The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive m...The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.展开更多
Let E be a uniformly convex Banach space which satisfies Opial's condition or has a Frechet differentiable norm,and C be a bounded closed convex subset of E. If T∶C→C is (asymptotically)nonexpans...Let E be a uniformly convex Banach space which satisfies Opial's condition or has a Frechet differentiable norm,and C be a bounded closed convex subset of E. If T∶C→C is (asymptotically)nonexpansive,then the modified Ishikawa iteration process defined byx n+1 =t nT ns nT nx n+1-s nx n+(1-t n)x n,converges weakly to a fixed point of T ,where {t n} and {s n} are sequences in [0,1] with some restrictions.展开更多
Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed poin...Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map- pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con- vex optimization problems, and split feasibility problems. Our result extends many recent important results.展开更多
It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gateaux differentiable norm are extendable to more general reflexive and strictly convex real Ban...It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G＆teaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.展开更多
The solution to evolution equations has developed an independent theory within nonlinear analysis dealing with the existence and approximation of such solution （ fixed point） of pseudocontractive operators and its v...The solution to evolution equations has developed an independent theory within nonlinear analysis dealing with the existence and approximation of such solution （ fixed point） of pseudocontractive operators and its variants. The object is to introduce a perturbed iteration method for proving the convergence of sequence of Lipschitzian pseudocontractive mapping using approximate fixed point technique. This iteration can be ued for nonlinear operators which are more general than Lipschitzian pseudocontractive operator and Bruck iteration fails for proving their convergence. Our results generalize the results of Chidume and Zegeye.展开更多
Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gateaux differentiable norm. Let T ： K →K be a uniformly continuous pseudocontractive mapping. Suppose every close...Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gateaux differentiable norm. Let T ： K →K be a uniformly continuous pseudocontractive mapping. Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Let {λn} C （0,1/2] be a sequence satisfying the conditions： （i） limn→∞λn=0; （ii） ∑n=0^∞ λn=∞. Let the sequence {xn} be generated from arbitrary x1∈K by xn＋1 = （1 -λn）xn ＋ λnTxn -λn（xn - x1）, n ≥ 1. Suppose limn→∞‖xn - Txn‖ = 0. Then {xn} converges strongly to a fixed point of T.展开更多
文摘The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.
基金Supported both by the National Natural Science Foundation(1 980 1 0 2 3 ) and the Teaching and ResearchAward Fund for Outstanding Young Teachers in Higher Education Institutions of MOEP.R.C
文摘Let E be a uniformly convex Banach space which satisfies Opial's condition or has a Frechet differentiable norm,and C be a bounded closed convex subset of E. If T∶C→C is (asymptotically)nonexpansive,then the modified Ishikawa iteration process defined byx n+1 =t nT ns nT nx n+1-s nx n+(1-t n)x n,converges weakly to a fixed point of T ,where {t n} and {s n} are sequences in [0,1] with some restrictions.
文摘Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map- pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con- vex optimization problems, and split feasibility problems. Our result extends many recent important results.
文摘It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G＆teaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.
文摘The solution to evolution equations has developed an independent theory within nonlinear analysis dealing with the existence and approximation of such solution （ fixed point） of pseudocontractive operators and its variants. The object is to introduce a perturbed iteration method for proving the convergence of sequence of Lipschitzian pseudocontractive mapping using approximate fixed point technique. This iteration can be ued for nonlinear operators which are more general than Lipschitzian pseudocontractive operator and Bruck iteration fails for proving their convergence. Our results generalize the results of Chidume and Zegeye.
基金the National Natural Science Foundation of China （No. 10771050）.
文摘Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gateaux differentiable norm. Let T ： K →K be a uniformly continuous pseudocontractive mapping. Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Let {λn} C （0,1/2] be a sequence satisfying the conditions： （i） limn→∞λn=0; （ii） ∑n=0^∞ λn=∞. Let the sequence {xn} be generated from arbitrary x1∈K by xn＋1 = （1 -λn）xn ＋ λnTxn -λn（xn - x1）, n ≥ 1. Suppose limn→∞‖xn - Txn‖ = 0. Then {xn} converges strongly to a fixed point of T.
