Strong convergence theorems of k -strict pseudo-contractions in Hilbert spaces

Xiao Long Qin; Shin Min Kang; Mei Juan Shang

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 3, page 695-706
  • ISSN: 0011-4642

Abstract

top
Let K be a nonempty closed convex subset of a real Hilbert space H such that K ± K ⊂ K , T K → H a k -strict pseudo-contraction for some 0 ≤ k < 1 such that F ( T ) = { x ∈ K x = T x } ≠ ∅ . Consider the following iterative algorithm given by ∀ x 1 ∈ K , x n + 1 = α n γ f ( x n ) + β n x n + ( ( 1 - β n ) I - α n A ) P K S x n , n ≥ 1 , where S K → H is defined by S x = k x + ( 1 - k ) T x , P K is the metric projection of H onto K , A is a strongly positive linear bounded self-adjoint operator, f is a contraction. It is proved that the sequence { x n } generated by the above iterative algorithm converges strongly to a fixed point of T , which solves a variational inequality related to the linear operator A . Our results improve and extend the results announced by many others.

How to cite

top

Qin, Xiao Long, Kang, Shin Min, and Shang, Mei Juan. "Strong convergence theorems of $k$-strict pseudo-contractions in Hilbert spaces." Czechoslovak Mathematical Journal 59.3 (2009): 695-706. <http://eudml.org/doc/37952>.

@article{Qin2009,
abstract = {Let $K$ be a nonempty closed convex subset of a real Hilbert space $H$ such that $K\pm K\subset K$, $T\: K\rightarrow H$ a $k$-strict pseudo-contraction for some $0\le k<1$ such that $F(T)=\lbrace x\in K\: x=Tx\rbrace \ne \emptyset $. Consider the following iterative algorithm given by \[ \forall x\_1\in K,\quad x\_\{n+1\}=\alpha \_n\gamma f(x\_n)+\beta \_nx\_n+((1-\beta \_n)I-\alpha \_n A)P\_KSx\_n,\quad n\ge 1, \] where $S\: K\rightarrow H$ is defined by $Sx=kx+(1-k)Tx$, $P_K$ is the metric projection of $H$ onto $K$, $A$ is a strongly positive linear bounded self-adjoint operator, $f$ is a contraction. It is proved that the sequence $\lbrace x_n\rbrace $ generated by the above iterative algorithm converges strongly to a fixed point of $T$, which solves a variational inequality related to the linear operator $A$. Our results improve and extend the results announced by many others.},
author = {Qin, Xiao Long, Kang, Shin Min, Shang, Mei Juan},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hilbert space; nonexpansive mapping; strict pseudo-contraction; iterative algorithm; fixed point; Hilbert space; nonexpansive mapping; strict pseudocontraction; iterative algorithm; fixed point; strong convergence},
language = {eng},
number = {3},
pages = {695-706},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strong convergence theorems of $k$-strict pseudo-contractions in Hilbert spaces},
url = {http://eudml.org/doc/37952},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Qin, Xiao Long
AU - Kang, Shin Min
AU - Shang, Mei Juan
TI - Strong convergence theorems of $k$-strict pseudo-contractions in Hilbert spaces
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 695
EP - 706
AB - Let $K$ be a nonempty closed convex subset of a real Hilbert space $H$ such that $K\pm K\subset K$, $T\: K\rightarrow H$ a $k$-strict pseudo-contraction for some $0\le k<1$ such that $F(T)=\lbrace x\in K\: x=Tx\rbrace \ne \emptyset $. Consider the following iterative algorithm given by \[ \forall x_1\in K,\quad x_{n+1}=\alpha _n\gamma f(x_n)+\beta _nx_n+((1-\beta _n)I-\alpha _n A)P_KSx_n,\quad n\ge 1, \] where $S\: K\rightarrow H$ is defined by $Sx=kx+(1-k)Tx$, $P_K$ is the metric projection of $H$ onto $K$, $A$ is a strongly positive linear bounded self-adjoint operator, $f$ is a contraction. It is proved that the sequence $\lbrace x_n\rbrace $ generated by the above iterative algorithm converges strongly to a fixed point of $T$, which solves a variational inequality related to the linear operator $A$. Our results improve and extend the results announced by many others.
LA - eng
KW - Hilbert space; nonexpansive mapping; strict pseudo-contraction; iterative algorithm; fixed point; Hilbert space; nonexpansive mapping; strict pseudocontraction; iterative algorithm; fixed point; strong convergence
UR - http://eudml.org/doc/37952
ER -

