Random fixed point theorems for a certain class of mappings in Banach spaces

Jong Soo Jung; Yeol Je Cho; Shin Min Kang; Byung-Soo Lee; Balwant Singh Thakur

Czechoslovak Mathematical Journal (2000)

  • Volume: 50, Issue: 2, page 379-396
  • ISSN: 0011-4642

Abstract

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Let ( Ω , Σ ) be a measurable space and C a nonempty bounded closed convex separable subset of p -uniformly convex Banach space E for some p > 1 . We prove random fixed point theorems for a class of mappings T Ω × C → C satisfying: for each x , y ∈ C , ω ∈ Ω and integer n ≥ 1 , ∥ T n ( ω , x ) - T n ( ω , y ) ∥ ≤ a ( ω ) · ∥ x - y ∥ + b ( ω ) { ∥ x - T n ( ω , x ) ∥ + ∥ y - T n ( ω , y ) ∥ } + c ( ω ) { ∥ x - T n ( ω , y ) ∥ + ∥ y - T n ( ω , x ) ∥ } , where a , b , c Ω → [ 0 , ∞ ) are functions satisfying certain conditions and T n ( ω , x ) is the value at x of the n -th iterate of the mapping T ( ω , · ) . Further we establish for these mappings some random fixed point theorems in a Hilbert space, 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 also extend the results of Xu [43] and randomize the corresponding deterministic ones of Casini and Maluta [5], Goebel and Kirk [13], Tan and Xu [37], and Xu [39, 41].

How to cite

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Jung, Jong Soo, et al. "Random fixed point theorems for a certain class of mappings in Banach spaces." Czechoslovak Mathematical Journal 50.2 (2000): 379-396. <http://eudml.org/doc/30569>.

@article{Jung2000,
abstract = {Let $(\Omega ,\Sigma )$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$-uniformly convex Banach space $E$ for some $p > 1$. We prove random fixed point theorems for a class of mappings $T\: \Omega \times C \rightarrow C$ satisfying: for each $x, y \in C$, $\omega \in \Omega $ and integer $n \ge 1$, \[\Vert T^n(\omega , x) - T^n(\omega , y) \Vert \le a(\omega )\cdot \Vert x - y \Vert + b(\omega )\lbrace \Vert x - T^n(\omega ,x) \Vert + \Vert y - T^n(\omega ,y) \Vert \rbrace + c(\omega )\lbrace \Vert x - T^n(\omega ,y) \Vert + \Vert y - T^n(\omega ,x) \Vert \rbrace , \] where $a,b,c\: \Omega \rightarrow [0, \infty )$ are functions satisfying certain conditions and $T^n(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,\cdot )$. Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^\{k,p\} $ for $1 < p < \infty $ and $k \ge 0$. As a consequence of our main result, we also extend the results of Xu [43] and randomize the corresponding deterministic ones of Casini and Maluta [5], Goebel and Kirk [13], Tan and Xu [37], and Xu [39, 41].},
author = {Jung, Jong Soo, Cho, Yeol Je, Kang, Shin Min, Lee, Byung-Soo, Thakur, Balwant Singh},
journal = {Czechoslovak Mathematical Journal},
keywords = {$p$-uniformly convex Banach space; normal structure; asymptotic center; random fixed points; generalized random uniformly Lipschitzian mapping; random fixed points; generalized random uniformly Lipschitzian mappings},
language = {eng},
number = {2},
pages = {379-396},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Random fixed point theorems for a certain class of mappings in Banach spaces},
url = {http://eudml.org/doc/30569},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Jung, Jong Soo
AU - Cho, Yeol Je
AU - Kang, Shin Min
AU - Lee, Byung-Soo
AU - Thakur, Balwant Singh
TI - Random fixed point theorems for a certain class of mappings in Banach spaces
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 2
SP - 379
EP - 396
AB - Let $(\Omega ,\Sigma )$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$-uniformly convex Banach space $E$ for some $p > 1$. We prove random fixed point theorems for a class of mappings $T\: \Omega \times C \rightarrow C$ satisfying: for each $x, y \in C$, $\omega \in \Omega $ and integer $n \ge 1$, \[\Vert T^n(\omega , x) - T^n(\omega , y) \Vert \le a(\omega )\cdot \Vert x - y \Vert + b(\omega )\lbrace \Vert x - T^n(\omega ,x) \Vert + \Vert y - T^n(\omega ,y) \Vert \rbrace + c(\omega )\lbrace \Vert x - T^n(\omega ,y) \Vert + \Vert y - T^n(\omega ,x) \Vert \rbrace , \] where $a,b,c\: \Omega \rightarrow [0, \infty )$ are functions satisfying certain conditions and $T^n(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,\cdot )$. Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{k,p} $ for $1 < p < \infty $ and $k \ge 0$. As a consequence of our main result, we also extend the results of Xu [43] and randomize the corresponding deterministic ones of Casini and Maluta [5], Goebel and Kirk [13], Tan and Xu [37], and Xu [39, 41].
LA - eng
KW - $p$-uniformly convex Banach space; normal structure; asymptotic center; random fixed points; generalized random uniformly Lipschitzian mapping; random fixed points; generalized random uniformly Lipschitzian mappings
UR - http://eudml.org/doc/30569
ER -

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