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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Existence, Uniqueness and Stability Results for Nonlinear Neutral Fractional Volterra-Fredholm Integro-Differential Equations

Discontinuity, Nonlinearity, and Complexity 12(2) (2023) 381--398 | DOI:10.5890/DNC.2023.06.011

$^1$ Department of Mathematics, Taiz University, Taiz P.O. Box 6803, Yemen

$^2$ Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen

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Abstract

In this paper, we established some new results concerning the existence and uniqueness of the solutions of nonlinear Volterra-Fredholm integro-differential equations of Caputo fractional order. These new results are obtained by using the Leray-Schauder nonlinear alternative, Krasnoselskii's and Banach fixed point theorems. In addition, we investigate generalized Ulam stability for this fractional system.

Acknowledgments

The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.

References

  1. [1]  Benson, D.A., Wheatcraft, S.W., and Meerschaert, M.M. (2000), Application of a fractional Advection-Dispersion equation, Water resources research, 36, 1403-1412.
  2. [2]  Hilfer, R. (2000), Applications of Fractional Calculus in Physics, World Scientific, Singapore.
  3. [3]  Metzler, R. and Klafter, J. (2000), The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics reports, 339, 1-77.
  4. [4]  Dung, N.T. (2013), Fractional stochastic differential equations with applications to finance, Journal of Mathematical Analysis and Applications, 397, 334-348
  5. [5]  Alsaedi, A., Nieto, J.J., and Venktesh, V. (2015), Fractional electrical circuits, Advances in Mechanical Engineering, 7, 1-7.
  6. [6]  Zhou, W.N., Zhou, X.H., Yang, J., Zhou, J., and Tong, D.B. (2018), Stability analysis and application for delayed neural networks driven by fractional Brownian noise, IEEE Transactions on Neural Networks and Learning Systems, 29(5), 1491-1502.
  7. [7]  Delouei, A., Emamian, A., Karimnejad, S., Sajjadi, H., and Jing, D. (2020), Two-dimensional analytical solution for temperature distribution in FG hollow spheres: general thermal boundary conditions, International Communications in Heat and Mass Transfer, 113, 1-10.
  8. [8]  Bhadane, P.R., Ghadle, K.P., and Hamoud, A.A. (2020), Approximate solution of fractional Black-Schole's European option pricing equation by using ETHPM, Nonlinear Functional Analysis and Applications, 25(2), 331-344.
  9. [9]  He, J.H. (1999), Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science, Technology $\&$ Society, 15(2), 86-90.
  10. [10]  Panda, R. and Dash, M. (2006), Fractional generalized splines and signal processing, Signal Process, 86, 2340-2350.
  11. [11]  Ahmad, B. and Sivasundaram, S. (2008), Some existence results for fractional integro-differential equations with nonlinear conditions, Communications in Applied Analysis, 12, 107-112.
  12. [12]  Bahuguna, D. and Dabas, J. (2008), 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, 1-12.
  13. [13]  Bani Issa, M.S., Hamoud, A.A., and Ghadle, K.P. (2021), Numerical solutions of fuzzy integro-differential equations of the second kind, Journal of Mathematics and Computer Science, 23(1), 67-74.
  14. [14]  Balachandran, K. and Trujillo, J. (2010), The nonlocal Cauchy problem for nonlinear fractional integro-differential equations in Banach spaces, Nonlinear Analysis: Theory, Methods $\&$ Applications, 72, 4587-4593.
  15. [15]  Dawood, L., Hamoud, A., and Mohammed, N. (2020), Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations, Journal of Mathematics and Computer Science, 21(2), 158-163.
  16. [16]  Hamoud, A. and Ghadle, K. (2018), Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind, Tamkang Journal of Mathematics, 49(4), 301-315.
  17. [17]  Hamoud, A. and Ghadle, K. (2018), The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Problemy Analiza -- Issues of Analysis, 7(25), 41-58.
  18. [18]  Hamoud, A. and Ghadle, K. (2019), Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, Journal of Applied and Computational Mechanics, 5(1), 58-69.
