Finite Family Of Nonexpansive Mappings Research Articles

Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce and study an iterative scheme by a hybrid method for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a real Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.

A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings

In this paper, we introduce and study a new mapping generated by a finite family of nonexpansive mappings and finite real numbers and introduce a general iterative method concerning the new mappings for finding a common element of the set of solutions of an equilibrium problem and of the set of common fixed points of a finite family of nonexpansive mappings in a Hilbert space. Then, we prove a strong convergence theorem of the proposed iterative method for a finite family of nonexpansive mappings to the unique solution of variational inequality which is the optimality condition for a minimization problem. Our main result can be applied to obtain strong convergence of the general iterative methods which are modifications of those in [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (1) (2006) 43–52; S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (1) (2007) 455–469; S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506–515] to a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.

Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings

In this paper, we introduce a new mapping and a Hybrid iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a Hilbert space. Then, we prove the strong convergence of the proposed iterative algorithm to a common fixed point of a finite family of nonexpansive mappings which is a solution of the generalized equilibrium problem. The results obtained in this paper extend the recent ones of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033].

A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rbe a finite family of nonexpansive self-mappings ofC. For a givenx1∈C, let{xn}and{xn(i)},i=1,2,…,r, be sequences definedxn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,…, xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1, whereani(j)∈[0,1]for allj∈{1,2,…,r},n∈ℕandi=1,2,…,j. In this paper, weak and strong convergence theorems of the sequence{xn}to a common fixed point of a finite family of nonexpansive mappingsTi (i=1,2,…,r)are established under some certain control conditions.

Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

Let E be a real uniformly convex Banach space, and let{Ti : i ∈ I} be N nonexpansive mappings from E into itself with F = {x ∈ E : Tix = x, i ∈ I} ≠ ϕ, where I = {1, 2, …, N}. From an arbitrary initial point x1 ∈ E, hybrid iteration scheme {xn} is defined as follows: xn+1 = αnxn + (1 − αn)(Tnxn − λn+1μA(Tnxn)), n ≥ 1, where A : E → E is an L‐Lipschitzian mapping, Tn = Ti, i = n(mod N), 1 ≤ i ≤ N, μ > 0, {λn}⊂[0, 1), and {αn}⊂[a, b] for some a, b ∈ (0, 1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of the mappings {Ti : i ∈ I} are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).

Some results on non-expansive mappings and relaxed cocoercive mappings in Hilbert spaces

In this article, we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of non-expansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Our results improve and extend the recent ones announced by Iiduka and Tahakshi [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), pp. 341–350] and many others.

A New Approximation Scheme Combining the Viscosity Method with Extragradient Method for Mixed Equilibrium Problems

AbstractWe introduce a new approximation scheme combining the viscosity method with extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature.

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak ⋆ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping J Φ with gauge ϕ. Let f be an α-contraction and { T n } a sequence of nonexpansive mappings, we study the strong convergence of explicit iterative schemes (1) x n + 1 = α n f ( x n ) + ( 1 − α n ) T n x n with a general theorem and then recover and improve some specific cases studied in the literature [K. Aoyoma, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed point of a countable family of nonexpansive mappings, Nonlinear Anal. 67 (8) (2007) 2350–2360; G. Lopez, V. Martin, H.-K. Xu, Perturbation techniques for nonexpansive mappings with applications, Nonlinear Anal. Real World Appl., in press, available online 4 May 2008; H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (1) (2004) 279–291; T.-H. Kim, H.-K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (1–2) (2005) 51–60; Y. Song, R. Chen, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. Math. Comput. 180 (2006) 275–287; Y. Song, R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl. 321 (1) (2006) 316–326; J. Chen, L. Zhang, T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2) (2007) 1450–1461; Y. Kimura, W. Takahashi, M. Toyoda, Convergence to common fixed points of a finite family of nonexpansive mappings, Arch. Math. 84 (2005) 350–363].

Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings

Let X be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. First purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self-mapping T and prove that this iterative process converges strongly to x ∗ ∈ F( T) ≔ { x ∈ X∣ x = T( x)}, where x ∗ is the unique solution in F( T) to the following variational inequality: 〈 f ( x ∗ ) - x ∗ , j ( v - x ∗ ) 〉 ⩽ 0 for all v ∈ F ( T ) . Second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self-mapping T and prove that these iterative schemes strongly converge to the same point x ∗ ∈ F( T). Basically, we show that if the perturbation mapping is nonexpansive, then the convergence property of the iterative process holds. In this respect, the results presented here extend, improve and unify some very recent theorems in the literature, see [L.C. Zeng, J.C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal. 64 (2006) 2507–2515; H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291; Y.S. Song, R.D. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. (2006)].

