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Behzad Djafari-Rouhani,Mohammad Farid,Kaleem Raza Kazmi 대한수학회 2016 대한수학회지 Vol.53 No.1
In this paper, we introduce and study an explicit hybrid relaxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.
DJAFARI-ROUHANI, BEHZAD,FARID, MOHAMMAD,KAZMI, KALEEM RAZA Korean Mathematical Society 2016 대한수학회지 Vol.53 No.1
In this paper, we introduce and study an explicit hybrid relaxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.
Nonlinear ergodic theorems of nonexpansive type mappings
Academic Press 2009 Journal of mathematical analysis and applications Vol.358 No.2
Let S be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banach space E whose norm is Frechet differentiable and @?={T<SUB>t</SUB>:t@?S} be a continuous representation of S as almost asymptotically nonexpansive type mapping of C into C such that the common fixed point set F(@?) of @? in C is nonempty. In this paper, we prove that if S is right reversible then for each x@?C, the closed convex set @?<SUB>s@?S</SUB>co@?{T<SUB>t</SUB>x:t@?s}@?F(@?) consists of at most one point. We also prove that if S is reversible, then the intersection @?<SUB>s@?S</SUB>co@?{T<SUB>t</SUB>x:t@?s}@?F(@?) is nonempty for each x@?C if and only if there exists a nonexpansive retraction P of C onto F(@?) such that PT<SUB>t</SUB>=T<SUB>t</SUB>P=P for all t@?S and Px is in the closed convex hull of {T<SUB>t</SUB>x:t@?S} for each x@?C.
STRONG CONVERGENCE OF PATHS FOR NONEXPANSIVE SEMIGROUPS IN BANACH SPACES
Kang, Shin Min,Cho, Sun Young,Kwun, Young Chel The Kangwon-Kyungki Mathematical Society 2011 한국수학논문집 Vol.19 No.3
Let E be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, C be a nonempty closed convex subset of E and f : $C{\rightarrow}C$ be a fixed bounded continuous strong pseudocontraction with the coefficient ${\alpha}{\in}(0,1)$. Let $\{{\lambda}_t\}_{0<t<1}$ be a net of positive real numbers such that ${\lim}_{t{\rightarrow}0}{\lambda}_t={\infty}$ and S = {$T(s)$ : $0{\leq}s$ < ${\infty}$} be a nonexpansive semigroup on C such that $F(S){\neq}{\emptyset}$, where F(S) denotes the set of fixed points of the semigroup. Then sequence {$x_t$} defined by $x_t=tf(x_t)+(1-t)\frac{1}{{\lambda}_t}{\int_{0}}^{{\lambda}_t}T(s)x{_t}ds$ converges strongly as $t{\rightarrow}0$ to $\bar{x}{\in}F(S)$, which solves the following variational inequality ${\langle}(f-I)\bar{x},\;p-\bar{x}{\rangle}{\leq}0$ for all $p{\in}F(S)$.
THUY, LE QUANG,MUU, LE DUNG The Kangwon-Kyungki Mathematical Society 2015 한국수학논문집 Vol.23 No.3
In this paper we propose an iteration hybrid method for approximating a point in the intersection of the solution-sets of pseudomonotone equilibrium and variational inequality problems and the fixed points of a semigroup-nonexpensive mappings in Hilbert spaces. The method is a combination of projection, extragradient-Armijo algorithms and Manns method. We obtain a strong convergence for the sequences generated by the proposed method.
