A Fixed-Point Theorem for Ordered Contraction-Type Decreasing Operators in Banach Space with Lattice Structure

In this work, we obtain a unique fixed point for a kind of ordered contraction-type decreasing operator in Banach space by using the iterative algorithm. The fixed-point study is mainly focused on two aspects. On the one hand, it is about the research of contraction-type mapping, for example, in [1–8]. On the other hand, it is about the study of monotone operators with concavity and convexity, for example, in [9– 15]. There is little research on operators that only satisfy the partial-order constrictions. Applications of operator theory in fractional differential equations can be seen in [16–43]. The following generalization of Banach’s contraction principle is due to Geraghty [44].


Introduction
In this work, we obtain a unique fixed point for a kind of ordered contraction-type decreasing operator in Banach space by using the iterative algorithm. The fixed-point study is mainly focused on two aspects. On the one hand, it is about the research of contraction-type mapping, for example, in [1][2][3][4][5][6][7][8]. On the other hand, it is about the study of monotone operators with concavity and convexity, for example, in [9][10][11][12][13][14][15]. There is little research on operators that only satisfy the partial-order constrictions. Applications of operator theory in fractional differential equations can be seen in .
The following generalization of Banach's contraction principle is due to Geraghty [44].
where ζ denotes the class of those functions β : ½0, ∞Þ ⟶ ½0, 1Þ which satisfy the condition βðt n Þ ⟶ 1 ⇒ t n ⟶ 0. Then f has a unique fixed-point z ∈ M, and f f n ðxÞg converges to z, for each x ∈ M.
Very recently, Amini-Harandi and Emami [6] proved the following existence theorem which is a version of Lemma 1 in the context of partially ordered complete metric spaces: Lemma 2. Let (M, ≼) be a partially ordered set and suppose that there exists a metric d in M such that ðM, dÞ is a complete metric space. Let f : M ⟶ M be an increasing map such that there exists an element Assume that either f is continuous or M is such that if an increasing sequence fx n g ⟶ x in M, then x n ≼ x, ∀n.
Besides, if for each x, y ∈ M, there exists z ∈ M, which is comparable to x and y.
Then f has a unique fixed point.
We found that in (2), the contraction is concerning a metric. But in fact, the relation of partial order does not play any role in (2). That is to say, in (1), the constriction y ≼ x is not effective because dðx, yÞ = dðy, xÞ. A question appears naturally in the authors' minds: "Can the contraction condition be merely about partial order so that y ≼ x can be effective?" The authors have been haunted by this question since it was found. Driven by this idea, we introduce a new kind of ordered contraction-type decreasing operator in Banach space with lattice structure and obtain a unique fixed point of the operator. Our results are helpful and meaningful for studies of fixed point. Comparing to [6], our improvements are in three aspects.
First, the contraction is merely about partial order, and the relation of partial-order y≼x does play an important role in the contraction condition. This has never been seen. Second, we consider the situation when the operator is decreasing. Third, we only use the iterative algorithm, and we can start the iterative process with any initial point, i.e., we do not need any assumptions of the existence of upper or lower solutions. An example is also presented to illustrate the theorem.
The outline of this paper is as follows. In the remainder of this section, we will give some preliminaries. In Section 2 of this paper, we present the existence and uniqueness theorem. In Section 3, an example is illustrated.
Definition 3 (see [45]). Let E be a real Banach space. A nonempty convex closed set P ⊂ E is called a cone if In the case that P is a given cone in a real Banach space ðE, k:kÞ, a partial order "≤" can be induced on E by x ≤ y ⇔ y − x ∈ P. The cone P is called normal if there exists a constant N > 0, such that for all x, y ∈ E, θ ≤ x ≤ y implies that kxk ≤ Nkyk. Details about cones and fixed point of operators can be found in [45,46]. Definition 4 (see [47,48]). We call a set X ⊂ E a lattice under the partial ordering ≤, if supfx, yg and inffx, yg exist for arbitrary x, y ∈ X.
Lemma 5 (see [45]). A cone P is normal if and only if there exists a norm k:k 1 in E which is equivalent to k:k such that for any θ ≤ x ≤ y, kxk 1 ≤ kyk 1 , i:e:, k:k 1 is monotone. The equivalence of k:k and k:k 1 means that there exist M > m > 0 such that m k:k 1 ≤ k:k ≤ Mk:k 1 .
Lemma 6 (see [45]). Let P be a normal cone in a real Banach space E. Suppose that fx n g is a monotone sequence which has a subsequence fx n i g converging to x * , then fx n g also converges to x * . Moreover, if fx n g is an increasing sequence, then fx n g ≤ x * ðn = 1, 2, 3, ⋯Þ; if fx n g is a decreasing sequence, then x * ≤fx n gðn = 1, 2, 3, ⋯Þ.

