Download Topological Fixed Point Theory for Singlevalued and by Afif Ben Amar, Donal O'Regan PDF

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. 

The authors proceed to offer a few mounted element theorems in a nonempty closed convex of any Banach algebras or Banach algebras pleasant a sequential (P) for the sum and the manufactured from nonlinear weakly sequentially non-stop operators, and illustrate the idea via contemplating sensible essential and partial differential equations. The life of mounted issues, nonlinear Leray–Schauder choices for various sessions of nonlinear (ws)-compact operators (weakly condensing, 1-set weakly contractive, strictly quasi-bounded) outlined on an unbounded closed convex subset of a Banach house also are mentioned. The authors additionally study the life of nonlinear eigenvalues and eigenvectors, in addition to the surjectivity of quasibounded operators. eventually, a few approximate mounted aspect theorems for multivalued mappings outlined on Banach areas. vulnerable and robust topologies play a task the following and either bounded and unbounded areas are thought of. The authors explicate a style built to point the right way to use approximate mounted aspect theorems to end up the lifestyles of approximate Nash equilibria for non-cooperative games.
Fixed element concept is a strong and fruitful instrument in smooth arithmetic and will be regarded as a center topic in nonlinear research. within the final 50 years, mounted element concept has been a flourishing sector of study. As such, the monograph starts off with an summary of those advancements sooner than gravitating in the direction of themes chosen to mirror the actual pursuits of the authors.

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Additional info for Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications

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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.

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