An iterative method for finding common solutions of equilibrium and fixed point problems

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

Strong Convergence of a Modified Iterative Algorithm for Mixed-Equilibrium Problems in Hilbert Spaces

The purpose of this paper is to study the strong convergence of a modified iterative scheme to find a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping, as well as the set of solutions of a mixed-equilibrium problem. Our results extend recent results of Takahashi and Takahashi (2007), Marino and Xu (2006), Combettes and Hirstoaga (2005), Iiduka and Takahashi (2005), and many others.

Weak and Strong Convergence Theorems for Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping and a finite family of nonexpansive mappings , respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others.

Convergence Theorems of Fixed Points for a Finite Family of Nonexpansive Mappings in Banach Spaces

We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.

Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variation inequalities for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend recent results announced by many others.

Approximation Methods for a Common Fixed Point of a Finite Family of Nonexpansive Mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: K → E be a continuous pseudocontractive mapping and f:K → E a contraction, both satisfying weakly inward condition. Then for t ∈ (0, 1), there exists a sequence {y t } ⊂ K satisfying the following condition: y t = (1 − t)f(y t ) + tT(y t ). Suppose further that {y t } is bounded or F(T) ≠ ∅ and E is a reflexive Banach space having weakly continuous duality mapping J ϕ for some gauge ϕ. Then it is proved that {y t } converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.

Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces

In this paper, we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Our results improve and extend the recent ones announced by Chen et al. [J.M. Chen, L.J. Zhang, T.G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, doi:10.1016/j.jmaa.2006.12.088], Iiduka and Tahakshi [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350], Yao and Yao [Y.H. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput, doi:10.1016/j.amc.2006.08.062] and Many others.

Implicit Iterations of Two Finite Families for Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to study the weak and strong convergence of an implicit iteration process to a common fixed point for two finite families of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Xu and Ori, Numer. Funct. Anal. Optim. 2001; 22:767–773, Zhou and Chang, Numer. Fund. Anal. Optim. 2002; 23:911–921, Chidume and Shahzad, Nonlinear. Anal. 2005; 62: 1149–1156.

Convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive Banach spaces

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty closed convex subset of E . Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive mappings. Let T 1 , T 2 , … , T N be a family of nonexpansive self-mappings of K , with F ≔ ⋂ i = 1 N Fix ( T i ) ≠ 0̸ , F = Fix ( T N T N − 1 … T 1 ) = Fix ( T 1 T N … T 2 ) = … = Fix ( T N − 1 T N − 2 … T 1 T N ) . Let { λ n } be a sequence in ( 0 , 1 ) satisfying the following conditions: C 1 : lim λ n = 0 ; C 2 : ∑ λ n = ∞ . For a fixed δ ∈ ( 0 , 1 ) , define S n : K → K by S n x ≔ ( 1 − δ ) x + δ T n x ∀ x ∈ K where T n = T n mod N . For an arbitrary fixed u , x 0 ∈ K , let B ≔ { x ∈ K : T N T N − 1 … T 1 x = γ x + ( 1 − γ ) u , for some γ > 1 } be bounded, and let the sequence { x n } be defined iteratively by x n + 1 = λ n + 1 u + ( 1 − λ n + 1 ) S n + 1 x n , for n ≥ 0 . Assume that lim n → ∞ ‖ T n x n − T n + 1 x n ‖ = 0 . Then, { x n } converges strongly to a common fixed point of the family T 1 , T 2 , … , T N . This convergence theorem is also proved for non-self maps.

Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T : K → E be a nonexpansive weakly inward mapping with F ( T ) ≠ ∅ and f : K → K be a contraction. Then for t ∈ ( 0 , 1 ) , there exists a sequence { y t } ⊂ K satisfying y t = ( 1 - t ) f ( y t ) + tT ( y t ) . Furthermore, if E is a strictly convex real reflexive Banach space having a uniformly Gâteaux differentiable norm, then { y t } converges strongly to a fixed point p of T such that p is the unique solution in F ( T ) to a certain variational inequality. Moreover, if { T i , i = 1 , 2 , … , r } is a family of nonexpansive mappings, then an explicit iteration process which converges strongly to a common fixed point of { T i , i = 1 , 2 , … , r } and to a solution of a certain variational inequality is constructed. Under the above setting, the family T i , i = 1 , 2 , … , r need not satisfy the requirment that ⋂ i = 1 r F ( T i ) = F ( T r T r - 1 , … , T 1 ) = F ( T 1 T r , … , T 2 ) = , ⋯ , = F ( T r - 1 T r - 2 , … , T 1 T r ) .

Strong Convergence Theorems for a Finite Family of Nonexpansive Mappings

AbstractWe modified the classic Mann iterative process to have strong convergence theorem for a finite family of nonexpansive mappings in the framework of Hilbert spaces. Our results improve and extend the results announced by many others.