Jingling Zhang

A note on ‘A best proximity point theorem for Geraghty-contractions’

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.

Fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces

The purpose of this paper is to present some definitions and basic concepts of best proximity point in a new class of probabilistic metric spaces and to prove the best proximity point theorems for the contractive mappings and weak contractive mappings. In order to get the best proximity point theorems, some new probabilistic contraction mapping principles have been proved. Meanwhile the error estimate inequalities have been established. Further, a method of the proof is also new and interesting, which is to use the mathematical expectation of the distribution function studying the related problems.

Best proximity point theorems for generalized contractions in partially ordered metric spaces

The purpose of this paper is to obtain four best proximity point theorems for generalized contractions in partially ordered metric spaces. Further, our P-operator technique, which changes a non-self mapping to a self-mapping, plays an important role. Some recent results in this area have been improved.MSC:47H05, 47H09, 47H10.

Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces

AbstractThe 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.

Convergence theorems of a three-step iteration method for a countable family of pseudocontractive mappings

AbstractThe 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, {zn=(1−γn)xn+γnTnxn,yn=(1−βn)xn+βnTnzn,xn+1=(1−αn)xn+αnTnyn, under suitable conditions, we prove that the sequence {xn} generated from above converges strongly to a common fixed point of {Tn}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.

Simple projection algorithm for a countable family of weak relatively nonexpansive mappings and applications

AbstractLet E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let {Tn}:C→C be a countable family of weak relatively nonexpansive mappings such that F=⋂n=1∞F(Tn)≠∅. For any given gauss x0∈C, define a sequence {xn} in C by the following algorithm: {C0=C,Cn+1={z∈Cn:ϕ(z,Tnxn)=ϕ(z,xn)},n=0,1,2,3,…,xn+1=ΠCn+1x0. Then {xn} converges strongly to q=ΠFx0.MSC:47H05, 47H09, 47H10.

Strong Convergence Theorems for a Common Fixed Point of Two Countable Families of Relatively Quasi Nonexpansive Mappings and Applications

The purpose of this paper is to prove strong convergence theorems for common fixed points of two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized -projection operator. In order to get the strong convergence theorems, a new iterative scheme by monotone hybrid method is presented and is used to approximate the common fixed points. Then, two examples of countable families of uniformly closed nonlinear mappings are given. The results of this paper modify and improve the results of Li et al. (2010), the results of Takahashi and Zembayashi (2008), and many others.

Uniformly closed replaced AKTT or ∗ AKTT condition to get strong convergence theorems for a countable family of relatively quasi-nonexpansive mappings and systems of equilibrium problems

AbstractThe purpose of this paper is to construct a new iterative scheme and to get a strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but does not satisfy condition AKTT and ∗AKTT. Our results can be applied to solve a convex minimization problem. In addition, this paper clarifies an ambiguity in a useful lemma. The results of this paper modify and improve many other important recent results. MSC:47H05, 47H09, 47H10.

Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications

The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.

Convergence theorems for modified generalized f -projections and generalized nonexpansive mappings

The purpose of this paper is to study a sequence of modified generalized f-projections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalized f-projection onto the limit set. Furthermore, we prove a strong convergence theorem for a countable family of α-nonexpansive mappings in a uniformly convex and smooth Banach space using the properties of a modified generalized f-projection operator. Our main results generalize the results of Ziming Wang, Yongfu Su, and Jinlong Kang and enrich the research contents of α-nonexpansive mappings.MSC:47H05, 47H09, 47H10.