Stability of the second order partial differential equations
<p>Abstract</p> <p>We say that a functional equation (<it>ξ</it>) is stable if any function <it>g </it>satisfying the functional equation (<it>ξ</it>) approximately is near to a true solution of (<it>ξ</it>)...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2011-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2011/1/81 |
| _version_ | 1811280649407954944 |
|---|---|
| author | Ghaemi MB Cho YJ Alizadeh B Gordji M Eshaghi |
| author_facet | Ghaemi MB Cho YJ Alizadeh B Gordji M Eshaghi |
| author_sort | Ghaemi MB |
| collection | DOAJ |
| description | <p>Abstract</p> <p>We say that a functional equation (<it>ξ</it>) is stable if any function <it>g </it>satisfying the functional equation (<it>ξ</it>) approximately is near to a true solution of (<it>ξ</it>).</p> <p>In this paper, by using Banach's contraction principle, we prove the stability of nonlinear partial differential equations of the following forms:</p> <p><display-formula><m:math name="1029-242X-2011-81-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mfenced separators="" open="{" close=""> <m:mrow> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>a</m:mi> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>b</m:mi> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>p</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>q</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">-</m:mo> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>p</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>q</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">.</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mrow> </m:math> </display-formula></p> <p><it>2000 Mathematics Subject Classification</it>. 26D10; 34K20; 39B52; 39B82; 46B99.</p> |
| first_indexed | 2024-04-13T01:19:46Z |
| format | Article |
| id | doaj.art-399d0a8d2ef441efa47eb8fb92092f10 |
| institution | Directory Open Access Journal |
| issn | 1025-5834 1029-242X |
| language | English |
| last_indexed | 2024-04-13T01:19:46Z |
| publishDate | 2011-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-399d0a8d2ef441efa47eb8fb92092f102022-12-22T03:08:49ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2011-01-012011181Stability of the second order partial differential equationsGhaemi MBCho YJAlizadeh BGordji M Eshaghi<p>Abstract</p> <p>We say that a functional equation (<it>ξ</it>) is stable if any function <it>g </it>satisfying the functional equation (<it>ξ</it>) approximately is near to a true solution of (<it>ξ</it>).</p> <p>In this paper, by using Banach's contraction principle, we prove the stability of nonlinear partial differential equations of the following forms:</p> <p><display-formula><m:math name="1029-242X-2011-81-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:mfenced separators="" open="{" close=""> <m:mrow> <m:mtable class="gathered"> <m:mtr> <m:mtd> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>a</m:mi> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>b</m:mi> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>p</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>q</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">-</m:mo> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mi>p</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>q</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>y</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-punc">.</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd/> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:mrow> </m:math> </display-formula></p> <p><it>2000 Mathematics Subject Classification</it>. 26D10; 34K20; 39B52; 39B82; 46B99.</p>http://www.journalofinequalitiesandapplications.com/content/2011/1/81generalized Hyers-Ulam stabilitylinear differential equationBanach's contraction principle |
| spellingShingle | Ghaemi MB Cho YJ Alizadeh B Gordji M Eshaghi Stability of the second order partial differential equations Journal of Inequalities and Applications generalized Hyers-Ulam stability linear differential equation Banach's contraction principle |
| title | Stability of the second order partial differential equations |
| title_full | Stability of the second order partial differential equations |
| title_fullStr | Stability of the second order partial differential equations |
| title_full_unstemmed | Stability of the second order partial differential equations |
| title_short | Stability of the second order partial differential equations |
| title_sort | stability of the second order partial differential equations |
| topic | generalized Hyers-Ulam stability linear differential equation Banach's contraction principle |
| url | http://www.journalofinequalitiesandapplications.com/content/2011/1/81 |
| work_keys_str_mv | AT ghaemimb stabilityofthesecondorderpartialdifferentialequations AT choyj stabilityofthesecondorderpartialdifferentialequations AT alizadehb stabilityofthesecondorderpartialdifferentialequations AT gordjimeshaghi stabilityofthesecondorderpartialdifferentialequations |