Nonspreading Mappings Research Articles

Existence and Approximation of Attractive Points of the Widely More Generalized Hybrid Mappings in Hilbert Spaces

We study the widely more generalized hybrid mappings which have been proposed to unify several well-known nonlinear mappings including the nonexpansive mappings, nonspreading mappings, hybrid mappings, and generalized hybrid mappings. Without the convexity assumption, we will establish the existence theorem and mean convergence theorem for attractive point of the widely more generalized hybrid mappings in a Hilbert space. Moreover, we prove a weak convergence theorem of Mann’s type and a strong convergence theorem of Shimizu and Takahashi’s type for such a wide class of nonlinear mappings in a Hilbert space. Our results can be viewed as a generalization of Kocourek, Takahashi and Yao, and Hojo and Takahashi where they studied the generalized hybrid mappings.

Weak and Strong Convergence Theorems for Strictly Pseudononspreading Mappings and Equilibrium Problem in Hilbert Spaces

The purpose of this paper is to propose an iterative algorithm for equilibrium problem and a class of strictly pseudononspreading mappings which is more general than the class of nonspreading mappings studied recently in Kurokawa and Takahashi (2010). We explored an auxiliary mapping in our theorems and proofs and under suitable conditions, some weak and strong convergence theorems are proved. The results presented in the paper extend and improve some recent results announced by some authors.

Strong convergence of the hybrid method for a finite family of nonspreading mappings and variational inequality problems

In this paper, we prove a strong convergence theorem by the hybrid method for finding a common element of the set of fixed points of a finite family of nonspreading mappings and the set of solutions of a finite family of variational inequality problems.

Fixed point theorems of ( a , b ) -monotone mappings in Hilbert spaces

We propose a new class of nonlinear mappings, called -monotone mappings, and show that this class of nonlinear mappings contains nonspreading mappings, hybrid mappings, firmly nonexpansive mappings, and -generalized hybrid mappings with . We also give an example to show that a -monotone mapping is not necessary to be a quasi-nonexpansive mapping. We establish an existence theorem of fixed points and the demiclosed principle for the class of -monotone mappings. As a special case of our result, we give an existence theorem of fixed points for -generalized hybrid mappings with . We also consider Mann’s type weak convergence theorem and CQ type strong convergence theorem for -monotone mappings. We give an example of -monotone mappings which assures the Mann’s type weak convergence.

WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS AND NONEXPANSIVE MAPPINGS AND NONSPREADING MAPPINGS IN HILBERT SPACES

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mappings and nonspreading mappings and the set of solution of an equilibrium problem on the setting of real Hilbert spaces.

Common fixed points for some generalized nonexpansive mappings and nonspreading-type mappings in uniformly convex Banach spaces

In this article, we study the fixed point theorems for nonspreading mappings, defined by Kohsaka and Takahashi, in Banach spaces but using the sense of norm instead of using the function ϕ. Furthermore, we prove a weak convergence theorem for finding a common fixed point of two quasi-nonexpansive mappings having demiclosed property in a uniformly convex Banach space. Consequently, such theorem can be deduced to the case of the nonspreading type mappings and some generalized nonexpansive mappings. MSC:49J40, 47J20.

Fixed point theorems and convergence theorems for generalized nonspreading mappings in Banach spaces

In this paper, we first prove a general fixed point theorem for nonlinear mappings in a Banach space. Then we prove a nonlinear mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for 2-generalized nonspreading mappings in a Banach space.

Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems

In this paper, we introduce some new iterative sequences for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of nonspreading mappings and a finite family of nonexpansive multivalued mappings in Hilbert space. We establish some weak and strong convergence theorems of the sequences generated by our iterative process. The results obtained in this paper extend and improve some recent known results.

Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mapping T and the solution sets of zero of a maximal monotone mapping and an α‐inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.

GENERALIZED HYBRID MAPPINGS IN HILBERT SPACES AND BANACH SPACES

In this paper, we deal with a broad class of nonlinear mappings in a Hilbert space and a Banach space called generalized hybrid which contains the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings. Then, we prove fixed point theorems for these nonlinear mappings in a Hilbert space and a Banach space. Furthermore, we obtain duality theorems for nonlinear mappings in a Banach space.

