Abstract

We prove a number of quadruple fixed point theorems under ๐œ™-contractive conditions for a mapping ๐นโˆถ๐‘‹4โ†’๐‘‹ in ordered metric spaces. Also, we introduce an example to illustrate the effectiveness of our results.

1. Introduction and Preliminaries

The notion of coupled fixed point was initiated by Gnana Bhaskar and Lakshmikantham [1] in 2006. In this paper, they proved some fixed point theorems under a set of conditions and utilized their theorems to prove the existence of solutions to some ordinary differential equations. Recently, Berinde and Borcut [2] introduced the notion of tripled fixed point and extended the results of Gnana Bhaskar and Lakshmikantham [1] to the case of contractive operator ๐นโˆถ๐‘‹ร—๐‘‹ร—๐‘‹โ†’๐‘‹, where ๐‘‹ is a complete ordered metric space. For some related works in coupled and tripled fixed point, we refer readers to [3โ€“32].

For simplicity we will denote the cross product of ๐‘˜โˆˆโ„• copies of the space ๐‘‹ by ๐‘‹๐‘˜.

Definition 1.1 (see [2]). Let ๐‘‹ be a nonempty set and ๐นโˆถ๐‘‹3โ†’๐‘‹ a given mapping. An element (๐‘ฅ,๐‘ฆ,๐‘ง)โˆˆ๐‘‹3 is called a tripled fixed point of ๐น if ๐น(๐‘ฅ,๐‘ฆ,๐‘ง)=๐‘ฅ,๐น(๐‘ฆ,๐‘ฅ,๐‘ฆ)=๐‘ฆ,๐น(๐‘ง,๐‘ฆ,๐‘ฅ)=๐‘ง.(1.1) Let (๐‘‹,๐‘‘) be a metric space. The mapping ๐‘‘โˆถ๐‘‹3โ†’๐‘‹, given by ๐‘‘((๐‘ฅ,๐‘ฆ,๐‘ง),(๐‘ข,๐‘ฃ,๐‘ค))=๐‘‘(๐‘ฅ,๐‘ฆ)+๐‘‘(๐‘ฆ,๐‘ฃ)+๐‘‘(๐‘ง,๐‘ค),(1.2) defines a metric on ๐‘‹3, which will be denoted for convenience by ๐‘‘.

Definition 1.2 (see [2]). Let (๐‘‹,โ‰ค) be a partially ordered set and ๐นโˆถ๐‘‹3โ†’๐‘‹ a mapping. One says that ๐น has the mixed monotone property if ๐น(๐‘ฅ,๐‘ฆ,๐‘ง) is monotone nondecreasing in ๐‘ฅ and ๐‘ง and is monotone nonincreasing in ๐‘ฆ; that is, for any ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹, ๐‘ฅ1,๐‘ฅ2โˆˆ๐‘‹,๐‘ฅ1โ‰ค๐‘ฅ2๎€ท๐‘ฅ,implies๐น1๎€ธ๎€ท๐‘ฅ,๐‘ฆ,๐‘งโ‰ค๐น2๎€ธ,๐‘ฆ,๐‘ฆ,๐‘ง1,๐‘ฆ2โˆˆ๐‘‹,๐‘ฆ1โ‰ค๐‘ฆ2๎€ท,implies๐น๐‘ฅ,๐‘ฆ2๎€ธ๎€ท,๐‘งโ‰ค๐น๐‘ฅ,๐‘ฆ1๎€ธ,๐‘ง,๐‘ง1,๐‘ง2โˆˆ๐‘‹,๐‘ง1โ‰ค๐‘ง2๎€ท,implies๐น๐‘ฅ,๐‘ฆ,๐‘ง1๎€ธ๎€ทโ‰ค๐น๐‘ฅ,๐‘ฆ,๐‘ง2๎€ธ.(1.3)

Let us recall the main results of [2] to understand our motivation toward our results in this paper.

Theorem 1.3 (see [2]). Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹3โ†’๐‘‹ be a continuous mapping such that ๐น has the mixed monotone property. Assume that there exist ๐‘—,๐‘˜,๐‘™โˆˆ[0,1) with ๐‘—+๐‘˜+๐‘™<1 such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง),๐น(๐‘ข,๐‘ฃ,๐‘ค))โ‰ค๐‘—๐‘‘(๐‘ฅ,๐‘ข)+๐‘˜๐‘‘(๐‘ฆ,๐‘ฃ)+๐‘™๐‘‘(๐‘ง,๐‘ค)(1.4) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ข,๐‘ฃ,๐‘คโˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, and ๐‘งโ‰ฅ๐‘ค. If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ฅ0,๐‘ฆ0), and ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ฆ0,๐‘ฅ0), then ๐น has a tripled fixed point.

Theorem 1.4 (see [2]). Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹3โ†’๐‘‹ be a mapping having the mixed monotone property. Assume that there exist ๐‘—,๐‘˜,๐‘™โˆˆ[0,1) with ๐‘—+๐‘˜+๐‘™<1 such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง),๐น(๐‘ข,๐‘ฃ,๐‘ค))โ‰ค๐‘—๐‘‘(๐‘ฅ,๐‘ข)+๐‘˜๐‘‘(๐‘ฆ,๐‘ฃ)+๐‘™๐‘‘(๐‘ง,๐‘ค)(1.5) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ข,๐‘ฃ,๐‘คโˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, and ๐‘งโ‰ฅ๐‘ค. Assume that ๐‘‹ has the following properties: (i) if a nondecreasing sequence ๐‘ฅ๐‘›โ†’๐‘ฅ, then ๐‘ฅ๐‘›โ‰ค๐‘ฅ for all ๐‘›โˆˆ๐, (ii) if a nonincreasing sequence ๐‘ฆ๐‘›โ†’๐‘ฆ, then ๐‘ฆ๐‘›โ‰ฅ๐‘ฆ for all ๐‘›โˆˆ๐. If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ฅ0,๐‘ฆ0), and ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ฆ0,๐‘ฅ0), then ๐น has a tripled fixed point.

Very recently, Karapฤฑnar introduced the notion of quadruple fixed point and obtained some fixed point theorems on the topic [33]. Extending this work, quadruple fixed point is developed and related fixed point theorems are proved in [34โ€“39].

Definition 1.5 (see [34]). Let ๐‘‹ be a nonempty set and ๐นโˆถ๐‘‹4โ†’๐‘‹ a given mapping. An element (๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค)โˆˆ๐‘‹ร—๐‘‹3 is called a quadruple fixed point of ๐น if ๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค)=๐‘ฅ,๐น(๐‘ฆ,๐‘ง,๐‘ค,๐‘ฅ)=๐‘ฆ,๐น(๐‘ง,๐‘ค,๐‘ฅ,๐‘ฆ)=๐‘ง,๐น(๐‘ค,๐‘ฅ,๐‘ฆ,๐‘ง)=๐‘ค.(1.6) Let (๐‘‹,๐‘‘) be a metric space. The mapping ๐‘‘โˆถ๐‘‹4โ†’๐‘‹, given by ๐‘‘((๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘(๐‘ฅ,๐‘ฆ)+๐‘‘(๐‘ฆ,๐‘ฃ)+๐‘‘(๐‘ง,โ„Ž)+๐‘‘(๐‘ค,๐‘™),(1.7) defines a metric on ๐‘‹4, which will be denoted for convenience by ๐‘‘.

