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Details van artikel 12 van 68 gevonden artikelen
| Titel: |
Convergence Theorems for Asymptotically Nonexpansive Mappings in Banach Spaces |
| Auteur: |
Yongfu Su Xiaolong Qin Meijuan Shang |
| Verschenen in: |
Acta mathematica Universitatis Comenianae |
| Paginering: | Jaargang LXXVII (2008) nr. 1 pagina's 31-42 |
| Jaar: |
2008 |
| Inhoud: |
Let <i>E</i> be a uniformly convex Banach space, and let <i>K</i> be a nonempty convex closed subset which is also a nonexpansive retract of <i>E</i>. Let <i>T</i>: <i>K</i> <FONT SIZE='3' FACE='Symbol'>®</FONT> <i>E</i> be an asymptotically nonexpansive mapping with {<i>k<sub>n</sub></i>} <FONT SIZE='3' FACE='Symbol'>Ì</FONT> [1, <FONT SIZE='3' FACE='Symbol'>¥</FONT>) such that (<FONT SIZE='3' FACE='Symbol'>å</FONT> from <i>n</i>=1 to <FONT SIZE='3' FACE='Symbol'>¥</FONT>)(<i>k<sub>n</sub></i> - 1) < <FONT SIZE='3' FACE='Symbol'>¥</FONT> and let <i>F</i>(<i>T</i>) be nonempty, where <i>F</i>(<i>T</i>) denotes the fixed points set of <i>T</i>. Let<FONT SIZE='3' FACE='Symbol'>{a</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>, <FONT SIZE='3' FACE='Symbol'>{b</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>,<FONT SIZE='3' FACE='Symbol'>{g</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>, <FONT SIZE='3' FACE='Symbol'>{a¢</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>, <FONT SIZE='3' FACE='Symbol'>{b¢</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>, <FONT SIZE='3' FACE='Symbol'>{g¢</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>, <FONT SIZE='3' FACE='Symbol'>{a¢¢</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>, <FONT SIZE='3' FACE='Symbol'>{b¢¢</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>and <FONT SIZE='3' FACE='Symbol'>{g¢¢</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT>be real sequences in [0, 1] such that <FONT SIZE='3' FACE='Symbol'>a</FONT><i><sub>n</sub></i> +<FONT SIZE='3' FACE='Symbol'>b</FONT><i><sub>n</sub></i> +<FONT SIZE='3' FACE='Symbol'>g</FONT><i><sub>n</sub></i> =<FONT SIZE='3' FACE='Symbol'>a¢</FONT><i><sub>n</sub></i> +<FONT SIZE='3' FACE='Symbol'>b¢</FONT><i><sub>n</sub></i> +<FONT SIZE='3' FACE='Symbol'>g¢</FONT><i><sub>n</sub></i> =<FONT SIZE='3' FACE='Symbol'>a¢¢</FONT><i><sub>n</sub></i> +<FONT SIZE='3' FACE='Symbol'>b¢¢</FONT><i><sub>n</sub></i> +<FONT SIZE='3' FACE='Symbol'>g¢¢</FONT><i><sub>n</sub></i> = 1 and <FONT SIZE='3' FACE='Symbol'> e £ a</FONT><i><sub>n</sub></i>,<FONT SIZE='3' FACE='Symbol'>a¢</FONT><i><sub>n</sub></i>, <FONT SIZE='3' FACE='Symbol'>a¢¢</FONT><i><sub>n</sub></i><FONT SIZE='3' FACE='Symbol'> £ 1 - e</FONT> for all <i>n</i> <FONT SIZE='3' FACE='Symbol'> Î</FONT> <i>N</i> and some <FONT SIZE='3' FACE='Symbol'> e > 0</FONT>, starting with arbitrary <i>x</i><sub>1</sub> <FONT SIZE='3' FACE='Symbol'> Î</FONT> <i>K</i>,define