In the present paper, we are concerned with a class of constrained vector optimization problems, where the objective functions and active constraint functions are locally Lipschitz at the referee point. Some second-order constraint qualifications of Zangwill type, Abadie type and Mangasarian–Fromovitz type as well as a regularity condition of Abadie type are proposed in a nonsmooth setting. The connections between these proposed conditions are established. They are applied to develop second-order Karush–Kuhn–Tucker necessary optimality conditions for local (weak, Geoffrion properly) efficient solutions to the considered problem. Examples are also given to illustrate the obtained results.

We establish sufficient conditions for the existence of invariant subspaces for operators on real Banach spaces, and we investigate the behaviour of operators without such subspaces. All proofs are elementary.

We prove a Korovkin-type approximation theorem using the relative uniform convergence of a sequence of functions at a point, which is a method stronger than the classical ones. We give some examples on this new convergence method and we study also rates of convergence.

We construct a linear operator R, between two Banach lattices, such that R has a modulus which does not satisfy the Riesz–Kantorovich formula, thereby answering a long-standing question in the theory of regular operators.

In the dual $$L_{\varPhi ^*}$$ of a $$\varDelta _2$$ -Orlicz space $$L_\varPhi $$ , that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology $$\tau (L_{\varPhi ^*},L_\varPhi )$$ if and only if on each order interval $$[-\zeta ,\zeta ]=\{\xi : -\zeta \le \xi \le \zeta \}$$ ( $$\zeta \in L_{\varPhi ^*}$$ ), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlos type result: every norm bounded sequence $$(\xi _n)_n$$ in $$L_{\varPhi ^*}$$ admits a sequence of forward convex combinations $${{\bar{\xi }}}_n\in \text {conv}(\xi _n,\xi _{n+1},\ldots )$$ such that $$\sup _n|{\bar{\xi }}_n|\in L_{\varPhi ^*}$$ and $${\bar{\xi }}_n$$ converges a.s.

In this paper, we obtain some Tauberian conditions in terms of slow oscillation and slow decreasing in certain senses, under which convergence of a double sequence in Pringsheim’s sense follows from its statistical (C, 1, 1) summability.

For a sequence of positive linear approximation operators defined by means of the squared Bernstein polynomials we derive a complete asymptotic expansion. The initial three coefficients are explicitly calculated. As a special case, we obtain a Voronovskaja type formula. Finally, we introduce two Durrmeyer-type variants and calculate the initial two coefficients of their asymptotic expansions.

In a self contained presentation we study some properties of the class $$\widetilde{K}_n$$ which is a slight variant of the usual Kato class $$K_n$$. As application we show the strong unique continuation property of the eigenfunction for Laplacian operator, also we consider the unique continuation for nonnegative solutions of an elliptic second order operator.

In this paper we obtain quantitative results for the convergence of the iterates of some positive linear operators which preserve certain functions. Some examples involving q-operators are given. We show that the considered q-operators, $$0<q<1$$, provide faster convergence for the iterates than those from the classical case ($$q=1$$).

Let $n\ge 2$ be the spatial dimension. The purpose of this note is to obtain some weighted estimates for the fractional maximal operator ${\mathfrak M}{\alpha}$ of order $\alpha$, $0\le\alpha<n$, on the weighted Choquet-Lorentz space $L^{p,q}(H_{w}^{d})$, where the weight $w$ is arbitrary and the underlying measure is the weighted $d$-dimensional Hausdorff content $H^{d}_{w}$, $0<d\le n$. Concerning a dependence of two parameters $\alpha$ and $d$, we establish a general form of the Fefferman-Stein type inequalities for ${\mathfrak M}_{\alpha}$. Our results contain the works of Adams, \cite{Ad} and of Orobitg and Verdera \cite{OV} as the special cases. Our results also imply the Tang result \cite{Ta}, if we assume the weight $w$ is in the Muckenhoupt $A_{1}$-class.