We establish some fixed point theorems for $\alpha$-admissible mappings in the context of metric-like space via various auxiliary functions. In particular, we prove the existence of a fixed point of the generalized Meir-Keeler type $\mathrm{\alpha -}\varphi$-contractive self-mapping $f$ defined on a metric-like space $X$. The given results generalize, improve, and unify several fixed point theorems for the generalized cyclic contractive mappings that have appeared recently in the literature.

We first introduce and analyze one multistep iterative algorithm by hybrid shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: the generalized mixed equilibrium problem, finitely many variational inclusions, the minimization problem for a convex and continuously Fréchet differentiable functional, and the fixed-point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another multistep iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions.

An iterative algorithm for finding a common element of the set of common fixed points of a finite family of asymptotically nonextensive nonself mappings and the set of solutions for equilibrium problems is discussed. A strong convergence theorem of common element is established in a uniformly smooth and uniformly convex Banach space.

The aim of this paper is to propose some fixed point theorems in complete parametric metric spaces. Using these theorems, we deduce as corollaries the recent results of Ionescu et al. Moreover, we suggest some new contractions and prove certain fixed point theorems in triangular intuitionistic fuzzy metric spaces. We also discuss some illustrative examples to highlight the realized improvements.

The purpose of this paper is to present two iterative schemes based on the relative resolvent and the generalized resolvent, respectively. And, it is shown that the iterative schemes converge weakly to common solutions for two finite families of maximal monotone operators in a real smooth and uniformly convex Banach space and one example is demonstrated to explain that some assumptions in the main results are meaningful, which extend the corresponding works by some authors.

We study the minimax inequalities for set-valued mappings with hierarchical process and propose two versions of minimax inequalities in topological vector spaces settings. As applications, we discuss the existent results of solutions for set equilibrium problems. Some examples are given to illustrate the established results.

We apply the extensions of the Abian-Brown fixed point theorem for set-valued mappings on chain-complete posets to examine the existence of generalized and extended saddle points of bifunctions defined on posets. We also study the generalized and extended equilibrium problems and the solvability of ordered variational inequalities on posets, which are equipped with a partial order relation and have neither an algebraic structure nor a topological structure.

The extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing oscillator. Meanwhile, the comparisons between the results of the EHAM and standard Runge-Kutta numerical method are also presented. The results demonstrate that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions. For EHAM as an analytical approximation method, we are not sure whether it can apply to all of the nonlinear systems; we can only verify its effectiveness through specific cases. As a result of the existence of nonlinear terms, we must study different types of systems, no matter from the complication of calculation and physical significance.

Using Bregman functions, we introduce the new concept of Bregman generalized $f$-projection operator ${\text{Proj}}_C^{f, g}:E^{*}\to C$, where $E$ is a reflexive Banach space with dual space $E^{*}; f: E\to ℝ\cup \left\{+\infty \right\}$ is a proper, convex, lower semicontinuous and bounded from below function; $g: E\to ℝ$ is a strictly convex and Gâteaux differentiable function; and $C$ is a nonempty, closed, and convex subset of $E$. The existence of a solution for a class of variational inequalities in Banach spaces is presented.

The intent of this paper is to introduce the notion of compatible mappings for $n$-tupled coincidence points due to (Imdad et al. (2013)). Related examples are also given to support our main results. Our results are the generalizations of the results of (Gnana Bhaskar and Lakshmikantham (2006), Lakshmikantham and Ćirić (2009), Choudhury and Kundu (2010), and Choudhary et al. (2013)).

Let $\mathrm{\Omega }\subset {ℝ}^n$ be a nonsmooth convex domain and let $f$ be a distribution in the atomic Hardy space $H_{at}^p(\mathrm{\Omega })$; we study the Schrödinger equations $-\mathrm{div}(A\nabla u)+Vu=f$ in $\mathrm{\Omega }$ with the singular potential $V$ and the nonsmooth coefficient matrix $A$. We will show the existence of the Green function and establish the $L^p$ integrability of the second-order derivative of the solution to the Schrödinger equation on $\mathrm{\Omega }$ with the Dirichlet boundary condition for . Some fundamental pointwise estimates for the Green function are also given.

We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method.

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solution c*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

We study a generalized equilibrium problem by using a nonsymmetric extension of Ky Fan’s inequality. As an application, we present a fixed point type algorithm inspired by a model from Tada and Takahashi (2007).

We propose some relaxed implicit and explicit viscosity approximation methods for hierarchical fixed point problems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These relaxed viscosity approximation methods are based on the well-known viscosity approximation method and hybrid steepest-descent method. We obtain some strong convergence theorems under mild conditions.

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.