By Afif Ben Amar, Donal O'Regan

It is a monograph overlaying topological mounted aspect conception for a number of sessions of unmarried and multivalued maps. The authors commence by way of providing simple notions in in the neighborhood convex topological vector areas. precise cognizance is then dedicated to vulnerable compactness, specifically to the theorems of Eberlein–Šmulian, Grothendick and Dunford–Pettis. Leray–Schauder possible choices and eigenvalue difficulties for decomposable single-valued nonlinear weakly compact operators in Dunford–Pettis areas are thought of, as well as a few variations of Schauder, Krasnoselskii, Sadovskii, and Leray–Schauder variety mounted aspect theorems for various sessions of weakly sequentially non-stop operators on common Banach areas. The authors then continue with an exam of Sadovskii, Furi–Pera, and Krasnoselskii fastened aspect theorems and nonlinear Leray–Schauder possible choices within the framework of vulnerable topologies and regarding multivalued mappings with weakly sequentially closed graph. those effects are formulated when it comes to axiomatic measures of vulnerable noncompactness.

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**Sample text**

Before proving the second inequality we introduce some auxiliary notations. x; b/ for any T > 0. x;b/ C xPŒT;1/ : In what follows assume that " > 0 and T > 0 are fixed and let b Next, let supŒkxk x2X . X/. Thus the proof is complete. t/j dt. t/jp dt < 1. 38. t; x/ D f W I R ! R is a given function. For an arbitrary function x W I ! t//. The operator Nf defined in such a way is said to be the superposition operator generated by the function f . The first contribution to the theory of the superposition operator dates back to Carathéodory [56].

A Hausdorff topological space X is said to be angelic space if for every relatively countably compact subset A of X the following two claims hold. 1. A is relatively compact. 2. If b 2 A, then there is a sequence in A that converges to b. Obviously, if K is a compact topological space, K is a Fréchet–Urysohn space if and only if it is angelic. It can be said that a Hausdorff topological space X is angelic if and only if X is a g-space for which any compact subspace is a Fréchet–Urysohn space. 7.

1 Leray–Schauder Alternatives 47 Hence K D K . Since K is weakly closed, it suffices to show that K is relatively weakly compact. K/; which is a contradiction. K/ D 0 and so K is relatively weakly compact. Now, F is a weakly sequentially continuous map of K into itself. 31, F has a fixed point in K . 3. Let be a closed convex subset of a Banach space E. In addition, let U be a weakly open subset of with Â 2 U, and F W U w ! U w / is bounded. 0; 1/ with Proof. A1 / occurs). U w / is bounded. D//.