On the Strong Convergence Theorems for Asymptotically Nonexpansive Semigroups in Banach Spaces
Shih-sen Chang,Liang Cai Zhao,Ding Ping Wu 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.1
Some strong convergence theorems of explicit iteration scheme for asymptotically nonexpansive semi-groups in Banach spaces are estab- lished. The results presented in this paper extend and improve some re- cent results in [T. Suzuki. On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131(2002)2133–2136; H. K. Xu. A strong convergence theoremfor contrac- tion semigroups in Banach spaces, Bull. Aust. Math. Soc. 72(2005)371– 379; N. Shioji and W. Takahashi. Strong convergence theorems for con- tinuous semigroups in Banach spaces, Math. Japonica. 1(1999)57–66; T. Shimizu and W. Takahashi. Strong convergence to common fixed points of families of nonexpansivemappings, J. Math. Anal. Appl. 211(1997)71–83; N. Shioji and W. Takahashi. Strong convergence theorems for asymptot- ically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. TMA, 34(1998)87–99; H. K. Xu. Approximations to fixed points of contraction semigroups in Hilbert space, Numer. Funct. Anal. Optim. 19(1998), 157–163.] Some strong convergence theorems of explicit iteration scheme for asymptotically nonexpansive semi-groups in Banach spaces are estab- lished. The results presented in this paper extend and improve some re- cent results in [T. Suzuki. On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131(2002)2133–2136; H. K. Xu. A strong convergence theoremfor contrac- tion semigroups in Banach spaces, Bull. Aust. Math. Soc. 72(2005)371– 379; N. Shioji and W. Takahashi. Strong convergence theorems for con- tinuous semigroups in Banach spaces, Math. Japonica. 1(1999)57–66; T. Shimizu and W. Takahashi. Strong convergence to common fixed points of families of nonexpansivemappings, J. Math. Anal. Appl. 211(1997)71–83; N. Shioji and W. Takahashi. Strong convergence theorems for asymptot- ically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. TMA, 34(1998)87–99; H. K. Xu. Approximations to fixed points of contraction semigroups in Hilbert space, Numer. Funct. Anal. Optim. 19(1998), 157–163.]
ON THE STRONG CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE SEMIGROUPS IN BANACH SPACES
Chang, Shih-Sen,Zhao, Liang Cai,Wu, Ding Ping The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.1
Some strong convergence theorems of explicit iteration scheme for asymptotically nonexpansive semi-groups in Banach spaces are established. The results presented in this paper extend and improve some recent results in [T. Suzuki. On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131(2002)2133-2136; H. K. Xu. A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72(2005)371-379; N. Shioji and W. Takahashi. Strong convergence theorems for continuous semigroups in Banach spaces, Math. Japonica. 1(1999)57-66; T. Shimizu and W. Takahashi. Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211(1997)71-83; N. Shioji and W. Takahashi. Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. TMA, 34(1998)87-99; H. K. Xu. Approximations to fixed points of contraction semigroups in Hilbert space, Numer. Funct. Anal. Optim. 19(1998), 157-163.]
Kim, Jong-Kyu Korean Mathematical Society 1997 대한수학회논문집 Vol.12 No.2
In this paper, we study the asymptotic behavior of orbits ${S(t)x}$ of an asymptotically nonexpansive semigroup $S = {S(t) : t \in G}$ for a right reversible semitopological semigroup G, defined on a weakly compact convex subset C of Banach spaces with Opial's condition for any $x \in C$.
Kim, J.K.,Nam, Y.M.,Jin, B.J. Korean Mathematical Society 1998 대한수학회논문집 Vol.13 No.3
In this paper, we deal with the asymptotic behavior for the almost-orbits {u(t)} of an asymptotically nonexpansive semigroup S = {S(t) : t $\in$ G} for a right reversible semitopological semigroup G, defined on a suitable subset C of Banach spaces with the Opial's condition, locally uniform Opial condition, or uniform Opial condition.
STRONG CONVERGENCE OF HYBRID METHOD FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS AND SEMIGROUPS
Li Liu,LijingWang,Yongfu Su 한국전산응용수학회 2011 Journal of applied mathematics & informatics Vol.29 No.3
In this paper, some strong convergence theorems are obtained for hybrid method for modified Ishikawa iteration process of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups in Hilbert spaces. The results presented in this article generalize and improve results of Tae-Hwa Kim and Hong-Kun Xu and others. The convergence rate of the iteration process presented in this article is faster than hybrid method of Tae-Hwa Kim and Hong-Kun Xu and others.