Consider the variational inequality $\text{V}\text{I}(C,F)$ of finding a point $x^{\mathrm{*}}\in C$ satisfying the property $\langle F{x}^{\mathrm{*}},x-x^{\mathrm{*}}\rangle \ge 0$ for all $x\in C$, where $C$ is a level set of a convex function defined on a real Hilbert space $H$ and $F:H\to H$ is a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets of $H$) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projection $P_C$ (calculating $P_C$ by computing a sequence of projections onto half-spaces containing the original domain $C$) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants of $F$ (i.e., Lipschitz constants on some bounded subsets of $H$)), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.

We introduce and study well-posedness in connection with the symmetric vector quasi-equilibrium problem, which unifies its Hadamard and Levitin-Polyak well-posedness. Using the nonlinear scalarization function, we give some sufficient conditions to guarantee the existence of well-posedness for the symmetric vector quasi-equilibrium problem.

Let $F\left(x, y\right)=a_s\left(x\right)y^s+a_{s-1}\left(x\right)y^{s-1}+\cdots +a_0\left(x\right)$ be a real-valued polynomial function in which the degree $s$ of $y$ in $F\left(x, y\right)$ is greater than or equal to 1. For any polynomial $y\left(x\right)$, we assume that $T:\mathbb{R}\left[x\right]\to \mathbb{R}\left[x\right]$ is a nonlinear operator with $T\left(y\left(x\right)\right)=F\left(x, y\left(x\right)\right)$. In this paper, we will find an eigenfunction $y\left(x\right)\in \mathbb{R}\left[x\right]$ to satisfy the following equation: $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ for some eigenvalue $a\in \mathbb{R}$ and we call the problem $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ a fixed point like problem. If the number of all eigenfunctions in $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ is infinitely many, we prove that (i) any coefficients of $F\left(x, y\right), a_s\left(x\right), a_{s-1}\left(x\right),\dots , a_0\left(x\right)$, are all constants in $\mathbb{R}$ and (ii) $y\left(x\right)$ is an eigenfunction in $F\left(x, y\left(x\right)\right)=ay\left(x\right)$ if and only if $y\left(x\right)\in \mathbb{R}$.

We use a modified S-iterative process to prove some strong and $\mathrm{\Delta }$-convergence results for asymptotically nonexpansive type mappings in uniformly convex hyperbolic spaces, which includes Banach spaces and CAT(0) spaces. Thus, our results can be viewed as extension and generalization of several known results in Banach spaces and CAT(0) spaces (see, e.g., Abbas et al. (2012), Abbas et al. (2013), Bruck et al. (1993), and Xin and Cui (2011)) and improve many results in the literature.