Let $c_0$ be the space of sequences converging to 0. A symmetric norming function (or briefly, s.n. function) is a function $\Phi$ from $c_0$ into nonnegative numbers with the properties of that in a norm, a normalizing criteria: $\Phi(1,0,0,\cdots) = 1$, and the symmetric condition: $\Phi(x_1,x_2,\cdots) = \Phi(x^*_1,x^*_2,\cdots)$, where $x^*_1, x^*_2, \cdots$ is the nonincreasing rearrangement of $|x_1|, |x_2|, \cdots$. In this paper, we will define spaces of analytic functions based on s.n. functions, which are generalization of the space $B^+_1$ in [2].

Assume that $F$ is a nonlinear operator on a real Hilbert space $H$ which is $\eta$-strongly monotone and $\kappa$-Lipschitzian on a nonempty closed convex subset $C$ of $H$. Assume also that $C$ is the intersection of the fixed point sets of a finite number of nonexpansive mappings on $H$. We develop a relaxed hybrid steepest-descent method which generates an iterative sequence $\{x_n\}$ from an arbitrary initial point $x_0 \in H$. The sequence $\{x_n\}$ is shown to converge in norm to the unique solution $u^*$ of the variational inequality \[ \langle F(u^*), v-u^* \rangle \geq 0 \quad \forall v \in C \] under the conditions which are more general than those in Ref. 19. Applications to constrained generalized pseudoinverse are included.

In this paper, we generalize the $(S)_+$-condition to bifunctions with values in an oredered Hausdorff topological vector space $\mathcal{Z}$, and define a weak $(S)_+$-condition for the bifunctions. These conditions extend naturally to operators from nonempty subsets of a topological vector space $X$ into the set $\mathcal{L}(X,\mathcal{Z})$ of all continuous linear mappings from $X$ into $\mathcal{Z}$. Then we derive some existence results for vector equilibrium problems and vector variational inequalities associated with bifunctions or operators satisfying the weak $(S)_+$-condition.

Let $H$ be a Hilbert space and $\mathcal{N}$ a nest in $H$. Denote by $S^a(H)$ the Jordan ring of all self-adjoint operators on $H$ and $\mathrm{Alg}\mathcal{N}$ the nest algebra associated to $\mathcal{N}$. We show that a bijective map $\Phi : S^a(H) \to S^a(H)$ satisfying (1) $\Phi(ABA) = \Phi(A) \Phi(B) \Phi(A)$ for every pair of $A,B$, or (2) $\Phi(AB + BA) = \Phi(A) \Phi(B) + \Phi(B) \Phi(A)$ for every pair of $A,B$, or (3) $\Phi(\frac{1}{2} (AB + BA)) = \frac{1}{2} (\Phi(A) \Phi(B) + \Phi(B) \Phi(A))$ for every pair of $A,B$ must be additive, that is, a Jordan ring isomorphism. We also show that if a bijective map $\Phi : \mathrm{Alg}\mathcal{N} \to \mathrm{Alg}\mathcal{N}$ satisfies the Jordan multiplicativity of the form (2) or (3), then $\Phi$ must be a Jordan isomorphism. Moreover, such Jordan multiplicative maps are characterized completely.

In [5], we gave a real reduction theory. It is the real analogue of J.von Neumann’s (complex) reduction theory. In [4], E.G.Effros gave a natural explanation for (complex) reduction theory by Borel maps. In this note, we also use Borel maps to give an explanation for real measurable fields of Hilbert spaces, von Neumann algebras, and etc.

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.

Let $q \gt 1$ and $E$ be a real $q−$uniformly smooth Banach space. Let $K$ be a nonempty closed convex subset of $E$ and $T : K \to K$ be a strictly pseudocontractive mapping in the sense of F. E. Browder and W. V. Petryshyn [1]. Let $\{u_n\}$ be a bounded sequence in $K$ and $\{\alpha_n\}, \{\beta_n\}, \{\gamma_n\}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the bounded sequence in $K$ generated from any given $x_1 \in K$ by the Ishikawa iteration method with errors: $y_n = (1 − \beta_n) x_n + \beta_n Tx_n$, $x_{n+1} = (1 − \alpha_n − \gamma_n) x_n + \alpha_n Ty_n + \gamma_n u_n$, $n \geq 1$. It is shown in this paper that if $T$ is compact or demicompact, then $\{x_n\}$ converges strongly to a fixed point of $T$.