Remark 1.6. In [33, 34, 38], the notion of quadruple fixed point is called quartet fixed point.

Definition 1.7 (see [34]). Let (๐‘‹,โ‰ค) be a partially ordered set and ๐นโˆถ๐‘‹4โ†’๐‘‹ a mapping. One says that ๐น has the mixed monotone property if ๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is monotone nondecreasing in ๐‘ฅ and ๐‘ง and is monotone nonincreasing in ๐‘ฆ and ๐‘ค; that is, for any ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘คโˆˆ๐‘‹, ๐‘ฅ1,๐‘ฅ2โˆˆ๐‘‹,๐‘ฅ1โ‰ค๐‘ฅ2๎€ท๐‘ฅ,implies๐น1๎€ธ๎€ท๐‘ฅ,๐‘ฆ,๐‘ง,๐‘คโ‰ค๐น2๎€ธ,๐‘ฆ,๐‘ฆ,๐‘ง,๐‘ค1,๐‘ฆ2โˆˆ๐‘‹,๐‘ฆ1โ‰ค๐‘ฆ2๎€ท,implies๐น๐‘ฅ,๐‘ฆ2๎€ธ๎€ท,๐‘ง,๐‘คโ‰ค๐น๐‘ฅ,๐‘ฆ1๎€ธ,๐‘ง,๐‘ง,๐‘ค1,๐‘ง2โˆˆ๐‘‹,๐‘ง1โ‰ค๐‘ง2๎€ท,implies๐น๐‘ฅ,๐‘ฆ,๐‘ง1๎€ธ๎€ท,๐‘คโ‰ค๐น๐‘ฅ,๐‘ฆ,๐‘ง2๎€ธ,๐‘ค,๐‘ค1,๐‘ค2โˆˆ๐‘‹,๐‘ค1โ‰ค๐‘ค2๎€ท,implies๐น๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค2๎€ธ๎€ทโ‰ค๐น๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค1๎€ธ.(1.8)

By following Matkowski [40], we let ฮฆ be the set of all nondecreasing functions ๐œ™โˆถ[0,+โˆž)โ†’[0,+โˆž) such that lim๐‘›โ†’+โˆž๐œ™๐‘›(๐‘ก)=0 for all ๐‘ก>0. Then, it is an easy matter to show that(1)๐œ™(๐‘ก)<๐‘ก for all ๐‘ก>0,(2)๐œ™(0)=0.

In this paper, we prove some quadruple fixed point theorems for a mapping ๐นโˆถ๐‘‹4โ†’๐‘‹ satisfying a contractive condition based on some ๐œ™โˆˆฮฆ.

2. Main Results

Our first result is the following.

Theorem 2.1. Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹4โ†’๐‘‹ be a continuous mapping such that ๐น has the mixed monotone property. Assume that there exists ๐œ™โˆˆฮฆ such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค๐œ™(max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)})(2.1) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™. If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ง0,๐‘ค0,๐‘ฅ0), ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ค0,๐‘ฅ0,๐‘ฆ0) and ๐‘ค0โ‰ฅ๐น(๐‘ค0,๐‘ฅ0,๐‘ฆ0,๐‘ง0), then ๐น has a quadruple fixed point.

