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Let be nonempty closed convex subset of real Hilbert space . Consider a nonexpansive semigroup with a common fixed point, a contraction with coefficient , and a strongly positive linear bounded operator with coefficient . Let . It is proved that the sequence generated iteratively by converges strongly to a common fixed point which solves the variational inequality for all .

X. Qin and Y. Su proved a strong convergence theorems of modified Ishikawa iteration by CQ method for relatively nonexpansive mappings in a Banach space [Xiaolong Qin, Yongfu Su, Nonlinear Anal. 67 (2007), no. 6, 1958-1965]. The result of this paper extends and improves the result of X. Qin and Y. Su in the two respects: (1). By using the monotone CQ method to modify the CQ method, so that the new method of proof is used. (2). Relax the restriction on $T$ from uniformly continuous to continuous. The result of this paper also extends and improves the recent ones announced by Nakajo, Takahashi, Kim, Martinez-Yanes, Xu and some others.

In this paper, we prove two strong convergence theorems of modified Ishikawa iteration and modified Halpern iteration for relatively nonexpansive mappings in a Banach space. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Martinez-Yanes, Xu and some others.

Let E be a uniformly convex Banach space and K a nonempty convex closed subset which is also a nonexpansive retract of E. Let T 1, T 2 and T 3: K → E be asymptotically nonexpansive mappings with {k n }, {l n } and {j n }. [1, ∞) such that Σ n=1 ∞ (k n − 1) 0. Starting from arbitrary x 1 ∈ K define the sequence {x n } by $$\left\{ \begin{gathered} z_n = P(\alpha ''_n T_3 (PT_3 )^{n - 1} x_n + (1 - \alpha ''_n )x_n ), \hfill \\ y_n = P(\alpha '_n T_2 (PT_2 )^{n - 1} z_n + (1 - \alpha '_n )x_n ), \hfill \\ x_{n + 1} = P(\alpha _n T_1 (PT_1 )^{n - 1} y_n + (1 - \alpha _n )x_n ). \hfill \\ \end{gathered} \right.$$ (i) If the dual E* of E has the Kadec-Klee property then {x n } converges weakly to a common fixed point p ∈ F; (ii) If T satisfies condition (A′) then {x n } converges strongly to a common fixed point p ∈ F.

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T1, T2 and T3 : K → E be nonexpansive mappings with nonempty common fixed points set. Let {αn}, {βn}, {γn}, {α′n}, {β′n}, {γ′n}, {α″n}, {β″n} and {γ″n} be real sequences in [0, 1] such that αn+βn+ γn=α′n + β′n + γ′n = α″n + β″n + γ″n = 1, starting from arbitrary X1 e K, define the sequence {xn} by {zn = P(α″nT1xn + β″nXn + γ″nwn) Yn = P(α′nT2zn + β′nxn + γ′nvn) xn+1 = PαnT3yn + βnxn + γnun) with the restrictions Σn=1∞ γn < ∞, Σn=1∞γ′n < ∞, Σn=1∞ γ″n < ∞. (i) If the dual E* of E has the Kadec-Klee property, then weak convergence of a {xn} to some x* ∈ F(T1) ∩ F(T2) ∩ (T3) is proved; (ii) If T1, T2 and T3 satisfy condition (A′), then strong convergence of {xn} to some x* ∈ F(T1) ∩ F(T2) ∩ (T3) is obtained.

This paper introduces a composite iteration scheme for approximating a zero point of accretive operator in the framework of uniformly smooth Banach spaces and the reflexive Banach space which has a weak continuous duality map, respectively. Strong convergence of the composite iteration scheme { x n } defined by { y n = β n x n + ( 1 − β n ) J r n x n , x n + 1 = α n u + ( 1 − α n ) y n , where J r n is the resolvent of m-accretive operator A and u ∈ C is an arbitrary (but fixed) element in C and sequences { α n } in ( 0 , 1 ) , { β n } in [ 0 , 1 ] is established. Under certain appropriate assumptions on the sequences { α n } , { β n } and { r n } , that { x n } defined by the above iteration scheme converges to a zero point of A is proved. The results improve and extend results of T.H. Kim, H.K. Xu and some others.

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. we establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the results announced by many others.