the sequence <FONT SIZE='3' FACE='Symbol'>{ </FONT><i>x<sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT> by setting <br> <i>z<sub>n</sub> = P</i>(<FONT SIZE='3' FACE='Symbol'>a¢¢</FONT><i><sub>n</sub></i><i>T</i>(<i>PT</i>)<sup><i>n</i>-1</sup><i>x<sub>n</sub></i>+ <FONT SIZE='3' FACE='Symbol'>b¢¢</FONT><i><sub>n</sub></i><i>x<sub>n</sub></i>+ <FONT SIZE='3' FACE='Symbol'>g¢¢</FONT><i><sub>n</sub></i><i>w<sub>n</sub></i>),<br> <i>y<sub>n</sub> =P</i>(<FONT SIZE='3' FACE='Symbol'>a¢</FONT><i><sub>n</sub></i><i>T</i>(<i>PT</i>)<sup><i>n</i>-1</sup><i>z<sub>n</sub></i>+ <FONT SIZE='3' FACE='Symbol'>b¢</FONT><i><sub>n</sub></i><i>x<sub>n</sub></i>+ <FONT SIZE='3' FACE='Symbol'>g¢</FONT><i><sub>n</sub></i><i>v<sub>n</sub></i>),<br> <i>x</i><sub><i>n</i>+1</sub> =<i>P</i>(<FONT SIZE='3' FACE='Symbol'>a</FONT><i><sub>n</sub></i><i>T</i>(<i>PT</i>)<sup><i>n</i>-1</sup><i>y<sub>n</sub></i>+ <FONT SIZE='3' FACE='Symbol'>b</FONT><i><sub>n</sub></i><i>x<sub>n</sub></i>+ <FONT SIZE='3' FACE='Symbol'>g</FONT><i><sub>n</sub></i><i>u<sub>n</sub></i>),<br><br>with the restrictions (<FONT SIZE='3' FACE='Symbol'>å</FONT> from <i>n</i>=1 to <FONT SIZE='3' FACE='Symbol'>¥</FONT>) (<FONT SIZE='3' FACE='Symbol'>g</FONT><i><sub>n</sub></i>) < <FONT SIZE='4' FACE='Symbol'>¥</FONT>,(<FONT SIZE='3' FACE='Symbol'>å</FONT> from <i>n</i>=1 to <FONT SIZE='3' FACE='Symbol'>¥</FONT>) (<FONT SIZE='3' FACE='Symbol'>g¢</FONT><i><sub>n</sub></i>) < <FONT SIZE='4' FACE='Symbol'>¥</FONT> and (<FONT SIZE='3' FACE='Symbol'>å</FONT> from <i>n</i>=1 to <FONT SIZE='3' FACE='Symbol'>¥</FONT>) (<FONT SIZE='3' FACE='Symbol'>g¢¢</FONT><i><sub>n</sub></i>) < <FONT SIZE='4' FACE='Symbol'>¥</FONT> where <FONT SIZE='3' FACE='Symbol'>{ </FONT><i>w<sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT> , <FONT SIZE='3' FACE='Symbol'>{ </FONT><i>v<sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT> and <FONT SIZE='3' FACE='Symbol'>{ </FONT><i>u<sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT> are bounded sequences in <i>K</i>. <br> (i) If <i>E</i> is realuniformly convex Banach space satisfying <i>Opial's</i> condition, then weak convergence of <FONT SIZE='3' FACE='Symbol'>{ </FONT><i>x<sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT> to some <i>p</i> <FONT SIZE='3' FACE='Symbol'>Î </FONT> <i>F</i>(<i>T</i>) is obtained; <br> (ii) If <i>T</i> satisfies condition (A), then <FONT SIZE='3' FACE='Symbol'>{ </FONT><i>x<sub>n</sub></i><FONT SIZE='3' FACE='Symbol'>}</FONT> convergence strongly to some <i>p</i> <FONT SIZE='3' FACE='Symbol'>Î </FONT> <i>F</i>(<i>T</i>). |
| Uitgever: |
Acta Mathematica Universitatis Comenianae, Institute of Applied Mathematics (provided by DOAJ) |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
Details van artikel 12 van 68 gevonden artikelen
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