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INDEX

(!O, A) uniformly locally contractive, 194

(p - f), 109 H-KKM, 157 H-compact, 157 H-convex, 156 H-space, 156 T-invariant, 195 T-invariant point, 195 T-orbitally lower semicontinuous,

18 1}-chainable, 194 a-compact, 107 aWo ,112 k-set contraction, 28 p-contractive, 195 p-nonexpansive, 195 I-set contraction, 29

T-orbitally complete, 18

acyclic, 107 acyclic map, 107 admissible, 97 almost affine, 93, 109 approximatively compact, 93 approximatively compact sets, 96 approximatively compact with respect

to p, 195 approximatively weakly compact,

96 asymptotically nonexpansive, 63 asymptotically regular, 24

Banach Contraction Principle, 12 Banach contraction theorem, 12 base of neighbourhoods, 5

217

best approximation, 73 best approximation operator, 76,

78 best simultaneous approximation,

198 bounded compact, 191 bounded linear functional, 5 boundedly (weakly) compact, 191 boundedly compact, 197 Brouwer, 11 Brouwer's Fixed Point Theorem,

11

canonical isomorphism, 68 canonical mapping, 5 Caristi, 16 Chebyshev map, 76 Chebyshev set, 76 dosed, 35 coincidence, 187 coincidence theorem, 174 compact, 2, 35 compact map, 12 compactly dosed, 157 complementarity problem, 200 complete, 1 completely continuous, 12 condensing, 103 cone, 88, 172 contractible, 156 contractible subsets, 156 contraction, 13 contraction map, 13 contractive, 13 converges weakly, 5 convex, 4, 43

218 INDEX

convex hull, 4 convex metric space, 64 convex space, 107 convex structure, 64 cover, 2

demiclosed, 6, 26, 39 demicompact, 33 demicontinuous, 6 densifying, 29 diameter, 2 diametral point, 9 dual, 172 duality mapping, 7

equilibrium problem, 203 expansive map, 21

fibres of F, 34 filter, 164 finite intersection property, 2 finite subcover, 2 finitely closed, 124 fixed point, 10 Fnkhet, 62 Fnkhet differentiable norm, 62 Fredholm equation, 46 Fredholm integral equation, 46

gauge function, 7 generalized contraction, 20, 22 generalized contraction (with respect

to Q, 22

Hammerstein integral equation, 47 Hausdorff metric, 29 Hausdorff topological space, 8 hemicontinuous, 6, 7, 132

I-scheme, 57 inner product, 3 inner product space, 3 inward, 37 inward map, 37

inward set, 37 Ishikawa iteration scheme, 57 Ishikawa scheme, 57 isometric embedding, 5 iteration process, 55

jointly continuous, 196

Kakutani factorizable multifunction, 37

Kakutani map, 107 Kakutani multifunction, 37 KKM principle, 122 KKM-map, 121 KKM-map, 122

LANE map, 83 Lipschitz class, 13, 67 Lipschitz mapping, 38 Lipschitzian mapping, 62 locally contractive, 194 locally convex space, 5, 8 locally uniformly convex, 7 lower (upper) semicontinuous, 16 lower semicontinuous, 16, 35 LUC, 7

Mann Iterative Process, 56 Mann iterative process, 56 maximal element, 150, 165 measure of noncompactness, 28 metric, 1 metric projection, 76 metric space, 1 minimizing sequence, 93 Minkowski functional, 9 monotone, 6 moon, 198 multifunction, 34

acyclic, 97 admissible, 97 closed, 35 compact, 35

Fixed Point Theory and Best Approximation: The KKM-Map Principle 219

condensing, 103 demiclosed, 39 inward, 42 Lipschitz mapping, 38 lower semicontinuous, 35 nonexpansive, 39 quasi-complete, 103 upper hemicontinuous, 187 upper semicontinuous, 35 weakly inward, 42

nonempty, 187 nonexpansive, 15, 22, 39 nonlinear ergodic theorem, 62 nonlinear integral equation, 47 norm, 3 norm topology, 5 normal structure, 10 normed linear space, 3 normed vector space, 3

open cover, 2 Opial's condition, 39 orbit, 18 orbitally complete, 18 orbitally lower semicontinuous, 18 Oshman property, 193 outward map, 37 outward set, 37

paracompact, 159 partition of unity, 159 polar, 172 polytopes, 107 precompact, 2 proper, 17, 97 proper map, 17 property (S), 113 proximinal, 76 proximinal with respect to p, 195 proximity map, 76

quasi-complete, 103, 195

quasi-concave, 126 quasi-convex, 126 quasi-nonexpansive, 27

radial retraction, 31, 76 reflexive, 5, 6

scalars, 3 Schauder, 12 Schauder Fixed Point Theorem, 12 Schauder Fixed Point Theorem (secon(

form), 12 selection, 159 semicontractive, 82 semiconvex, 42 semiconvexity, 43 seminorm, 8 separating, 9 sequence

Cauchy, 1 sequence of iterates, 15 sequentially complete, 195 sequentially strongly continuous,

set 134

absolutely convex, 8 absorbing, 8 balanced, 8 circled, 8 inward, 37

set-valued map, 34, 76 smooth, 198 space

C[a, b], 1 Banach, 2, 3 dual, 5 Hausdorff topological, 8 Hilbert, 3, 4 inner product, 4 linear, 2 normed, 3 normed linear, 3 normed vector, 3

220

topological, 8 vector, 3

star center, 4 starshaped, 4 strictly contractive, 14 strictly convex, 4 strictly monotone, 6 strong topology, 5 strongly continuous, 6, 82 strongly monotone, 67

INDEX

sufficiently many linear functionals, 183

sun, 197 supremum norm, 4

topological vector spaces, 7 totally bounded, 2, 103 two point boundary value problem,

46

ueED, 7 uniformly convex, 4

in every direction, 7 unit ball, 6 unit sphere, 6 upper semicontinuous, 16, 35 Urysohn integral equation, 47

variational inequality, 199 vectors, 3 Volterra integral equation, 45

weak closure, 5 weak topology, 5, 9 weak* -topology, 9 weakly H-convex, 156 weakly asymptotically regular, 62 weakly closed, 5 weakly compact, 5 weakly continuous, 6, 183 weakly inward, 37, 43 weakly outward, 37

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