Document Type: Research Paper

Authors

1 Department of Mathematics, Bonab Branch, Islamic Azad University, Bonab, Iran.

2 Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.

Abstract

In this paper, we generalize the Meir-Keeler condensing  operators  via a concept of the class of operators  $ O (f;.)$, that was given by Altun and Turkoglu [4], and apply this extension to obtain some tripled fixed point theorems.  As an application of this extension, we  analyze the existence of solution for a system of nonlinear functional integral equations of Volterra type. Finally,  we present an example  to show the effectiveness of our results. We use the technique of measure of noncompactness to obtain our results.

Keywords

Main Subjects

[1] R. Agarwal, M. Meehan, and D. O'Regan, Fixed point theory and applications, Cambridge University Press, 2004.

[2] A. Aghajani, J. Banas, and Y. Jalilian, Existence of solution for a class nonlinear Voltrra sigular integral, Appl. Math. Comput., 62 (2011), pp. 1215-1227.

[3] A. Aghajani, M. Mursaleen, and A. Shole Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measures of noncompactness, Acta Mathematica Scientia., 35 (2015), pp. 552-566.

[4] I. Altun and D. Turkoglu, A fixed point theorem for mappings satisfying a general contractive condition of operator type, Journal of Computational Analysis and Applications., 9 (2007), pp. 9-14.

[5] R. Arab, R. Allahyari, and A. Shole Haghighi, Construction of a Measure of Noncompactness on BC($Omega$) and its Application to Volterra Integral Equations, Mediterr. J. Math., 13 (2016), pp. 1197-1210.

[6] Sh. Banaei, M.B. Ghaemi, and R. Saadati, An extension of Darbo's theorem and its application to system of neutral diferential equations with deviating argument, Miskolc Mathematical Notes, 18 (2017), pp. 83-94.

[7] J. Banas, M. Jleli, M. Mursaleen, and B. Samet, Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer, Singapore, 2017.

[8] J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, New York, 1980.

[9] J. Banas, D. O'regan, and K. Sadarangani, On solutions of a quadratic hammerstein integral equation on an unbounded interval, Dynam. Systems Appl., 18 (2009), pp. 251-264.

[10] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova., 24 (1955), pp. 84-92.

[11] M.A. Darwish, Monotonic solutions of a convolution functional integral equation, Appl. Math. Comput., 219 (2013), pp. 10777-10782.

[12] V. Karakaya, N. El Houda Bouzara, and Y. Atalan, Existence of tripled fixed points for a class of condensing operators in banach spaces, The Scientific World Journal, (2014), pp. 1-9.

[13] K. Kuratowski, Sur les espaces, Fund. Math., 15 (1930), pp. 301-309.

[14] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math Anal Appl., 28 (1969), pp. 326-329.

[15] M. Mursaleen and S.A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in $l_p$ spaces, Nonlinear Analysis., 75 (2012), pp. 2111-2115.

[16] L. Olszowy, Solvability of infinite systems of singular integral equations in Frechet space of continuous functions, Computers and Mathematics with Applications, 59 (2010), pp. 2794-2801.

[17] A. Samadi and Mohammad B. Ghaemi, An extension of Darbo fixed point theorem and its applications to coupled fixed point and integral equations, Filomat., 28 (2014), pp. 879-886.

[18] B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Analysis, 72 (2010), pp. 4508-4517.