Local monotonicity coefficients in Orlicz sequence spaces equipped with the p-Amemiya norm

In this paper, the monotonicity is investigated with respect to Orlicz sequence space lΦ,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l_{\varPhi , p}$\end{document} equipped with the p-Amemiya norm, and the necessary and sufficient condition is obtained to guarantee the uniform monotonicity, locally uniform monotonicity, and strict monotonicity for lΦ,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l_{\varPhi , p}$\end{document}. This completes the results of the paper (Cui et al. in J. Math. Anal. Appl. 432:1095–1105, 2015) which were obtained for the non-atomic measure space. Local upper and lower coefficients of monotonicity at any point of the unit sphere are calculated, lΦ,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l_{\varPhi , p}$\end{document} is calculated.


Introduction
The role of monotonicity in Banach lattices is similar to the role of rotundity in Banach spaces. It is well known that monotonicity properties of Banach lattices has various applications in the fields of ergodic theory (see [2]) and approximation theory; in particular, they are very useful for estimating the errors of the approximation. Also, they have been introduced and studied in the context of their geometric structure by Birkhoff in [3]. Moreover, Betiuk-Pilarska and Prus showed recently that if X is a weakly orthogonal Banach lattice with ε m (X) < 1, then X has the weak normal structure. Consequently, X has the weak fixed point property (see [4]).
In this paper, monotonicity properties and the coefficient of monotonicity for Orlicz sequence spaces equipped with the p-Amemiya norm are investigated.
Let X be a Banach lattice with a lattice norm · and X + be the positive cone of X. We denote by B(X) the unit ball of X, by S(X) the unit sphere of X, and S(X + ) = S(X) ∩ X + . We begin with auxiliary definitions and results that are used in the sequel.
A Banach lattice X is strictly monotonic (STM) if, for all x, y ∈ X + , the conditions x ≥ y, y = 0, and x = y imply x = y. X is uniformly monotone (UM) if, for any sequence {x n }, {y n } in X + , y n ≥ x n , the equalities lim n→∞ x n = lim n→∞ y n imply lim n→∞ y nx n = 0.
Obviously, each UM Banach lattice is both (upper and lower) locally uniformly monotone, and both these properties imply strict monotonicity. In an UM Banach lattice, the norm is order continuous and monotonically complete. For example, the lattice L p (1 ≤ p < ∞) is an UM, but the lattice L ∞ is not even a STM.
For a given Orlicz function Φ, we define a convex functional I Φ on l 0 by We define supp(x) = {i ∈ N : |x(i)| = 0} and the Orlicz sequence space l Φ generated by an Orlicz function Φ by the formula The Orlicz space l Φ is usually equipped with the Luxemburg norm or with an equivalent one called the Orlicz norm. For any 1 ≤ p ≤ ∞ and u ≥ 0, define , u} for p = ∞ and next define s Φ,p (x) = s p • I Φ (x) for all 1 ≤ p ≤ ∞ and all x ∈ l 0 . Note that the functions s p and s Φ,p are convex. Moreover, the function s p is increasing on R + for 1 ≤ p < ∞, but the function s ∞ is only increasing on the interval [1, ∞).
For any x ∈ l 0 , define the p-Amemiya norm by the formula In this paper, the Orlicz sequence space equipped with the p-Amemiya norm will be denoted by l Φ,p . It is known that [34]). Orlicz sequence spaces l Φ,p are Banach lattices.
For a given Orlicz function Φ, define For every Orlicz function Φ, we define its complementary function Ψ : R → [0, ∞] by the formula Ψ (v) = sup{u|v| -Φ(u) : u ≥ 0}. The complementary function Ψ is also an Orlicz function. Let p + be the right-hand side derivative of Φ on [0, b Φ ) and set p where k * p (x) < ∞, and at k * * p (x) whenever this number is finite.

Monotonicity in l Φ,p spaces
We begin this section with some useful lemmas.

