Linear Operations among Summable Functions

Essential Supremum and Supremum of Summable Functions

Let D lR N , 0 < (D) < +1 and f : D ! lR is an arbitrary summable function. Then the function F() := R fx2D:f(x)g (f(x) ?) dd (2 lR) is continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative F 0 () = ?f ]. Further on, it holds ess supf = supf 2 lR : F() > 0g, where ess supf denotes the essential supremum of f. These properties can be used for computing esssup...

Second-order Linear Systems with Summable Coefficients*

Sturm and his successors! have examined in detail properties of the zeros of solutions of linear differential systems similar to (1), (3) below, and of linear combinations similar to (2). They have commonly employed the continuity and existence of derivatives of the coefficients. Under more lenient assumptions, the writer will obtain sufficient conditions that the zeros of the linear combinatio...

Banach on Linear Operations 13 Banach on Linear Operations

Théorie des Opérations Linéaires. By Stefan Banach. Warsaw, Monografje Matematyczne, Vol. I, 1932. vii+254 pp. The present book, the first of the series published by the Monografje Matematyzcne, is an enlarged and considerably modified translation of the Teorja Operacyi (in Polish) by Banach. It represents a noteworthy climax of long series of researches started by Volterra, Fredholm, Hubert, H...

Integral Representation of Linear Continuous Operators from the Space of Lebesgue-bochner Summable Functions into Any Banach Space.

SOLVING FUZZY LINEAR PROGRAMMING PROBLEMS WITH LINEAR MEMBERSHIP FUNCTIONS-REVISITED

Recently, Gasimov and Yenilmez proposed an approach for solving two kinds of fuzzy linear programming (FLP) problems. Through the approach, each FLP problem is first defuzzified into an equivalent crisp problem which is non-linear and even non-convex. Then, the crisp problem is solved by the use of the modified subgradient method. In this paper we will have another look at the earlier defuzzifi...