A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f\left(Ū\right)\subseteq \overline{f\left(U\right)}$ for any open connected set U ⊆ X. We prove that if X is a locally connected hereditarily Baire space and Y is a T₁-space then an $F_{\sigma }$-measurable mapping f: X → Y is weakly Gibson if and only if for any connected set C ⊆ X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson $F_{\sigma }$-measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph.

A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that   $li{m}_{x\mapsto x_0}\left(\right|f\left(x\right)-f\left(x_0\right)\left|\right)/\left(\right|\varphi \left(x\right)-\varphi \left(x_0\right)\left|\right)=0$ for every $x_0$. G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying   $li{m}_{t\mapsto t_0}\left(\right|t-t_0\left|\right)/\left(\right|\varphi \left(t\right)-\varphi \left(t_0\right)\left|\right)=0$. We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.

Classes of functions continuous in various senses, in particular $\theta$-continuous, $\alpha$-continuous, feeblz continuous a.o., and relations between the classes, are studied.

In Orlicz spaces theory some strengthened version of the Jensen inequality is often used to obtain nice geometrical properties of the Orlicz space generated by the Orlicz function satisfying this inequality. Continuous functions satisfying the classical Jensen inequality are just convex which means that such functions may be described geometrically in the following way: a segment joining every pair of points of the graph lies above the graph of such a function. In the current paper we try to obtain...

2000 Mathematics Subject Classification: 26A33 (main), 44A40, 44A35, 33E30, 45J05, 45D05In the paper, the machinery of the Mellin integral transform is applied to deduce and prove some operational relations for a general operator of the Erdélyi-Kober type. This integro-differential operator is a composition of a number of left-hand sided and right-hand sided Erdélyi-Kober derivatives and integrals. It is referred to in the paper as a mixed operator of the Erdélyi-Kober type. For special values of...

The abstract Perron-Stieltjes integral in the Kurzweil-Henstock sense given via integral sums is used for defining convolutions of Banach space valued functions. Basic facts concerning integration are preseted, the properties of Stieltjes convolutions are studied and applied to obtain resolvents for renewal type Stieltjes convolution equations.

In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal...

Let $A_{\alpha }f\left(x\right)={\left|B\right(0,\left|x\right|\left)\right|}^{-\alpha /n}{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{\alpha }$ is bounded from $L^p$ to $L^{p_{\alpha }}$ with $p_{\alpha }=np/(\alpha p-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_{\alpha ,Y}$, which is strictly larger than X, and a ’target’ space $T_Y$, which is strictly smaller than Y, under the assumption that $A_{\alpha }$ is bounded from X into Y and the Hardy-Littlewood maximal operator...

We prove some quantitatively sharp estimates concerning the Δ₂ and ∇₂ conditions for functions which generalize known ones. The sharp forms arise in the connection between Orlicz space theory and the theory of elliptic partial differential equations.