On some set-valued iteration semigroups generated by interval-valued functions

Let X be an arbitrary set. We characterize all interval-valued functions \({A:X\to 2^\mathbb{R}}\) for which a multifunction \({F:(0,\infty)\times X\to 2^X}\) of the form \({F(t,x)=A^{-}\big(A(x)+\min \{t,q-\inf A(x)\}\big)}\), where \({q=\sup A(X)}\), is an iteration semigroup. The multifunction F is the set-valued counterpart of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.     

On lower semicontinuity of some set-valued iteration semigroups

We study a lower semicontinuity of set-valued iteration semigroups which are the counterparts of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.     

On upper semicontinuity of some set-valued iteration semigroups

We study an upper semicontinuity of set-valued iteration semigroups which are the counterparts of the fundamental form of continuous iteration semigroups of single-valued functions on an interval.     

Selections of set-valued functions satisfying the general linear inclusion

Applying the classical Banach fixed point theorem we prove that a set-valued function satisfying a general linear functional inclusion admits a unique selection fulfilling the corresponding functional equation. We also adopt the method of the proof for investigating the Rassias stability of general linear equation.     

On uniformly continuous Nemytskij operators generated by set-valued functions

Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all Hölder functions ${\varphi : I \to C}Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all H?lder functions j: I ? C{\varphi : I \to C} into the set H _β (I, clb(Z)) of all H?lder set-valued functions f: I ? clb(Z){\phi : I \to clb(Z)} and is uniformly continuous, then

F(x,y)=A(x,y) \text+* B(x),       x ? I, y ? CF(x,y)=A(x,y) \stackrel{*}{\text{+}} B(x),\qquad x \in I, y \in C      14. On multivalued iteration semigroups      Magdalena Piszczek 《Aequationes Mathematicae》2011,81(1-2):97-108 We will give a necessary and sufficient condition for the family {F t :?t??? 0} of multifunctions ${F_t(x) = \sum_{i=0}^{\infty} \frac{t^i}{i!}G^i(x)}$ , where G is a continuous and additive multifunction, to be an iteration semigroup.      15. d-isomorphic semigroups of continuous functions in locally compact spaces      D. T. Thành 《Acta Mathematica Hungarica》1992,60(1-2):103-105       16. u-Isomorphic semigroups of continuous functions in locally compact spaces      Á. Császár 《Acta Mathematica Hungarica》1988,51(3-4):337-340       17. Fitting piecewise linear continuous functions      Alejandro Toriello  Juan Pablo Vielma 《European Journal of Operational Research》2012,219(1):86-95 We consider the problem of fitting a continuous piecewise linear function to a finite set of data points, modeled as a mathematical program with convex objective. We review some fitting problems that can be modeled as convex programs, and then introduce mixed-binary generalizations that allow variability in the regions defining the best-fit function’s domain. We also study the additional constraints required to impose convexity on the best-fit function.      18. Isomorphism of the multiplicative semigroups of rings of continuous functions      E. M. Vechtomov 《Siberian Mathematical Journal》1978,19(4):536-545       19. Quadratic set-valued functions      Dagmar Renate Henney 《Arkiv f?r Matematik》1962,4(4):377-378       20. Steklov operators and semigroups in weighted spaces of continuous real functions      M. Campiti  I. Rasa  C. Tacelli 《Acta Mathematica Hungarica》2008,120(1-2):103-125 We consider Steklov operators in weighted spaces of continuous functions on the whole real line and on a bounded interval. We study the connections of these operators with some second order degenerate parabolic problems establishing a general Voronovskaja type formula.

On multivalued iteration semigroups

We will give a necessary and sufficient condition for the family {F _t :?t??? 0} of multifunctions ${F_t(x) = \sum_{i=0}^{\infty} \frac{t^i}{i!}G^i(x)}$ , where G is a continuous and additive multifunction, to be an iteration semigroup.     

Fitting piecewise linear continuous functions

We consider the problem of fitting a continuous piecewise linear function to a finite set of data points, modeled as a mathematical program with convex objective. We review some fitting problems that can be modeled as convex programs, and then introduce mixed-binary generalizations that allow variability in the regions defining the best-fit function’s domain. We also study the additional constraints required to impose convexity on the best-fit function.     

Steklov operators and semigroups in weighted spaces of continuous real functions

We consider Steklov operators in weighted spaces of continuous functions on the whole real line and on a bounded interval. We study the connections of these operators with some second order degenerate parabolic problems establishing a general Voronovskaja type formula.