The concept of a t-balancedCayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Širáň, M. Škoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Širáň, Skewmorphisms of regular Cayley maps, Discrete Math. 244 (1-3) (2002) 167-179], and apply our results to the specific case of regular Cayley maps of abelian groups.

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Introduction. In his recent papers [23], [24], [25] Petryshyn has introduced and studied a class of projectionally compact (F-compact) nonlinear mappings of a real Banach space with property (n)K into itself. The main result of this study was an elementary and essentially a constructive proof of two fixed point theorems for bounded [23] [28]. One of our objectives is to show that many properties and results which are true for completely continuous operators carry over to generalized F-compact operators.We now outline briefly the main results of this paper. In §1 we introduce and discuss some basic definitions to be used in this paper. In §2 we discuss further properties of F-compact and generalized F-compact operators. The main result of this section is Theorem 2.4 which asserts that under certain conditions "the Fréchet derivative A'(x) is F-compact if and only if A(x) is F-compact."In §3 we discuss certain properties of linear F-compact operators. The main result of this section is the characterization Theorem 3.3 which asserts that "a symmetric linear mapping A of a Hubert space into itself is F-compact if and only if A = 5+F where SSO and Fis completely continuous."

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