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Random fixed points for a certain class of asymptotically regular mappings

Balwant Singh ThakurJong Soo JungDaya Ram SahuYeol Je Cho — 1998

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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...

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

Jong Soo JungYeol Je ChoShin Min KangByung-Soo LeeBalwant Singh Thakur — 2000

Czechoslovak Mathematical Journal

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 ...

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