Proof. Suppose ๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0โˆˆ๐‘‹ are such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ง0,๐‘ค0,๐‘ฅ0), ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ค0,๐‘ฅ0,๐‘ฆ0), and ๐‘ค0โ‰ฅ๐น(๐‘ค0,๐‘ฅ0,๐‘ฆ0,๐‘ง0). Define ๐‘ฅ1๎€ท๐‘ฅ=๐น0,๐‘ฆ0,๐‘ง0,๐‘ค0๎€ธ,๐‘ฆ1๎€ท๐‘ฆ=๐น0,๐‘ง0,๐‘ค0,๐‘ฅ0๎€ธ,๐‘ง1๎€ท๐‘ง=๐น0,๐‘ค0,๐‘ฅ0,๐‘ฆ0๎€ธ,๐‘ค1๎€ท๐‘ค=๐น0,๐‘ฅ0,๐‘ฆ0,๐‘ง0๎€ธ.(2.2) Then, ๐‘ฅ0โ‰ค๐‘ฅ1, ๐‘ฆ0โ‰ฅ๐‘ฆ1, ๐‘ง0โ‰ค๐‘ง1, and ๐‘ค0โ‰ฅ๐‘ค1. Again, define ๐‘ฅ2=๐น(๐‘ฅ1,๐‘ฆ1,๐‘ง1,๐‘ค1), ๐‘ฆ2=๐น(๐‘ฆ1,๐‘ง1,๐‘ค1,๐‘ฅ1), ๐‘ง2=๐น(๐‘ง1,๐‘ค1,๐‘ฅ1,๐‘ฆ1), and ๐‘ค2=๐น(๐‘ค1,๐‘ฅ1,๐‘ฆ1,๐‘ง1). Since ๐น has the mixed monotone property, we have ๐‘ฅ0โ‰ค๐‘ฅ1โ‰ค๐‘ฅ2, ๐‘ฆ2โ‰ค๐‘ฆ1โ‰ค๐‘ฆ0, ๐‘ง0โ‰ค๐‘ง1โ‰ค๐‘ง2, and ๐‘ค2โ‰ค๐‘ค1โ‰ค๐‘ค0. Continuing this process, we can construct four sequences (๐‘ฅ๐‘›), (๐‘ฆ๐‘›), (๐‘ง๐‘›), and (๐‘ค๐‘›) in ๐‘‹ such that ๐‘ฅ๐‘›๎€ท๐‘ฅ=๐น๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1,๐‘ง๐‘›โˆ’1,๐‘ค๐‘›โˆ’1๎€ธโ‰ค๐‘ฅ๐‘›+1๎€ท๐‘ฅ=๐น๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›๎€ธ,๐‘ฆ๐‘›+1๎€ท๐‘ฆ=๐น๐‘›,๐‘ง๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›๎€ธโ‰ค๐‘ฆ๐‘›๎€ท๐‘ฆ=๐น๐‘›โˆ’1,๐‘ง๐‘›โˆ’1,๐‘ค๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1๎€ธ,๐‘ง๐‘›๎€ท๐‘ง=๐น๐‘›โˆ’1,๐‘ค๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธโ‰ค๐‘ง๐‘›+1๎€ท๐‘ง=๐น๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธ,๐‘ค๐‘›+1๎€ท๐‘ค=๐น๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›๎€ธโ‰ค๐‘ค๐‘›๎€ท๐‘ค=๐น๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1,๐‘ง๐‘›โˆ’1๎€ธ.(2.3) If, for some integer ๐‘›, we have (๐‘ฅ๐‘›+1,๐‘ฆ๐‘›+1,๐‘ง๐‘›+1,๐‘ค๐‘›+1)=(๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›), then ๐น(๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›)=๐‘ฅ๐‘›, ๐น(๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›)=๐‘ฆ๐‘›, ๐น(๐‘ง๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›)=๐‘ง๐‘›, and ๐น(๐‘ค๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›)=๐‘ค๐‘›; that is, (๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›) is a quadruple fixed point of ๐น. Thus, we will assume that (๐‘ฅ๐‘›+1,๐‘ฆ๐‘›+1,๐‘ง๐‘›+1,๐‘ค๐‘›+1)โ‰ (๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›) for all ๐‘›โˆˆโ„•; that is, we assume that ๐‘ฅ๐‘›+1โ‰ ๐‘ฅ๐‘›,๐‘ฆ๐‘›+1โ‰ ๐‘ฆ๐‘›, or ๐‘ง๐‘›+1โ‰ ๐‘ง๐‘› or ๐‘ค๐‘›+1โ‰ ๐‘ค๐‘›. For any ๐‘›โˆˆโ„•, we have ๐‘‘๎€ท๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฅโˆถ=๐‘‘๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›๎€ธ๎€ท๐‘ฅ,๐น๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1,๐‘ง๐‘›โˆ’1,๐‘ค๐‘›โˆ’1๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅ๎€ธ๎€ธโ‰ค๐œ™max๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›โˆ’1,๐‘‘๎€ท๐‘ฆ๎€ธ๎€พ๎€ธ๐‘›,๐‘ฆ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฆโˆถ=๐‘‘๐‘›โˆ’1,๐‘ง๐‘›โˆ’1,๐‘ค๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐น๐‘›,๐‘ง๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆ๎€ธ๎€ธโ‰ค๐œ™max๐‘›โˆ’1,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐‘‘๐‘›โˆ’1,๐‘ฅ๐‘›,๐‘‘๎€ท๐‘ง๎€ธ๎€พ๎€ธ๐‘›+1,๐‘ง๐‘›๎€ธ๎€ท๐น๎€ท๐‘งโˆถ=๐‘‘๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ง,๐น๐‘›โˆ’1,๐‘ค๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ท๎€ฝ๐‘‘๎€ท๐‘ง๎€ธ๎€ธโ‰ค๐œ™max๐‘›,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐‘‘๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘›โˆ’1,๐‘‘๎€ท๐‘ค๎€ธ๎€พ๎€ธ๐‘›,๐‘ค๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘คโˆถ=๐‘‘๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐น๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆ๎€ธ๎€ธโ‰ค๐œ™max๐‘›โˆ’1,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐‘‘๐‘›โˆ’1,๐‘ฅ๐‘›.๎€ธ๎€พ๎€ธ(2.4) From (2.4), it follows that ๎€ฝ๐‘‘๎€ท๐‘ฅmax๐‘›+1,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ง,๐‘‘๐‘›+1,๐‘ง๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›+1๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅ๎€ธ๎€พโ‰ค๐œ™max๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›โˆ’1.๎€ธ๎€พ๎€ธ(2.5) By repeating (2.5) ๐‘› times, we get that ๎€ฝ๐‘‘๎€ท๐‘ฅmax๐‘›+1,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ง,๐‘‘๐‘›+1,๐‘ง๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›+1๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅ๎€ธ๎€พโ‰ค๐œ™max๐‘›,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›โˆ’1๎€ธ๎€พ๎€ธโ‰ค๐œ™2๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅmax๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ง,๐‘‘๐‘›โˆ’1,๐‘ง๐‘›โˆ’2๎€ธ๎€ท๐‘ค,๐‘‘๐‘›โˆ’1,๐‘ค๐‘›โˆ’1โ‹ฎ๎€ธ๎€พ๎€ธโ‰ค๐œ™๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅmax1,๐‘ฅ0๎€ธ๎€ท๐‘ฆ,๐‘‘1,๐‘ฆ0๎€ธ๎€ท๐‘ง,๐‘‘1,๐‘ง0๎€ธ๎€ท๐‘ค,๐‘‘1,๐‘ค0.๎€ธ๎€พ๎€ธ(2.6) Now, we will show that (๐‘ฅ๐‘›), (๐‘ฆ๐‘›), (๐‘ง๐‘›), and (๐‘ค๐‘›) are Cauchy sequences in ๐‘‹. Let ๐œ–>0. Since lim๐‘›โ†’+โˆž๐œ™๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅmax1,๐‘ฅ0๎€ธ๎€ท๐‘ฆ,๐‘‘1,๐‘ฆ0๎€ธ๎€ทz,๐‘‘1,๐‘ง0๎€ธ๎€ท๐‘ค,๐‘‘1,๐‘ค0๎€ธ๎€พ๎€ธ=0(2.7) and ๐œ–>๐œ™(๐œ–), there exist ๐‘›0โˆˆโ„• such that ๐œ™๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅmax1,๐‘ฅ0๎€ธ๎€ท๐‘ฆ,๐‘‘1,๐‘ฆ0๎€ธ๎€ท๐‘ง,๐‘‘1,๐‘ง0๎€ธ๎€ท๐‘ค,๐‘‘1,๐‘ค0๎€ธ๎€พ๎€ธ<๐œ–โˆ’๐œ™(๐œ–)โˆ€๐‘›โ‰ฅ๐‘›0.(2.8) This implies that ๎€ฝ๐‘‘๎€ท๐‘ฅmax๐‘›+1,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ง,๐‘‘๐‘›+1,๐‘ง๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘›+1๎€ธ๎€พ<๐œ–โˆ’๐œ™(๐œ–)โˆ€๐‘›โ‰ฅ๐‘›0.(2.9) For ๐‘š,๐‘›โˆˆโ„•, we will prove by induction on ๐‘š that ๎€ฝ๐‘‘๎€ท๐‘ฅmax๐‘›,๐‘ฅ๐‘š๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘š๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘š๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘š๎€ธ๎€พ<๐œ–โˆ€๐‘šโ‰ฅ๐‘›โ‰ฅ๐‘›0.(2.10) Since ๐œ–โˆ’๐œ™(๐œ–)<๐œ–, then by using (2.9) we conclude that (2.10) holds when ๐‘š=๐‘›+1. Now suppose that (2.10) holds for ๐‘š=๐‘˜. For ๐‘š=๐‘˜+1, we have ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘˜+1๎€ธ๎€ท๐‘ฅโ‰ค๐‘‘๐‘›,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›+1,๐‘ฅ๐‘˜+1๎€ธ๎€ท๐น๎€ท๐‘ฅโ‰ค๐œ–โˆ’๐œ™(๐œ–)+๐‘‘๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›๎€ธ๎€ท๐‘ฅ,๐น๐‘˜,๐‘ฆ๐‘˜,๐‘ง๐‘˜,๐‘ค๐‘˜๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅ๎€ธ๎€ธโ‰ค๐œ–โˆ’๐œ™(๐œ–)+๐œ™max๐‘›,๐‘ฅ๐‘˜๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘˜๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘˜๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘˜๎€ธ๎€พ๎€ธ<๐œ–โˆ’๐œ™(๐œ–)+๐œ™(๐œ–)=๐œ–.(2.11) Similarly, we show that ๐‘‘๎€ท๐‘ฆ๐‘›,๐‘ฆ๐‘˜+1๎€ธ๐‘‘๎€ท๐‘ง<๐œ–,๐‘›,๐‘ง๐‘˜+1๎€ธ๐‘‘๎€ท๐‘ค<๐œ–,๐‘›,๐‘ค๐‘˜+1๎€ธ<๐œ–.(2.12) Hence, we have ๎€ฝ๐‘‘๎€ท๐‘ฅmax๐‘›,๐‘ฅ๐‘˜+1๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฆ๐‘˜+1๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ง๐‘˜+1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ค๐‘˜+1๎€ธ๎€พ<๐œ–.(2.13) Thus, (2.10) holds for all ๐‘šโ‰ฅ๐‘›โ‰ฅ๐‘›0. Hence, (๐‘ฅ๐‘›), (๐‘ฆ๐‘›), (๐‘ง๐‘›), and (๐‘ค๐‘›) are Cauchy sequences in ๐‘‹.
Since ๐‘‹ is a complete metric space, there exist ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘คโˆˆ๐‘‹ such that (๐‘ฅ๐‘›), (๐‘ฆ๐‘›), (๐‘ง๐‘›) and (๐‘ค๐‘›) converge to ๐‘ฅ, ๐‘ฆ, ๐‘ง, and ๐‘ค, respectively. Finally, we show that (๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is a quadruple fixed point of ๐น. Since ๐น is continuous and (๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›)โ†’(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค), we have ๐‘ฅ๐‘›+1=๐น(๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›)โ†’๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค). By the uniqueness of limit, we get that ๐‘ฅ=๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค). Similarly, we show that ๐‘ฆ=๐น(๐‘ฆ,๐‘ง,๐‘ค,๐‘ฅ), ๐‘ง=๐น(๐‘ง,๐‘ค,๐‘ฅ,๐‘ฆ), and ๐‘ค=๐น(๐‘ค,๐‘ฅ,๐‘ฆ,๐‘ง). So, (๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is a quadruple fixed point of ๐น.

