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Year 2020, Volume: 4 Issue: 2, 77 - 91, 30.06.2020

Md. Asaduzzaman Md. Zulfikar Ali

https://doi.org/10.31197/atnaa.647561

Abstract

References

  • [1] S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin, Heidelberg (2011).
  • [2] K. Diethelm, The Analysis of Fractional Differential Equations, Springer (2010).
  • [3] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, 84. Springer, Dordrecht (2011).
  • [4] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing (2010).
  • [5] V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK (2009).
  • [6] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, Berlin, Heidelberg (2008).
  • [7] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands (2007).
  • [8] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands (2006).
  • [9] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).
  • [10] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., Academic Press, San Diego, New York, 1999.
  • [11] S. Abbas, M. Benchohra and G.M. N’Gu´er´ekata, Topics in Fractional Differential Equations, Springer, New York (2012).
  • [12] S. Abbas and M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009), 62-72.
  • [13] S. Abbas and M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay, Nonlinear Anal. Hybrid Syst. 3 (2009), 597-604.
  • [14] R.P. Agarwal, M. Benchohra and S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (3) (2010), 973-1033.
  • [15] M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York 2 (2006).
  • [16] M. Benchohra and B. A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential Equations 2009 (2009), No. 10, 11 pp.
  • [17] A. N. Vityuk and A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (3) (2004), 318-325.
  • [18] M. Benchohra and Z. Boutefal, Impulsive Differential Equations of Fractional Order with Infinite Delay, J. Frac. Cal. Appl. 4(2) (2013), 209-223.
  • [19] S. Abbas and M. Benchohra, Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 51, 2 (2012) 5-18.
  • [20] S. Abbas and M. Benchohra, Darboux problem for implicit impulsive partial hyperbolic fractional order differential equations. Electron. J. Differential Equations (2011), No. 150, 14 pp.
  • [21] S. Abbas and M. Benchohra, Upper and Lower Solutions Method for Partial Hyperbolic Differential Equations with Caputo’s Fractional Derivative, LIBERTAS MATHEMATICA, vol XXXI (2011), pp. 103-110.
  • [22] S. Abbas and M. Benchohra, Darboux problem for partial functional differential equations with infinite delay and Caputo’s fractional derivative. Adv. Dyn. Syst. Appl. 5 (2010), no. 1,1-19.
  • [23] J. K. Hale and S. Verduyn Lunel, Introduction to Functional -Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York (1993).
  • [24] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin (1991).
  • [25] V. Kolmanovskii, and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, Kluwer Academic Publishers, Dordrecht (1999).
  • [26] V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (1994).
  • [27] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer (2011).
  • [28] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York (1996).
  • [29] C. Corduneanu and V. Lakshmikantham, Equations with unbounded delay, Nonlinear Anal.4 (1980), 831-877.
  • [30] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21, (1978),11-41.
  • [31] T. Czlapinski, On the Darboux problem for partial differential-functional equations with infinite delay at derivatives. Nonlinear Anal. 44 (2001), 389-398.
  • [32] T. Czlapinski, Existence of solutions of the Darboux problem for partial differential-functional equations with infinite delay in a Banach space. Comment. Math. Prace Mat. 35 (1995), 111-122.
  • [33] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989).
  • [34] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).
  • [35] T. A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998), 23-31.
  • [36] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York (2003).
  • [37] D. Henry, Geometric theory of Semilinear Parabolic Partial Differential Equations, Springer-Verlag, Berlin-New York (1989).
  • [38] E. Hernandez, A. Anguraj and M. Mallika Arjunan, Existence results for an impulsive second order differential equation with state-dependent delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17 (2010), 287-301.
  • [39] M. Fréchet, Sur quelques points du calculfonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1-74.

Existence of Solution to Fractional Order Impulsive Partial Hyperbolic Differential Equations with Infinite Delay

Year 2020, Volume: 4 Issue: 2, 77 - 91, 30.06.2020

Md. Asaduzzaman Md. Zulfikar Ali

https://doi.org/10.31197/atnaa.647561

Abstract



In this paper, we
investigate the existence of solutions to a class of initial value problem for fractional
order impulsive partial hyperbolic differential equations (for short FOIPHDEs) with
infinite delay. Here we use Mixed Riemann-Liouville fractional derivative to
construct the considered FOIPHDEs. The analysis of this paper is based on Burton-Krik fixed point theorem. A new
existence result for FOIPHDEs with
infinite delay has been obtained. To support the analytic proof, we give an illustrative example.




