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On the Riemann integral of set-valued functions

Piątek B.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

2012

z. 2

5--18

In the paper we present a generalization of the Riemann integral of set-valued function introduced by Dinghas and independently by Hukuhara to the case of nonempty bounded closed convex subsets of Banach spaces. Moreover, we compare properties of this integral to the Riemann integral of real function and consider its relation with the Hukuhara derivative.

Praca zawiera uogólnienie definicji całki Riemanna z funkcji wielowartościowej wprowadzonej przez Dinghasa i niezależnie przez Hukuharę na przypadek niepustych domkniętych, ograniczonych i wypukłych podzbiorów przestrzeni Banacha. Następnie porównujemy własności tak zdefiniowanej całki z całką Riemanna z funkcji rzeczywistej oraz rozważamy jej związek z pochodną Hukuhary.

Integral representation of set-valued sine families

Mainka-Niemczyk E.

Journal of Applied Analysis

2012

Vol. 18, nr 2

243-258

Let K be a closed convex cone with nonempty interior in a real Banach space. The main objective of this paper is to show that if {Gt : t ≥ 0} is a regular sine family of continuous additive set-valued functions associated with {Ft : t ≥ 0}, then Gt are of the form Gt(x) = ∫ t 0 Fu(x)du for x ∈ K,t ≥ 0.

Certain classes of multivalent functions with negative coefficients defined by using a differential operator

Aouf M. K.

Journal of Mathematics and Applications

2008

Vol. 30

5-21

In this paper, we investigate the various important prop-erties and characteristics of the subclasses Sn (p, q, α, β) and Cn (p, q, α, β) of multivalent functions with negative coefficients defined by using a differential operator. We also derive many results for the modified Hadamard products of functions belongingto the classes Sn(p, q, α, β) and Cn(p, q, α, β). Finally several applications involving an integral operator and certain fractional calculus operators are also considered.

Some inequalities involving differential operators and their applications to certain multivalent functions

Dziok J., Huseyin I., Piejko K.

Journal of Mathematics and Applications

2007

Vol. 29

109-120

The main object of the present paper is to investigate several results of certain differential operators which were recently introduced and (or) studied in a series of papers by Chen et et al. [1-3], Irmak et al. [8, 10, 11], Dziok et al. [5, 6] and Liu et al. [14]. In addition, some applications of our results involving certain differential inequalities of multivalently analytic and (or) multivalently raeromorphic functions are given. Our certain results also include some recent results in [5, 6, 9, 11, 12].