Exploring the fractional Volterra-Fredholm integro-differential equation: An iterative approach

Doaa Filali; Mohammad Akram; Mohammad Dilshad; Doaa Filali; Mohammad Akram; Mohammad Dilshad

doi:10.3934/math.20251042

AIMS Mathematics

2025, Volume 10, Issue 10: 23467-23495. doi: 10.3934/math.20251042

Previous Article Next Article

Research article Special Issues

Exploring the fractional Volterra-Fredholm integro-differential equation: An iterative approach

1.
Department of Mathematical Science, College of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Islamic University of Madinah, Madinah 42351, Saudi Arabia
3.
Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Received: 21 May 2025 Revised: 24 September 2025 Accepted: 30 September 2025 Published: 15 October 2025
MSC : 47H05, 47H09, 47H10, 47H22, 47H25, 49J40

In this paper, we construct a new four-step iterative method and show that our newly designed scheme converges faster than a number of iterative methods. We corroborate our claims by performing numerical experiments. We analyze the strong convergence result to approximate the fixed point of a contractive-like mapping and establish the weak $ \omega^{2} $-stability of our scheme. Further, strong and weak convergence results are incorporated for a generalized $ \alpha $-Reich-Suzuki nonexpansive mapping under some mild assumptions. We also illustrate numerical examples to validate our theoretical claims. Finally, we set forth our scheme to explore a Caputo-type nonlinear fractional Volterra-Fredholm integro-differential equation and a fractional diffusion equation.
- contractive-like mapping,
- $ G_\alpha $-RSNEM,
- fixed point,
- $ \omega^{2} $-stability,
- nonlinear fractional Volterra-Fredholm integro-differential equation
Citation: Doaa Filali, Mohammad Akram, Mohammad Dilshad. Exploring the fractional Volterra-Fredholm integro-differential equation: An iterative approach[J]. AIMS Mathematics, 2025, 10(10): 23467-23495. doi: 10.3934/math.20251042

Related Papers:

Abstract

In this paper, we construct a new four-step iterative method and show that our newly designed scheme converges faster than a number of iterative methods. We corroborate our claims by performing numerical experiments. We analyze the strong convergence result to approximate the fixed point of a contractive-like mapping and establish the weak $ \omega^{2} $-stability of our scheme. Further, strong and weak convergence results are incorporated for a generalized $ \alpha $-Reich-Suzuki nonexpansive mapping under some mild assumptions. We also illustrate numerical examples to validate our theoretical claims. Finally, we set forth our scheme to explore a Caputo-type nonlinear fractional Volterra-Fredholm integro-differential equation and a fractional diffusion equation.

