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Fixed points of Lipschitzian semigroups in Banach spaces

100%

Górnicki J.

Studia Mathematica

1997

tom 126

nr 2

101-113

We prove the following theorem: Let p > 1 and let E be a real p-uniformly convex Banach space, and C a nonempty bounded closed convex subset of E. If $T = {T_s: C → C: s ∈ G = [0,∞)}$ is a Lipschitzian semigroup such that $g = lim inf_{G ∋ α → ∞} inf_{G ∋ δ ≥ 0} 1/α ʃ^α_0 ∥T_{β+δ}∥^p dβ < 1 + c$, where c > 0 is some constant, then there exists x ∈ C such that $T_sx = x$ for all s ∈ G.

Random fixed points for a certain class of asymptotically regular mappings

88%

Thakur B., Jung J., Sahu D., Cho Y.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1998

tom 18

nr 1-2

27-43

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.

Ograniczanie wyników

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Studia Mathematica

1 Cho Y.

1 Górnicki J.

1 Jung J.

1 Sahu D.

1 Thakur B.

1 1998

1 1997

Wyniki wyszukiwania

Fixed points of Lipschitzian semigroups in Banach spaces

Random fixed points for a certain class of asymptotically regular mappings