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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 of Mild Solutions for Hilfer Fractional Volterra-Fredholm Integro-Differential Inclusions via almost Sectorial Operators

Discontinuity, Nonlinearity, and Complexity 13(3) (2024) 531--542 | DOI:10.5890/DNC.2024.09.011

C. S. Varun Bose, R. Udhayakumar

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632 014, Tamil Nadu, India

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Abstract

In this manuscript, we are mainly investigating the existence of the mild solution for the Hilfer fractional Volterra-Fredholm differential systems via almost sectorial operator. The nonlinear alternative Leray-Schauder fixed point technique for multivalued maps is applied to the fractional calculus concept to demonstrate the conclusions. Finally, an application is provided to demonstrate how the major results might be applied.

References

  1. [1]  Agarwal, R.P., Lakshmikanthan, V., and Nieto, J.J. (2010), On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods and Applications, 75(6), 2859-2862.
  2. [2]  Lakshmikantham, V. and Vatsala, A.S. (2008), Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods and Applications, 69, 2677-2682.
  3. [3]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, 204, 1-523.
  4. [4]  Zhou, Y. (2015), Fractional evolution equations and Inclusions, Analysis and Control, Elsevier, New York.
  5. [5]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego.
  6. [6]  Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer, 44.
  7. [7]  Alqahtani, R.T., Ahmad, S., and Akgül, A. (2021), Dynamical analysis of bio-ethanol production model under generalized nonlocal operator in Caputo sense, Mathematics, 9(19), 2370.
  8. [8]  Yong, W., Ahmad, S., Ullah, A., and Shah, K. (2022), Study of the fractional-order HIV-1 infection model with uncertainty in initial data, Mathematical Problems in Engineering, 2022(2022), 1-16.
  9. [9]  Ahmad, S., Aman, U., Ali, A., and Dumitru, B. (2022), On Solution of fractional model of human liver under hybrid fractional derivative, Discontinuity, Nonlinearity, and Complexity, 11(1), 183-190.
  10. [10]  Dineshkumar, C. and Udhayakumar, R. (2022), Results on approximate controllability of nondensely defined fractional neutral stochastic differential systems, Numerical Methods for Partial Differential Equations, 38(4), 733-759.
  11. [11]  Du, J., Jiang, W., and Khan Niazi, A.U. (2017), Approximate controllability of impulsive Hilfer fractional differential inclusions, Journal of Nonlinear Sciences and Applications, 10, 595-611.
  12. [12]  Ganesh, R., Sakthivel, R., Mahmudov, N.I., and Anthoni, S.M. (2013), Approximate controllability of fractional integro-differential evolution equations, Journal of Applied Mathematics, 2013, 291816.
  13. [13]  Yang, M. and Wang, Q. (2017), Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions, Mathematical Methods in the Applied Sciences, 40(4), 1126-1138.
  14. [14]  Chang, Y.K. and Chalishajar, D.N. (2008), Controllability of mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces, Journal of the Franklin Institute, 345, 499-507.
  15. [15]  Diemling, K. (1992), Multivalued Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications, Berlin.
  16. [16]  Diethelm, K. (2010), The Analysis of Fractional Differential Equations, An Application-Oriented Exposition using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Springer-Verlag, Berlin.
  17. [17]  Jose, S.A., Tom, A., Syed Ali, M., Abinaya, S., and Sudsutad, W. (2021), Existence, uniqueness and stability results of semilinear functional special random impulsive differential equations, Dynamic of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 28, 269-293.
  18. [18]  Ding, X.L. and Ahmad, B. (2016), Analytical solutions to fractional evolution equations with almost sectorial operators, Advances in Differential Equations, 203(1), 1-25.
  19. [19]  Varun Bose, C.S. and Udhayakumar, R. (2022), A note on the existence of Hilfer fractional differential inclusions with almost sectorial operators, Mathematical Methods in the Applied Sciences, 45, 2530-2541.