  19. [19]  Hamoud, A.A. (2021), Uniqueness and stability results for Caputo fractional Volterra–Fredholm integro-differential equations, Journal of Siberian Federal University, Mathematics \& Physics, 14(3), 313-325.
  20. [20]  Wu, J. and Liu, Y. (2010), Existence and Uniqueness Results for Fractional Integro-Differential Equations with Nonlocal Conditions, 2nd IEEE International Conference on Information and Financial Engineering, 91-94.
  21. [21]  Matar, M. (2010), Controllability of fractional semilinear mixed Volterra-Fredholm integro-differential equations with nonlocal conditions, International Journal of Mathematical Analysis, 4(23), 1105-1116.
  22. [22]  Wu, J. and Liu, Y. (2009), Existence and uniqueness of solutions for the fractional integro-differential equations in Banach spaces, Electronic Journal of Differential Equations, 2009, 1-8.
  23. [23]  Ibrahim, R. and Momani, S. (2007), On the existence and uniqueness of solutions of a class of fractional differential equations, Journal of Mathematical Analysis and Applications, 334(1), 1-10.
  24. [24]  Hamoud, A., Hussain, K., and Ghadle, K. (2019), The reliable modified Laplace Adomian decomposition method to solve fractional Volterra-Fredholm integro differential equations, DCDIS Series B: Applications $\&$ Algorithms, 26, 171-184.
  25. [25]  Hamoud, A. and Ghadle, K. (2018), Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations, Indian Journal of Mathematics, 60(3), 375-395.
  26. [26]  Hamoud, A. and Ghadle, K. (2018), Existence and uniqueness of the solution for Volterra- Fredholm integro-differential equations, Journal of Siberian Federal University. Mathematics \& Physics, 11(6), 692-701.
  27. [27]  Hamoud, A., Ghadle, K., and Pathade, P. (2019), An existence and convergence results for Caputo fractional Volterra integro-differential equations, Jordan Journal of Mathematics and Statistics, 12(3), 307-327.
  28. [28]  Salem, S., Ahmed, B., Alsaedi, B., and Ntouyas, S. (2019), Fractional differential equation involving mixed nonlinearities with nonlocal multi-point and Reimann-Steiljes integral-multi-strip conditions, Fractal and Fractional Journal, 34(3), 1-16.
  29. [29]  De Oliveira, E.C. and Sousa, J.V.D.C. (2018), Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations, Results in Mathematics, 73(3), 1-16.
  30. [30]  Vivek, D., Kanagarajan, K., and Elsayed, E. (2018), Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterranean Journal of Mathematics, 15, 1-15.
  31. [31]  Ahmad, B., Alruwaily, Y., Alsaedi, A., and Ntouyas, S.K. (2019), Existence and stability results for a fractional order differential equation with non-conjugate Riemann-Stieltjes integro multi point boundary conditions, Mathematics, 249(7), 1-14.
  32. [32]  Karthikeyan, K. and Trujillo, J. (2012), Existence and uniqueness results for fractional integro-differential equations with boundary value conditions, Communications in Nonlinear Science and Numerical Simulation, 17, 4037-4043.
  33. [33]  Hamoud, A., Sharif, A., and Ghadle, K. (2021), Existence, uniqueness and stability results of fractional Volterra-Fredholm integro differential equations of $\psi$-Hilfer type, Discontinuity, Nonlinearity, and Complexity, 10(3), 535-545, DOI: 10.5890/DNC.2021.09.013.
  34. [34]  Hussain, K.H., Hamoud, A.A., and Mohammed, N.M. (2019), Some new uniqueness results for fractional integro-differential equations, Nonlinear Functional Analysis and Applications, 24(4), 827-836.
  35. [35]  Lakshmikantham, V. and Rao, M. (1995), Theory of Integro-Differential Equations, Gordon \& Breach, London.
  36. [36]  Kilbas, A., Srivastava, H. and Trujillo, J. (2006),Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 204.
  37. [37]  Miller, K. and Ross, B. (1993), An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York.
  38. [38]  Samko, S., Kilbas, A., and Marichev, O. (1993), Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon.
  39. [39]  Zhou, Y. (2014), Basic Theory of Fractional Differential Equations, Singapore: World Scientific.

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