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In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.

We study the minimax theorems for set-valued mappings with several hierarchical process, and propose three versions for minimax theorems in topological vector spaces setting. These problems arise from some minimax theorems in the vector settings. As applications, we discuss the existences of two kinds of saddle points. Our results are new or include as special cases recent existing results.

We study some properties for parametric generalized vector equilibrium problems and the convergent behavior for the correspondent solution sets of this problem under some suitable conditions. Several existence results and the topological structures of the efficient solutions set are established. Some new results of existence for weak solutions and strong solutions are derived. Finally, we give some examples to illustrate our theory including the example studied by Fang (1992), who established the perturbed nonlinear program $({P}_{\mu })$ and described successfully that the optimal solution of $({P}_{\mu })$ will approach the optimal solution of linear program (P).

The aim of this paper is to study the minimax theorems for set-valued mappings with or without linear structure. We define several kinds of cone-convexities for set-valued mappings, give some examples of such set-valued mappings, and study the relationships among these cone-convexities. By using our minimax theorems, we derive some existence results for saddle points of set-valued mappings. Some examples to illustrate our results are also given.

In this paper, we study the iterative algorithm and convergence theorems for $F$-implicit generalized variational inequalities problem ($F$-IGVIP). By employing our earlier works ([6], Theorem 2.2), we establish several iterative convergence results for $F$-IGVIP. The algorithm and convergence results are new for solving the strong solution of $F$-IGVIP. Furthermore, new algorithms and convergence theorems for $F$-implicit generalized complementarity problem ($F$-IGCP) are also discussed.

Let $q \gt 1$ and $E$ be a real $q$-uniformly smooth Banach space, $K$ be a nonempty closed convex subset of $E$ and $T : K \to K$ be a Lipschitz continuous mapping. Let $\{u_n\}$ and $\{v_n\}$ be bounded sequences in $K$ and $\{\alpha_n\}$ and $\{\beta_n\}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the sequence generated from an arbitrary $x_1 \in K$ by the Ishikawa iteration process with errors: $y_n = (1 − \beta n) x_n + \beta_n Tx_n + v_n$, $x_{n+1} = (1 − \alpha_n) x_n + \alpha_n Ty_n + u_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of $T$ is established.

Let $q \gt 1$ and $E$ be a real $q$-uniformly smooth Banach space, $K$ be a nonempty closed convex subset of $E$ and $T : K \to K$ be a single-valued mapping. Let $\{u_n\}^{\infty}_{n=1}$, $\{v_n\}^{\infty}_{n=1}$, $\{w_n\}^{\infty}_{n=1}$ be three sequences in $K$ and $\{\alpha_n\}^{\infty}_{n=1}$, $\{\beta_n\}^{\infty}_{n=1}$ and $\{\gamma_n\}^{\infty}_{n=1}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the sequence generated from an arbitrary $x_1 \in K$ by the three-step iteration process with errors: $x_{n+1} = (1 − \alpha_n) x_n + \alpha_n Ty_n + u_n$, $y_n = (1 − \beta_n) x_n + \beta_n Tz_n + v_n$, $z_n = (1 − \gamma_n) x_n + \gamma_n Tx_n + w_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of $T$ is established. We also derive the corresponding new results on the strong convergence of the three-step iterative process.

The purpose of this paper is to introduce and study two new hybrid proximal-point algorithms for finding a common element of the set of solutions to a generalized equilibrium problem and the sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex Banach space. We established strong and weak convergence theorems for these two modified hybrid proximal-point algorithms, respectively.

The purpose of this paper is to investigate the problems of the well-posedness for a system of mixed quasivariational-like inequalities in Banach spaces. First, we generalize the concept ofα-well-posedness to the system of mixed quasivariational-like inequalities, which includes symmetric quasi-equilibrium problems as a special case. Second, we establish some metric characterizations ofα-well-posedness for the system of mixed quasivariational-like inequalities. Under some suitable conditions, we prove that theα-well-posedness is equivalent to the existence and uniqueness of solution for the system of mixed quasivariational-like inequalities. The corresponding concept ofα-well-posedness in the generalized sense is also considered for the system of mixed quasivariational-like inequalities having more than one solution. The results presented in this paper generalize and improve some known results in the literature.

In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.

In this research, a multihierarchical methodology works by taking a whole intersection property to establish a fixed point theorem. Using such a whole intersection property to let multihierarchical procedures of set-valued mappings can work on a Hausdorff topological vector space. Some examples are proposed in order to illustrate our theory.

We study new types for minimax theorems with a couple of set-valued mappings, and we propose several versions for minimax theorems in topological vector spaces setting. These problems arise naturally from some minimax theorems in the vector settings. Both the types of scalar minimax theorems and the set minimax theorems are discussed. Furthermore, we propose three versions of minimax theorems for the last type. Some examples are also proposed to illustrate our theorems.