By taking ๐œ™(๐‘ก)=๐‘˜๐‘ก, where ๐‘˜โˆˆ[0,1), in Theorem 2.1, we have the following.

Corollary 2.2. Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹4โ†’๐‘‹ be a continuous mapping such that ๐น has the mixed monotone property. Assume that there exists ๐‘˜โˆˆ[0,1) such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค๐‘˜max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}(2.14) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™. If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ง0,๐‘ค0,๐‘ฅ0), ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ค0,๐‘ฅ0,๐‘ฆ0), and ๐‘ค0โ‰ฅ๐น(๐‘ค0,๐‘ฅ0,๐‘ฆ0,๐‘ง0), then ๐น has a quadruple fixed point.

As a consequence of Corollary 2.2, we have the following.

Corollary 2.3. Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹4โ†’๐‘‹ be a continuous mapping such that ๐น has the mixed monotone property. Assume that there exist ๐‘Ž1,๐‘Ž2,๐‘Ž3,๐‘Ž4โˆˆ[0,1) with ๐‘Ž1+๐‘Ž2+๐‘Ž3+๐‘Ž4<1 such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค๐‘Ž1๐‘‘(๐‘ฅ,๐‘ข)+๐‘Ž2๐‘‘(๐‘ฆ,๐‘ฃ)+๐‘Ž3๐‘‘(๐‘ง,โ„Ž)+๐‘Ž4๐‘‘(๐‘ค,๐‘™)(2.15) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™. If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ง0,๐‘ค0,๐‘ฅ0), ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ค0,๐‘ฅ0,๐‘ฆ0) and ๐‘ค0โ‰ฅ๐น(๐‘ค0,๐‘ฅ0,๐‘ฆ0,๐‘ง0), then ๐น has a quadruple fixed point.

By adding an additional hypothesis, the continuity of ๐น in Theorem 2.1 can be dropped.

Theorem 2.4. Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹4โ†’๐‘‹ be a mapping having the mixed monotone property. Assume that there exists ๐œ™โˆˆฮฆ such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค๐œ™(max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)})(2.16) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™. Assume also that ๐‘‹ has the following properties: (i) if a nondecreasing sequence ๐‘ฅ๐‘›โ†’๐‘ฅ, then ๐‘ฅ๐‘›โ‰ค๐‘ฅ for all ๐‘›โˆˆ๐, (ii) if a nonincreasing sequence ๐‘ฆ๐‘›โ†’๐‘ฆ, then ๐‘ฆ๐‘›โ‰ฅ๐‘ฆ for all ๐‘›โˆˆ๐.
If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ง0,๐‘ค0,๐‘ฅ0), ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ค0,๐‘ฅ0,๐‘ฆ0), and ๐‘ค0โ‰ฅ๐น(๐‘ค0,๐‘ฅ0,๐‘ฆ0,๐‘ง0), then ๐น has a quadruple fixed point.

