Let $F:{ℝ}^n\times ℝ\to ℝ$ be a real-valued polynomial function of the form $F(\overline{x},y)=a_s(\overline{x})y^s+a_{s-1}(\overline{x})y^{s-1}+\cdots +a_0(\overline{x})$ where the degree $s$ of $y$ in $F(\overline{x},y)$ is greater than $1$. For arbitrary polynomial function $f(\overline{x})\in ℝ[\overline{x}]$, $\overline{x}\in {ℝ}^n$, we will find a polynomial solution $y(\overline{x})\in ℝ[\overline{x}]$ to satisfy the following equation ($\mathrm{\star }$): $F(\overline{x},y(\overline{x}))=af(\overline{x})$ where $a\in ℝ$ is a constant depending on the solution $y(\overline{x})$, namely a quasi-coincidence (point) solution of ($\mathrm{\star }$), and $a$ is called a quasi-coincidence value of ($\mathrm{\star }$). In this paper, we prove that $(i)$ the number of all solutions in ($\mathrm{\star }$) does not exceed ${\mathrm{deg}}_{y}F(\overline{x},y)\left((2^{\mathrm{deg}f(\overline{x})}+s+3)·2^{\mathrm{deg}f(\overline{x})}+1\right)$ provided those solutions are of finitely many exist, $(ii)$ if all solutions are of infinitely many exist, then any solution is represented as the form $y(\overline{x})=-a_{s-1}(\overline{x})/s{a}_s(\overline{x})+\lambda p(\overline{x})$ where $\lambda$ is arbitrary and $p(\overline{x})=(f(\overline{x})/a_s(\overline{x}){)}^{1/s}$ is also a factor of $f(\overline{x})$, provided the equation ($\mathrm{\star }$) has infinitely many quasi-coincidence (point) solutions.

We introduce the notion of generalized ${\alpha }_{*}$-admissible mappings. By using this notion, we prove a fixed point theorem. Our result generalizes Mizoguchi-Takahashi’s fixed point theorem. We also provide some examples to show the generality of our work.

We first introduce a new class of mappings called Bregman asymptotic pointwise nonexpansive mappings and investigate the existence and the approximation of fixed points of such mappings defined on a nonempty, bounded, closed, and convex subset C of a real Banach space E. Without using the original Opial property of a Banach space E, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes for Bregman asymptotic pointwise nonexpansive mappings in a reflexive Banach space E. Our results are applicable in the function spaces $L^p$, where is a real number.

We introduce and study new types of mixed monotone multivalued mappings in partially ordered complete metric spaces. We give relationships between those two types of mappings and prove their coupled fixed point and coupled common fixed point theorems in partially ordered complete metric spaces. Some examples of each type of mappings satisfying the conditions of the main theorems are also given. Our main result includes several recent developments in fixed point theory of mixed monotone multivalued mappings.

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

We suggest and analyze an iterative scheme for finding the approximate element of the common set of solutions of a system of variational inequalities, a mixed equilibrium problem, and a hierarchical fixed point problem in a real Hilbert space. Strong convergence of the proposed method is proved under some conditions. The results presented in this paper extend and improve some well-known results in the literature.

Based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method, we propose implicit and explicit iterative schemes for computing a common element of the solution set of a system of variational inequalities and the set of zeros of an accretive operator, which is also a unique solution of a variational inequality. Under suitable assumptions, we study the strong convergence of the sequences generated by the proposed algorithms. The results of this paper improve and extend several known results in the literature.

Recently, fixed point theory on graphs has been considered by many authors. In this paper, by combining some ideas in some published papers and introducing $G$-type quasi-contractions, we give some fixed point results for $G$-type quasi-contractions on graphs. The results improve some old results in the literature.

We introduce two new concepts of weakly relaxed $\eta \text{-}\alpha$ monotone mappings and weakly relaxed $\eta \text{-}\alpha$ semimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxed $\eta \text{-}\alpha$ monotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxed $\eta \text{-}\alpha$ semimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.

