For a subset $K$ of a metric space $(X,d)$ and $x ∈ X$,$$P_K(x)=\Bigg\{y\in K : d(x, y)= d(x, K) ≡\text{inf}\{d(x, k) : k \in K\}\Bigg\}$$ is called the set of best $K$-approximant to $x$. An element $g◦\in K$ is said to be a best simultaneous approximation of the pair $y_1, y_2 \in X$ if $$max\Bigg\{d(y_1, g◦), d(y_2, g◦)\Bigg\}= \inf\limits_{g\in K} max\Bigg\{d(y_1, g), d(y_2, g)\Bigg\}.$$ In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings $T$ and $S$ on $K$, results are proved on both $T$- and $S$- invariant points for a set of best simultaneous approximation. Some results on best $K$-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas^[1], S. Chandok and T. D. Narang^[2], T. D. Narang and S. Chandok^[11], S. A. Sahab, M. S.  Khan and S. Sessa^[14], P. Vijayaraju^[20] and P. Vijayaraju and M. Marudai^[21].

We characterize polynomial growth of cosine functions in terms of the resolvent of its generator and give a necessary and sufficient condition for a cosine function with an infinitesimal generator which is polynomially bounded.

In this article we introduce the difference sequence spaces $W_0[ f, \Delta m]$, $W_1[ f ,\Delta m]$,$W_\infty[ f ,\Delta m]$ and $S[f,\Delta m]$, defined by a modulus function $f$. We obtain a relation between $W_1[ f ,\Delta m]\cap l_\infty[ f ,\Delta m]$ and $S[ f ,\Delta m]\cap l_\infty[ f ,\Delta m]$ and prove some inclusion results.

In this paper, we address the problem of exact computation of the Hausdorff measure of a class of Sierpinski carpets E − the self-similar sets generating in a unit regular pentagon on the plane. Under some conditions, we show the natural covering is the best one, and the Hausdorff measures of those sets are equal to $|E|^s$, where $ s = dim_{H}E$.

In this paper we will show that if an approximation process $\{L_n\}_{n∈N}$ is shape-preserving relative to the cone of all $k$-times differentiable functions with non-negative $k$-th derivative on [0,1], and the operators $L_n$ are assumed to be of finite rank $n$, then the order of convergence of $D^kL_n f$ to $D^k f$ cannot be better than $n^{−2}$ even for the functions $x^k$, $x^{k+1}$, $x^{k+2}$ on any subset of [0,1] with positive measure. Taking into account this fact, we will be able to find some asymptotic estimates of linear relative $n$-width of sets of differentiable functions in the space $L^p[0,1], p \in N$.

In this paper, we present a more simple and much shorter proof for the main result in the paper “An negative answer to a conjecture on the self-similar sets satisfying the open set condition”, which was published in the journal Analysis in Theory and Applications in 2009.

In the present paper, we study certain differential inequalities involving $p$-valent functions and obtain sufficient conditions for uniformly $p$-valent starlikeness and uniformly $p$-valent convexity. We also offer a correct version of some known criteria for uniformly $p$-valent starlike and uniformly $p$-valent convex functions.

We obtain weak type $(1,q)$ inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces. The proofs use some properties of maximal operators. Our results are closely related to the strong type inequalities in [13, 14, 15].

In this paper, by providing some different conditions respect to another works, we shall present two results on absolute retractivity of some sets related to some multifunctions of the form $F : X\times X \to P_{b,cl}(X)$, on complete metric spaces.

In this paper, we establish two weighted integral inequalities for commutators of fractional Hardy operators with Besov-Lipschitz functions. The main result is that this kind of commutator, denoted by $H^{\alpha}_b$, is bounded from $L^p_{x^\gamma} (R_+)$ to $L^q_{x^\delta} (R_+)$ with the bound explicitly worked out.

In this paper, we research the Müntz rational approximation of two kinds of special function classes, and give the corresponding estimates of approximation rates of these classes.

In this paper using an argument from [1] , we prove one of the probabilistic version of Hardy’s inequality.