Research Article
BibTex RIS Cite

On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations

Year 2019, , 39 - 46, 30.04.2019
https://doi.org/10.36753/mathenot.559244

Abstract

In this paper we provide sufficient condition guaranteeing existence and the asymptotic behavior of
solutions of a class of Hadamard–Volterra integral equations in the Banach space of continuous and
bounded functions on unbounded interval. The main tools used in our considerations are the concept of
measure of noncompactness in conjunction with the Darbo and Mönch fixed point theorems.

References

  • [1] Abbas, S., Benchohra, M., Graef, J. and Henderson, J., Implicit Fractional Differential and Integral Equations; Existence and Stability, De Gruyter, Berlin, 2018.
  • [2] Abbas, S., Benchohra, M., and N’Guérékata, G. M., Topics in Fractional Differential Equations, Springer, New York, 2012.
  • [3] Abbas, S., Benchohra, M. and N’Guérékata, G. M., Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
  • [4] Abbas, S., Benchohra, M. and Henderson, J., Global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order Comm. Appl. Nonlinear Anal.19 (2012), no. 1, 79–89.
  • [5] Abbas, S., Alaidarous, E., Benchohra, M. and Nieto, J.J., Existence and stability of solutions for Hadamard- Stieltjes fractional integral equations. Discrete Dyn. Nat. Soc. 2015, Art. ID 317094, 6 pp.
  • [6] Appell, J., Measure of noncompactness, condensing operators and fixed points: an application-oriented survey Fixed Point Theory, 6 (2) (2005), 157-229.
  • [7] Agarwal, R.P. and O’Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations Kluwer Academic Publishers, Dordrecht, 2001.
  • [8] Ahmad, B. and Ntouyas, S.K., Initial value problems of fractional order Hadamard-Type functional differential equations Electron. J. Differential Equations 2015, No. 77, 9 pp.
  • [9] Banas, J. and O’Regan, D., On existence and local attractivity of solutions of a quadratic Volterra integral equations of fractional order J. Math. Anal. Appl. 345 (2008), no. 1, 573–582.
  • [10] Banas, J. and Rzepka, B., Nondecreasing solutions of a quadratic singular Volterra integral equation Math. Comput. Modelling 49 (2009), no. 3-4, 488–496.
  • [11] Banas, J., Rocha Martin, J. and Sadarangani, K., On solutions of a quadratic integral equations of Hammerstein type Math. Comput. Modelling 43 (2006), no. 1-2, 97–104.
  • [12] Banas, J., Rocha Martin, J. and Sadarangani, K., On existence and asymptotic stability of solutions of a nonlinear integral equation J. Math. Anal. Appl. 284 (2003), no. 1, 165–173.
  • [13] Banas, J., Caballero, J., Rocha, J. and Sadarangani, K., Global asymptotic stability of solutions of a functional integral equation Nonlinear Anal. 69 (2008), no. 7, 1945–1952.
  • [14] Banas, J. and Dhage, B.C., Monotonic solutions of a class of quadratic integral equations of Volterra type Comput. Math. Appl. 49 (2005), no. 5-6, 943–952.
  • [15] Banas, J. and Rzepka, B., On existence and asymptotic stability of solutions of a nonlinear integral equation J. Math. Anal. Appl 284 (2003), no. 1, 165–173.
  • [16] Banas, J., Cabrera, J. and Sadarangani, K., On existence and asymptotic behaviour of solutions of a functional integral equation Nonlinear Anal. 66 (2007), no. 10, 2246–2254.
  • [17] Baskonus, H. M. and Bulut, H., On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method. Open Math. 13 (2015), 547–556.
  • [18] Benchohra, M., Henderson, J. and Seba, D., Measure of noncompactness and fractional differential equations in Banach spaces, PanAmer. Math. J. 20 (2010), no. 3, 27–37.
  • [19] Corduneanu, C., Integral Equations and Applications Cambridge Univ. Press, Cambridge, 1991.
  • [20] González, C., Melado, A.J. and Fuster, E.L., A Mönch type fixed point theorem under the interior condition J. Math. Anal. Appl. 352 (2009), 816-821.
  • [21] Katugampola, U., New approach to generalized fractional integral Appl. Math. Comput. 218 (2011), no. 3, 860–865.
  • [22] Kilbas, A., Hadamard-Type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191–1204.
  • [23] Samko, S., Kilbas, A. and Marichev, O.I., Fractional Integrals and Derivatives (Theorie and Applications) Gordon and Breach Science Publishers. Yverdon, 1993.
  • [24] Thiramanus, P., Tariboon, J. and Ntouyas, S.K., Integrals inequalities with ’maxima’ and their applications to Hadamard-Type fractional differential equations J. Inequal. Appl. 2014, 2014:398, 15 pp.
Year 2019, , 39 - 46, 30.04.2019
https://doi.org/10.36753/mathenot.559244

