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Proximal normal structure and relatively nonexpansive mappings

A. Anthony Eldred, W. A. Kirk, P. Veeramani (2005)

Studia Mathematica

The notion of proximal normal structure is introduced and used to study mappings that are "relatively nonexpansive" in the sense that they are defined on the union of two subsets A and B of a Banach space X and satisfy ∥ Tx-Ty∥ ≤ ∥ x-y∥ for all x ∈ A, y ∈ B. It is shown that if A and B are weakly compact and convex, and if the pair (A,B) has proximal normal structure, then a relatively nonexpansive mapping T: A ∪ B → A ∪ B satisfying (i) T(A) ⊆ B and T(B) ⊆ A, has a proximal point in the sense that...

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 (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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