Some Geometric Properties of Lacunary Sequence Spaces Related to Fixed Point Property

and Applied Analysis 3 the fixed point theory and that a Banach space X has property L if and only if it is reflexive and has the uniform Opial property. For a real vector space X, a function ρ : X → 0,∞ is called amodular if it satisfies the following conditions: i ρ x 0 if and only if x 0; ii ρ αx ρ x for all scalar α with |α| 1; iii ρ αx βy ≤ ρ x ρ y , for all x, y ∈ X and all α, β ≥ 0 with α β 1; the modular ρ is called convex if iv ρ αx βy ≤ αρ x βρ y , for all x, y ∈ X and all α, β ≥ 0 with α β 1. For modular ρ on X, the space Xρ { x ∈ X : ρ λx −→ 0 as λ −→ 0 } 1.6 is called the modular space. A sequence xn in Xρ is called modular convergent to x ∈ Xρ if there exists a λ > 0 such that ρ λ xn − x → 0 as n → ∞. A modular ρ is said to satisfy the Δ2-condition ρ ∈ Δ2 if for any ε > 0 there exist constants K ≥ 2 and a > 0 such that ρ 2u ≤ Kρ u ε 1.7 for all u ∈ Xρ with ρ u ≤ a. If ρ satisfies the Δ2-condition for any a > 0 with K ≥ 2 dependent on a, we say that ρ satisfies the strong Δ2-condition ρ ∈ Δs2 . By a lacunary sequence θ kr , where k0 0, we will mean an increasing sequence of nonnegative integers with kr −kr−1 → ∞ as r → ∞. The intervals determined by θ will be denote by Ir kr−1, kr . We write hr kr − kr−1 and the ratio kr/kr−1, will be denoted by qr . The space of lacunary strongly convergent sequence Nθ was defined by Freedman et al. 19 as Nθ { x xk : lim r→∞ 1 hr ∑ k∈Ir |xk − l| 0, for some l } . 1.8 It is well known that there is very closed connection between the space of lacunary strongly convergent sequence and the space of strongly Cesaro summability sequences. This connection can be found in 18–23 , because a lot of these connection, a lot of geometric property of Cesaro sequence spaces can generalize the lacunary sequence spaces. Let w be the space of all real sequences. Let p pr be a bounded sequence of the positive real numbers. In 2007, Karakaya 24 introduced the new sequence spaces l p, θ involving lacunary sequence as follows: l ( p, θ ) { x x i : ∞ ∑ r 1 ( 1 hr ∑ i∈Ir |x i | )pr < ∞ } 1.9 4 Abstract and Applied Analysis and paranorm on l p, θ is given by