Abstract

References

  • [1] Abbas, S., Benchohra, M., Graef, J. and Henderson, J., Implicit Fractional Differential and Integral Equations; Existence and Stability, De Gruyter, Berlin, 2018.
  • [2] Abbas, S., Benchohra, M., and N’Guérékata, G. M., Topics in Fractional Differential Equations, Springer, New York, 2012.
  • [3] Abbas, S., Benchohra, M. and N’Guérékata, G. M., Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
  • [4] Abbas, S., Benchohra, M. and Henderson, J., Global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order Comm. Appl. Nonlinear Anal.19 (2012), no. 1, 79–89.
  • [5] Abbas, S., Alaidarous, E., Benchohra, M. and Nieto, J.J., Existence and stability of solutions for Hadamard- Stieltjes fractional integral equations. Discrete Dyn. Nat. Soc. 2015, Art. ID 317094, 6 pp.
  • [6] Appell, J., Measure of noncompactness, condensing operators and fixed points: an application-oriented survey Fixed Point Theory, 6 (2) (2005), 157-229.
  • [7] Agarwal, R.P. and O’Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations Kluwer Academic Publishers, Dordrecht, 2001.
  • [8] Ahmad, B. and Ntouyas, S.K., Initial value problems of fractional order Hadamard-Type functional differential equations Electron. J. Differential Equations 2015, No. 77, 9 pp.
  • [9] Banas, J. and O’Regan, D., On existence and local attractivity of solutions of a quadratic Volterra integral equations of fractional order J. Math. Anal. Appl. 345 (2008), no. 1, 573–582.
  • [10] Banas, J. and Rzepka, B., Nondecreasing solutions of a quadratic singular Volterra integral equation Math. Comput. Modelling 49 (2009), no. 3-4, 488–496.
  • [11] Banas, J., Rocha Martin, J. and Sadarangani, K., On solutions of a quadratic integral equations of Hammerstein type Math. Comput. Modelling 43 (2006), no. 1-2, 97–104.
  • [12] Banas, J., Rocha Martin, J. and Sadarangani, K., On existence and asymptotic stability of solutions of a nonlinear integral equation J. Math. Anal. Appl. 284 (2003), no. 1, 165–173.
  • [13] Banas, J., Caballero, J., Rocha, J. and Sadarangani, K., Global asymptotic stability of solutions of a functional integral equation Nonlinear Anal. 69 (2008), no. 7, 1945–1952.
  • [14] Banas, J. and Dhage, B.C., Monotonic solutions of a class of quadratic integral equations of Volterra type Comput. Math. Appl. 49 (2005), no. 5-6, 943–952.
  • [15] Banas, J. and Rzepka, B., On existence and asymptotic stability of solutions of a nonlinear integral equation J. Math. Anal. Appl 284 (2003), no. 1, 165–173.
  • [16] Banas, J., Cabrera, J. and Sadarangani, K., On existence and asymptotic behaviour of solutions of a functional integral equation Nonlinear Anal. 66 (2007), no. 10, 2246–2254.
  • [17] Baskonus, H. M. and Bulut, H., On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method. Open Math. 13 (2015), 547–556.
  • [18] Benchohra, M., Henderson, J. and Seba, D., Measure of noncompactness and fractional differential equations in Banach spaces, PanAmer. Math. J. 20 (2010), no. 3, 27–37.
  • [19] Corduneanu, C., Integral Equations and Applications Cambridge Univ. Press, Cambridge, 1991.
  • [20] González, C., Melado, A.J. and Fuster, E.L., A Mönch type fixed point theorem under the interior condition J. Math. Anal. Appl. 352 (2009), 816-821.
  • [21] Katugampola, U., New approach to generalized fractional integral Appl. Math. Comput. 218 (2011), no. 3, 860–865.
  • [22] Kilbas, A., Hadamard-Type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191–1204.
  • [23] Samko, S., Kilbas, A. and Marichev, O.I., Fractional Integrals and Derivatives (Theorie and Applications) Gordon and Breach Science Publishers. Yverdon, 1993.
  • [24] Thiramanus, P., Tariboon, J. and Ntouyas, S.K., Integrals inequalities with ’maxima’ and their applications to Hadamard-Type fractional differential equations J. Inequal. Appl. 2014, 2014:398, 15 pp.
There are 24 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Said Baghdad This is me

Mouffak Benchohra

Publication Date April 30, 2019
Submission Date May 8, 2018
Published in Issue Year 2019

Cite

APA Baghdad, S., & Benchohra, M. (2019). On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations. Mathematical Sciences and Applications E-Notes, 7(1), 39-46. https://doi.org/10.36753/mathenot.559244
AMA Baghdad S, Benchohra M. On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations. Math. Sci. Appl. E-Notes. April 2019;7(1):39-46. doi:10.36753/mathenot.559244
Chicago Baghdad, Said, and Mouffak Benchohra. “On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations”. Mathematical Sciences and Applications E-Notes 7, no. 1 (April 2019): 39-46. https://doi.org/10.36753/mathenot.559244.
EndNote Baghdad S, Benchohra M (April 1, 2019) On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations. Mathematical Sciences and Applications E-Notes 7 1 39–46.
IEEE S. Baghdad and M. Benchohra, “On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations”, Math. Sci. Appl. E-Notes, vol. 7, no. 1, pp. 39–46, 2019, doi: 10.36753/mathenot.559244.
ISNAD Baghdad, Said - Benchohra, Mouffak. “On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations”. Mathematical Sciences and Applications E-Notes 7/1 (April 2019), 39-46. https://doi.org/10.36753/mathenot.559244.
JAMA Baghdad S, Benchohra M. On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations. Math. Sci. Appl. E-Notes. 2019;7:39–46.
MLA Baghdad, Said and Mouffak Benchohra. “On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations”. Mathematical Sciences and Applications E-Notes, vol. 7, no. 1, 2019, pp. 39-46, doi:10.36753/mathenot.559244.
Vancouver Baghdad S, Benchohra M. On Existence and Asymptotic Behavior of Solutions of Hadamard-Volterra Integral Equations. Math. Sci. Appl. E-Notes. 2019;7(1):39-46.

20477

The published articles in MSAEN are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.