We define the notion of ${\alpha }^{*}\text{-}\psi$-contractive mappings for cone metric space and obtain fixed points of multivalued mappings in connection with Hausdorff distance function for closed bounded subsets of cone metric spaces. We obtain some recent results of the literature as corollaries of our main theorem. Moreover, a nontrivial example of ${\alpha }^{*}\text{-}\psi$-contractive mapping satisfying all conditions of our main result has been constructed.

Several authors have introduced various conditions which can be used in order to prove common fixed point results. However, it became clear recently that some of these conditions, though formally distinct from each other, actually coincide in the case when the given mappings have a unique point of coincidence. Hence, in fact, new common fixed point results cannot be obtained in this way. We make a review of such connections and results in this paper.

We introduce the concept of the generalized $w_{\alpha }$-contraction mappings and establish the existence of fixed point theorem for such mappings by using the properties of $w$-distance and $\alpha$-admissible mappings. We also apply our result to coincidence point and common fixed point theorems in metric spaces. Further, the fixed point theorems endowed with an arbitrary binary relation are also derived from our results. Our results generalize the result of Kutbi, 2013, and several results in the literature.

We consider the multiquadratic functional equation. We establish its general solution and provide a characterization for this functional equation. Finally, we prove the Hyers-Ulam-Rassias stability of this functional equation.

The purpose of this paper is to prove some weak and strong convergence theorems for solving the multiple-set split feasibility problems for $\kappa$-strictly pseudononspreading mapping in infinite-dimensional Hilbert spaces by using the proposed iterative method. The main results presented in this paper extend and improve the corresponding results of Xu et al. (2006), of Osilike et al. (2011), and of many other authors.

We introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Banach space. We prove strong convergence theorems for the sequences produced by the methods and approximate a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. It is important to state clearly that the contribution of this paper in relation with the previous works (Marino and Xu, 2006) is a technical method to prove strong convergence theorems of a general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces. Our results improve and generalize many known results in the current literature.

In this paper, we introduce some almost generalized $(\psi ,\varphi )$-contractions in the setting of $G$-metric spaces. We prove some fixed points results for such contractions. The presented theorems improve and extend some known results in the literature. An example is also presented.

By taking a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

We introduce a $b$-metric on the cone $b$-metric space and then prove some equivalences between them. As applications, we show that fixed point theorems on cone $b$-metric spaces can be obtained from fixed point theorems on $b$-metric spaces.

Under the $\varphi$-contraction conditions, we prove common fixed point theorems for self-mappings in the space $𝒞ℬ(X)$ of the bounded closed sets in the complete stationary fuzzy metric space with the $H_M$-fuzzy metric for the bounded closed sets.

We establish a new equilibrium existence theorem of generalized abstract economies with general preference correspondences. As an application, we derive an existence theorem of generalized quasi-variational inequalities in the general setting of $l.c.$-spaces without any linear structure.

We prove some new existence theorems of fixed points for Caristi type maps and some suitable generalized distances without lower semicontinuity assumptions on dominated functions. As applications of our results, some new fixed point theorems and new generalizations of the Banach contraction principle are given.

The purpose of this paper is to prove some common point theorems for the generalized cyclic Meir-Keeler-type ($\alpha ,\phi ,A,B$)-contraction in partially ordered metric spaces. Our results generalize many recent common point theorems in the literature.

The objective of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed point in fuzzy metric spaces. Some illustrative examples are furnished which demonstrate the validity of the hypotheses and degree of utility of our results. We derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. As an application to our main result, we prove an integral-type fixed point theorem in fuzzy metric space. Our results improve and extend a host of previously known results including the ones contained in Imdad et al. (2012).

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of an $m$-accretive operator $A$ in the setting of real uniformly convex and 2-uniformly smooth Banach space $X$. Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the set $A^{-\mathrm{1}}(\mathrm{0})$ of zeros of $A$ in $X$. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

New fixed-point theorems on metric spaces are established, and analogous results on partial metric spaces are deduced. This work can be considered as a continuation of the paper Samet et al. (2013).