Random fixed points for a certain class of asymptotically regular mappings

Balwant Singh Thakur; Jong Soo Jung; Daya Ram Sahu; Yeol Je Cho

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1998)

  • Volume: 18, Issue: 1-2, page 27-43
  • ISSN: 1509-9407

Abstract

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Let (Ω, σ) be a measurable space and K a nonempty bounded closed convex separable subset of a p-uniformly convex Banach space E for p > 1. We prove a random fixed point theorem for a class of mappings T:Ω×K ∪ K satisfying the condition: For each x, y ∈ K, ω ∈ Ω and integer n ≥ 1, ⃦Tⁿ(ω,x) - Tⁿ(ω,y) ⃦ ≤ aₙ(ω)· ⃦x - y ⃦ + bₙ(ω) ⃦x -Tⁿ(ω,x) ⃦ + ⃦y - Tⁿ(ω,y) ⃦ + cₙ(ω) ⃦x - Tⁿ(ω,y) ⃦ + ⃦y - Tⁿ(ω,x) ⃦, where aₙ, bₙ, cₙ: Ω → [0, ∞) are functions satisfying certain conditions and Tⁿ(ω,x) is the value at x of the n-th iterate of the mapping T(ω,·). Further we establish some random fixed point theorems for these mappings in Hilbert spaces, in L p spaces, in Hardy spaces H p and in Sobolev spaces H k , p for 1 < p < ∞ and k ≥ 0. As a consequence of our main result, we extend and randomize the corresponding deterministic ones of Górnicki [14, 15] and others.

How to cite

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Balwant Singh Thakur, et al. "Random fixed points for a certain class of asymptotically regular mappings." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 18.1-2 (1998): 27-43. <http://eudml.org/doc/275963>.

@article{BalwantSinghThakur1998,
abstract = {Let (Ω, σ) be a measurable space and K a nonempty bounded closed convex separable subset of a p-uniformly convex Banach space E for p > 1. We prove a random fixed point theorem for a class of mappings T:Ω×K ∪ K satisfying the condition: For each x, y ∈ K, ω ∈ Ω and integer n ≥ 1, ⃦Tⁿ(ω,x) - Tⁿ(ω,y) ⃦ ≤ aₙ(ω)· ⃦x - y ⃦ + bₙ(ω) ⃦x -Tⁿ(ω,x) ⃦ + ⃦y - Tⁿ(ω,y) ⃦ + cₙ(ω) ⃦x - Tⁿ(ω,y) ⃦ + ⃦y - Tⁿ(ω,x) ⃦, where aₙ, bₙ, cₙ: Ω → [0, ∞) are functions satisfying certain conditions and Tⁿ(ω,x) is the value at x of the n-th iterate of the mapping T(ω,·). Further we establish some random fixed point theorems for these mappings in Hilbert spaces, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^\{k,p\}$ for 1 < p < ∞ and k ≥ 0. As a consequence of our main result, we extend and randomize the corresponding deterministic ones of Górnicki [14, 15] and others.},
author = {Balwant Singh Thakur, Jong Soo Jung, Daya Ram Sahu, Yeol Je Cho},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {p-uniformly convex Banach space; normal structure; asymptotic center; random fixed point; generalized random Lipschitzian mapping; random fixed points; generalized random Lipschitzian mappings; -uniformly convex spaces; Hardy spaces; Sobolev spaces},
language = {eng},
number = {1-2},
pages = {27-43},
title = {Random fixed points for a certain class of asymptotically regular mappings},
url = {http://eudml.org/doc/275963},
volume = {18},
year = {1998},
}

TY - JOUR
AU - Balwant Singh Thakur
AU - Jong Soo Jung
AU - Daya Ram Sahu
AU - Yeol Je Cho
TI - Random fixed points for a certain class of asymptotically regular mappings
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1998
VL - 18
IS - 1-2
SP - 27
EP - 43
AB - Let (Ω, σ) be a measurable space and K a nonempty bounded closed convex separable subset of a p-uniformly convex Banach space E for p > 1. We prove a random fixed point theorem for a class of mappings T:Ω×K ∪ K satisfying the condition: For each x, y ∈ K, ω ∈ Ω and integer n ≥ 1, ⃦Tⁿ(ω,x) - Tⁿ(ω,y) ⃦ ≤ aₙ(ω)· ⃦x - y ⃦ + bₙ(ω) ⃦x -Tⁿ(ω,x) ⃦ + ⃦y - Tⁿ(ω,y) ⃦ + cₙ(ω) ⃦x - Tⁿ(ω,y) ⃦ + ⃦y - Tⁿ(ω,x) ⃦, where aₙ, bₙ, cₙ: Ω → [0, ∞) are functions satisfying certain conditions and Tⁿ(ω,x) is the value at x of the n-th iterate of the mapping T(ω,·). Further we establish some random fixed point theorems for these mappings in Hilbert spaces, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{k,p}$ for 1 < p < ∞ and k ≥ 0. As a consequence of our main result, we extend and randomize the corresponding deterministic ones of Górnicki [14, 15] and others.
LA - eng
KW - p-uniformly convex Banach space; normal structure; asymptotic center; random fixed point; generalized random Lipschitzian mapping; random fixed points; generalized random Lipschitzian mappings; -uniformly convex spaces; Hardy spaces; Sobolev spaces
UR - http://eudml.org/doc/275963
ER -

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