References

top
  1. Acedo, G. L., Xu, H. K., 10.1016/j.na.2006.08.036, Nonlinear Anal. 67 (2007), 2258-2271. (2007) Zbl1133.47050MR2331876DOI10.1016/j.na.2006.08.036
  2. Browder, F. E., 10.1073/pnas.53.6.1272, Proc. Natl. Acad. Sci. USA 53 (1965), 1272-1276. (1965) MR0178324DOI10.1073/pnas.53.6.1272
  3. Browder, F. E., 10.1007/BF00251595, Arch. Ration. Mech. Anal. 24 (1967), 82-90. (1967) Zbl0148.13601MR0206765DOI10.1007/BF00251595
  4. Browder, F. E., Petryshyn, W. V., 10.1016/0022-247X(67)90085-6, J. Math. Anal. Appl. 20 (1967), 197-228. (1967) Zbl0153.45701MR0217658DOI10.1016/0022-247X(67)90085-6
  5. Halpern, B., 10.1090/S0002-9904-1967-11864-0, Bull. Amer. Math. Soc. 73 (1967), 957-961. (1967) MR0218938DOI10.1090/S0002-9904-1967-11864-0
  6. Lions, P. L., Approximation de points fixes de contractions, C.R. Acad. Sci. Paris Ser. A--B 284 (1977), A1357--A1359. (1977) Zbl0349.47046MR0470770
  7. Marino, G., Xu, H. K., 10.1016/j.jmaa.2006.06.055, J. Math. Anal. Appl. 329 (2007), 336-349. (2007) MR2306805DOI10.1016/j.jmaa.2006.06.055
  8. Marino, G., Xu, H. K., 10.1016/j.jmaa.2005.05.028, J. Math. Anal. Appl. 318 (2006), 43-52. (2006) Zbl1095.47038MR2210870DOI10.1016/j.jmaa.2005.05.028
  9. Moudafi, A., 10.1006/jmaa.1999.6615, J. Math. Anal. Appl. 241 (2000), 46-55. (2000) Zbl0957.47039MR1738332DOI10.1006/jmaa.1999.6615
  10. Suzuki, T., Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl. (2005), 103-123. (2005) Zbl1123.47308MR2172156
  11. Wittmann, R., 10.1007/BF01190119, Arch. Math. 58 (1992), 486-491. (1992) Zbl0797.47036MR1156581DOI10.1007/BF01190119
  12. Xu, H. K., 10.1023/A:1023073621589, J. Optim. Theory Appl. 116 (2003), 659-678. (2003) Zbl1043.90063MR1977756DOI10.1023/A:1023073621589
  13. Xu, H. K., 10.1112/S0024610702003332, J. London Math. Soc. 66 (2002), 240-256. (2002) Zbl1013.47032MR1911872DOI10.1112/S0024610702003332
  14. Xu, H. K., 10.1017/S0004972700020116, Bull. Austral. Math. Soc. 65 (2002), 109-113. (2002) Zbl1030.47036MR1889384DOI10.1017/S0004972700020116
  15. Xu, H. K., 10.1016/j.jmaa.2004.04.059, J. Math. Anal. Appl. 298 (2004), 279-291. (2004) Zbl1061.47060MR2086546DOI10.1016/j.jmaa.2004.04.059
  16. Zhou, H., 10.1016/j.na.2007.05.032, Nonlinear Analysis 69 (2008), 456-462. (2008) MR2426262DOI10.1016/j.na.2007.05.032

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.