By Afif Ben Amar, Donal O'Regan
ISBN-10: 3319319477
ISBN-13: 9783319319476
This can be a monograph protecting topological mounted element thought for numerous sessions of unmarried and multivalued maps. The authors commence by way of featuring simple notions in in the community convex topological vector areas. unique awareness is then dedicated to susceptible 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 style fastened aspect theorems for various periods of weakly sequentially non-stop operators on normal Banach areas. The authors then continue with an exam of Sadovskii, Furi–Pera, and Krasnoselskii mounted aspect theorems and nonlinear Leray–Schauder possible choices within the framework of vulnerable topologies and related to multivalued mappings with weakly sequentially closed graph. those effects are formulated when it comes to axiomatic measures of vulnerable noncompactness.
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Extra resources for Topological Fixed Point Theory for Singlevalued and Multivalued Mappings and Applications
Example text
1 to prove that F has a fixed point in C  . 4. Let E be a Dunford–Pettis lattice space, a nonempty closed convex subset of E, U a relatively open subset of and z 2 U \ EC . 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.
X implies that Txn ! Tx for fxn g X and x 2 X. 27. Observe that every continuous operator is also demicontinuous. It is also easy to show that every weakly sequentially continuous operator is demicontinuous. In order to show that the converse implication is not true we give an example connected with the theory of the superposition operator. 8. The superposition operator Nf transforms the space L1 into itself and is continuous. Obviously Nf is also demicontinuous in this setting. 24). a; b/ be a given interval.
Y/ a multivalued mapping. x/ is closed for all x 2 X. T/ is closed in X Y. The Kakutani-fixed point theorem was the first fixed point result concerning multivalued mappings. It is a generalization of the fixed point theorem by Brouwer. 37 (Kakutani Fixed Point Theorem). Let convex subset of Rn . Let T W ! P. / satisfy 1 Basic Concepts be a nonempty compact 1. x/ is nonempty closed and convex, 2. T is upper semicontinuous. Then T has a fixed point. 38 (Fan–Glicksberg Fixed Point Theorem [95]). Let X be a locally convex topological vector space and let  X be nonempty compact and convex.