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Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets

Józef Banaś, Afif Ben Amar (2013)

Commentationes Mathematicae Universitatis Carolinae

In this paper we prove a collection of new fixed point theorems for operators of the form T + S on an unbounded closed convex subset of a Hausdorff topological vector space ( E , Γ ) . We also introduce the concept of demi- τ -compact operator and τ -semi-closed operator at the origin. Moreover, a series of new fixed point theorems of Krasnosel’skii type is proved for the sum T + S of two operators, where T is τ -sequentially continuous and τ -compact while S is τ -sequentially continuous (and Φ τ -condensing, Φ τ -nonexpansive...

Measures of non-compactness in Orlicz modular spaces.

A. G. Aksoy, J.-B. Baillon (1993)

Collectanea Mathematica

In this paper we show that the ball-measure of non-compactness of a norm bounded subset of an Orlicz modular space L-Psi is equal to the limit of its n-widths. We also obtain several inequalities between the measures of non-compactness and the limit of the n-widths for modular bounded subsets of L-Psi which do not have Delta-2-condition. Minimum conditions on Psi to have such results are specified and an example of such a function Psi is provided.

Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed F σ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed G δ set G ⊆ ℝ such that every weak contraction...

Minimax theorems without changeless proportion

Liang-Ju Chu, Chi-Nan Tsai (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: i n f y ∈ Y s u p x ∈ X f ( x , y ) ≤ s u p x ∈ X i n f y ∈ Y g ( x , y ) . We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: s u p x ∈ X f ( x , y ) ≤ s u p x ∈ X g ( x , y ) , ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some properties...

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