References

[1]	A. A. N. Abdou, Fixed point theorems: Exploring applications in fractional differential equations for economic growth, Fractal Fract., 8 (2024), 243. https://doi.org/10.3390/fractalfract8040243 doi: 10.3390/fractalfract8040243
[2]	R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), 61–79.
[3]	J. Ahmad, K. Ullah, H. A. Hammad, R. George, A solution of a fractional differential equation via novel fixed-point approaches in Banach spaces, AIMS Math., 8 (2023), 12657–12670. https://doi.org/10.3934/math.2023636 doi: 10.3934/math.2023636
[4]	K. H. Alam, Y. Rohen, Convergence of a refined iterative method and its application to fractional Volterra–Fredholm integro-differential equations, Comput. Appl. Math., 44 (2025), https://doi.org/10.1007/s40314-024-02964-4
[5]	J. Ali, F. Ali, A new iterative scheme for approximating fixed points with an application to delay differential equation, J. Nonlinear Convex Anal., 21 (2020), 2151–2163.
[6]	F. Ali, J. Ali, Convergence, stability and data dependence of a new iterative algorithm with an application, Comput. Appl. Math., 39 (2020), 267.
[7]	D. Ali, S. Ali, D. Pompei-Cosmin, T. Antoniu, A. A. Zaagan, A. M. Mahnashi, A quicker iteration method for approximating the fixed point of generalized $\alpha$-Reich-Suzuki nonexpansive mappings with applications, Fractal Fract., 7 (2023), 790. https://doi.org/10.3390/fractalfract7110790 doi: 10.3390/fractalfract7110790
[8]	V. Berinde, On the approximation of fixed points of weak contractive mapping, Carpath J. Math., 17 (2003), 7–22.
[9]	V. Berinde, On the stability of some fixed point procedures, Bul. Ştiinţ. Univ. Baia Mare, Ser. B, Matematică Informatică, 18 (2002), 7–14.
[10]	A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., 82 (1976), 641–657. https://doi.org/10.1090/S0002-9904-1976-14091-8 doi: 10.1090/S0002-9904-1976-14091-8
[11]	F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci., 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041 doi: 10.1073/pnas.54.4.1041
[12]	T. Cardinali, P. Rubbioni, A generalization of the Caristi fixed point theorem in metric spaces, Fixed Point Theory, 11 (2010), 3–10.
[13]	D. Filali, N. H. E. Eljaneid, A. Alatawi, E. Alshaban, M. S. Ali, F. A. Khan, A novel and efficient iterative approach to approximating solutions of fractional differential equations, Mathematics, 13 (2025), 33. https://doi.org/10.3390/math13010033 doi: 10.3390/math13010033
[14]	K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, UK, 1990. https://doi.org/10.1017/CBO9780511526152
[15]	H. A. Hammad, S. F. Aljurbua, Solving fractional random differential equations by using fixed point methodologies under mild boundary conditions, Fractal Fract., 8 (2024), 384. https://doi.org/10.3390/fractalfract8070384 doi: 10.3390/fractalfract8070384
[16]	H. A. Hammad, H. Rehman, M. De la Sen, A new four-step iterative procedure for approximating fixed points with application to 2D volterra integral equations, Mathematics, 10 (2022), 4257. https://doi.org/10.3390/math10224257 doi: 10.3390/math10224257
[17]	H. A. Hammad, H. Rehman, M. Zayed, Applying faster algorithm for obtaining convergence, stability, and data dependence results with application to functional-integral equations, AIMS Math., 7 (2022), 19026–19056. https://doi.org/10.3934/math.20221046 doi: 10.3934/math.20221046
[18]	O. Hans, Reduzierende zufallige Transformationen, Czechoslovak Math. J., 7 (1957), 154–158. https://doi.org/10.21136/CMJ.1957.100238 doi: 10.21136/CMJ.1957.100238
[19]	A. M. Harder, T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, 33 (1998), 693–706.
[20]	G. H. Ibraheem, M. Turkyilmazoglu, M. A. AL-Jawary, Novel approximate solution for fractional differential equations by the optimal variational iteration method, J. Comput. Sci., 64 (2022), 101841. https://doi.org/10.1016/j.jocs.2022.101841 doi: 10.1016/j.jocs.2022.101841
[21]	C. O. Imoru, M. O. Olatinwo, On the stability of Picard and Mann iteration processes, Carpathian J. Math., 19 (2003), 155–160.
[22]	S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. https://doi.org/10.1090/S0002-9939-1974-0336469-5 doi: 10.1090/S0002-9939-1974-0336469-5
[23]	S. Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl., 67 (1979), 261–273. https://doi.org/10.1016/0022-247X(79)90023-4 doi: 10.1016/0022-247X(79)90023-4
[24]	S. Khatoon, I. Uddin, D. Baleanu, Approximation of fixed point and its application to fractional differential equation, J. Appl. Math. Comput., 66 (2021), 507–525. https://doi.org/10.1007/s12190-020-01445-1 doi: 10.1007/s12190-020-01445-1
[25]	C. Kongban, P. Kumam, J. Martinez-Moreno, A. F. R. L. Hierro, On random fixed point for generalizations of Suzuki's type with application to stochastic dynamic programming, Res. Fixed Point Theory Appl., 2019, 2018026. https://doi.org/10.30697/rfpta-2018-026
[26]	A. Krogh, The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue, J. Phys., 52 (1919). https://doi.org/10.1113/jphysiol.1919.sp001839
[27]	R. M. Leach, D. F. Treacher, ABC of oxygen, Oxygen transport-2, Tissue hypoxia, BMJ, 317 (1998), 1370–1373. https://doi.org/10.1136/bmj.317.7169.1370
[28]	R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2010), 1586–1593. https://doi.org/10.1016/j.camwa.2009.08.039 doi: 10.1016/j.camwa.2009.08.039
