Let $E$ be a smooth, strictly convex and reflexive Banach space, let $C^*$ be a closed convex subset of the dual space $E^*$ of $E$ and let $\Pi_{C^*}$ be the generalized projection of $E^*$ onto $C^*$. Then the mapping $R_{C^*}$ defined by $R_{C^*} = J^{-1} \Pi_{C^*}J$ is a sunny generalized nonexpansive retraction of $E$ onto $J^{-1}C^{*}$, where $J$ is the normalized duality mapping on $E$. In this paper, we first prove that if $K$ is a closed convex cone in $E$ and $P$ is the nonexpansive retaction of $E$ onto $K$, then $P$ a sunny generalized nonexpansive retraction of $E$ onto $K$. Using this result, we obtain an equivalent condition for a closed half-space of $E$ to be a nonexpansive retract of $E$.

In 2003, Araujo and Jarosz showed that every bijective linear map $\theta: A \to B$ between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both $A,B$ are unital standard algebras on Hilbert spaces and $\theta$ preserves range or domain orthogonality. In particular, such maps are automatically bounded. In this paper, we will study linear orthogonality preservers in a unified way. We will show that every surjective linear map between standard operator algebras preserving range/domain orthogonality carries a standard form, and is thus automatically bounded.

In this article, after establishing weighted Plancherel-Pôlya-type inequalities, we introduce a new class of weighted Hardy spaces $H^p_{b,w}$ by using $g$-function, where $w$ is a Muckenhoupt's weight and $b$ is a para-accretive function. Then we show the atomic decomposition and molecular characterization of $H^p_{b,w}$. As applications, we prove the boundedness of Calderón-Zygmund operators between $H^p_{b,w}$ and classical weighted Hardy spaces $H^p_w$.

We show mean ergodic theorems for vector-valued weakly almost periodic functions (in the sense of Eberlein) defined on a semigroup which take values in a locally convex topological vector space. Next, motivated by Fréchet [10], we study the relationship between almost periodicity of semigroups of mappings and their equicontinuity, and also prove mean ergodic theorems for equicontinuous semigroups.

Let $A$ be the generator of a nondegenerate local $\alpha$-times integrated $C$-cosine function $C(\cdot)$ on a Banach space $X$ for some $\alpha \geq 0$, $f \in L_{loc}^1([0,T_0),X) \cap C((0,T_0),X)$, and $x,y \in X$. We first show that the abstract Cauchy problem : $ACP (A,C f, C x,C y) \quad u''(t) = A u(t) + C f(t)$, $u(0) = Cx$ and $u'(0) = C y$, has a strong solution is equivalent to the function $v(\cdot) = C(\cdot)x + j_0*C(\cdot) y + j_0*C*f(\cdot) \in C^{\alpha+1}([0,T_0),X)$ and $D^{\alpha+1} v(\cdot) \in C^1((0,T_0),X)$, and then use it to prove some new existence and approximation theorems concerning strong solutions of $ACP(A, C z + j_{\alpha-1}*C g,C x, C y)$ and mild solutions of $ACP(A, C x + j_1 C y + j_2 C z + j_{\alpha-1}*C g,0,0)$ (for $\alpha\geq 2$) in $C^2([0,T_0),X)$ when $C(\cdot)$ is locally Lipschitz continuous, and vectors $x$, $y$ and $z$ satisfy some suitable regularity assumptions. Here $0 \lt T_0 \leq \infty$ is fixed.

We study the growth orders of $\gamma$-th order Cesàro means $C_{n}^{\gamma}(T)$ ($\gamma \geq 0$) and Abel means $A_{r}(T)$ of the discrete semigroup $\{T^{n}: n \geq 0\}$ generated by a bounded linear operator $T$ on a Banach space. Let $T$ be of the form $T = -(I+N)$, where $N$ is a nilpotent operator of order $k+1$ with $k \in \mathbb{N}$. Thus $N^{k+1} = 0$ and $N^{k} \not = 0$. Then we prove that (a) $\|C_{n}^{\gamma}(T)\|\sim n^{k-\gamma}\; (n \to \infty)$ if $0 \leq \gamma \leq k+1$, and $\|C_{n}^{\gamma}(T)\|\sim n^{-1}\; (n \to \infty)$ if $\gamma \geq k+1$; (b) $\|A_{r}(T)\|\sim 1-r\; (r \uparrow 1)$. Here $a(n) \sim b(n)\; (n \to \infty)$ [resp. $a(r) \sim b(r)\; (r \uparrow 1)$] means that $0 \lt \liminf_{n \to \infty} a(n)/b(n) \leq \limsup_{n \to \infty} a(n)/b(n) \lt \infty$ [resp. $0 \lt \liminf_{r \uparrow 1} a(r)/b(r) \leq \limsup_{r \uparrow 1} a(r)/b(r) \lt \infty$].

We study the mean ergodicity of resolvent families and give a general theorem for nondensely defined generator. In particular, it is applied to $n$-times integrated semigroups

Robert has established a global convergence theorem in $\{0,1\}^{n}$: If a map $\hat{F}$ from $\{0,1\}^{n}$ to itself is contracting relative to the boolean vector distance $d$, then there exists a positive integer $p \leq n$ such that $\hat{F}^{p}$ is constant. In other words, $\hat{F}$ has a unique fixed point $\xi$ such that for any $x$ in $\{0,1\}^{n}$, we have $\hat{F}^{p}(x) = \xi$. The structure ($\{0,1\},+,\cdot,-,0,1$) may be regarded as the two-element boolean algebra. In this paper, this result is extended to any map $F$ from the product $X$ of $n$ finite boolean algebras to itself.

This paper is devoted to the numerical analysis of inverse problems for abstract hyperbolic differential equations. The presentation exploits a general approximation scheme and is based on $C_0$-cosine and $C_0$-semigroup theory within a functional analysis approach. We consider both discretizations in space as well as in time. The discretization in time is considered under the Krein-Fattorini conditions.

Combinatorics of complexes plays an important role in topology, nonlinear analysis, game theory, and mathematical economics. In 1967, Ky Fan used door-to-door principle to prove a combinatorial Stokes’ theorem on pseudomanifolds. In 1993, Shih and Lee developed the geometric context of general position maps, $\pi$-balanced and $\pi$-subbalanced sets and used them to prove a combinatorial formula for multiple set-valued labellings on simplexes. On the other hand, in 1998, Lee and Shih proved a multiple combinatorial Stokes’ theorem, generalizing the Ky Fan combinatorial formula to multiple labellings. That raises a question : Does there exist a unified theorem underlying Ky Fan’s theorem and Shih and Lee’s results? In this paper, we prove a multiple combinatorial Stokes’ theorem with balanced structure. Our method of proof is based on an incidence function. As a consequence, we obtain a multiple combinatorial Sperner’s lemma with balanced structure.

For continuous vector-valued functions, we discuss relations among exponential and polynomial growth orders of the $\gamma$-Cesàro mean ($\gamma \ge 0$) and of the Abel mean. In general, the Abel mean has growth order not larger than those of Cesàro means, and a higher-order Cesàro mean has a smaller growth order than a lower-order Cesàro mean. But, for a positive function in a Banach lattice, the Abel mean and all $\gamma$-Cesàro means with $\gamma \ge 1$ (but not with $0 \le \gamma \lt 1$) have the same polynomial growth order. The possibility of non-equal growth orders for these means is illustrated by some examples of $C_0$-semigroups and cosine operator functions.