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Year 2020, Volume: 4 Issue: 4, 321 - 331, 30.12.2020
https://doi.org/10.31197/atnaa.799854

Abstract

References

  • [1] B. Ahmad, S.K. Ntouyas, A. Alsaedi, M. Alnahdi, Existence theory for fractional-order neutral boundary value problems, Frac. Differ. Calc. 8 (2018), 111-126.
  • [2] B. Ahmad, S.K. Ntouyas, A. Alsaedi, Caputo-type fractional boundary value problems for differential equations and inclusions with multiple fractional derivatives, J. Nonlinear Funct. Anal. 2017 (2017), Art. ID 52.
  • [3] B. Ahmad and S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlinear conditions, Communications Appl. Anal. 12, (2008) 107-112.
  • [4] A. Ardjouni and A. Djoudi, Existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions, Adv. Theory Nonlinear Anal. Appl. 3, (2019) 46-52.
  • [5] D. Bahuguna and J. Dabas, Existence and uniqueness of a solution to a partial integro-differential equation by the method of Lines, Electronic Journal of Qualitative Theory of Differential Equations, 4, (2008) 1-12.
  • [6] K. Balachandran, J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Anal. Theory Meth. Applic. 72, (2010) 4587-4593.
  • [7] K. Balachandran, K. Uchiyama, Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces, Proceedings of the Indian Academy of Science, 110, (2000) 225-232.
  • [8] A. Hamoud and K. Ghadle, Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the econd kind, Tamkang J. Math. 49(4), (2018) 301-315.
  • [9] A. Hamoud, K. Hussain and K. Ghadle, The reliable modified Laplace Adomian decomposition method to solve fractional Volterra-Fredholm integro differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 26, (2019) 171-184.
  • [10] A. Hamoud and K. Ghadle, Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, J. Appl. Comput. Mech. 5(1), (2019) 58-69.
  • [11] R. Ibrahim and S. Momani, On the existence and uniqueness of solutions of a class of fractional differential equations, Journal of Mathematical Analysis and Applications, 334(1), (2007) 1-10.
  • [12] K. Karthikeyan and J. Trujillo, Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., 17, (2012) 4037-4043.
  • [13] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006.
  • [14] A. Lachouri, A. Ardjouni, A. Djoudi, Positive solutions of a fractional integro-differential equation with integral boundary conditions, Commun. Optim. Theory, 2020 (2020) 1-9.
  • [15] V. Lakshmikantham and M. Rao, Theory of Integro-Differential Equations, Gordon & Breach, London, 1995.
  • [16] M. Matar, Controllability of fractional semilinear mixed VolterraFredholm integro differential equations with nonlocal conditions, Int. J. Math. Anal., 4(23), (2010) 1105-1116.
  • [17] S. Muthaiah, M. Murugesan and N. Thangaraj, Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations, Adv. Theory Nonlinear Anal. Appl. 3 (2019) No. 3, 162-173.
  • [18] S. Momani, A. Jameel and S. Al-Azawi, Local and global uniqueness theorems on fractional integro-differential equations via Bihari’s and Gronwall’s inequalities, Soochow Journal of Mathematics, 33(4), (2007) 619-627.
  • [19] K. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
  • [20] S.K. Ntouyas, Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions, Discuss. Math. Differ. Incl. Control Optim., 33 (2013) 17-39.
  • [21] M. Reed and B. Simon, Functional Analysis, Academic Press, Inc., New York, 1980.
  • [22] S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • [23] X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equations, Appl. Math. J. Chinese Univ. Ser. B, 22(3), (2007) 291-298.
  • [24] X. Wang, L. Wang, Q. Zeng, Fractional differential equations with integral boundary conditions, J. of Nonlinear Sci. Appl. 8 (2015), 309-314.
  • [25] J. Wu and Y. Liu, Existence and uniqueness of solutions for the fractional integro-differential equations in Banach spaces, Electronic Journal of Differential Equations, 2009, (2009) 1-8.
  • [26] J. Wu and Y. Liu, Existence and uniqueness results for fractional integrodifferential equations with nonlocal conditions, 2nd IEEE International Conference on Information and Financial Engineering, (2010) 91-94.
  • [27] S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252, (2000) 804-812.
  • [28] Y. Zhou, Basic Theory of Fractional Differential Equations, Singapore: World Scientific, 2014.

Existence and Uniqueness of Solutions for Fractional Neutral Volterra-Fredholm Integro Differential Equations

Year 2020, Volume: 4 Issue: 4, 321 - 331, 30.12.2020
https://doi.org/10.31197/atnaa.799854

Abstract

The topic fractional calculus can be measured as an old as well as a new subject. In the fractional calculus the various integral inequalities plays an important role in the study of qualitative and quantitative properties of solution of differential and integral equations.
In this paper, we study the existence and uniqueness of solutions for the neutral Caputo fractional Volterra-Fredholm integro differential equations with fractional integral boundary conditions by means of the Arzela-Ascoli's theorem, Leray-Schauder nonlinear alternative and the Krasnoselskii fixed point theorem. New conditions on the nonlinear terms are given to pledge the equivalence. An example is provided to illustrate the results.

