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Common fixed points of one-parameter nonexpansive semigroups in strictly convex Banach spaces
Tomonari Suzuki
Abstract and Applied Analysis , 2006,
Abstract: One of our main results is the following convergence theorem for one-parameter nonexpansive semigroups: let C be a bounded closed convex subset of a Hilbert space E , and let { T( t ):t∈ + } be a strongly continuous semigroup of nonexpansive mappings on C . Fix u∈C and t 1 , t 2 ∈ + with t 1 < t 2 . Define a sequence { x n } in C by x n = ( 1 α n ) / ( t 2 t 1 ) ∫ t 1 t 2 T( s ) x n ds+ α n u for n∈ , where { α n } is a sequence in ( 0,1 ) converging to 0 . Then { x n } converges strongly to a common fixed point of { T( t ):t∈ + } .
A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces  [PDF]
Pongsakorn Sunthrayuth,Poom Kumam
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/560671
Abstract: We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others. 1. Introduction Throughout this paper we denoted by and the set of all positive integers and all positive real numbers, respectively. Let be a real Banach space, and let be a nonempty closed convex subset of . A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . We know that is nonempty if is bounded; for more detail see [1]. A one-parameter family from of into itself is said to be a nonexpansive semigroup on if it satisfies the following conditions: (i) ;(ii) for all ;(iii)for each the mapping is continuous;(iv) for all and . We denote by the set of all common fixed points of , that is, . We know that is nonempty if is bounded; see [2]. Recall that a self-mapping is a contraction if there exists a constant such that for each . As in [3], we use the notation to denote the collection of all contractions on , that is, . Note that each has a unique fixed point in . In the last ten years, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3–5]. Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a strongly positive bounded linear operator on : that is, there is a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is the fixed point set of a nonexpansive mapping on and is a given point in . In 2003, Xu [3] proved that the sequence generated by converges strongly to the unique solution of the minimization problem (1.2) provided that the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] for further developments in both Hilbert and Banach spaces) and proved that if is a real Hilbert space, the sequence generated by the following algorithm: where is a contraction mapping with constant and satisfies certain conditions, converges strongly to a fixed point of in which is unique solution of the variational inequality: In 2006, Marino and Xu [8] combined the iterative method (1.3)
Perron conditions for exponential expansiveness of one-parameter semigroup
Bogdan Sasu
Le Matematiche , 2003,
Abstract: We present a new approach for the theorems of Perron type for exponential expansiveness of one-parameter semigroups in terms of l^p(N, X) spaces. We prove that an exponentially bounded semigroup is exponentially expansive if and only if the pair (l^p (N, X), l^q(N, X)) is completely admissible relative to a discrete equation associated to the semigroup, where p, q ∈ [1, ∞), p ≥ q. We apply our results in order to obtain very general characterizations for exponential expansiveness of C_0-semigroups in terms of the complete admissibility of the pair (L^ p (R_+ , X), L^ q (R_+ , X)) and for exponential dichotomy, respectively, in terms of the admissibility of the pair (L^p(R_+,X), L^q(R_+,X)).
A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions
Kamonrat Nammanee,Suthep Suantai,Prasit Cholamjiak
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/506976
Abstract: We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mapping , where is a gauge function on [0,∞). Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.
Common fixed points of one-parameter nonexpansive semigroups in strictly convex Banach spaces  [PDF]
Tomonari Suzuki
Abstract and Applied Analysis , 2006, DOI: 10.1155/aaa/2006/58684
Abstract: One of our main results is the following convergencetheorem for one-parameter nonexpansive semigroups: let C be a bounded closed convex subset of a Hilbert space E, and let {T(t):t∈ℝ
The set of common fixed points of an n-parameter continuous semigroup of mappings  [PDF]
Tomonari Suzuki
Mathematics , 2004,
Abstract: In this paper, using Kronecker's theorem, we discuss the set of common fixed points of an n-parameter continuous semigroup of mappings. We also discuss convergence theorems to a common fixed point of an n-parameter nonexpansive semigroup.
Semigroup Growth Bounds  [PDF]
E B Davies
Mathematics , 2003,
Abstract: We prove lower bounds on $||T_t||$, where $T_t$ is a one-parameter semigroup, starting from information on the resolvent norms, i.e. the pseudospectra. We provide a physically important example in which the growth of the semigroup norm cannot be derived from spectral information. The generator is a self-adjoint Schr\"odinger operator, but the semigroup growth is measured using the $L^1$ norm rather than the $L^2$ norm. Numerical confirmation of the results is provided.
The set of common fixed points of a one-parameter continuous semigroup of mappings is F(T(1)) cap F(T(sqrt 2))  [PDF]
Tomonari Suzuki
Mathematics , 2004,
Abstract: In this paper, we prove the following theorem: Let {T(t) : t >= 0} be a one-parameter continuous semigroup of mappings on a subset C of a Banach space E. The set of fixed points of T(t) is denoted by F(T(t)) for each t >= 0. Then cap_{t >= 0} F(T(t)) = F(T(1)) cap F(T(sqrt 2)) holds. Using this theorem, we discuss convergence theorems to a common fixed point of {T(t) : t >= 0}.
A Method for a Solution of Equilibrium Problem and Fixed Point Problem of a Nonexpansive Semigroup in Hilbert's Spaces  [cached]
Buong Nguyen,Duong NguyenDinh
Fixed Point Theory and Applications , 2011,
Abstract: We introduce a viscosity approximation method for finding a common element of the set of solutions for an equilibrium problem involving a bifunction defined on a closed, convex subset and the set of fixed points for a nonexpansive semigroup on another one in Hilbert's spaces.
Necessary and Sufficient Conditions Where One Γ-Semigroup is a Γ-Group
Sabri Sadiku
Journal of Modern Mathematics and Statistics , 2012, DOI: 10.3923/jmmstat.2010.44.49
Abstract: In this study, researchers have studied the Γ-algebraic structures and some characteristics of them. According to Sen and Saha, we defined algebraic structures: Γ-semigroup, Γ-regular semigroup, Γ-idempotent semigroup, Γ-invers semigroup and Γ-group. Theorem 2, 3 and 4 proves the existence of Γ-group and gives necessary and sufficient conditions where one Γ-semigroup is a Γ-group. Finally, theorem 5 shows necessary and sufficient conditions where one Γ-regular semigroup is a Γ-group. In addition, for every Γ- algebraic structure that we mentioned before we give an original example.
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