Keywords

FOIPHDEs Burton-Krik fixed point theorem

References

  • [1] S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin, Heidelberg (2011).
  • [2] K. Diethelm, The Analysis of Fractional Differential Equations, Springer (2010).
  • [3] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering, 84. Springer, Dordrecht (2011).
  • [4] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing (2010).
  • [5] V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK (2009).
  • [6] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, Berlin, Heidelberg (2008).
  • [7] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands (2007).
  • [8] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands (2006).
  • [9] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).
  • [10] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng., Academic Press, San Diego, New York, 1999.
  • [11] S. Abbas, M. Benchohra and G.M. N’Gu´er´ekata, Topics in Fractional Differential Equations, Springer, New York (2012).
  • [12] S. Abbas and M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009), 62-72.
  • [13] S. Abbas and M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay, Nonlinear Anal. Hybrid Syst. 3 (2009), 597-604.
  • [14] R.P. Agarwal, M. Benchohra and S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (3) (2010), 973-1033.
  • [15] M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York 2 (2006).
  • [16] M. Benchohra and B. A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential Equations 2009 (2009), No. 10, 11 pp.
  • [17] A. N. Vityuk and A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (3) (2004), 318-325.
  • [18] M. Benchohra and Z. Boutefal, Impulsive Differential Equations of Fractional Order with Infinite Delay, J. Frac. Cal. Appl. 4(2) (2013), 209-223.
  • [19] S. Abbas and M. Benchohra, Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 51, 2 (2012) 5-18.
  • [20] S. Abbas and M. Benchohra, Darboux problem for implicit impulsive partial hyperbolic fractional order differential equations. Electron. J. Differential Equations (2011), No. 150, 14 pp.
  • [21] S. Abbas and M. Benchohra, Upper and Lower Solutions Method for Partial Hyperbolic Differential Equations with Caputo’s Fractional Derivative, LIBERTAS MATHEMATICA, vol XXXI (2011), pp. 103-110.
  • [22] S. Abbas and M. Benchohra, Darboux problem for partial functional differential equations with infinite delay and Caputo’s fractional derivative. Adv. Dyn. Syst. Appl. 5 (2010), no. 1,1-19.
  • [23] J. K. Hale and S. Verduyn Lunel, Introduction to Functional -Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York (1993).
  • [24] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin (1991).
  • [25] V. Kolmanovskii, and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, Kluwer Academic Publishers, Dordrecht (1999).
  • [26] V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (1994).
  • [27] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer (2011).
  • [28] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York (1996).
  • [29] C. Corduneanu and V. Lakshmikantham, Equations with unbounded delay, Nonlinear Anal.4 (1980), 831-877.
  • [30] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21, (1978),11-41.
  • [31] T. Czlapinski, On the Darboux problem for partial differential-functional equations with infinite delay at derivatives. Nonlinear Anal. 44 (2001), 389-398.
  • [32] T. Czlapinski, Existence of solutions of the Darboux problem for partial differential-functional equations with infinite delay in a Banach space. Comment. Math. Prace Mat. 35 (1995), 111-122.
  • [33] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989).
  • [34] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).
  • [35] T. A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998), 23-31.
  • [36] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York (2003).
  • [37] D. Henry, Geometric theory of Semilinear Parabolic Partial Differential Equations, Springer-Verlag, Berlin-New York (1989).
  • [38] E. Hernandez, A. Anguraj and M. Mallika Arjunan, Existence results for an impulsive second order differential equation with state-dependent delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 17 (2010), 287-301.
  • [39] M. Fréchet, Sur quelques points du calculfonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1-74.
There are 39 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Md. Asaduzzaman 0000-0001-7133-9317

Md. Zulfikar Ali This is me

Publication Date June 30, 2020
Published in Issue Year 2020 Volume: 4 Issue: 2

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