This paper present definition and basic concepts of random structure spaces, therefore, study relations of random metric spaces, random normed spaces, random inner product spaces and probabilistic metric spaces, probabilistic normed spaces, probabilistic inner product spaces. Finally, pose some questions for study or discussion.

The purpose of this paper is to present a new class of probabilistic normed spaces and to study fixed point problems in this class of probabilistic normed spaces. This paper includes the following two contents: (1) The definition of a new class of probabilistic normed spaces, the so-called S-probabilistic normed spaces, is given. In order to study the fixed point problems, some relevant properties of S-probabilistic normed spaces are discussed and some basic useful results are obtained; (2) The notion of probabilistic weak convergence is firstly presented in this paper. Therefore the probabilistic weak and strong convergence theorems of fixed points for nonexpansive mappings, asymptotically nonexpansive mappings and strongly pseudocontractive mappings are also proved by using the new methods and techniques.

The aim of this paper is to prove a more generalized contraction mapping principle. By using this more generalized contraction mapping principle, a further generalized best proximity theorem was established. Some concrete results have been derived by using the above two theorems. The results of this paper improve many important results published recently in the literature.

The purpose of this paper is to introduce some basic definitions about fixed point and best proximity point in two classes of probabilistic metric spaces and to prove contraction mapping principle and relevant best proximity point theorems. The first class is the so-called S-probabilistic metric spaces. In S-probabilistic metric spaces, the generalized contraction mapping principle and generalized best proximity point theorems have been proved by authors. These results improve and extend the recent results of Su and Zhang (Fixed Point Theory Appl. 2014:170, 2014). The second class is the so-called Menger probabilistic metric spaces. In Menger probabilistic metric spaces, the contraction mapping principle and relevant best proximity point theorems have been proved by authors. These results also improve and extend the results of many authors. In order to get the results of this paper, some new methods have been used. Meanwhile some error estimate inequalities have been established.

In this paper, we study the convergence of two type iteration processes as follows: (I) x n + 1 = α n x + ( 1 - α n ) T ( β n x + ( 1 - β n ) Tx n ) , (II) x n + 1 = α n ( β n x + ( 1 - β n ) Tx n ) + ( 1 - α n ) A n x n , where A n = 1 n + 1 ∑ j = 0 n T j : C → C in uniformly convex Banach space X, which possesses a weakly sequentially continuous duality mapping J and in uniformly convex Banach space X with a uniformly Gateaux differentiable norm, respectively. And prove that above sequences converge strongly to Px when the real sequence {αn}, {βn} satisfies appropriate conditions, where P is sunny non-expansive from C onto F(T).

Let E be a real Banach space and let K be a nonempty closed convex subset of E. Let { T i } i = 1 N be N strictly pseudocontractive self-maps of K such that F = ⋂ i = 1 N F ( T i ) ≠ ∅ , where F ( T i ) = { x ∈ K : T i x = x } and let { α n } n = 1 ∞ , { β n } n = 1 ∞ ⊂ [ 0 , 1 ] be two real sequence satisfying the conditions: ( i ) ∑ n = 1 ∞ ( 1 − α n ) = + ∞ ; ( ii ) ∑ n = 1 ∞ ( 1 − α n ) 2 + ∞ ; ( iii ) ∑ n = 1 ∞ ( 1 − β n ) + ∞ ; ( iv ) ( 1 − α n ) ( 1 − β n ) L 2 1 , ∀ n ⩾ 1 , where L ⩾ 1 is common Lipschitz constant of { T i } i = 1 N . For x 0 ∈ K , let { x n } n = 1 ∞ be new implicit process defined by x n = α n x n − 1 + ( 1 − α n ) T n y n , y n = β n x n − 1 + ( 1 − β n ) T n x n where T n = T n mod N , then ( i ) lim n → ∞ ‖ x n − p ‖ exists, for all p ∈ F ; ( ii ) lim inf n → ∞ ‖ x n − T n x n ‖ = 0 . The results of this paper generalize and improve the results of Osilike in 2004. In this paper, the proof methods of the main results are also different from that of Osilike.

