Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra
The aim of this article is to establish the existence of the solution of non-linear functional integral equations <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-05-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/11/5/674 |
| _version_ | 1798026016649641984 |
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| author | Hari M. Srivastava Anupam Das Bipan Hazarika S. A. Mohiuddine |
| author_facet | Hari M. Srivastava Anupam Das Bipan Hazarika S. A. Mohiuddine |
| author_sort | Hari M. Srivastava |
| collection | DOAJ |
| description | The aim of this article is to establish the existence of the solution of non-linear functional integral equations <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mi>U</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>l</mi> </msubsup> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>h</mi> </msubsup> <mi>P</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>r</mi> <mi>d</mi> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> </mfenced> </mfenced> <mo>×</mo> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>a</mi> </msubsup> <mi>Q</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mfenced> <mi>d</mi> <mi>r</mi> <mi>d</mi> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> </mfenced> </mrow> </semantics> </math> </inline-formula> of two variables, which is of the form of two operators in the setting of Banach algebra <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mfenced separators="" open="(" close=")"> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> <mo>×</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> </mfenced> <mo>,</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mfenced separators="" open="(" close=")"> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> <mo>×</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> </mfenced> </mrow> </semantics> </math> </inline-formula> and a fixed point theorem, which is a generalization of Darbo’s fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example. |
| first_indexed | 2024-04-11T18:28:23Z |
| format | Article |
| id | doaj.art-22e6f6f6bedb45e5935d6c4909035ee9 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T18:28:23Z |
| publishDate | 2019-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-22e6f6f6bedb45e5935d6c4909035ee92022-12-22T04:09:32ZengMDPI AGSymmetry2073-89942019-05-0111567410.3390/sym11050674sym11050674Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach AlgebraHari M. Srivastava0Anupam Das1Bipan Hazarika2S. A. Mohiuddine3Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, Arunachal Pradesh, IndiaDepartment of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, Arunachal Pradesh, IndiaOperator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaThe aim of this article is to establish the existence of the solution of non-linear functional integral equations <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mi>U</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>l</mi> </msubsup> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>h</mi> </msubsup> <mi>P</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>r</mi> <mi>d</mi> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> </mfenced> </mfenced> <mo>×</mo> <mi>G</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>a</mi> </msubsup> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>a</mi> </msubsup> <mi>Q</mi> <mfenced separators="" open="(" close=")"> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mfenced> <mi>d</mi> <mi>r</mi> <mi>d</mi> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>h</mi> <mo>)</mo> </mrow> </mfenced> </mrow> </semantics> </math> </inline-formula> of two variables, which is of the form of two operators in the setting of Banach algebra <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mfenced separators="" open="(" close=")"> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> <mo>×</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> </mfenced> <mo>,</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mfenced separators="" open="(" close=")"> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> <mo>×</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>]</mo> </mfenced> </mrow> </semantics> </math> </inline-formula> and a fixed point theorem, which is a generalization of Darbo’s fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example.https://www.mdpi.com/2073-8994/11/5/674functional integral equationsBanach algebrafixed point theoremmeasure of noncompactness |
| spellingShingle | Hari M. Srivastava Anupam Das Bipan Hazarika S. A. Mohiuddine Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra Symmetry functional integral equations Banach algebra fixed point theorem measure of noncompactness |
| title | Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra |
| title_full | Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra |
| title_fullStr | Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra |
| title_full_unstemmed | Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra |
| title_short | Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra |
| title_sort | existence of solution for non linear functional integral equations of two variables in banach algebra |
| topic | functional integral equations Banach algebra fixed point theorem measure of noncompactness |
| url | https://www.mdpi.com/2073-8994/11/5/674 |
| work_keys_str_mv | AT harimsrivastava existenceofsolutionfornonlinearfunctionalintegralequationsoftwovariablesinbanachalgebra AT anupamdas existenceofsolutionfornonlinearfunctionalintegralequationsoftwovariablesinbanachalgebra AT bipanhazarika existenceofsolutionfornonlinearfunctionalintegralequationsoftwovariablesinbanachalgebra AT samohiuddine existenceofsolutionfornonlinearfunctionalintegralequationsoftwovariablesinbanachalgebra |