By Afif Ben Amar, Donal O'Regan

It is a monograph masking topological mounted aspect idea for numerous sessions of unmarried and multivalued maps. The authors start by way of offering simple notions in in the community convex topological vector areas. exact consciousness is then dedicated to vulnerable compactness, particularly 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 versions of Schauder, Krasnoselskii, Sadovskii, and Leray–Schauder kind mounted aspect theorems for various periods 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 concerning 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**

Suppose G W E ! E is a positive bounded linear weakly compact operator and T W U ! U// . 1 /z C GTx: ) and Proof. Consider GT W U ! A2 / does not hold. A1 / occurs). 1 /z C GTx; for some ł 2 Œ0; 1 : Now D ¤ ; since z 2 D. Because E is a normed lattice, EC is closed, and so, U \ EC is a closed subset of . 2, we prove that D is compact. 11). Since D\. 1, there is a continuous function ' W ! x/ D 0 for x 2 n U. Since is convex, c W ! x/GTx; if x 2 U; : z; if x 2 n U: c is positive. 3, we prove that there exists a positive element x0 2 with c 0 D x0 .

1, there is a continuous function ' W ! x/ D 0 for x 2 n U. Since is convex, z 2 , we can c W ! H1 /. Indeed, we have @ U D We first check that GT c is continuous. H1 /. H1 /. U/ is Since Œ0; 1 is compact, it follows that GT weakly relatively compact. U/ [ fzg/ is convex and weakly compact. D / D . 1 are satisfied c Therefore, there exists x0 2 c 0 D x0 . From the for the operator GT. with GTx c x0 must be an element of U. x0 / D 1. Accordingly, GTx0 D x0 and the proof is complete. 1. Let E be a Dunford–Pettis space, a nonempty closed convex subset of E, U a relatively open subset of and z 2 U.

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]. 39. t; x/ satisfies Carathéodory conditions if it is measurable in t for each x 2 R and is continuous in x for almost all t 2 I.