Proof. By following the same process in Theorem 2.1, we construct four Cauchy sequences (๐‘ฅ๐‘›), (๐‘ฆ๐‘›), (๐‘ง๐‘›), and (๐‘ค๐‘›) in ๐‘‹ with ๐‘ฅ1โ‰ค๐‘ฅ2โ‰คโ‹ฏโ‰ค๐‘ฅ๐‘›๐‘ฆโ‰คโ€ฆ,1โ‰ฅ๐‘ฆ2โ‰ฅโ‹ฏโ‰ฅ๐‘ฆ๐‘›zโ‰ฅโ‹ฏ,1โ‰ค๐‘ง2โ‰คโ‹ฏโ‰ค๐‘ง๐‘›๐‘คโ‰คโ‹ฏ,1โ‰ฅ๐‘ค2โ‰ฅโ‹ฏโ‰ฅ๐‘ค๐‘›โ‰ฅโ‹ฏ,(2.17) such that ๐‘ฅ๐‘›โ†’๐‘ฅโˆˆ๐‘‹, ๐‘ฆ๐‘›โ†’๐‘ฆโˆˆ๐‘‹, ๐‘ง๐‘›โ†’๐‘งโˆˆ๐‘‹, and ๐‘ค๐‘›โ†’๐‘คโˆˆ๐‘‹. By the hypotheses on ๐‘‹, we have ๐‘ฅ๐‘›โ‰ค๐‘ฅ, ๐‘ฆ๐‘›โ‰ฅ๐‘ฆ, ๐‘ง๐‘›โ‰ค๐‘ง, and ๐‘ค๐‘›โ‰ฅ๐‘ค for all ๐‘›โˆˆ๐. From (2.16), we have ๐‘‘๎€ท๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐‘ฅ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฅโˆถ=๐‘‘(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›๎€ท๎€ฝ๐‘‘๎€ท๎€ธ๎€ธโ‰ค๐œ™max๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ท,๐‘‘๐‘ฆ,๐‘ฆ๐‘›๎€ธ๎€ท,๐‘‘๐‘ง,๐‘ง๐‘›๎€ธ๎€ท,๐‘‘๐‘ค,๐‘ค๐‘›,๐‘‘๎€ท๐‘ฆ๎€ธ๎€พ๎€ธ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฆ,๐น(๐‘ฆ,๐‘ง,๐‘ค,๐‘ฅ)โˆถ=๐‘‘๐‘›,๐‘ง๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›๎€ธ๎€ธ๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆ,๐น(๐‘ฆ,๐‘ง,๐‘ค,๐‘ฅ)โ‰ค๐œ™max๐‘›๎€ธ๎€ท๐‘ง,๐‘ฆ,๐‘‘๐‘›๎€ธ๎€ท๐‘ค,๐‘ง,๐‘‘๐‘›๎€ธ๎€ท๐‘ฅ,๐‘ค,๐‘‘๐‘›,๐‘‘๎€ท๐น,๐‘ฅ๎€ธ๎€พ๎€ธ(๐‘ง,๐‘ค,๐‘ฅ,๐‘ฆ),๐‘ง๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘งโˆถ=๐‘‘(๐‘ง,๐‘ค,๐‘ฅ,๐‘ฆ),๐น๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ท๎€ฝ๐‘‘๎€ท๎€ธ๎€ธโ‰ค๐œ™max๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ท,๐‘‘๐‘ฆ,๐‘ฆ๐‘›๎€ธ๎€ท,๐‘‘๐‘ง,๐‘ง๐‘›๎€ธ๎€ท,๐‘‘๐‘ค,๐‘ค๐‘›,๐‘‘๎€ท๐‘ค๎€ธ๎€พ๎€ธ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ค,๐น(๐‘ค,๐‘ฅ,๐‘ฆ,๐‘ง)โˆถ=๐‘‘๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›๎€ธ๎€ธ๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆ,๐น(๐‘ค,๐‘ฅ,๐‘ฆ,๐‘ง)โ‰ค๐œ™max๐‘›๎€ธ๎€ท๐‘ง,๐‘ฆ,๐‘‘๐‘›๎€ธ๎€ท๐‘ค,๐‘ง,๐‘‘๐‘›๎€ธ๎€ท๐‘ฅ,๐‘ค,๐‘‘๐‘›.,๐‘ฅ๎€ธ๎€พ๎€ธ(2.18) From (2.18), we have โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๐‘‘๎€ทmax๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐‘ฅ๐‘›+1๎€ธ,๐‘‘๎€ท๐‘ฆ๐‘›+1๎€ธ,๐‘‘๎€ท,๐น(๐‘ฆ,๐‘ง,๐‘ค,๐‘ฅ)๐น(๐‘ง,๐‘ค,๐‘ฅ,๐‘ฆ),๐‘ง๐‘›+1๎€ธ,๐‘‘๎€ท๐‘ค๐‘›+1๎€ธโŽซโŽชโŽชโŽชโŽฌโŽชโŽชโŽชโŽญโŽ›โŽœโŽœโŽโŽงโŽชโŽจโŽชโŽฉ๐‘‘๎€ท,๐น(๐‘ค,๐‘ฅ,๐‘ฆ,๐‘ง)โ‰ค๐œ™max๐‘ฅ,๐‘ฅ๐‘›๎€ธ๎€ท,๐‘‘๐‘ฆ,๐‘ฆ๐‘›๎€ธ,๐‘‘๎€ท๐‘ง,๐‘ง๐‘›๎€ธ๎€ท,๐‘‘๐‘ค,๐‘ค๐‘›๎€ธโŽซโŽชโŽฌโŽชโŽญโŽžโŽŸโŽŸโŽ .(2.19) Letting ๐‘›โ†’+โˆž in (2.19), it follows that ๐‘ฅ=๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค), ๐‘ฆ=๐น(๐‘ฆ,๐‘ง,๐‘ค,๐‘ฅ), ๐‘ง=๐น(๐‘ง,๐‘ค,๐‘ฅ,๐‘ฆ), and ๐‘ค=๐น(๐‘ค,๐‘ฅ,๐‘ฆ,๐‘ง). Hence, (๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is a quadruple fixed point of ๐น.

By taking ๐œ™(๐‘ก)=๐‘˜๐‘ก, where ๐‘˜โˆˆ[0,1), in Theorem 2.4, we have the following result.

Corollary 2.5. Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹4โ†’๐‘‹ be a mapping having the mixed monotone property. Assume that there exists ๐‘˜โˆˆ[0,1) such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,z,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค๐‘˜max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}(2.20) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™. Assume also that ๐‘‹ has the following properties: (i) if a nondecreasing sequence ๐‘ฅ๐‘›โ†’๐‘ฅ, then ๐‘ฅ๐‘›โ‰ค๐‘ฅ for all ๐‘›โˆˆ๐, (ii) if a nonincreasing sequence ๐‘ฆ๐‘›โ†’๐‘ฆ, then ๐‘ฆ๐‘›โ‰ฅ๐‘ฆ for all ๐‘›โˆˆ๐. If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ง0,๐‘ค0,๐‘ฅ0), ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ค0,๐‘ฅ0,๐‘ฆ0), and ๐‘ค0โ‰ฅ๐น(๐‘ค0,๐‘ฅ0,๐‘ฆ0,๐‘ง0), then ๐น has a quadruple fixed point.

As a consequence of Corollary 2.5, we have the following.

Corollary 2.6. Let (๐‘‹,โ‰ค) be a partially ordered set and (๐‘‹,๐‘‘) a complete metric space. Let ๐นโˆถ๐‘‹4โ†’๐‘‹ be a mapping having the mixed monotone property. Assume that there exist ๐‘Ž1,๐‘Ž2,๐‘Ž3,๐‘Ž4โˆˆ[0,1) with ๐‘Ž1+๐‘Ž2+๐‘Ž3+๐‘Ž4<1 such that ๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค๐‘Ž1๐‘‘(๐‘ฅ,๐‘ข)+๐‘Ž2๐‘‘(๐‘ฆ,๐‘ฃ)+๐‘Ž3๐‘‘(๐‘ง,โ„Ž)+๐‘Ž4๐‘‘(๐‘ค,๐‘™)(2.21) for all ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™. Assume that ๐‘‹ has the following properties: (i) if a nondecreasing sequence ๐‘ฅ๐‘›โ†’๐‘ฅ, then ๐‘ฅ๐‘›โ‰ค๐‘ฅ for all ๐‘›โˆˆ๐, (ii) if a nonincreasing sequence ๐‘ฆ๐‘›โ†’๐‘ฆ, then ๐‘ฆ๐‘›โ‰ฅ๐‘ฆ for all ๐‘›โˆˆ๐. If there exist ๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0โˆˆ๐‘‹ such that ๐‘ฅ0โ‰ค๐น(๐‘ฅ0,๐‘ฆ0,๐‘ง0,๐‘ค0), ๐‘ฆ0โ‰ฅ๐น(๐‘ฆ0,๐‘ง0,๐‘ค0,๐‘ฅ0), ๐‘ง0โ‰ค๐น(๐‘ง0,๐‘ค0,๐‘ฅ0,๐‘ฆ0), and ๐‘ค0โ‰ฅ๐น(๐‘ค0,๐‘ฅ0,๐‘ฆ0,๐‘ง0), then ๐น has a quadruple fixed point.

Now we prove the following result.

Theorem 2.7. In addition to the hypotheses of Theorem 2.1 (resp., Theorem 2.4), suppose that ๐‘ฅ๎€บ๎€ท0โ‰ค๐‘ฆ0๎€ธโˆง๎€ท๐‘ง0โ‰ค๐‘ฆ0๎€ธโˆง๎€ท๐‘ฅ0โ‰ค๐‘ค0๎€ธโˆง๎€ท๐‘ง0โ‰ค๐‘ค0โˆจ๐‘ฆ๎€ธ๎€ป๎€บ๎€ท0โ‰ค๐‘ฅ0๎€ธโˆง๎€ท๐‘ฆ0โ‰ค๐‘ง0๎€ธโˆง๎€ท๐‘ค0โ‰ค๐‘ฅ0๎€ธโˆง๎€ท๐‘ค0โ‰ค๐‘ง0.๎€ธ๎€ป(2.22) Then, ๐‘ฅ=๐‘ฆ=๐‘ง=๐‘ค.

