26-33.
  • [29] X. G. Zhang, C. L. Mao, Y. H. Wu, H. Su, Positive solutions of a singular nonlocal fractional order differential system via Schauder’s fixed point theorem. Abstr. Appl. Anal. 2014.
There are 29 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Articles
Authors

Djourdem Habib 0000-0002-7992-581X

Publication Date June 30, 2020
Acceptance Date June 26, 2020
Published in Issue Year 2020 Volume: 2 Issue: 1

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