[29]	S. R. Mahmoud, A. O. Nazek, A. Hala, New class of nonlinear fractional integro-differential equations with theoretical analysis via fixed point approach: Numerical and exact solutions, J. Appl. Anal. Comput., 13 (2023), 2767–2787. https://doi.org/10.11948/20220575 doi: 10.11948/20220575
[30]	A. Mamoud, K. P. Ghadle, M. S. B. Isa, G. Giniswami, Existence and uniqueness theorem for fractional Volterra-Fredholm integro-differential equations, Int. J. Appl. Math., 31 (2018), 333–348. https://doi.org/10.12732/ijam.v31i3.3 doi: 10.12732/ijam.v31i3.3
[31]	W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.1090/S0002-9939-1953-0054846-3 doi: 10.1090/S0002-9939-1953-0054846-3
[32]	K. S. Miller, B. Ross, An introduction to fractional calculus and fractional differential equations, Wiley, New York, 1993.
[33]	M. A. Noor, New approximation scheme for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042 doi: 10.1006/jmaa.2000.7042
[34]	A. E. Ofem, J. A. Abuchu, G. C. Ugwunnadi, H. Isik, O. K. Narain, On a four-step iterative algorithm and its application to delay integral equations in hyperbolic spaces, Rend. Circ. Mat. Palermo. Series 2, 73 (2024), 189–224. https://doi.org/10.1007/s12215-023-00908-1 doi: 10.1007/s12215-023-00908-1
[35]	M. O. Osilike, Some stability results for fixed point iteration procedures, J. Nigerian Math. Soc., 14 (1995), 17–29.
[36]	R. Pandey, R. Pant, V. Rakocevic, R. Shukla, Approximating fixed points of a general class of nonexpansive mappings in Banach spaces with applications, Results Math., 74 (2019), https://doi.org/10.1007/s00025-018-0930-6
[37]	R. Pant, R. Pandey, Existence and convergence results for a class of non-expansive type mappings in hyperbolic spaces, Appl. Gen. Topol., 20 (2019), 281–295. https://doi.org/10.4995/agt.2019.11057 doi: 10.4995/agt.2019.11057
[38]	H. Piri, B. Daraby, S. Rahrovi, M. Ghasemi, Approximating fixed points of generalized $\alpha$-non-expansive mappings in Banach spaces by new faster iteration process, Numer. Algorithms, 81 (2018), 1129–1148. https://doi.org/10.1007/s11075-018-0588-x doi: 10.1007/s11075-018-0588-x
[39]	M. Rahman, S. Ahmad, M. Arfan, A. Akgul, F. Jarad, Fractional order mathematical model of serial killing with different choices of control strategy, Fractal Fract., 6 (2022), 162. https://doi.org/10.3390/fractalfract6030162 doi: 10.3390/fractalfract6030162
[40]	B. E. Rhoades, Some fixed point iteration procedures, Int. J. Math. Math. Sci., 14 (1991), 1–16. https://doi.org/10.1155/S0161171291000017 doi: 10.1155/S0161171291000017
[41]	B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures Ⅱ, Indian J. Pure Appl. Math., 24 (1993), 691–703.
[42]	J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mapping, Bull. Aust. Math. Soc., 43 (1991), 153–159. https://doi.org/10.1017/S0004972700028884 doi: 10.1017/S0004972700028884
[43]	H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380. https://doi.org/10.1090/S0002-9939-1974-0346608-8 doi: 10.1090/S0002-9939-1974-0346608-8
[44]	V. Srivastava, K. N. Rai, A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues, Math. Comput. Model., 51 (2010), 616–624.
[45]	A. Spacek, Zufallige Gleichungen, Czechoslovak Math. J., 5 (1955), 462–466. https://doi.org/10.21136/CMJ.1955.100162
[46]	X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64–69. https://doi.org/10.1016/j.aml.2008.03.001 doi: 10.1016/j.aml.2008.03.00
[47]	T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340 (2008), 1088–1095. https://doi.org/10.1016/j.jmaa.2007.09.023 doi: 10.1016/j.jmaa.2007.09.023
[48]	T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), 1861–1869. https://doi.org/10.1090/S0002-9939-07-09055-7 doi: 10.1090/S0002-9939-07-09055-7
[49]	B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147–155. https://doi.org/10.1016/j.amc.2015.11.065 doi: 10.1016/j.amc.2015.11.065
[50]	I. Timis, On the weak stability of Picard iteration for some contractive type mappings, Ann. Univ. Craiova Math. Comput. Sci. Ser., 37 (2010), 106–114.
[51]	K. Ullah, M. Arshad, Numerical reckoning fixed points for Suzuki's generalized nonexpansive mappings via new iteration process, Filomat, 32 (2018), 187–196. https://doi.org/10.2298/FIL1801187U doi: 10.2298/FIL1801187U
[52]	J. Yu, Y. Feng, On the generalized time fractional reaction-diffusion equation: Lie symmetries, exact solutions and conservation laws, Chaos Soliton. Fract., 182 (2024), 114855. https://doi.org/10.1016/j.chaos.2024.114855 doi: 10.1016/j.chaos.2024.114855
[53]	J. Yu, Y. Feng, Symmetry analysis, optimal system, conservation laws and exact solutions of time fractional diffusion-type equation, Int. J. Geom. Methods Mod. Phys., 2024, 2450286. https://doi.org/10.1142/S0219887824502864
[54]	H. Yuan, Q. Zhu, The well-posedness and stabilities of mean-field stochastic differential equations driven by $G$-Brownian motion, SIAM J. Control Optim., 63 (2025). https://doi.org/10.1137/23M159368
[55]	Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Lévy processes, IEEE Trans. Autom. Control., 70 (2025), 1176–1183. https://doi.org/10.1109/TAC.2024.3448128 doi: 10.1109/TAC.2024.3448128

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)