Proof. Without loss of generality, we may assume that ๐‘ฅ0โ‰ค๐‘ฆ0, ๐‘ง0โ‰ค๐‘ฆ0, ๐‘ฅ0โ‰ค๐‘ค0, and ๐‘ง0โ‰ค๐‘ค0. By the mixed monotone property of ๐น, we have ๐‘ฅ๐‘›โ‰ค๐‘ฆ๐‘›, ๐‘ง๐‘›โ‰ค๐‘ฆ๐‘›, ๐‘ฅ๐‘›โ‰ค๐‘ค๐‘›, and ๐‘ง๐‘›โ‰ค๐‘ค๐‘› for all ๐‘›โˆˆ๐. Thus, by (2.1), we have ๐‘‘๎€ท๐‘ฆ๐‘›+1,๐‘ฅ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฆโˆถ=๐‘‘๐‘›,๐‘ง๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆ๎€ธ๎€ธโ‰ค๐œ™max๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ง๐‘›๎€ธ๎€ท๐‘ฅ,๐‘‘๐‘›,๐‘ค๐‘›,๐‘‘๎€ท๐‘ฆ๎€ธ๎€พ๎€ธ(2.23)๐‘›+1,๐‘ง๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฆโˆถ=๐‘‘๐‘›,๐‘ง๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ง,๐น๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆ๎€ธ๎€ธโ‰ค๐œ™max๐‘›,๐‘ง๐‘›๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ค๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘‘๐‘›,๐‘ฆ๐‘›,๐‘‘๎€ท๐‘ค๎€ธ๎€พ๎€ธ(2.24)๐‘›+1,๐‘ฅ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘คโˆถ=๐‘‘๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›๎€ธ๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›,๐‘ค๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ฅ๎€ธ๎€ธโ‰ค๐œ™max๐‘›,๐‘ค๐‘›๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ง,๐‘‘๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ง๐‘›,๐‘‘๎€ท๐‘ค๎€ธ๎€พ๎€ธ(2.25)๐‘›+1,๐‘ง๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘คโˆถ=๐‘‘๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›,๐‘ง๐‘›๎€ธ๎€ท๐‘ง,๐น๐‘›,๐‘ค๐‘›,๐‘ฅ๐‘›,๐‘ฆ๐‘›๎€ท๎€ฝ๐‘‘๎€ท๐‘ง๎€ธ๎€ธโ‰ค๐œ™max๐‘›,๐‘ค๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ,๐‘‘๐‘›,๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ง๐‘›.๎€ธ๎€พ๎€ธ(2.26) By (2.23) and (2.26), we have ๎€ฝ๐‘‘๎€ท๐‘ฆmax๐‘›+1,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›+1,๐‘ง๐‘›+1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›+1,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›+1,๐‘ง๐‘›+1๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆ๎€ธ๎€พโ‰ค๐œ™max๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›,๐‘ง๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ค,๐‘‘๐‘›,๐‘ง๐‘›๎€ธ๎€พ๎€ธโ‰ค๐œ™2๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆmax๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐‘‘๐‘›โˆ’1,๐‘ง๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ค,๐‘‘๐‘›โˆ’1,๐‘ง๐‘›โˆ’1โ‹ฎ๎€ธ๎€พ๎€ธโ‰ค๐œ™๐‘›+1๎€ท๎€ฝ๐‘‘๎€ท๐‘ฆmax0,๐‘ฅ0๎€ธ๎€ท๐‘ฆ,๐‘‘0,๐‘ง0๎€ธ๎€ท๐‘ค,๐‘‘0,๐‘ฅ0๎€ธ๎€ท๐‘ค,๐‘‘0,๐‘ง0.๎€ธ๎€พ๎€ธ(2.27) By letting ๐‘›โ†’+โˆž in (2.27) and using the property of ๐œ™ and the fact that ๐‘‘ is continuous on its variable, we get that max{๐‘‘(๐‘ฆ,๐‘ฅ),๐‘‘(๐‘ฆ,๐‘ง),๐‘‘(๐‘ค,๐‘ฅ),๐‘‘(๐‘ค,๐‘ง)}=0. Hence, ๐‘ฆ=๐‘ง=๐‘ฅ=๐‘ค.

Corollary 2.8. In addition to the hypotheses of Corollary 2.3 (resp., Corollary 2.5), suppose that ๐‘ฅ๎€บ๎€ท0โ‰ค๐‘ฆ0๎€ธโˆง๎€ท๐‘ง0โ‰ค๐‘ฆ0๎€ธโˆง๎€ท๐‘ฅ0โ‰ค๐‘ค0๎€ธโˆง๎€ท๐‘ง0โ‰ค๐‘ค0โˆจ๐‘ฆ๎€ธ๎€ป๎€บ๎€ท0โ‰ค๐‘ฅ0๎€ธโˆง๎€ท๐‘ฆ0โ‰ค๐‘ง0๎€ธโˆง๎€ท๐‘ค0โ‰ค๐‘ฅ0๎€ธโˆง๎€ท๐‘ค0โ‰ค๐‘ง0.๎€ธ๎€ป(2.28) Then, ๐‘ฅ=๐‘ฆ=๐‘ง=๐‘ค.

Example 2.9. Let ๐‘‹=[0,1] with usual order. Define ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ by ๐‘‘(๐‘ฅ,๐‘ฆ)=|๐‘ฅโˆ’๐‘ฆ|. Define ๐นโˆถ๐‘‹4โ†’๐‘‹ by ๐น๎ƒฏ1(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค)=0,max{๐‘ฆ,๐‘ค}โ‰ฅmin{๐‘ฅ,๐‘ง},4(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),max{๐‘ฆ,๐‘ค}<min{๐‘ฅ,๐‘ง}.(2.29) Then, (a)(๐‘‹,๐‘‘,โ‰ค) is a complete ordered metric space, (b)for ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™, we have that 1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)},(2.30)(c)holds for all ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™,(d)๐น has the mixed monotone property.

