{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T21:55:01Z","timestamp":1782856501549,"version":"3.54.5"},"reference-count":29,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T00:00:00Z","timestamp":1557964800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aim of this article is to establish the existence of the solution of non-linear functional integral equations     x  ( l , h )  =  U  ( l , h , x  ( l , h )  )  + F  l , h ,  \u222b  0  l   \u222b  0  h  P  ( l , h , r , u , x  ( r , u )  )  d r d u , x  ( l , h )    \u00d7 G  l , h ,  \u222b  0  a   \u222b  0  a  Q  l , h , r , u , x ( r , u )  d r d u , x  ( l , h )       of two variables, which is of the form of two operators in the setting of Banach algebra     C  [ 0 , a ] \u00d7 [ 0 , a ]  , a &gt; 0 .     Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on     C  [ 0 , a ] \u00d7 [ 0 , a ]      and a fixed point theorem, which is a generalization of Darbo\u2019s fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example.<\/jats:p>","DOI":"10.3390\/sym11050674","type":"journal-article","created":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T11:21:22Z","timestamp":1558005682000},"page":"674","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":34,"title":["Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari M.","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anupam","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, Arunachal Pradesh, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bipan","family":"Hazarika","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, Arunachal Pradesh, India"},{"name":"Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9050-9104","authenticated-orcid":false,"given":"S. A.","family":"Mohiuddine","sequence":"additional","affiliation":[{"name":"Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,16]]},"reference":[{"key":"ref_1","unstructured":"Chandrasekhar, S. (1950). Radiative Transfer, Oxford University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1080\/00036818908839899","article-title":"Integral equations arising in the kinetic theory of grass","volume":"34","author":"Hu","year":"1989","journal-title":"Appl. Anal."},{"key":"ref_3","first-page":"221","article-title":"Approximation of solution of some quadratic integral equations in transport theory","volume":"4","author":"Kelly","year":"1982","journal-title":"J. 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