The Main Results
We suppose that E is a partially ordered Banach space. P is a normal cone. The partial-order "≤" on E is induced by the cone P.
Let ζ denote the class of those functionals β : P ⟶ ½0, 1Þ which satisfy the condition β w n ð Þ⟶ 1 ⇒ w n ⟶ θ for any monotonic sequence w n f g⊂ P ð3Þ Theorem 7 (main theorem). Suppose that X ⊂ E is a closed subset, P ⊂ X. X is a lattice. A : X ⟶ X is a decreasing operator and satisfies the following ordered contraction condition: Then A has unique fixed-point u * ∈ X. Moreover, constructing successively sequence for any initial u 0 ∈ X, we have Remark 8. Here, we study the decreasing operator while most of the contractions are about increasing operators. And the contraction condition (4) is merely about the partial order, while most of the contractions are about metric.

Remark 9.
Two elements x and y in an ordered set ðX, ≤Þ are said to be comparable if either x ≤ y or y ≤ x, and we denote it as x ≤ ≥y.
Proof. Let u 0 ∈ X, we have Au 0 ∈ X. So we have the following two cases.
Case 1. When u 0 is comparable with Au 0 . Firstly, without loss of generality, we suppose that If u 0 = Au 0 , then the proof is finished. Suppose that u 0 < Au 0 . Since A is decreasing, we obtain Au 0 ≥ A 2 u 0 , and it is easy to prove that A 2 is increasing. Using the contractive condition (4), we have hence, From (4), we have Journal of Function Spaces From (9) and (10), we have the following two conclusions: (b) There exists u 0 ∈ X such that u 0 ≤ Bu 0 We assert that the operator B has unique fixed point in X.
And the unique fixed point of B is also the unique fixed point of A. In order to be clear, we divide the process of proof into three steps.
Step 1. We will use the method of iteration to construct a fixed point of B. In fact, consider the iterative sequence u n+1 = Bu n , n = 0, 1, 2, ⋯: Since u 0 ≤ Bu 0 and the operator B is increasing, we have This means that fu n g is an increasing sequence. It follows from (12) that Since P is normal, from the equivalence of k:k and k:k 1 in Lemma 5 we have Then, fku n+1 − u n k 1 g is a decreasing sequence and bounded as follows. So Assume r > 0. Then, from (16), we have The above inequality yields Since lim sup m,n→∞ ku n − u m k 1 > 0 and lim n→∞ ku n − u n+1 k 1 = 0, then lim sup m,n→∞ This contradicts (21) and shows that fu n g is a Cauchy sequence in X.

Journal of Function Spaces
Step 2. We will obtain the uniqueness of the fixed point of B. On the contrary, if u is another fixed point of B, we will get u = u * .
In fact, the first case, when uis comparable withu 0 , without loss of generality, we suppose that u ≤ u 0 . Since B is increasing, Moreover, by (12) and so Consequently, the sequence is nonnegative and decreasing, and so lim n→∞ γ n = γ ≥ 0. Now we show that γ = 0: On the contrary, assume that γ > 0. By passing to subsequences, if necessary, we may assume that lim The second case, when u cannot compare with u 0 . FromXwhich is a lattice, we obtain satisfying i.e., u is comparable with y 1 and u 0 is comparable with y 1 . Since B is increasing, we know B n y 1 ≤ B n u, B n y 1 ≤ B n u 0 , n = 1, 2, ⋯: Moreover, by (12) So we have Similar to the process of (31), (32), and (33), So from (39) and (40), we get u = u * : ð41Þ (33) together with (41) implies that u * is unique fixed point of B.
Step 3. We will point that the unique fixed point of B is also the unique fixed point of A. Since Thus, i.e., BðAu * Þ = Au * . From the uniqueness of the fixed point of B, we know So u * is the unique fixed point of A in X.
Case 2. Another case, when u 0 is not comparable to Au 0 . From X which is a lattice, we know there For any initial v 0 ∈ X, constructing successively sequence v n = Av n−1 : ð47Þ Journal of Function Spaces From (28), we know Since v 2n+1 = A 2n+1 v 0 = B n ðAv 0 Þ and from the arbitrary of v 0 in (28), we obtain (49) and (50) imply that holds. Similarly to the proof of Step 2 and Step 3 in case 1, we get that u * is the unique fixed point of A.

An Example
Let E = R, equipped with usual normal k·k = | · | and usual partial order ≤. X = ½0, ∞Þ, P = ½0, ∞Þ. Then, P ⊂ X ⊂ E. P is a normal cone in X. ðX, k·k, ≤Þ is a partially ordered Banach space. And X is a lattice under the partial order ≤ induced by the cone P.
Then, X = ½0, ∞Þ, P = ½0, ∞Þ satisfying the assumptions of Theorem 7. Define the mapping A : X ⟶ X by Ax = ðm + 1 + mxÞ/ð1 + xÞ, x ∈ X where m is a fixed real number. Then, A is nonincreasing. Define β : P ⟶ ½0, ∞Þ by βðwÞ = 1/ð1 + wÞ, then β ∈ ζ. Now, for all x, y ∈ X with x ≤ y, we have so that A and β satisfy the assumption of Theorem 7. Observing that all the other conditions of Theorem 7 are also satisfied, A has a unique fixed-point x * > 0. Moreover, constructing successively sequence for any initial x 0 ≥ 0, we have lim x→∞ x n = x * : ð54Þ

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.