In this paper, we introduce an iteration scheme of Mann's type for general commutative nonexpansive semigroups and then obtain a strong convergence theorem in compact sets of general Banach spaces by using the theory of means of abstract semigroups. Using this result, we prove some strong con- vergence theorems in cases of discrete and one-parameter semigroups.

The concept of periodic deflating subspaces of regular periodic matrix pairs $\{(A_j, B_j)\}_{j=1}^K$ is a generalization of deflating subspaces of a regular matrix pair $(A,B)$. In this paper we derive perturbation bounds for each individual subspace of simple periodic deflating subspaces.

In this paper we work with the ordinary differential equation $u'' - u^{p} = 0$ for some well-defined functions $u^{p}$. We obtain some interesting phenomena concerning blow-up, blow-up rate, life-span, stability, instability for solutions.

For certain analytic functions $\psi$, a lower Riesz bound for a finite wavelet system generated by $\psi$, yields an upper bound for the number of zeros on a bounded interval of the corresponding wavelet sums. In particular, we show that if the Fourier transform of $\psi$ is compactly supported, say on $[-\Omega,\Omega]$, and if $B \gt 2e \Omega$, then any finite sum $\sum_{|k| \leq \alpha/2} a_{k} \psi(x-k)$ cannot have more than $B \alpha$ zeros in $[-\alpha,\alpha]$ for $\alpha \gt 0$ sufficiently large.

We consider the boundedness of Calderón--Zygmund operators on Lipschitz space and Sobolev space without assuming cancellation condition $T1 = 0$, and we apply our results to Calderón's commutator.

This paper deals with a class of third order multi-point boundary value problem at resonance case. Some existence theorems are obtained by using the coincidence degree theory of Mawhin.

Some inequalities and approximations for the finite Hilbert transform by the use of Taylor's formula with the integral remainder are given.

In this paper we study the Hájeck-Rènyi type inequality of asymptotically almost negatively associated (AANA) random variables and derive strong laws of large numbers for weighted sums of AANA sequences.

For the random design nonparametric regression, cross-validation is a popular bandwidth selector. It is constructed by using the criterion of ``weighted" integrated square error. In practice, however, the weighting scheme by the design density in the criterion causes that its associated cross-validation function puts more emphasis in regions with more data, gives little attention to regions with few data, but has no consideration for regions without data. In such a case, the value of the cross-validated bandwidth depends on the distribution of the design points, but is independent of the location of the interval on which the regression function value is estimated. Hence, if there are sparse regions in the realization of the design, then the resulting cross-validated bandwidth is usually not large enough in magnitude such that its corresponding kernel regression function estimate has rough appearance in these sparse regions. To avoid this drawback to cross-validation, we suggest using the criterion of ''unweighted'' integrated square error to construct the bandwidth selector. Under the criterion, a bandwidth selector called integrated cross-validation is proposed, and the resulting bandwidth is shown to be asymptotically optimal. Empirical studies demonstrate that the kernel regression function estimate obtained by using our proposed bandwidth is better than that employing the ordinary cross-validated bandwidth, in both senses of having smoother appearance and yielding smaller sample unweighted integrated square error.

By introducing a parameter $\lambda \in [0,1]$, we give an inequality relating Carleman's inequality with Van der Corput's inequality. In particular, a generalization of Carleman's inequality with a best constant factor $e^{\frac{1}{1-\lambda}}$, $\lambda \in [0,1)$ is considered.