Existence of solutions of an infinite Mixed Volterra-Fredholm integral system in ℓ1 space
Abstract
Applying the FPT, we offer an existence result for an infinite mixed Volterra-Fredholm type nonlinear integral system in the sequence space ℓ1. Here Meir-Keeler FPT is used and we use the concept of measure of noncompactness. To further demonstrate the given existence result, we provide some examples.
Downloads
References
A. Aghajani, M. Mursaleen, A. S. Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta Math. Sci. 35,552–566,(2015).
R. R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A. E. Rodkina, B. N. Sadovskii, Measures of noncompactness and condensing operators, J. Operator Theory, 55, 1–244,(1992).
A. R. Arab, R. Allahyari, A. S. Haghighi, Existence of solutions of infinite systems of integral equations in two variables via measure of noncompactness, J. Appl. Math. Comput. 246, 283–291,(2014).
S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3, 133–181,(1922).
J. Banas, M. Lecko, Solvability of infinite systems of differential equations in Banach sequence spaces, J. Comput. Appl. Math. 137, 363–375, (2001).
Z. Chen, W. Jiang, An approximate solution for a mixed linear Volterra–Fredholm integral equation, Appl. Math. Lett. 25, 1131–1134, (2012).
A. Das, B. Hazarika, R. Arab, M. Mursaleen, Solvability of the infinite system of integral equations in two variables in the sequence spaces c0 and ℓ1, J. Comput. Appl. Math. 326, 183–192, (2017).
G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Semin. Mat. Univ. Padova, 24, 84–92, (1955).
R. Ezzati, S. Najafalizadeh, Numerical methods for solving linear and nonlinear Volterra-Fredholm integral equations by using CAS wavelets, World Appl. Sci. J. 18, 1847–1854, (2012).
P. M. A. Hasan, N. A. Sulaiman, F. Soleymani, A. Akg¨ul, The existence and uniqueness of solution for linear system of mixed Volterra-Fredholm integral equations in Banach space, AIMS Math. 5, 226–235,(2020).
S. A. Hosseini, S. Shahmorad, F. Talati, A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations, Numer. Algorithms, 68,511–529,(2015).
M. Kazemi, H. Chaudhary, A. Deep, Existence and approximate solutions for Hadamard fractional integral equations in a Banach space, J. Integral Equations Appl. 35, 27–40,(2023).
M. Kazemi, R. Ezzati, A. Deep, On the solvability of non-linear fractional integral equations of product type, J. Pseudo-Differ. Oper. Appl. 14, 39,(2023).
M. Kazemi, On existence of solutions for some functional integral equations in Banach algebra by fixed point theorem, Int. J. Nonlinear Anal. Appl. 13, 451–466,(2022).
E. Keeler, A. Meir, A theorem on contraction mappings, J. Math. Anal. Appl. 28, 326–329,(1969).
K. Kuratowski, Sur les espaces complets, Fund. Math. 15, 301–309,(1930).
M. Mursaleen, S.A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in lp spaces, Nonlinear Anal. 75, 2111–2115,(2012).
C. Nwaigwe, D.N. Benedict, Generalized Banach fixed-point theorem and numerical discretization for nonlinear Volterra–Fredholm equations, J. Comput. Appl. Math. 425, 115019,(2023).
K. Wang, Q. Wang, K. Guan, Iterative method and convergence analysis for a kind of mixed nonlinear Volterra–Fredholm integral equation, J. Appl. Math. Comput. Mech.225, 631–637,(2013).
A. M. Wazwaz, Linear and nonlinear integral equations: Methods and applications, Springer Berlin, Heidelberg (2011).
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática

This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).