The aim of this paper is to present the definition of a generalized altering distance function and to extend the results of Yan et al. (Fixed Point Theory Appl. 2012:152, 2012) and some others, and to prove a new fixed point theorem of generalized contraction mappings in a complete metric space endowed with a partial order by using generalized altering distance functions. The results of this paper can be used to investigate a large class of nonlinear problems. As an application, we discuss the existence of a solution for a periodic boundary value problem.

Novel bismuth nanoparticle-modified mesoporous carbon (MPC) was successfully prepared on a glassy carbon electrode (Bi@MPC/GCE) for the adsorptive stripping voltammetric determination of nickel by complexing with dimethylglyoxime (DMG). The presence of MPC obviously improved the properties of Bi particles like the electron transfer ability, particle size and hydrophicility, important parameters to achieve preferable analytical performances of Bi@MPC/GCE toward Ni(II). The best electrochemical behaviors of Bi@MPC/GCE was obtained for the stripping determination of Ni(II), compared with electrodes individually modified with Bi and MPC. The synergic effect between metallic Bi and ordered MPC (forming a 3D array like Bi microelectrodes) made major contribution to such improved electrochemical properties of Bi@MPC/GCE for Ni(II) sensing. The good linear analytical curve was achieved in a Ni(II) concentration range from 0.1[Formula: see text][Formula: see text]M to 5.0[Formula: see text][Formula: see text]M with a correlation coefficient of 0.9995. The detection limit and sensitivity were calculated to be 1.2[Formula: see text]nM ([Formula: see text]) and 1410[Formula: see text][Formula: see text]A[Formula: see text]mM[Formula: see text][Formula: see text]cm[Formula: see text], respectively. The new method was successfully applied to Ni(II) determination in soybean samples with recoveries higher than 99% and proved to be a simple, efficient alternative for Ni(II) monitoring in real samples.

The purpose of this paper is to discus the normalizable problem of a class of fuzzy normed spaces. Some su±cient conditions for normalizable have been given based on the analyzing to the topological properties of fuzzy normed spaces. Meanwhile some examples of normalizable fuzzy normed spaces have been given. Finally we discuss the continuity and boundedness of linear operator between two fuzzy normed spaces. Keywords: Fuzzy normed spaces; Normalizable; t-norm; Convergence; Bound- edness; Continuity.

The purpose of this paper is to obtain best proximity point theorems for a weakly contractive mapping and a weakly Kannan mapping in partial metric spaces. In this paper, the P-operator technique, which changes a non-self mapping to a self mapping, provides a key method. Many recent results in this area have been improved. MSC:47H05, 47H09, 47H10.

In this paper, a hybrid projection algorithm for a total quasi-asymptotically pseudo-contractive mapping is introduced in a Hilbert space. A strong convergence theorem of the proposed algorithm to a fixed point of a total quasi-asymptotically pseudo-contractive mapping is proved. Our main result extends and improves many corresponding results.

The purpose of this paper is to construct a three-step iteration method (as follows) and obtain the convergence theorem for a countable family of Lipschitz pseudocontractive mappings in Hilbert space H. For the iteration format, { z n = ( 1 − γ n ) x n + γ n T n x n , y n = ( 1 − β n ) x n + β n T n z n , x n + 1 = ( 1 − α n ) x n + α n T n y n , under suitable conditions, we prove that the sequence { x n } generated from above converges strongly to a common fixed point of { T n } n ≥ 1 . The results obtained in this paper improve and extend previous results that have been proved for this class of nonlinear mappings. MSC:47H05, 47H09, 47H10.

In Caballero et al. (Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231), the authors prove a best proximity point theorem for Geraghty nonself contraction. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results. MSC:47H05, 47H09, 47H10.

The aim of this paper is to extend the results of Harjani and Sadarangani and some other authors and to prove a new fixed point theorem of a contraction mapping in a complete metric space endowed with a partial order by using altering distance functions. Our theorem can be used to investigate a large class of nonlinear problems. As an application, we discuss the existence of a solution for a periodic boundary value problem.

Saewan and Kumam (2010) have proved the convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach spaceEwith Kadec-Klee property. In this paper, authors prove the convergence theorems and do not need the Kadec-Klee property of Banach space and some other conditions used in the paper of S. Saewan and P. Kumam. Therefore, the results presented in this paper improve and extend some recent results.

The authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemirelatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool (2010) and many others.

The following main results have been given. (1) LetEbe ap-uniformly convex Banach space and letT:E→E*be a(p-1)-L-Lipschitz mapping with condition0<(pL/c2)1/(p-1)<1. ThenThas a unique generalized duality fixed pointx*∈Eand (2) letEbe ap-uniformly convex Banach space and letT:E→E*be aq-α-inverse strongly monotone mapping with conditions1/p+1/q=1,0<(q/(q-1)c2)q-1<α. ThenThas a unique generalized duality fixed pointx*∈E. (3) LetEbe a2-uniformly smooth and uniformly convex Banach space with uniformly convex constantcand uniformly smooth constantband letT:E→E*be aL-lipschitz mapping with condition0<2b/c2<1. ThenThas a unique zero pointx*. These main results can be used for solving the relative variational inequalities and optimal problems and operator equations.

In this paper we prove some fixed point theorems of contractive mappings in a complete cone metric space. We replace the real numbers by ordering Banach space and define cone metric space. Our results generalized some fixed point theorems in partially ordered metric spaces.

The purpose of this article is to prove a new fixed point theorem and to establish a simple approximative method of fixed point for nonexpansive mappings in the reflexive Banach space. The new fixed point theorem presented in this article improved and extend the classical results of Goebel-Kirk, Kim-Xu and many others.

The purpose of this paper is to introduce hybrid projection algorithms for finding a common element which is a zero of a monotone mapping and a fixed point of a weak relatively nonexpansive mapping in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.

The purpose of this paper is to prove the strong convergence of the sequence {x n } generated by (1.5) to a common zero of a finite family of accretive operators in a strictly convex and reflexive Banach space having a uniformly Gâteaux differentiable norm under certain conditions on the control parameters {α n },{β n },{γ n },{δ n } and {r n }. Next, we study the viscosity approximation method with weakly contractive mappings for finding a common zero of a finite family of accretive operators. The results presented in this paper improve and extend the recent ones by Aoyama et al. [3], Jung [6], Zegeye and Shahzad [7], Hu and Liu [9] and many others.

The purpose of this paper is to propose a hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for an inverse strongly monotonemapping. We obtain a strong convergence theorem. The results of this paper improved and extended the results of W. Takahashi and Toyoda [W. Takahashi, Toyoda W, Weak convergence theorems for nonexpansive mappings and monotone mappings, Journal of Optimization Theory And Applications:vol 118, 417-428 (2003), ] and some others in some respects.

We consider the monotone variational inequality of finding $x^*\in C$ such that $\langle (I-T)x^*,x-x^*\rangle \ge 0$ for $x\in C$, where $C$ is a closed convex subset of a real Hilbert space and $T$ is a nonexpansive self-mapping of $C$. Techniques of nonexpansive mappings are applied to regularize this variational inequality. The regularized solutions and an iteration process are shown to converge in norm to a solution of this variational inequality.

We consider a new algorithm for a generalized system for relaxed cocoercive nonlinear inequalities involving three different operators in Hilbert spaces by the convergence of projection methods. Our results include the previous results as special cases extend and improve the main results of R. U. Verma (2004), S. S. Chang et al. (2007), Z. Y. Huang and M. A. Noor (2007), and many others.

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T:K→E be an asymptotically nonexpansive mapping with {kn}⊂[1,∞) such that ∑n=1∞(kn−1)<∞ and F(T) is nonempty, where F(T) denotes the fixed points set of T. Let {αn}, {αn'}, and {αn''} be real sequences in (0,1) and ε≤αn,αn',αn''≤1−ε for all n∈ℕ and some ε>0. Starting from arbitrary x1∈K, define the sequence {xn} by x1∈K, zn=P(αn''T(PT)n−1xn+(1−αn'')xn), yn=P(αn'T(PT)n−1zn+(1−αn')xn), xn+1=P(αnT(PT)n−1yn+(1−αn)xn). (i) If the dual E* of E has the Kadec-Klee property, then { xn} converges weakly to a fixed point p∈F(T); (ii) if T satisfies condition (A), then {xn} converges strongly to a fixed point p∈F(T).