Volume 49, Issue 1

Let f ∈ C r ([− 1,1 ]), r ≥ 0 and let L * be a linear right fractional differential operator such that L * ( f ) ≥ 0 throughout [−1,0]. We can find a sequence of polynomials Q n of degree ≤ n such that L * ( Q n ) ≥ 0 over [−1,0], furthermore f is approximated right fractionally and simultaneously by Q n on [−1,1]. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for f (r) .

Let X be a real normed space, V be a subset of X and α : [0, ∞ ) → [0, ∞ ] be a nondecreasing function. We say that a function f : V → [− ∞ , ∞ ] is conditionally α-convex if for each convex combination ∑i=0ntivi$\sum\nolimits_{i = 0}^n {t_i v_i }$ of elements from V such that ∑i=0ntivi∈V$\sum\nolimits_{i = 0}^n {t_i v_i \in V}$ , the following inequality holds true f(∑i=0ntivi)≤∑i=0ntif(vi)+α(max⁡i∈{0,…,n}ti‖vi−∑i=0ntivi‖).$$f\left( {\sum\limits_{i = 0}^n {t_i v_i } } \right) \le \sum\limits_{i = 0}^n {t_i f(v_i )} + \alpha (\mathop {\max }\limits_{i \in \{ 0, \ldots ,n\} } \left. {t_i } \right\|v_i - \sum\limits_{i = 0}^n {t_i v_i } \left\| ) \right..$$ We present some necessary and some suffcient conditions for f to be conditionally α-convex.

In this paper, we first investigate the Hyers–Ulam stability of the generalized Cauchy–Jensen functional equation of p -variable f(∑i=1paixi)=∑i=1paif(xi)$f\left(\sum\nolimits_{i = 1}^p {a_i x_i } \right) = \sum\nolimits_{i = 1}^p {a_i f(x_i )}$ in an intuitionistic fuzzy Banach space. Then, we conclude the results for Cauchy–Jensen functional equation of p -variable f(x1+⋯+xpp)=1p(f(x1)+⋯+f(xp))$f\left( {{\textstyle{{x_1 + \cdots + x_p } \over p}}} \right) = {1 \over p}(f(x_1 ) + \cdots + f(x_p ))$ . Next, we discuss the intuitionistic fuzzy continuity of Cauchy–Jensen mappings.

The main result of this note is that if I is an ideal generated by a regular double summability matrix summability method T that is the product of two nonnegative regular matrix methods for single sequences, then I -statistical convergence and convergence in I -density are equivalent. In particular, the method T generates a density µ T with the additive property (AP) and hence, the additive property for null sets (APO). The densities used to generate statistical convergence, lacunary statistical convergence, and general de la Vallée-Poussin statistical convergence are generated by these types of double summability methods. If a matrix T generates a density with the additive property then T -statistical convergence, convergence in T -density and strong T -summabilty are equivalent for bounded sequences. An example is given to show that not every regular double summability matrix generates a density with additve property for null sets.

In this paper, we obtain strong localization results and local direct results in the approximation of continuous functions by the non-truncated max-product sampling operators based on Fejér and sinc (Wittaker)-type kernels. These operators present potential applications in signal theory.

This note investigates weaker conditions than a Poincaré inequality in analysis on metric measure spaces. We discuss two resistance conditions which are stated in terms of capacities. We show that these conditions can be characterized by versions of Sobolev–Poincaré inequalities. As a consequence, we obtain so-called Lip-lip condition related to pointwise Lipschitz constants. Moreover, we show that the pointwise Hardy inequalities and uniform fatness conditions are equivalent under an appropriate resistance condition.

A general common fixed point theorem for two pairs of weakly subsequentially continuous mappings (recently introduced) satisfying a significant estimated implicit function is proved. An extension of this result is thereby obtained. Our results assert the existence and uniqueness of common fixed points in several cases.

In this paper, we introduce a new condition namely: condition (W.C.C.) and utilize the same to prove a Suzuki type unique common fixed point theorem for two hybrid pairs of mappings in partial metric spaces employing the partial Hausdorff metric which generalizes several known results of the existing literature proved in metric and partial metric spaces.

Let A (𝔻) denote the disk algebra. Every endomorphism of A (𝔻) is induced by some ϕ ∈ A (𝔻) with ‖ ϕ ‖ ≤ 1. In this paper, it is shown that if ϕ is not an automorphism of 𝔻 and ϕ has a fixed point in the open unit disk then the endomorphism induced by ϕ is decomposable if and only if the fixed set of ϕ is singleton. Further, we determine the local spectra of the endomorphism induced by ϕ in the cases when the fixed set of ϕ either includes unit circle or is a singleton.

The purpose of this paper is to study convergence of a newly defined modified S -iteration process to a common fixed point of two asymptotically quasi-nonexpansive type mappings in the setting of CAT(0) space. We give a suffcient condition for convergence to a common fixed point and establish some strong convergence theorems for the said iteration process and mappings under suitable conditions. Our results extend and improve many known results from the existing literature.