Lemma 2.4
Let Φ be an Orlicz function. For any subset A ⊂ N and any p with 1 ≤ p < ∞, the following conditions are equivalent: For any k ≥ 1, we get I Next, let us discuss the strict monotonicity, upper and lower local uniform monotonicities, and uniform monotonicity of l Φ,p . Theorem 2.1 If 1 ≤ p < ∞, the space l Φ,p is strictly monotone if and only if one of the following conditions holds: Proof Here, we only discuss the case where 1 ≤ p < ∞, see [22] for detailed discussion whenever p = ∞. Necessity Moreover, for any k ∈ (0, 1], we get Therefore, x n Φ,p = inf k>0 In the same way we obtain that x Φ,p = x n + x Φ,p = 1. This shows that l Φ,p is not STM. Therefore, x Φ,p = y Φ,p , which implies that l φ,p is not STM. Sufficiency. If a Φ = 0, for any x, y ∈ (l Φ,p ) + satisfying x Φ,p = 1 and y = 0, the following cases are considered, 2. If K p (x) = ∅, then K p (x + y) = ∅, and according to Lemma 2.3, When K p (x + y) = ∅, taking k ∈ K p (x + y), the proof can be proceeded in the same way as in case 1. In conclusion, l Φ,p is strictly monotone.
In the following we consider (2) was established. For any x ∈ S(l Φ,p ) and any j ∈ supp(x), we have xχ j Φ,p > 0. According to Lemma 2.4, we know that K p (xχ N\{j} ) = ∅. If K p (x) = ∅, then If K p (x) = ∅, taking h ∈ K p (x), we get According to Lemma 2.1, l Φ,p is strictly monotone.

Lemma 2.5
If x ∈ S(l Φ,p ), x ≥ 0, and K p (x) = ∅, then x is a point of upper local uniform monotonicity as well as a point of lower local uniform monotonicity.
Proof By Lemma 2.3, we have This shows that x is a point of lower local uniform monotonicity.
Assume that x is not a point of upper local uniform monotonicity. Then there exists a sequence {x n } in (l Φ,p ) + such that x n Φ,p ≥ ε > 0 and lim n→∞ x + x n Φ,p = 1. Next, we consider some cases under this assumption.
1. There exists an infinite number of n such that K p (x n + x) = ∅. Due to Lemma 2.3, we have This inequality holds for infinite n ∈ N, which is contradictory obviously. 2. There exists an infinite number of n ∈ N such that K p (x n + x) = ∅. In this case, applying the double extract subsequence theorem, we may assume that K p (x n + x) = ∅ for any n ∈ N. Let k n ∈ K p (x n + x), n = 1, 2, . . . . Here, we consider two subcases as follows.
(1) If lim n→∞ k n = k 0 < ∞, then Since K p (x) = ∅, lim n→∞ x n + x Φ,p = 1 and by the Fatou lemma, we have which is a contradiction.
This is a contradiction, which finishes the proof.

Theorem 2.2 For the Orlicz sequence space, the following conditions are equivalent:
(1) l Φ,p is uniformly monotone.
for all x ∈ l Φ,p and all k > 0, so we have k * x = ∞ and K p (x) = ∅. For any ε > 0 and any supp(x) ⊂ A such that xχ A Φ,p ≥ ε, it is easy to see that K p (xχ A ) = ∅, K p (xχ N\A ) = ∅. Therefore, By Lemma 2.2, l Φ,p is uniformly monotone.

Corollary 2.1 l Φ,p is order continuous if and only if
|u| for u = 0 and Φ(0) = 0. Then the Orlicz space l Φ,p is STM for p = 1 but it is not STM for 1 < p ≤ ∞.
A Banach lattice X is said to be weakly orthogonal if, for every weakly null sequence {x n }, it follows that lim n→∞ |x n | ∧ |x| = 0 for all x ∈ X. Dalby [36] proved that weakly compact convex subsets of a weakly orthogonal Banach lattice with uniformly monotone norm have a weak normal structure. And Yunan Cui [37] proved that the Köthe sequence spaces X are weakly orthogonal if and only if X is order continuous. So we obtain the following result.

Corollary 2.2
If a Φ = 0, Φ ∈ 2 (0), then each nonexpansive mapping of a nonempty convex weakly compact set in l Φ,p has a fixed point.

Coefficients of local uniform monotonicities of l Φ,p
For a given Banach lattice X, the upper (lower) modulus of monotonicity of X for all ε > 0 (resp. 0 < ε ≤ 1) is defined by the formula In 1993, Kurc [14] proved the following equality: Obviously, X is uniformly monotone if and only if η X (ε) > 0 (or δ X (ε) > 0) for every ε ∈ Namely, X is uniformly monotonic if and only if ε m (X) = ε m (X) = 0. Similarly, for any x ∈ S(X + ) and any ε > 0 (resp. ε ∈ [0, 1]), the functions defined by For more information about the characteristics of monotonicity and the modulus of monotonicity, see [7,9,13,20] and the references therein.
Let us discuss the local characteristic of monotonicity at the points from the unit sphere. where θ (u) = inf{λ > 0 : I Φ ( u λ ) < ∞}.
Proof Only the case 1 ≤ p < ∞ should be considered. If condition (2) is satisfied, then l Φ,p is upper locally uniformly monotone, so ε m (x) = 0 according to Theorem 2.2.