The System of Mixed Equilibrium Problems for Quasi‐Nonexpansive Mappings in Hilbert Spaces

We first introduce the iterative procedure to approximate a common element of the fixed‐point set of two quasinonexpansive mappings and the solution set of the system of mixed equilibrium problem (SMEP) in a real Hilbert space. Next, we prove the weak convergence for the given iterative scheme under certain assumptions. Finally, we apply our results to approximate a common element of the set of common fixed points of asymptotic nonspreading mapping and asymptotic TJ mapping and the solution set of SMEP in a real Hilbert space.

Fixed point theorems for some new nonlinear mappings in Hilbert spaces

AbstractIn this paper, we introduced two new classes of nonlinear mappings in Hilbert spaces. These two classes of nonlinear mappings contain some important classes of nonlinear mappings, like nonexpansive mappings and nonspreading mappings. We prove fixed point theorems, ergodic theorems, demiclosed principles, and Ray's type theorem for these nonlinear mappings.Next, we prove weak convergence theorems for Moudafi's iteration process for these nonlinear mappings. Finally, we give some important examples for these new nonlinear mappings.

Fixed point theorems and Δ-convergence theorems for generalized hybrid mappings on CAT(0) spaces

In this paper, we introduce generalized hybrid mapping on CAT(0) spaces. The class of generalized hybrid mappings contains the class of nonexpansive mappings, nonspreading mappings, and hybrid mappings. We study the fixed point theorems of generalized hybrid mappings on CAT(0) spaces. We also consider some iteration processes for generalized hybrid mappings on CAT(0) spaces, and our results generalize some results of fixed point theorems on CAT(0) spaces and Hilbert spaces.

A weak convergence theorem for common fixed points of some generalized nonexpansive mappings and nonspreading mappings in a Hilbert space

We study the iterative methods for approximation of fixed points of some generalized nonexpansive mappings and nonspreading mappings in a Hilbert space. Consequently, we obtain the weak convergence theorem for common fixed points of two such mappings.

Convergence theorem of common fixed points for a family of nonspreading mappings in Hilbert space

We introduce the concept of K-mapping of a finite family of nonspreading mappings \({\{T_i\}_{i=1}^N}\) and we show that the fixed point set of the K-mapping is the set of common fixed points of \({\{T_i\}_{i=1}^N}\). Moreover, we prove strong convergence theorem of the Ishikawa iterative process to a common fixed point of a finite family of nonspreading mappings in Hilbert space under certain control conditions.

Strong convergence theorems by Halpern–Mann iterations for relatively nonexpansive mappings in Banach spaces

In this paper, we modify Halpern and Mann’s iterations for finding a fixed point of a relatively nonexpansive mapping in a Banach space. Consequently, a strong convergence theorem for a nonspreading mapping is deduced. Using a concept of duality theorems, we also obtain analogue results for certain generalized nonexpansive and generalized nonexpansive type mappings. Finally, we discuss two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a Banach space.

Weak Convergence Theorems for Maximal Monotone Operators with Nonspreading mappings in a Hilbert space

Sea C un subconjunto convexo cerrado de un espacio real de Hilbert H. Sea T una asignacion de C en si mismo, sea A una asignacion monotona α-inversa de C en H y sea B un operador monotono maximal en H tal que el dominio de B esta incluido en C. Se introduce una secuencia iterativa para encontrar un punto de F(T) ∩ (A + B)-10, donde F(T) es el conjunto de puntos fijos de T y (A + B)-10 es el conjunto de los puntos cero de A + B. Entonces, se obtiene el resultado principal que se relaciona con la convergencia debil de la secuencia. Utilizando este resultado, obtenemos un teorema de convergencia para encontrar un punto comun de una asignacion fija y una asignacion en un espacio de Hilbert. Ademas, consideramos el problema para encontrar un elemento comun del conjunto de soluciones de un problema de equilibrio y el conjunto de puntos fijos de una asignacion.

Fixed Point Theorems and Weak Convergence Theorems for Generalized Hybrid Mappings in Hilbert Spaces

In this paper, we first consider a broad class of nonlinear mappings containing the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings in a Hilbert space. Then, we deal with fixed point theorems and weak convergence theorems for these nonlinear mappings in a Hilbert space.

Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces

Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in Kurokawa and Takahashi [Y. Kurokawa, W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 1562–1568]. We explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of Halpern’s type for our class of mappings and hence resolved in the affirmative an open problem posed by Kurokawa and Takahashi in their final remark for the case where the mapping T is averaged.

Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces

In this paper, we first obtain a weak mean convergence theorem of Baillon’s type for nonspreading mappings in a Hilbert space. Further, using an idea of mean convergence, we prove a strong convergence theorem for nonspreading mappings in a Hilbert space.