Proof. To prove (๐‘), given ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค,๐‘ข,๐‘ฃ,โ„Ž,๐‘™โˆˆ๐‘‹ with ๐‘ฅโ‰ฅ๐‘ข, ๐‘ฆโ‰ค๐‘ฃ, ๐‘งโ‰ฅโ„Ž, and ๐‘คโ‰ค๐‘™, we examine the following cases.
Case 1. If max{๐‘ฆ,๐‘ค}โ‰ฅmin{๐‘ฅ,๐‘ง}, and max{๐‘ฃ,๐‘™}โ‰ฅmin{๐‘ข,๐‘ค}. Here, we have 1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=0โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.31)
Case 2. If max{๐‘ฆ,๐‘ค}โ‰ฅmin{๐‘ฅ,๐‘ง} and max{๐‘ฃ,๐‘™}<min{๐‘ข,โ„Ž}. This case is impossible since ๐‘ฆโ‰ค๐‘ฃ<min{๐‘ข,โ„Ž}โ‰คmin{๐‘ฅ,๐‘ง},๐‘คโ‰ค๐‘™<min{๐‘ข,โ„Ž}โ‰คmin{๐‘ฅ,๐‘ง}.(2.32) So, max{๐‘ฆ,๐‘ค}<min{๐‘ฅ,๐‘ง}.(2.33)
Case 3. If max{๐‘ฆ,๐‘ค}<min{๐‘ฅ,๐‘ง} and max{๐‘ฃ,๐‘™}โ‰ฅmin{๐‘ข,โ„Ž}.
This case will have different possibilities.
(i) Let max{๐‘ฆ,๐‘ค}=๐‘ฆ and max{๐‘ฃ,๐‘™}=๐‘ฃ. Suppose that โ„Žโ‰ค๐‘ฃ; then โ„Žโˆ’๐‘ฆโ‰ค๐‘ฃโˆ’๐‘ฆ and hence min{๐‘ฅ,z}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆโ‰ค๐‘งโˆ’๐‘ฆ=๐‘งโˆ’โ„Ž+โ„Žโˆ’๐‘ฆโ‰ค๐‘งโˆ’โ„Ž+๐‘ฃโˆ’๐‘ฆ=๐‘‘(๐‘ง,โ„Ž)+๐‘‘(๐‘ฆ,๐‘ฃ)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.34) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆ)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.35) Suppose that ๐‘ขโ‰ค๐‘ฃ; then ๐‘ขโˆ’๐‘ฆโ‰ค๐‘ฃโˆ’๐‘ฆ and hence min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆโ‰ค๐‘ฅโˆ’๐‘ฆ=๐‘ฅโˆ’๐‘ข+๐‘ขโˆ’๐‘ฆโ‰ค(๐‘ฅโˆ’๐‘ข)+(๐‘ฃโˆ’๐‘ฆ)=๐‘‘(๐‘ฅ,๐‘ข)+๐‘‘(๐‘ฃ,๐‘ฆ)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.36) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆ)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.37)
(ii) Let max{๐‘ฆ,๐‘ค}=๐‘ฆ and max{๐‘ฃ,๐‘™}=๐‘™. Suppose that โ„Žโ‰ค๐‘™; then โ„Žโˆ’๐‘ฆโ‰ค๐‘™โˆ’๐‘ฆ and (since ๐‘คโ‰ค๐‘ฆ) hence min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆโ‰ค๐‘งโˆ’๐‘ฆ=๐‘งโˆ’โ„Ž+โ„Žโˆ’๐‘ฆโ‰ค๐‘งโˆ’โ„Ž+๐‘™โˆ’๐‘ฆโ‰ค๐‘งโˆ’โ„Ž+๐‘™โˆ’๐‘ค=๐‘‘(๐‘ง,โ„Ž)+๐‘‘(๐‘ค,๐‘™)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.38) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆ)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.39)
Suppose that ๐‘ขโ‰ค๐‘™; then ๐‘ขโˆ’๐‘ฆโ‰ค๐‘™โˆ’๐‘ฆ and (since ๐‘คโ‰ค๐‘ฆ ) hence min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆโ‰ค๐‘ฅโˆ’๐‘ฆ=๐‘ฅโˆ’๐‘ข+๐‘ขโˆ’๐‘ฆโ‰ค(๐‘ฅโˆ’๐‘ข)+(๐‘™โˆ’๐‘ฆ)โ‰ค๐‘ฅโˆ’๐‘ข+๐‘™โˆ’๐‘ค=๐‘‘(๐‘ฅ,๐‘ข)+๐‘‘(๐‘ค,๐‘™)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.40) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ฆ)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.41)
(iii) Let max{๐‘ฆ,๐‘ค}=๐‘ค and max{๐‘ฃ,๐‘™}=๐‘ฃ. Suppose that โ„Žโ‰ค๐‘ฃ; then โ„Žโˆ’๐‘คโ‰ค๐‘ฃโˆ’๐‘ค, but ๐‘ฆโ‰ค๐‘ค, and hence min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘คโ‰ค๐‘งโˆ’๐‘ค=๐‘งโˆ’โ„Ž+โ„Žโˆ’๐‘คโ‰ค๐‘งโˆ’โ„Ž+๐‘ฃโˆ’๐‘คโ‰ค๐‘งโˆ’โ„Ž+๐‘ฃโˆ’๐‘ฆ=๐‘‘(๐‘ง,โ„Ž)+๐‘‘(๐‘ฆ,๐‘ฃ)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.42) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ค)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.43)
Suppose that ๐‘ขโ‰ค๐‘ฃ; then ๐‘ขโˆ’๐‘คโ‰ค๐‘ฃโˆ’๐‘ค and hence min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘คโ‰ค๐‘ฅโˆ’๐‘ค=๐‘ฅโˆ’๐‘ข+๐‘ขโˆ’๐‘คโ‰ค(๐‘ฅโˆ’๐‘ข)+(๐‘ฃโˆ’๐‘ค)โ‰ค๐‘ฅโˆ’๐‘ข+๐‘ฃโˆ’๐‘ฆ=๐‘‘(๐‘ฅ,๐‘ข)+๐‘‘(๐‘ฃ,๐‘ฆ)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.44) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ค)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.45)
(iv) Let max{๐‘ฆ,๐‘ค}=๐‘ค and max{๐‘ฃ,๐‘™}=๐‘™. Suppose that โ„Žโ‰ค๐‘™; then โ„Žโˆ’๐‘คโ‰ค๐‘™โˆ’๐‘ค and hence min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘คโ‰ค๐‘งโˆ’๐‘ค=๐‘งโˆ’โ„Ž+โ„Žโˆ’๐‘คโ‰ค๐‘งโˆ’โ„Ž+๐‘™โˆ’๐‘ค=๐‘‘(๐‘ง,โ„Ž)+๐‘‘(๐‘ค,๐‘™)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.46) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ค)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.47)
