In this paper, we introduce a new class of $h$-$\eta$-maximal monotone mappings and a new system of generalized mixed implicit quasi-variational inclusions involving set-valued mappings and $h$-$\eta$-maximal monotone mappings. By using resolvent operator technique of $h$-$\eta$-maximal monotone mappings, A new iterative algorithm to compute approximate solutions of the system is suggested and analyzed. The convergence of the iterative sequence generated by the new algorithm is also proved. These results generalize many known results in literature.

In this paper, we extend the scalar-valued $B$-semipreinvex functions and vector-valued preinvex functions to the cases of vector-valued $B$-semipreinvex functions in Banach spaces. We investigate some properties for the vector-valued $B$-semipreinvex functions and consider a new class of vector-valued nonsmooth programming problems in which functions are locally Lipschitz. In terms of the Ralph vector sub-gradient, we obtain the generalized Kuhn-Tucker type sufficient optimality conditions and saddle point condition. Also, a generalized Mond-Weir type dual is formulated and some duality theorems are established involving locally Lipschitz $B$-semipreinvex functions for the pair of primal and dual programming. The results presented in this paper generalize some main results of Kuang and Batista Dos Santoset, Osuna-Gomez, Rojas-Medar and Rufian-Lizana.

A not necessarily continuous, linear or multiplicative function $\theta$ from an algebra $\mathcal A$ into itself is called a local automorphism if $\theta$ agrees with an automorphism of $\mathcal A$ at each point in $\mathcal A$. In this paper, we study the question when a local automorphism of a $C$*-algebra, or a W*-algebra, is an automorphism.

In this paper, two generalized vector implicit variational inequalities and three generalized vector implicit complementarity problems are introduced with a general order in ordered Banach spaces and the equivalence between them is studied under certain assumptions. Furthermore, some existence theorems for the generalized vector implicit variational inequalities are derived by using Lin’s result (1986) and Nadler’s result (1969). We affirmatively answer the open question proposed by Rapcsak (2000). Our results are the extension and improvement of the corresponding results in Huang and Li (2006).

We study a third-order differential equation arising in the study of falling film on a coated vertical fibre. We first provide some properties of solutions to this third-order equation. Then we prove some nonexistence results of global solution to this equation under certain boundary condition at infinity.

The purpose of this paper is to use viscosity approximation methods for a finite family of asymptotically nonexpansive mappings to establish the necessary and sufficient conditions for the iterative sequence to converge to a common fixed point of those mappings in uniformly convex Banach spaces.

Recently, H. K. Xu [J. Math. Anal. Appl. 314 (2006) 631-643] considered the iterative method for approximation to zeros of an $m$-accretive operator $A$ in a Banach space $X$. In this paper, we propose a hybrid viscosity approximation method with perturbed mapping that generates the sequence $\{x_n\}$ by the algorithm $x_{n+1} = \alpha_n(u+f(x_n)) + (1-\alpha_n) [J_{r_n} x_n - \lambda_n F(J_{r_n} x_n)]$, where $\{\alpha_n\}$, $\{r_n\}$ and $\{\lambda_n\}$ are three sequences satisfying certain conditions, $f$ is a contraction on $X$, $J_r$ denotes the resolvent $(I+rA)^{-1}$ for $r \gt 0$, and $F$ is a perturbed mapping which is both $\delta$-strongly accretive and $\lambda$-strictly pseudocontractive with $\delta + \lambda \geq 1$. Under the assumption that $X$ either has a weakly continuous duality map or is uniformly smooth, we establish some strong convergence theorems for this hybrid viscosity approximation method with perturbed mapping.

In this paper, we introduce a new concept of $(A,\eta)$-monotone operators, which generalizes the $(H,\eta)$-monotonicity and $A$-monotonicity in Hilbert spaces and other existing monotone operators as special cases. We study some properties of $(A,\eta)$-monotone operators and define the resolvent operators associated with $(A,\eta)$-monotone operators. Further, by using the new resolvent operator technique, we construct some new iterative algorithms for solving a new class of nonlinear $(A,\eta)$-monotone operator inclusion systems involving non-monotone set-valued mappings in Hilbert spaces. We also prove the existence of solutions for the nonlinear operator inclusion systems and the convergence of iterative sequences generated by the algorithm. Our results improve and generalize the corresponding results of recent works.

In this paper, we consider a more general form of variational inclusions, called generalized variational inclusion (for short, GVI). In connection with GVI, we also consider a generalized resolvent equation with $H$-resolvent operator, called $H$-resolvent equation (for short, $H$-RE). We suggest iterative algorithms to compute the approximate solutions of GVI and $H$-RE. The existence of a unique solution of GVI and $H$-RE and convergence of iterative sequences generated by the proposed algorithms are also studied. Several special cases are also discussed.

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.

The purpose of this paper is to give some necessary and sufficient conditions for the iterative sequence of Reich type to converging to a fixed point. The results presented in this paper extend and improve some recent results ([2-4]).

We introduce a pair of multiobjective generalized second order symmetric dual programs where the objective function contains a support function. Weak, strong and converse duality theorems for these second order problems are established under suitable generalized second order convexity assumptions. Also, we give some special cases of our second order symmetric duality results.

