Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve a quadratic minimization problem.

We give a unified approach for studying the existence of multiple positive solutions of nonlinear differential equations of the form

$$-u''(t)=g(t)f(t,u(t)),\quad \text{for almost every } t \in (0,1),$$

where $g, f$ are non-negative functions, subject to various nonlocal boundary conditions. We study these problems via new results for a perturbed integral equation, in the space $C[0,1]$, of the form

$$u(t)=\gamma(t){\alpha}[u]+\delta(t){\beta}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds$$

where $\alpha[u]$, $\beta[u]$ are linear functionals given by Stieltjes integrals but are not assumed to be positive for all positive $u$. This means we actually cover many more differential equations than the simple equation written above. Previous results have studied positive functionals only, but even for positive functionals our methods give improvements on previous work. The well-known $m$-point boundary value problems are special cases and we obtain sharp conditions on the coefficients, which allows some of them to have opposite signs. We also use some optimal assumptions on the nonlinear term.

By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation $$\Delta^2 x_{n-1} +f(n, x_n)\,{=}\,0,$$ where $f\in C({\bf R}\times {\bf R}^m, {\bf R}^m), f(t+M,z)=f(t,z)$ for any $(t, z)\in {\bf R}\times{\bf R}^m$ and $M$ is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.This project is supported by TRATOYT of China and by the State Education Commission Trans-Century Training Program Foundation for the Talents.

Multilinear commutators with vector symbol $\vec{b}=(b_1,\ldots,b_m)$ defined by \[ T_{\vec{b}}(f)(x)=\int_{{\bb R}^n}\Bigg[\prod\limits^m_{j=1}(b_j(x)-b_j(y))\Bigg]K(x,y)f(y)dy \] are considered, where $K$ is a Calderón–Zygmund kernel. The following a priori estimates are proved for $w\in A_\infty$ . For $0 < p < \infty$ , there exists a constant $C$ such that \[ \|\dot{T}_{{\vec{b}}}(f)\|_{L^P(w)}\le C\|\vec{b}\|\|M_{L(\log\,L)^{1/r}}(f)\|_{L^P(w)} \] and \[ \sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:|T_{\vec{b}}f(y)|>t\})\le C\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:M_{L(\log\,L)^{1/r}}(\|\vec{b}\|f)(y)>t\}), \] where \begin{eqnarray*} &\|\vec{b}\|=\prod\limits^m_{j=1}\|b_j\|_{osc_{\exp L}^r j},\\ &\Phi(t)=t\log^{1/r}(e+t),\quad \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_m}, \end{eqnarray*} and $M_{L(\log L)^{\alpha}}$ is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.

As a corollary, it is deduced that the operators $T_{\vec{b}}$ are bounded on $L^p(w)$ when $w\in A_p$ , and that they satisfy corresponding weighted $L(\log\,L)^{1/r}$ -type estimates with $w\in A_1$ .

The restrictions Bspq(Ω) and Fspq(Ω) of the Besov and Triebel–Lizorkin spaces of tempered distributions Bspq(ℝn) and Fspq(ℝn) to Lipschitz domains Ω⊂ℝn are studied. For general values of parameters (s∈ℝ, p>0, q>0) a ‘universal’ linear bounded extension operator from Bspq(Ω) and Fspq(Ω) into the corresponding spaces on ℝn is constructed. The construction is based on a new variant of the Calderón reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of Bspq(Ω) and Fspq(Ω) in terms of their values in Ω are also obtained.

The essential norm of weighted composition operators on weighted Banach spaces of analytic functions is computed in terms of the weights and the inducing symbols. As a consequence the boundedness and compactness of these operators is characterized. As another consequence the essential norm of composition operators on weighted Bloch spaces is obtained and, consequently, the boundedness and compactness of composition operators on these spaces is also characterized. Particular instances of weighted Bloch spaces are the Lipschitz spaces. The method used allows a unified treatment of the problem of boundedness and compactness on these spaces.

Commuting maps on a class of algebras called triangular algebras are investigated. In particular, sufficient conditions are given such that every commuting map L on such an algebra is of the form L(a) = ax+h(a), where x lies in the center of the algebra and h is a linear map from the algebra to its center.

We study, on the entire space ℝN(N [ges ] 1), the diffusive logistic equation

and its generalizations. Here p > 1 is a constant. Problem (1.1) plays an important role in understanding various population models and some other problems in applied mathematics. When λ = 1 and p = 2, it is also known as the Fisher equation and KPP equation, due to the pioneering works of Fisher [8] and Kolmogoroff, Petrovsky and Piscounoff [18].

Let ([sum ]A, T) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Φ:[sum ]A → Rd be a continuous function. Denote by σΦ(x) the ergodic limit limn→∞n−1 [sum ]n−1j=0 Φ(Tjx) when the limit exists. Possible ergodic limits are just mean values ∫ Φdμ for all T-invariant measures. For any possible ergodic limit α, the following variational formula is proved:

[formula here]

where hμ denotes the entropy of μ and htop denotes topological entropy. It is also proved that unless all points have the same ergodic limit, then the set of points whose ergodic limit does not exist has the same topological entropy as the whole space [sum ]A

Our first result is a ‘sum-product’ theorem for subsets A of the finite field ${{\mathbb F}_p}$, p prime, providing a lower bound on $\max (|A+A|, |A\cdot A|)$. The second and main result provides new bounds on exponential sums

\[\sum_{x_1,\dots,x_k\in A} \exp(2\pi ix_1\dotsc x_k\xi/p),\]

where $A\subset{{\mathbb F}_p}$.