Let $A$ be the generator of an $n$-times integrated semigroup $T(\cdot)$ and let $r \in \mathbb{N}$. We first prove the equivalence of Riemann, Peano, and Taylor operators, which are three different expressions of the $r$-th power of $A_1$, the part of $A$ in the closure of the domain $D(A)$ of $A$. Then we discuss optimal and non-optimal rates of approximation of $T(\cdot)x$ for $x \in D(A^{r−1}_{1})$, via the $(n+r)$-th Taylor expansion of $T(\cdot)$ in terms of $A^k_1$, $k = 0, \ldots, r−1$.

Let $A$ be an $n \times n$ (complex) matrix. Recall that the numerical range $W(A)$ of $A$ is the set $\{ \langle Ax,x \rangle : x \in \mathbb{C}^n, \|x\| = 1 \}$ in the plane, where $\langle \cdot,\cdot \rangle$ denotes the usual inner product in $\mathbb{C}^n$. In this paper a series of tests is given, allowing one to determine when the numerical range of a $4 \times 4$ matrix $A$ is an elliptic disc.

Let $X$ and $Y$ be Hausdorff and locally convex topological vector spaces. In this paper, we prove that a convex subset of $X$ is closed if and only if it is closed in the topology on $X$ induced by the set of continuous linear mappings from $X$ into $Y$ . As applications, some existence results for vector equilibrium problems and vector variational inequalities associated with discontinuous mappings are given.

This paper is to establish an existence result (Theorem 3.1) for the implicit quasivariational inequality without continuity assumptions in infinitedimensional normed spaces.

In this paper, we investigate the perturbation problem for local $C$-regularized semigroups on a Banach space and establish a Miyadera type perturbation theorem.

In this paper, we obtain two new existence theorems for mild solutions and classical solutions to nonautonomous functional equations with infinite delay in Banach spaces.

In this paper, we introduce and study the bilevel optimization problems in topological spaces without linear structure. For these problems, we establish two models which are different in the feasible region setting of lower-level problems. Some new existence results are obtained in rather weak conditions. These theorems improve and generalize the corresponding results of bilevel optimization in the literatures.

A retrievel of a Sperner map from a Sperner matroid is illustrated. As an application, a new proof of a completion theorem of Lovász’s matroid version of Sperner’s lemma is given.

We prove that the ultrapower and the central sequence algebra of the hyperfinite II$_1$ factor are prime factors which implies that the central sequence algebra of the tensor product of two factors may not be the tensor product of the central sequence algebras of the factors respectively. This answers negatively a question of D. McDuff. Isomorphisms of ultrapowers of factors are also studied.

We use unitary operators in a finite von Neumann algebra as orthonormal basis for certain Hilbert space. Possible applications in noncommutative wavelets and computer graphics are discussed.

We introduce concepts of $\sigma$-lim sup and $\sigma$-lim inf for bounded sequences of real numbers and show a Cauchy criterion for sequences of vectors which converge in the sense of $a\sigma$-limit (i.e., absolute almost convergence). Then a sufficient condition on a bounded sequence $\{ \{ x^{(m)}_n \}^{\infty}_{n=1} \}^{\infty}_{m=1} \subset \ell^{\infty}(X)$ is given for the following equality to hold: \[ a\sigma - \lim_{m \to \infty} \sigma - \lim_{n \to \infty} x_{n}^{(m)} = \sigma - \lim_{n \to \infty} a\sigma - \lim_{m \to \infty} x_{n}^{(m)}. \] Finally, applying this result we show that $\sigma - \lim\limits_{n \to \infty} f(\sin(n\theta))$ and $\sigma - \lim\limits_{n \to \infty} f(\cos(n\theta))$ exist whenever $f$ is a weakly continuous function on $[−1,1]$ with values in a reflexive Banach space.

In this paper, we study some problems about approximate isomorphisms between Banach *-algebras or $C^*$-algebras.

The Banach-Stone problem for $L^p$-spaces is to assert when a linear isometry between $L^p$-spaces is a weighted composition operator. We shall show that every $\sigma$-finite measure space with Sikorski’s property solves the Banach-Stone probelm. In addition, we show that if $X$ is a totally ordered and Dedekind complete, then every $\sigma$-finite $\mu$-separable measure space $(X,\mathcal{B},\mu)$ has Sikorski’s property.

In this survey article, we introduce the isometric extension problem of isometric mapping between the unit spheres and the distance one preserving problem. Some important results in the development of the related problems are outlined in this paper and some recent advancement and open problems are repointed.

We consider the null boundary controllability for a one-dimensional fourth order semilinear equation. We show that if the initial data is continuous and sufficiently small, then the fourth order semilinear equation is controllable.