In this paper, by using the techniques of measures of non-compactness and the Petryshyn fixed point theorem, we investigate the existence of solutions of a Caputo fractional functional integro-differential equation and obtain some new results. These existing results involve particular results gained from earlier studies under weaker conditions.
[1] N. Adjimi, A. Boutiara, and M. E. Samei, et al., On solutions of a hybrid generalized Caputo-type problem via the noncompactness measure in the generalized version of Darbo’s criterion, Journal of Inequalities and Applications, 2023 (2023), 34.
[2] P. Amiri and M. E. Samei, Existence of Urysohn and Atangana-Baleanu fractional integral inclusion systems solutions via common fixed point of multi-valued operators, Chaos, Solitons & Fractals, 165 (2022), 112822.
[3] R. P. Agarwal, Y. Zhou, and Y. He, Existence of fractional neutral functional differential equations, Computers & Mathematics with Applications, 59(3) (2010), 1095–1100.
[4] A. Anguraj, P. Karthikeyan, and J. J. Trujillo, Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition, Advances in Difference Equations, 2011 (2011), 1–12.
[5] R. Arab, The existence of fixed points via the measure of noncompactness and its application to functional-integral equations, Mediterr. J. Math., 13(2) (2016), 759-773.
[6] R. Azimi, M. Mohagheghy Nezhad, and R. Pourgholi, Legendre Spectral Tau Method for Solving the Fractional Integro-Differential Equations with A Weakly Singular Kernel, Global Analysis and Discrete Mathematics, 4(4) (2022), 361-372.
[7] C. Baishya, R. N. Premakumari, M. E. Samei, and M. K. Naik, Chaos control of fractional order nonlinear Bloch equation by utilizing sliding mode controller, Chaos, Solitons & Fractals, 174 (2023), 113773.
[8] S. P. Bhairat and M. E. Samei, Nonexistence of global solutions for a HilferKatugampola fractional differential problem, Partial Differential Equations in Applied Mathematics, 7 (2023), 100495.
[9] S. Dadsetadi, K. Nouri, and L. Torkzadeh, Solvability of some nonlinear Integro-Differential equations of fractional order via measure of non-compactness, The Pure and Applied Mathematics, 27(1) (2020), 13–24.
[10] S. Dadsetadi and K. Nouri, Study on existence of solution for some fractional integro differential equations via the iterative process, Advances in Modelling and Analysis A, 55(2) (2018), 57–61.
[11] A. Deep, Deepmala, and M. Rabbani, A numerical method for solvability of some non-linear functional integral equations, Appl. Math. Comput., 402 (2021), 125637.
[12] A. M. El-Sayed, E. M. Hamdallah, and M. M. Ba-Ali, Qualitative Study for a Delay Quadratic Functional Integro-Differential Equation of Arbitrary (Fractional) Orders, Symmetry, 14(4) (2022), 748.
[13] L. S. Goldenshtejn and A. S. Markus, On the measure of non-compactness of bounded sets and of linear operators, In Studies in Algebra and Math. Anal. (Russian), pages 45–54. Izdat. “Karta Moldovenjaske”, Kishinev, 1965.
[14] F. Haddouchi, M. E. Samei, and S. Rezapour, Study of a sequential ψ-Hilfer fractional integro-differential equations with nonlocal BCs, J. Pseudo-Differ. Oper. Appl. 14 (2023), 61.
[15] M. Houas and M. E. Samei. Existence and Stability of Solutions for Linear and Nonlinear Damping of q-Fractional Duffing-Rayleigh Problem, Mediterr. J. Math. 20 (2023), 148.
[16] K. Karthikeyan and J. J. Trujillo, Existence and Uniqueness Results for Fractional Integro-Differential Equations With Boundary Value Conditions, Commun. Nonlinear Sci. Numer. Simulat, 17 (2012), 4037–4043.
[17] M. Kazemi and R. Ezzati, Existence of solution for some nonlinear two-dimensional volterra integral equations via measures of noncompactness, Appl. Math. Comput., 275 (2016), 165–171.
[18] M. Kazemi, On existence of solutions for some functional integral equations in Banach algebra by fixed point theorem, Int. J. Nonlinear Anal. Appl., 13 (2022), 451–466.
[19] M. Kazemi, A. Deep, and A. R. Yaghoobnia, Application of fixed point theorem on the study of the existence of solutions in some fractional stochastic functional integral equations, Math. Sci., 18 (2024), 125–136.