Suppose that ๐‘ขโ‰ค๐‘™; then ๐‘ขโˆ’๐‘คโ‰ค๐‘™โˆ’๐‘ค and hence min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}=min{๐‘ฅ,๐‘ง}โˆ’๐‘คโ‰ค๐‘ฅโˆ’๐‘ค=๐‘ฅโˆ’๐‘ข+๐‘ขโˆ’๐‘คโ‰ค(๐‘ฅโˆ’๐‘ข)+(๐‘™โˆ’๐‘ค)=๐‘‘(๐‘ฅ,๐‘ข)+๐‘‘(๐‘ค,๐‘™)โ‰ค2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.48) Therefore, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))=๐‘‘4๎‚=1(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),04โ‰ค1(min{๐‘ฅ,๐‘ง}โˆ’๐‘ค)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.49)
Case 4. (i) If max{๐‘ฆ,๐‘ค}<min{๐‘ฅ,๐‘ง} and max{๐‘ฃ,๐‘™}<min{๐‘ข,โ„Ž}.
Since ๐‘ฅโ‰ฅ๐‘ข and ๐‘งโ‰ฅโ„Ž, then min{๐‘ฅ,๐‘ง}โ‰ฅmin{๐‘ข,โ„Ž}, and also since ๐‘ฆโ‰ฅ๐‘ฃ and ๐‘คโ‰ฅ๐‘™, then max{๐‘ฃ,๐‘™}โ‰ฅmax{๐‘ฆ,๐‘ค}. Thus, ๎‚€1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง),๐น(๐‘ข,๐‘ฃ,๐‘ค))=๐‘‘41(min{๐‘ฅ,๐‘ง}โˆ’max{๐‘ฆ,๐‘ค}),4๎‚=1(min{๐‘ข,โ„Ž}โˆ’max{๐‘ฃ,๐‘™})4||||.(min{๐‘ฅ,๐‘ง}โˆ’min{๐‘ข,โ„Ž})+(max{๐‘ฃ,๐‘™}โˆ’max{๐‘ฆ,๐‘ค})(2.50)
(ii) If min{๐‘ข,โ„Ž}=๐‘ข and max{๐‘ฃ,๐‘™}=๐‘ฃ, then min{๐‘ฅ,๐‘ง}โˆ’min{๐‘ข,โ„Ž}โ‰ค๐‘ฅโˆ’๐‘ข and max{๐‘ฃ,๐‘™}โˆ’max{๐‘ฆ,๐‘ค}โ‰ค๐‘ฃโˆ’๐‘ฆ. Thus, 1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค4[]=1(๐‘ฅโˆ’๐‘ข)+(๐‘ฃโˆ’๐‘ฆ)4[]โ‰ค1๐‘‘(๐‘ฅ,๐‘ข)+๐‘‘(๐‘ฆ,๐‘ฃ)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.51)
(iii) If min{๐‘ข,โ„Ž}=โ„Ž and max{๐‘ฃ,๐‘™}=๐‘ฃ, then min{๐‘ฅ,๐‘ง}โˆ’min{๐‘ข,โ„Ž}โ‰ค๐‘งโˆ’โ„Ž and max{๐‘ฃ,๐‘™}โˆ’max{๐‘ฆ,๐‘ค}โ‰ค๐‘ฃโˆ’๐‘ฆ, hence 1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค4[]=1(๐‘งโˆ’โ„Ž)+(๐‘ฃโˆ’๐‘ฆ)4[]โ‰ค1๐‘‘(๐‘ง,โ„Ž)+๐‘‘(๐‘ฆ,๐‘ฃ)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.52)
(iv) If min{๐‘ข,โ„Ž}=๐‘ข and max{๐‘ฃ,๐‘™}=๐‘™, then min{๐‘ฅ,๐‘ง}โˆ’min{๐‘ข,โ„Ž}โ‰ค๐‘ฅโˆ’๐‘ข and max{๐‘ฃ,๐‘™}โˆ’max{๐‘ฆ,๐‘ค}โ‰ค๐‘™โˆ’๐‘ค, and hence 1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค4[]=1(๐‘ฅโˆ’๐‘ข)+(๐‘™โˆ’๐‘ค)4[]โ‰ค1๐‘‘(๐‘ฅ,๐‘ข)+๐‘‘(๐‘ค,๐‘™)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.53)
(v) If min{๐‘ข,โ„Ž}=โ„Ž and max{๐‘ฃ,๐‘™}=๐‘™, then min{๐‘ฅ,๐‘ง}โˆ’min{๐‘ข,โ„Ž}โ‰ค๐‘งโˆ’โ„Ž and max{๐‘ฃ,๐‘™}โˆ’max{๐‘ฆ,๐‘ค}โ‰ค๐‘™โˆ’๐‘ค, and hence 1๐‘‘(๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค),๐น(๐‘ข,๐‘ฃ,โ„Ž,๐‘™))โ‰ค4[]=1(๐‘งโˆ’โ„Ž)+(๐‘™โˆ’๐‘ค)4[]โ‰ค1๐‘‘(๐‘ง,โ„Ž)+๐‘‘(๐‘ค,๐‘™)2max{๐‘‘(๐‘ฅ,๐‘ข),๐‘‘(๐‘ฆ,๐‘ฃ),๐‘‘(๐‘ง,โ„Ž),๐‘‘(๐‘ค,๐‘™)}.(2.54)
To prove (c), let ๐‘ฅ,๐‘ฆ,๐‘ง,๐‘คโˆˆ๐‘‹. To show that ๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is monotone nondecreasing in ๐‘ฅ, let ๐‘ฅ1,๐‘ฅ2โˆˆ๐‘‹ with ๐‘ฅ1โ‰ค๐‘ฅ2.
If max{๐‘ฆ,๐‘ค}โ‰ฅmin{๐‘ฅ1,๐‘ง}, then ๐น(๐‘ฅ1,๐‘ฆ,๐‘ง,๐‘ค)=0โ‰ค๐น(๐‘ฅ2,๐‘ฆ,๐‘ง,๐‘ค). If max{๐‘ฆ,๐‘ค}<min{๐‘ฅ1,๐‘ง}, then ๐น๎€ท๐‘ฅ1๎€ธ=1,๐‘ฆ,๐‘ง,๐‘ค4๎€ท๎€ฝ๐‘ฅmin1๎€พ๎€ธโ‰ค1,๐‘งโˆ’max{๐‘ฆ,๐‘ค}4๎€ท๎€ฝ๐‘ฅmin2๎€พ๎€ธ๎€ท๐‘ฅ,๐‘งโˆ’max{๐‘ฆ,๐‘ค}=๐น2๎€ธ.,๐‘ฆ,๐‘ง,๐‘ค(2.55) Therefore, ๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is monotone nondecreasing in ๐‘ฅ. Similarly, we may show that ๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is monotone nondecreasing in ๐‘ง.
To show that ๐น(๐‘ฅ,y,๐‘ง,๐‘ค) is monotone nonincreasing in ๐‘ฆ, let ๐‘ฆ1,๐‘ฆ2โˆˆ๐‘‹ with ๐‘ฆ1โ‰ค๐‘ฆ2. If max{๐‘ฆ2,๐‘ค}โ‰ฅmin{๐‘ฅ,๐‘ง}, then ๐น(๐‘ฅ,๐‘ฆ2,๐‘ง,๐‘ค)=0โ‰ค๐น(๐‘ฅ1,๐‘ฆ,๐‘ง,๐‘ค). If max{๐‘ฆ2,๐‘ค}<min{๐‘ฅ,๐‘ง}, then ๐น๎€ท๐‘ฅ,๐‘ฆ2๎€ธ=1,๐‘ง,๐‘ค4๎€ท๎€ฝ๐‘ฆmin{๐‘ฅ,๐‘ง}โˆ’max2โ‰ค1,๐‘ค๎€พ๎€ธ4๎€ท๎€ฝ๐‘ฆmin{๐‘ฅ,๐‘ง}โˆ’max1๎€ท,๐‘ค๎€พ๎€ธ=๐น๐‘ฅ,๐‘ฆ2๎€ธ.,๐‘ง,๐‘ค(2.56) Therefore, ๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is monotone nonincreasing in ๐‘ฆ. Similarly, we may show that ๐น(๐‘ฅ,๐‘ฆ,๐‘ง,๐‘ค) is monotone nonincreasing in ๐‘ค.
Thus, by Theorem 2.1 (let ๐œ™(๐‘ก)=(๐‘ก/2)), ๐น has a unique quadruple fixed point, namely, (0,0,0,0). Since the condition of Theorem 2.7 is satisfied, (0,0,0,0) is the unique quadruple fixed point of ๐น.

Remark 2.10. We notice that for, ๐นโˆถ๐‘‹2๐‘›โ†’๐‘‹,(๐‘›โˆˆโ„•), it is very natural to consider the analog of Theorem 2.1โ€“Theorem 2.7 to get fixed points. Moreover, for ๐นโˆถ๐‘‹2๐‘›+1โ†’๐‘‹(๐‘›โˆˆโ„•), the analog of Theorem 7โ€“Theorem 11 of Berinde and Borcut [2] yields fixed points.