References

  • [1] B. Ahmad, S.K. Ntouyas, A. Alsaedi, M. Alnahdi, Existence theory for fractional-order neutral boundary value problems, Frac. Differ. Calc. 8 (2018), 111-126.
  • [2] B. Ahmad, S.K. Ntouyas, A. Alsaedi, Caputo-type fractional boundary value problems for differential equations and inclusions with multiple fractional derivatives, J. Nonlinear Funct. Anal. 2017 (2017), Art. ID 52.
  • [3] B. Ahmad and S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlinear conditions, Communications Appl. Anal. 12, (2008) 107-112.
  • [4] A. Ardjouni and A. Djoudi, Existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions, Adv. Theory Nonlinear Anal. Appl. 3, (2019) 46-52.
  • [5] D. Bahuguna and J. Dabas, Existence and uniqueness of a solution to a partial integro-differential equation by the method of Lines, Electronic Journal of Qualitative Theory of Differential Equations, 4, (2008) 1-12.
  • [6] K. Balachandran, J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Anal. Theory Meth. Applic. 72, (2010) 4587-4593.
  • [7] K. Balachandran, K. Uchiyama, Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces, Proceedings of the Indian Academy of Science, 110, (2000) 225-232.
  • [8] A. Hamoud and K. Ghadle, Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the econd kind, Tamkang J. Math. 49(4), (2018) 301-315.
  • [9] A. Hamoud, K. Hussain and K. Ghadle, The reliable modified Laplace Adomian decomposition method to solve fractional Volterra-Fredholm integro differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications & Algorithms, 26, (2019) 171-184.
  • [10] A. Hamoud and K. Ghadle, Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, J. Appl. Comput. Mech. 5(1), (2019) 58-69.
  • [11] R. Ibrahim and S. Momani, On the existence and uniqueness of solutions of a class of fractional differential equations, Journal of Mathematical Analysis and Applications, 334(1), (2007) 1-10.
  • [12] K. Karthikeyan and J. Trujillo, Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., 17, (2012) 4037-4043.
  • [13] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006.
  • [14] A. Lachouri, A. Ardjouni, A. Djoudi, Positive solutions of a fractional integro-differential equation with integral boundary conditions, Commun. Optim. Theory, 2020 (2020) 1-9.
  • [15] V. Lakshmikantham and M. Rao, Theory of Integro-Differential Equations, Gordon & Breach, London, 1995.
  • [16] M. Matar, Controllability of fractional semilinear mixed VolterraFredholm integro differential equations with nonlocal conditions, Int. J. Math. Anal., 4(23), (2010) 1105-1116.
  • [17] S. Muthaiah, M. Murugesan and N. Thangaraj, Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations, Adv. Theory Nonlinear Anal. Appl. 3 (2019) No. 3, 162-173.
  • [18] S. Momani, A. Jameel and S. Al-Azawi, Local and global uniqueness theorems on fractional integro-differential equations via Bihari’s and Gronwall’s inequalities, Soochow Journal of Mathematics, 33(4), (2007) 619-627.
  • [19] K. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
  • [20] S.K. Ntouyas, Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions, Discuss. Math. Differ. Incl. Control Optim., 33 (2013) 17-39.
  • [21] M. Reed and B. Simon, Functional Analysis, Academic Press, Inc., New York, 1980.
  • [22] S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
  • [23] X. Su and L. Liu, Existence of solution for boundary value problem of nonlinear fractional differential equations, Appl. Math. J. Chinese Univ. Ser. B, 22(3), (2007) 291-298.
  • [24] X. Wang, L. Wang, Q. Zeng, Fractional differential equations with integral boundary conditions, J. of Nonlinear Sci. Appl. 8 (2015), 309-314.
  • [25] J. Wu and Y. Liu, Existence and uniqueness of solutions for the fractional integro-differential equations in Banach spaces, Electronic Journal of Differential Equations, 2009, (2009) 1-8.
  • [26] J. Wu and Y. Liu, Existence and uniqueness results for fractional integrodifferential equations with nonlocal conditions, 2nd IEEE International Conference on Information and Financial Engineering, (2010) 91-94.
  • [27] S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252, (2000) 804-812.
  • [28] Y. Zhou, Basic Theory of Fractional Differential Equations, Singapore: World Scientific, 2014.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ahmed Hamoud 0000-0002-8877-7337

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

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