[20] T. Kherraz, M. Benbachir, M. Lakrib, M. E. Samei, M. K. Kaabar, and S. A. Bhanotar, Existence and uniqueness results for fractional boundary value problems with multiple orders of fractional derivatives and integrals, Chaos, Solitons & Fractals, 166 (2023), 113007.
[21] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204 (2006).
[22] K. Kuratowski, Sur les espaces completes, Fund. Math., 15 (1934), 301–335.
[23] A. Lachouri, M. E. Samei, and A. Ardjouni, Existence and stability analysis for a class of fractional pantograph q-difference equations with nonlocal boundary conditions, Boundary Value Problems, 2023 (2023), 2.
[24] L. M. Mishra and R. V. Agarwal, On existence theorems for some nonlinear functional-integral equations, Dynamic systems and Applications, 25(3) (2016), 303–320.
[25] H. K. Nashine and R. Arab, Existence of solutions to nonlinear functional-integral equations via the measure of noncompactness, J. Fixed Point Theory Appl., 20(2) (2018), 66.
[26] K. Nouri, M. Nazari, and B. Keramati, Existence results for a coupled system of fractional integro-differential equations with time-dependent delay, Journal of Fixed Point Theory and Applications, 19(4) (2017), 2927–2943.
[27] R. D. Nussbaum, The fixed-point index and fixed point theorem for k-set contractions, Pro Quest LLC, Ann Arbor, MI, 1969. Thesis (Ph.D.)–The University of Chicago.
[28] P. R. Patle, M. Gabeleh, V. Rakoˇcevi´c, and M. E. Samei, New best proximity point (pair) theorems via MNC and application to the existence of optimum solutions for a system of ψ-Hilfer fractional differential equations, Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales. Serie A. Matem´aticas, 117(3) (2023), 124.
[29] W. V. Petryshyn, Structure of the fixed points sets of k-set-contractions, Arch. Rational Mech. Anal., 40 (1970/1971), 312–328.
[30] M. E. Samei, H. Zanganeh, and S. M. Aydogan, Investigation of a class of the singular fractional integrodifferential quantum equations with multi-step methods, Journal of Mathematical Extension, 15(15) (2021), 1–54.
[31] M. E. Samei, Employing Kuratowski measure of non-compactness for positive solutions of system of singular fractional q-differential equations with numerical effects, Filomat, 34(9) (2020), 2971–2989.
[32] S. Singh, B. Watson, and P. Srivastava, Fixed point theory and best approximation: the KKM-map principle, Mathematics and its Applications, 424 (1997).
[33] D. N. Susahab, S. Shahmorad, and M. Jahanshahi, Efficient quadrature rules for solving nonlinear fractional integro-differential equations of the Hammerstein type, Applied Mathematical Modelling, 39(18) (2015), 5452– 5458.
[34] S. Unhaley and S. Kendre, On existence and uniqueness results for iterative fractional integro-differential equation with deviating arguments, Appl. Math. E-Notes, 19 (2019), 116–127.
[35] C. Vetro and F. Vetro, On the existence of at least a solution for functional integral equations via measure of noncompactness, Banach J. Math. Anal., 11(3) (2017), 497–512.
[36] X. G. Yue, M. E. Samei, A. Fathipour, M. K. Kaabar, and A. Kashuri, Using Krasnoselskii’s theorem to investigate the Cauchy and neutral fractional q-integro-differential equation via numerical technique, Nonlinear Engineering, 11(1) (2022), 186–206.
[37] L. Zhang, B. Ahmad, G. Wang, and R. P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math., 249 (2013), 51-56.
Kazemi, M. (2025). Existence of solutions of Caputo fractional integro-differential equations. Computational Methods for Differential Equations, 13(2), 568-577. doi: 10.22034/cmde.2024.57398.2399
MLA
Kazemi, M. . "Existence of solutions of Caputo fractional integro-differential equations", Computational Methods for Differential Equations, 13, 2, 2025, 568-577. doi: 10.22034/cmde.2024.57398.2399
HARVARD
Kazemi, M. (2025). 'Existence of solutions of Caputo fractional integro-differential equations', Computational Methods for Differential Equations, 13(2), pp. 568-577. doi: 10.22034/cmde.2024.57398.2399
CHICAGO
M. Kazemi, "Existence of solutions of Caputo fractional integro-differential equations," Computational Methods for Differential Equations, 13 2 (2025): 568-577, doi: 10.22034/cmde.2024.57398.2399
VANCOUVER
Kazemi, M. Existence of solutions of Caputo fractional integro-differential equations. Computational Methods for Differential Equations, 2025; 13(2): 568-577. doi: 10.22034/cmde.2024.57398.2399