Coupled fixed point theorems for mixed monotone mappings and an application to integral equations

In this paper, we extend the coupled fixed point theorems for a mixed monotone mapping F:X×X→X in partially ordered metric spaces established by Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379-1393]. An application to nonlinear integral equations is also given to illustrate our results.     

区间值度量空间中的不动点定理

研究了一种特殊的模糊度量[ρ],称为区间值度量。区间数的运算（如加减乘除运算）在相关文献中已有定义,对区间数的减法运算进行新的定义,得到相应的不等式性质,接着给出了区间值度量的定义;介绍了区间值度量空间中相关的定义,如收敛序列、Cauchy序列以及完备性等;讨论了区间值度量空间中的不动点定理和公共不动点定理。     

Split common fixed point and null point problems for demicontractive operators in Hilbert spaces

In this article, we consider a split common fixed point and null point problem which includes the split common fixed point problem, the split common null problem and other problems related to the fixed point problem and the null point problem. We introduce an algorithm for studying the split common fixed point and null problem for demicontractive operators and maximal monotone operators in real Hilbert spaces. We establish a strong convergence result under some suitable conditions and reduce our main result to above-mentioned problems. Moreover, we also apply our main results to the split equilibrium problem. Finally, we give numerical results to demonstrate the convergence of our algorithms.     

Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings

This paper generalizes a classical result about the space of bounded closed sets with the Hausdorff metric, and establishes the completeness of CB(X) with respect to the completeness of the metric space X, where CB(X) is the class of fuzzy sets with nonempty bounded closed α-cut sets, equipped with the supremum metric d_∞ which takes the supremum on the Hausdorff distances between the corresponding α-cut sets. In addition, some common fixed point theorems for fuzzy mappings are proved and two examples are given to illustrate the validity of the main results in fixed point theory.     

Some common fixed point theorems for generalized nonlinear contractive mappings

Sufficient conditions for the existence of a common fixed point for generalized nonlinear contractive mappings are derived. As applications, some results on the set of best approximation for this class of mappings are also obtained. The proved results generalize and extend various known results in the literature.     

Some new fixed point theorems of generalized nonlinear contractive multivalued maps in complete metric spaces

In this paper, we present some new fixed point theorems for generalized nonlinear contractive multivalued maps and these results generalize or improve the corresponding recent fixed point results in the literature.     

Approximate fixed point sequence for a finite family of asymptotically pseudocontractive mappings

Let E be a real Banach space and K be a nonempty, closed, convex, and bounded subset of E. Let T_i:K→K, i=1,2,…,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε_n}, and asymptotically pseudocontractive mappings with sequences , where {ε_n} and , i=1,2,…,N, satisfy certain mild conditions. Let a sequence {x_n} be generated from x₁K by

for all integers n1, where T_n=T_n(modN), {u_n} be a sequence in K, and {λ_n}, {θ_n} and {μ_n} are three real sequences in [0,1] satisfying appropriate conditions; then x_n−T_lx_n→0 as n→∞ for each l{1,2,…,N}.     

Existence and uniqueness of common fixed points for two multivalued operators in ordered metric spaces

In this paper, we discuss some new fixed point theorems for a pair of multivalued operators which satisfy weakly generalized contractive conditions. Our results are the extension and improvement of corresponding results of [J. Harjani and K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Analysis, 71 (2009) 3403–3410] and [X. Zhang, Common fixed point theorems for some new generalized contractive type mappings, J. Math. Anal. Appl. 333 (2007) 780–786]. Finally, some examples are given to illustrate the usability of our results.     

On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces

The purpose of this paper is to introduce the concept of non-self asymptotically quasi-nonexpansive-type mappings and to construct a iterative sequence to converge to a common fixed point for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces. The results presented in this paper improve and extend the corresponding results in Alber, Chidume and Zegeye [Ya.I. Alber, C.E. Chidume, H. Zegeye, Approximating of total asymptotically nonexpansive mappings, Fixed Point Theory and Applications (2006) 1–20. Article ID10673], Ghosh and Debnath [M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 207 (1997) 96–103], Liu [Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive type mappings, Journal of Mathematical Analysis and Applications 259 (2001) 1–37; Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications 259 (2001) 18–24; Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space, Journal of Mathematical Analysis and Applications 266 (2002) 468–471], Petryshyn [W.V. Petryshyn, T.E. Williamson Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 43 (1973) 459–497], Quan and Chang [J. Quan, S.S. Chang, X.J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications V (2006) 1–38. Article ID 70830], Shahzad and Udomene [N. Shahzad, A. Udomene, Approximating common fixed point of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2006) 1–10. Article ID 18909] Xu [B.L. Xu, M.A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 267 (2002) 444–453], Zhang [S.S. Zhang, Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Mathematicae Applicatae Sinica 24 (2001) 236–241] and Zhou and Chang [Y.Y. Zhou, S.S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numerical Functional Analysis and Optimization 23 (2002) 911–921].     

Hyperbolicity of the fixed point set for the simple genetic algorithm

We study an infinite population model for the genetic algorithm, where the iteration of the algorithm corresponds to an iteration of a map G. The map G is a composition of a selection operator and a mixing operator, where the latter models effects of both mutation and crossover. We examine the hyperbolicity of fixed points of this model. We show that for a typical mixing operator all the fixed points are hyperbolic.     

Common fixed points of a countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces

In this paper, we introduce a one-step iterative scheme for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mappings in a real uniformly convex Banach space. We establish weak and strong convergence theorems of the proposed iterative scheme under some control conditions.     

Some fixed point theorems in 1-M-complete fuzzy metric-like spaces

Very recently, in order to unify the notions of fuzzy metric space and metric-like space, Shukla and Abbas introduced the concept of fuzzy metric-like space and proved some fixed-point results in this setting. In this article, we modify the notion of Cauchy sequence and completeness to generalize their results. Thus, we extend their theorems to a more general framework, which is also appropriate to generalize some recent, well-known results in this line of research. Furthermore, several examples are presented to illustrate the significance of our results.     

Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces

In this paper, we introduce a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of a general variational inequality and the set of common fixed points of an infinite family of k_i-strict pseudocontractions in a Banach space. We obtain some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others.     

Strong convergence theorems for a new iterative method of -strictly pseudo-contractive mappings in Hilbert spaces

In this paper, we introduce a new iterative method of a k-strictly pseudo-contractive mapping for some 0≤k<1 and prove that the sequence {x_n} converges strongly to a fixed point of T, which solves a variational inequality related to the linear operator A. Our results have extended and improved the corresponding results of Y.J. Cho, S.M. Kang and X. Qin [Some results on k-strictly pseudo-contractive mappings in Hilbert spaces, Nonlinear Anal. 70 (2008) 1956–1964], and many others.     

On solving the minimization problem and the fixed-point problem for a finite family of non-expansive mappings in CAT(0) spaces

In this paper, we propose a new modified proximal point algorithm for a finite family of non-expansive mappings in the framework of CAT(0) spaces. We establish Δ-convergence and strong convergence theorems under some mild conditions. Our results extend some known results which appeared in the literature.