  20. [20]  Varun Bose, C.S., Udhayakumar, R., Elshenhab, A.M., Sathish Kumar, M., and Ro, J.S. (2022), Discussion on the approximate controllability of Hilfer fractional neutral integro-differential inclusions via almost sectorial operators, Fractal and Fractional, 6(607), 1-22.
  21. [21]  Sivasankar, S. and Udhayakumar, R. (2022), Hilfer fractional neutral stochastic volterra Integro-Differential inclusions via almost sectorial operators, Mathematics, 10, 2074.
  22. [22]  Wang, R.N., Chen, D.H., and Xiao, T.J. (2012), Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252(1), 202-235.
  23. [23]  Zhang, L. and Zhou, Y. (2014), Fractional Cauchy problems with almost sectorial operators, Applied Mathematics and Computation, 257, 145-157.
  24. [24]  Hilfer, R. (2000), Application of Fractional Calculus in Physics, World Scientific, Singapore.
  25. [25]  Jaiswal, J. and Bahuguna, D. (2020), Hilfer fractional differantial equations with almost sectorial operators, Differental Equations and Dynamical Systems, 1-17.
  26. [26]  Karthikeyan, K., Debbouche, A., and Torres, D.F.M. (2021), Analysis of Hilfer fractional integro-differential equations with almost sectorial operators, Fractal and Fractional, 5(1), 1-14.
  27. [27]  Bedi, P., Kumar, A., Abdeljawad, T., Khan, Z.A., and Khan, A. (2020), Existence and approaximate controllability of Hilfer fractional evolution equations with almost sectorial operators, Advances in Differential Equations, 615, 1-15.
  28. [28]  Kavitha, K., Vijayakumar, V., Udhayakumar, R., Sakthivel, N., and Nissar, K.S. (2021), A note on approaximate controllability of the Hilfer fractional neutral differential inclusions with infinite delay, Mathematical Methods in Applied Sciences, 44(6), 4428-4447.
  29. [29]  Mohan Raja, M., Vijayakumar, V., and Udhayakumar, R. (2020), A new approach on the approximate controllability of fractional evolution inclusion of order $1
  30. [30]  Wang, J. and Zhou, Y. (2011), Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Analysis, 12(6), 3642-3653.
  31. [31]  Sakthivel, R., Ganesh, R., and Anthoni, S.M. (2013), Approximate controllability of fractional nonlinear differential inclusions, Applied Mathematics and Computation, 225, 708-717.
  32. [32]  Ahmad, B., Garout, D., Ntouyas, S.K., and Alsaedi, A. (2019), Caputo fractional differential inclusions of arbitrary order with non-local integro-multipoint boundary conditions, Miskolc Mathematical Notes, 20(2), 683-699.
  33. [33]  Falguni, A. and Jitendra, P. (2018), Contrallability of fractional impulsive differential inclusions with sectorial operators in Banach space, Journal of Applied Science and Computations, 5(6), 184-196.
  34. [34]  Sakthivel, R., Ren, Y., Debbouche, A., and Mahmudov, N.I. (2016), Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions, Applicable Analysis, 95(11), 2361-2382.
  35. [35]  Zhou, Y. (2014), Basic Theory of Fractional Differential Equations, World Scientific, Singapore.
  36. [36]  Periago, F. and Straub, B. (2002), A functional caculus for almost sectorial operators and applications to abstract evoution equations, Journal of Evolution Equations, 2, 41-62.
  37. [37]  Gu, H. and Trujillo, J.J. (2015), Existence of integral solution for evolution equation with Hilfer fractional derivative, Applied Mathematics and Computation, 257, 344-354.
  38. [38]  Zhou, M., Li, C., and Zhou, Y. (2022), Existence of mild solutions for Hilfer fractional evolution equations with almost sectorial operators, Axioms, 11(2), 1-13.
  39. [39]  Lasota, A. and Opial, Z. (1965), An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations or noncompact acyclic-valued map, Bulletin L'Academie Polonaise des Science, Series des Sciences Mathematiques, Astronomiques et Physiques, 13, 781-786.
  40. [40]  Ahmad, B., Alsaedi, A., Ntouyas, S.K., and Tariboon, J. (2017), Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer International Publishing AG.

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