The aim of this paper is to prove the existence and uniqueness of a solution for a Volterra-Fredholm nonlinear integral equation in two variables under certain conditions, for example, Using the Lipschitz condition with Banach's space contraction principle, under assumptions (i)-(iv), a unique solution exists in Banach's space Z. Moreover, we examine some fundamental characteristics of the solutions for a Volterra-Fredholm nonlinear integral equation in two variables which occur in applications using the inequality established in [6, Theorem 1]. It has also been demonstrated that the functions and parameters included in the equation under investigation continuously influence the behavior of solutions under perturbations in parameters. This investigation may allow for the extension of findings. and the last section contains an illustrative example for the validity of the obtained results. making the method much more attractive for practical applications. The examples show the method is straightforward and effective, and the method can also be extended to other nonlinear integral equation problems.
Keywords
Integral equationVolterra-FredholmExistence and uniqueness of solutionProperties of solutionsBanach fixed-point.
References
B. L. Moiseiwitsch, Integral Equations. London, UK: Longman, 1977.
B. G. Pachpatte, “On mixed Volterra-Fredholm type integral equations,” Indian J. Pure Appl. Math., vol. 17, no. 4, pp. 488-496, 1986.
H. R. Thieme, “A model for the spatial spread of an epidemic,” J. Math. Biol., vol. 4, no. 4, pp. 337-351, 1977, [Online]. Available: https://doi.org/10.1007/BF00275030.
O. Diekmann, “Thresholds and traveling waves for the geographical spread of infection,” J. Math. Biol., vol. 6, no. 2, pp. 109-130, 1978, [Online]. Available: https://doi.org/10.1007/BF00277864.
C. Corduneanu, “Abstract Volterra equations: a survey,” Math. Comput. Model., vol. 32, no. 11-13, pp. 1503-1528, 2000, [Online]. Available: https://doi.org/10.1016/S0895-7177(00)00216-9.
A. M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications. Beijing, China: Higher Education Press and Berlin: Springer, 2011, [Online]. Available: https://doi.org/10.1007/978-3-642-21449-3.
B. G. Pachpatte, “On Volterra-Fredholm integral equation in two variables,” Demonstratio Math., vol. 40, no. 4, pp. 839-852, 2007, [Online]. Available: https://doi.org/10.1515/dema-2007-0410.
B. G. Pachpatte, “New bounds on certain fundamental integral inequalities,” J. Math. Inequal., vol. 4, no. 3, pp. 405-412, 2010, [Online]. Available: https://doi.org/10.7153/jmi-04-37.
B. G. Pachpatte, “Inequalities applicable to mixed Volterra-Fredholm type integral equations,” An. Sti. Univ. Al. I. Cuza Iasi Math., vol. 56, no. 1, pp. 17-24, 2010.
B. G. Pachpatte, Inequalities for Differential and Integral Equations. San Diego, CA, USA: Academic Press, 1998, [Online]. Available: https://doi.org/10.1016/S0076-5392(98)X8001-X.
B. G. Pachpatte, “Growth estimates on mixed Volterra-Fredholm type integral inequalities,” Fasc. Math., vol. 42, pp. 63-72, 2009.
B. G. Pachpatte, “On a general mixed Volterra-Fredholm integral equation,” An. Sti. Univ. Al. I. Cuza Iasi Math., vol. 56, pp. 17-24, 2010, [Online]. Available: https://doi.org/10.2478/v10157-010-0002-z.
B. G. Pachpatte, Multidimensional Integral Equations and Inequalities. Amsterdam, Netherlands: Elsevier (Springer), 2011, [Online]. Available: https://doi.org/10.1016/S0076-5392(11)X0001-3.
C. Bacotiu, “On mixed nonlinear integral equations of Volterra-Fredholm type with modified argument,” Studia Univ. Babes-Bolyai, Math., vol. 54, no. 1, pp. 29-41, 2009.
H. Vu and L. S. Dong, “Existence and Uniqueness of Solution for Two-Dimensional Fuzzy Volterra-Fredholm Integral Equation,” Thai J. Math., vol. 19, no. 4, pp. 1355-1365, 2021.
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