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.

This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the $\mathrm{\Delta }$-convergence results of the implicit Ishikawa iteration sequences for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces. The results presented in this paper extend and generalize some previous results.

We analyse the sensitivity of differential programs of the form $\text{Min}f(x)$ subject to $g(x)=b,x\in D$ where $f$ and $g$ are ${𝒞}^1$ maps whose respective images lie in ordered Banach spaces. Following previous works on multiobjective programming, the notion of $T$-optimal solution is used. The behaviour of some nonsingleton sets of $T$-optimal solutions according to changes of the parameter $b$ in the problem is analysed. The main result of the work states that the sensitivity of the program is measured by a Lagrange multiplier plus a projection of its derivative. This sensitivity is measured by means of the paratingent derivative.

This paper is devoted to investigate the fixed points and best proximity points of multivalued cyclic self-mappings on a set of subsets of complete metric spaces endowed with a partial order under a generalized contractive condition involving a Hausdorff distance. The existence and uniqueness of fixed points of both the cyclic self-mapping and its associate composite self-mappings on each of the subsets are investigated, if the subsets in the cyclic disposal are nonempty, bounded and of nonempty convex intersection. The obtained results are extended to the existence of unique best proximity points in uniformly convex Banach spaces.

Let $X$ be a real reflexive Banach space with a weakly continuous duality mapping $J_{\phi }$. Let $C$ be a nonempty weakly closed star-shaped (with respect to $u$) subset of $X$. Let $ℱ$ = $\{T(t):t\in [0,+\infty )\}$ be a uniformly continuous semigroup of asymptotically nonexpansive self-mappings of $C$, which is uniformly continuous at zero. We will show that the implicit iteration scheme: $y_n={\alpha }_{n}u+(1-{\alpha }_n)T(t_n)y_n$, for all $n\in \mathbb{N}$, converges strongly to a common fixed point of the semigroup $ℱ$ for some suitably chosen parameters $\{{\alpha }_n\}$ and $\{t_n\}$. Our results extend and improve corresponding ones of Suzuki (2002), Xu (2005), and Zegeye and Shahzad (2009).

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping $S$ in the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, viscosity approximation method, and gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mapping $S$. The implicit hybrid method with regularization is based on four well-known methods: the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with regularization.

Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operator $F:x_n={\alpha }_n\gamma V(x_n)+{\beta }_n{x}_{n-\mathrm{1}}+((\mathrm{1}-{\beta }_n)I-{\alpha }_n\mu F)T_n{x}_n$ and get strong convergence under some mild assumptions. Our results improve and extend the corresponding conclusions announced by many others.

We discuss the existence of solutions for a periodic boundary value problem and for some polynomials. For this purpose, we present some fixed point theorems for weakly and generalized weakly contractive mappings in the setting of partially ordered complete metric spaces.

We study the existence and uniqueness of a fixed point of the multidimensional operators which satisfy Meir-Keeler type contraction condition. Our results extend, improve, and generalize the results mentioned above and the recent results on these topics in the literature.

Let $F:ℝ\times ℝ\to ℝ$ be a real-valued polynomial function of the form $F(x,y)={\sum }_{i=\mathrm{0}}^s\mathrm{‍}{f}_i(x)y^i$, with degree of $\mathrm{}y\mathrm{}\mathrm{}$ in $F(x,y)=s\ge \mathrm{}\mathrm{}\mathrm{1},\mathrm{}\mathrm{}x\in ℝ\mathbb{}.$ An irreducible real-valued polynomial function $p(x)$ and a nonnegative integer $m$ are given to find a polynomial function $y(x)\in ℝ[x]$ satisfying the following expression: $F(x,y(x))=c{p}^m(x)$ for some constant $c\in ℝ$. The constant $c$ is dependent on the solution $y(x)$, namely, a quasi-fixed (polynomial) solution of the polynomial-like equation $(\mathrm{*})$. In this paper, we will provide a non-NP-complete algorithm to solve all quasi-fixed solutions if the equation $(\mathrm{*})$ has only a finite number of quasi-fixed solutions.

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point $x^{*}$ such that $x^{*}\in \Omega ,\langle (A-\gamma f)x^{*}-(I-B)S{x}^{*},x-x^{*}\rangle \ge 0,\forall x\in \Omega$, where $\Omega$ is the set of the solutions of the following variational inequality: $x^{*}\in Ϝ,\langle (A-S)x^{*},x-x^{*}\rangle \ge 0,\forall x\in Ϝ$, where $A,B$ are two strongly positive bounded linear operators, $f$ is a $\rho$-contraction, $S$ is a nonexpansive mapping, and $Ϝ$ is the fixed points set of a nonexpansive semigroup $\{T(s){\}}_{s\ge 0}$. We present a double-net convergence hierarchical to some elements in $Ϝ$ which solves the above hierarchical constrained variational inequalities.

We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. The results of the paper are generalization and continuation of the recent results on this issue.

The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution set $\Xi$ of a general system of variational inequalities, the solution set $\mathrm{\Gamma }$ of a split feasibility problem, and the fixed point set $\mathrm{Fix}(S)$ of a strictly pseudocontractive mapping $S$ in the setting of the Hilbert spaces. These iterative algorithms are based on the regularization method, the Mann-type iteration method, and the extragradient method due to Nadezhkina and Takahashi (2006). Furthermore, we prove that the sequences generated by the proposed algorithms converge weakly to an element of $\text{Fix}(S)\cap \mathrm{\Xi }\cap \mathrm{\Gamma }$ under mild conditions.

A new fixed point theorem is established under the setting of a generalized finitely continuous topological space (GFC-space) without the convexity structure. As applications, a weak KKM theorem and a minimax inequalities of Ky Fan type are also obtained under suitable conditions. Our results are different from known results in the literature.

Content-addressable memory (CAM) has been described by collective dynamics of neural networks and computing with attractors (equilibrium states). Studies of such neural network systems are typically based on the aspect of energy minimization. However, when the complexity and the dimension of neural network systems go up, the use of energy functions might have its own limitations to study CAM. Recently, we have proposed the decirculation process in neural network dynamics, suggesting a step toward the reshaping of network structure and the control of neural dynamics without minimizing energy. Armed with the decirculation process, a sort of decirculating maps and its structural properties are built here, dedicated to showing that circulation breaking taking place in the connections among many assemblies of neurons can collaborate harmoniously toward the completion of network structure that generates CAM.

We introduce the concepts of ordered variational inequalities and ordered complementarity problems with both domain and range in Banach lattices. Then we apply the Fan-KKM theorem and KKM mappings to study the solvability of these problems.

We consider generalized Berinde-type contractions in the context of partial metric spaces. Such contractions are also known as generalized almost contractions in the literature. In this paper, we extend, generalize, and enrich the results in this direction. Some examples are presented to illustrate our results.

Introducing a general split feasibility problem in the setting of infinite-dimensional Hilbert spaces, we prove that the sequence generated by the purposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

Let $K$ be a nonempty compact convex subset of a topological vector space. In this paper-sufficient conditions are given for the existence of $x\in K$ such that $F(T)\cap \text{V}\text{E}\text{P}(F)\ne \varnothing$, where $F(T)$ is the set of all fixed points of the multivalued mapping $T$ and $\text{VEP}(F)$ is the set of all solutions for vector equilibrium problem of the vector-valued mapping $F$. This leads us to generalize and improve some existence results in the recent references.