In this paper, we discuss $\epsilon$-optimality conditions and $\epsilon$-saddle point theorems for $\epsilon$-approximate solutions for convex semidefinite optimization problem which hold under a weakened constraint qualification or which hold without any constraint qualification.

Moreover, we formulate a Wolfe type dual problem for the convex semidefinite optimization problem, and prove $\epsilon$-weak duality and $\epsilon$-strong duality between the primal problem and the dual problem, which hold under a weakened constraint qualification.

In this paper, several classes of vector $F$-complementarity problems (in short, VF-CP) and vector $F$-variational inequalities (in short, VF-VI) with relations determined by nonconvex preferences are introduced in Banach spaces. Some characterizations of solution sets for (VF-CP) and (VF-VI) are also presented. Furthermore, the results obtained are applied to vector optimization problems.

In this paper, we introduce an implicit iterative sequence to approximate zero points of accretive operators in Banach spaces and then prove weak convergence theorems for resolvents of accretive operators in Banach spaces satisfying Opial’s condition. Further, we discuss the strong convergence of the iterative sequences for resolvents of accretive operators with compact domains in general Banach spaces. Using these results, we consider the variational inequality problem of finding a solution of a variational inequality.

We study dynamical systems of the projection gradient type for convex constrained minimization problems, clearance type dynamical system for fixed point problems with nonexpansive self-mappings and descent-like dynamical system for variational inequalities with maximal monotone operators in Banach spaces. We prove the weak convergence of dynamical trajectories and establish the estimates of the convergence rate with respect to functionals of the problems. We also produce some strong convergence theorem. The results presented in the paper are new even in Hilbert spaces.

Suppose $E$ is a real Banach space with uniform normal structure and suppose $E$ has a uniformly Gateaux differentiable norm. Let $C$ be a nonempty closed convex and bounded subset of $E$. Let $T_1,T_2,\cdots T_r: C \to C$ be a finite family of asymptotically nonexpansive mappings. In this paper, we suggest and analyze an iterative algorithm for a finite family of asymptotically nonexpansive mappings $\{T_i\}_{i=1}^r$. We show the convergence of the proposed algorithm to a common fixed point $p \in \cap_{i=1}^{r} F(T_i)$ which is the unique solution of some variational inequality. Our results can be considered as an refinement and improvement of many known results.

The purpose of this paper is to study the existence of solution and approximation solvability for a new system of generalized nonlinear mixed quasivariational inequalities with two-variable operators in Hilbert spaces. We prove the convergence of iterative sequences generated by the algorithm for the system of generalized nonlinear mixed quasivariational inequality problems. The main results of this paper are generalizations, improvements and unifications of recent results in Chang et al. ([2], [3]), Kim-Kim [4], Kim-Kim [5], Nie et al. [9], Noor ([10]-[16]), Verma ([18]-[22]) and others ([6]-[8]).

Let $E$ be a Banach space, $C$ a nonempty closed convex subset of $E$, $f : C \to C$ a contraction, and $T_i : C \to C$ a nonexpansive mapping with nonempty $F := \bigcap_{i = 1}^N Fix(T_i)$, where $N \ge 1$ is an integer and $Fix(T_i)$ is the set of fixed points of $T_i$. Let $\{x_t^n\}$ be the sequence defined by $x_t^n = tf(x_t^n) + (1-t) T_{n+N} T_{n+N-1} \cdots T_{n+1} x_t^n$ ($0 \lt t \lt 1$). First, it is shown that as $t \to 0$, the sequence $\{x_t^n\}$ converges strongly to a solution in $F$ of certain variational inequality provided $E$ is reflexive and has a weakly sequentially continuous duality mapping. Then it is proved that the iterative algorithm $x_{n+1} = \lambda_{n+1} f(x_n) + (1-\lambda_{n+1}) T_{n+1} x_n$ ($n \ge 0$) converges strongly to a solution in $F$ of certain variational inequality in the same Banach space provided the sequence $\{\lambda_n\}$ satisfies certain conditions and the sequence $\{x_n\}$ is weakly asymptotically regular. Applications to the convex feasibility problem are included.

The purpose of this paper is to define the concept of well-posedness in the generalized sense of a fixed point problem for multivalued operators. Several conditions under which the fixed point problem is well-posed in the generalized sense are given. Some new fixed point theorems are also proved.

In this paper, we first introduce an iterative sequence of Mann’s type and Halpern’s type for finding a zero point of an m-accretive operator in a real Banach space. Then we obtain the strong and weak convergence by changing control conditions of the sequence. The result improves and extends a strong convergence theorem and a weak convergence theorem obtained by Kamimura and Takahashi [9], simultaneously.

A new nonexpansive mapping in a Banach space which is called generalized nonexpansive was introduced by the authors [4]. In this paper, we prove a weak convergence theorem for finding a fixed point of a generalized nonexpansive mapping in a Banach space. Moreover, using this result, we consider a proximal-type algorithm and the feasibility problem.

The aim of this paper is to devote to the sensitivity analysis in vector optimization problems. We prove that, under some suitable qualification conditions, the efficient solution map and efficient frontier of a parameterized vector optimization problem are proto-differentiable. It also provide two sufficient conditions for inner and outer approximation of the proto-derivative of the efficient frontier map.