The notion of ‘finite type’ iterated function systems of contractive similitudes is introduced, and a scheme for computing the exact Hausdorff dimension of their attractors in the absence of the open set condition is described. This method extends a previous one by Lalley, and applies not only to the classes of self-similar sets studied by Edgar, Lalley, Rao and Wen, and others, but also to some new classes that are not covered by the previous ones.

Monotonicity with respect to the order v of the magnitude of general Bessel functions [Cscr ]v(x) = aJv(x)+bYv(x) at positive stationary points of associated functions is derived. In particular, the magnitude of [Cscr ]v at its positive stationary points is strictly decreasing in v for all positive v. It follows that supx[mid ]Jv(x)[mid ] strictly decreases from 1 to 0 as v increases from 0 to ∞. The magnitude of x1/2[Cscr ]v(x) at its positive stationary points is strictly increasing in v. It follows that supx[mid ]x1/2[Cscr ]v(x)[mid ] equals √2/π for 0 [les ] v [les ] 1/2 and strictly increases to ∞ as v increases from 1/2 to ∞.

It is shown that v1/3supx[mid ]Jv(x)[mid ] strictly increases from 0 to b = 0.674885… as v increases from 0 to ∞. Hence for all positive v and real x,

formula here

where b is the best possible such constant. Furthermore, for all positive v and real x,

formula here

where c = 0.7857468704… is the best possible such constant.

Additionally, errors in work by Abramowitz and Stegun and by Watson are pointed out.

By means of generalized Riccati transformation techniques and generalized exponential functions, some oscillation criteria are given for the nonlinear dynamic equation \[ (p(t)x^{\Delta} (t))^{\Delta}+q(t)(f\circ x^{\sigma})=0 \] on time scales. The results are also applied to linear and nonlinear dynamic equations with damping, and some sufficient conditions are obtained for the oscillation of all solutions.

This work was motivated in part by the following general question: given an ideal I in a Cohen–Macaulay (abbreviated to CM) local ring R such that dim R/I=0, what information about I and its associated graded ring can be obtained from the Hilbert function and Hilbert polynomial of I? By the Hilbert (or Hilbert–Samuel) function of I, we mean the function HI(n) =λ(R/In) for all n[ges ]1, where λ denotes length. Samuel [23] showed that for large values of n, the function HI(n) coincides with a polynomial PI(n) of degree d=dim R. This polynomial is referred to as the Hilbert, or Hilbert–Samuel, polynomial of I. The Hilbert polynomial is often written in the form

formula here

where e0(I), [ctdot ], ed(I) are integers uniquely determined by I. These integers are known as the Hilbert coefficients of I.

A method is developed for obtaining Ramanujan's mock theta functions from ordinary theta functions by performing certain operations on their q-series expansions. The method is then used to construct several new mock theta functions, including the first ones of eighth order. Summation and transformation formulae for basic hypergeometric series are used to prove that the new functions actually have the mock theta property. The modular transformation formulae for these functions are obtained.

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system.

Let $K$ and $\mu$ be the self-similar set and the self-similar measure associated with an IFS (iterated function system) with probabilities $(S_i, p_i)_{i=1,\ldots,N}$ satisfying the open set condition. Let $\Sigma=\{1,\ldots,N\}^{\bb N}$ denote the full shift space and let $\pi : \Sigma \longrightarrow K$ denote the natural projection. The (symbolic) local dimension of $\mu$ at $\omega \in \Sigma$ is defined by $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ , where $K_{\omega\mid n}=S_{\omega 1}\circ\ldots\circ S_{\omega_n}(K)$ for $\omega = (\omega_1, \omega_2,\ldots) \in \Sigma$ . A point $\omega$ for which the limit $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ does not exist is called a divergence point. In almost all of the literature the limit $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ is assumed to exist, and almost nothing is known about the set of divergence points. In the paper a detailed analysis is performed of the set of divergence points and it is shown that it has a surprisingly rich structure. For a sequence $(x_n)_n$ , let ${\sf A}(x_n)$ denote the set of accumulation points of $(x_n)_n$ . For an arbitrary subset $I$ of ${\bb R}$ , the Hausdorff and packing dimension of the set

\[ \left\{\omega\in\Sigma\left\vert {\sf A}\left(\frac{\log\mu K_{\omega\mid n}}{\log\hbox{ diam }K_{\omega\mid n}}\right)\right.=I\right\} \]

and related sets is computed. An interesting and surprising corollary to this result is that the set of divergence points is extremely ‘visible’; it can be partitioned into an uncountable family of pairwise disjoint sets each with full dimension.

In order to prove the above statements the theory of normal and non-normal points of a self-similar set is formulated and developed in detail. This theory extends the notion of normal and non-normal numbers to the setting of self-similar sets and has numerous applications to the study of the local properties of self-similar measures including a detailed study of the set of divergence points.

Spectrum-preserving linear mappings were studied for the first time by G. Frobenius [18]. He proved that a linear mapping Φ from Mn([Copf ]) onto Mn([Copf ]) which preserves the spectrum has one of the forms Φ(x) = axa−1 or Φ(x) = atxa−1, for some invertible matrix a. (Incidentally the hypothesis that Φ is onto is superfluous; see Proposition 2.1(i).) This result was extended by J. Dieudonné [17] supposing Φ onto and satisfying SpΦ(x) ⊂ Sp x, for every n × n matrix x.

Several results of M. Nagasawa, S. Banach and M. Stone, R. V. Kadison, A. Gleason and J. P. Kahane and W. Żelazko led I. Kaplansky in [22] to the following problem: given two Banach algebras with unit and Φ a linear mapping from A into B such that Φ(1) = 1 and SpΦ(x) ⊂ Sp x, for every x ∈ A, is it true that Φ is a Jordan morphism? With this general formulation, this question cannot be true (see [2], p. 28).

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S-curvature are classified.