Global Attractivity Results on Complete Ordered Metric Spaces for Third-Order Difference Equations

Abstract

We establish fixed-point theorems for mixed monotone mappings in the setting of ordered metric spaces which satisfy a contractive condition for all points that are related by a given ordering. We also give a global attractivity result for all solutions of the difference equation 𝑢_𝑛+1=𝐹(𝑢_𝑛,𝑢_𝑛−1,𝑢_𝑛−2),𝑛=2,3,…, where 𝐹 satisfies certain monotonicity conditions with respect to the given ordering. As an application of our obtained results, we present some iterative algorithms to solve a class of matrix equations. A numerical example is also presented to test the validity of the algorithms.

1. Introduction

The following global attractivity result from [1] (see also [2]) is very useful in establishing convergence results in many situations.

Theorem 1 (see [1]). Let [𝑎,𝑏] be a closed and bounded interval of real numbers and let 𝐹∈𝐶([𝑎,𝑏]³,[𝑎,𝑏]) satisfy the following conditions.(i)The function 𝐹(𝑧₁,𝑧₂,𝑧₃) is monotonic in each of its arguments.(ii)For each 𝑚,𝑀∈[𝑎,𝑏] and for each 𝑖∈{1,2,3}, one defines 𝑀_𝑖(𝑚,𝑀)=𝑀,if𝐹isincreasingin𝑧_𝑖,𝑚,if𝐹isdecreasingin𝑧_𝑖,𝑚_𝑖(𝑚,𝑀)=𝑀_𝑖(𝑀,𝑚) (1)
and assume that if (𝑚,𝑀) is a solution of the system 𝑀=𝐹𝑀₁(𝑚,𝑀),𝑀₂(𝑚,𝑀),𝑀₃(),𝑚𝑚,𝑀𝑚=𝐹₁(𝑚,𝑀),𝑚₂(𝑚,𝑀),𝑚₃(𝑚,𝑀), (2)
then 𝑀=𝑚.
Then there exists exactly one equilibrium 𝑥 of the equation 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−2,𝑛=0,1,… (3) and every solution of the above equation converges to 𝑥.

The above result in Theorem 1 attracted considerable attention of the leading specialists in difference equations and discrete dynamic systems and has been generalized and extended to the case of maps in ℝ^𝑛, see [3], and maps in Banach space with cone, see [4–6].

Moreover, there has been recent interest in establishing fixed-point theorems in partially ordered complete metric spaces with a contractivity condition which holds for all points that are related by partial ordering, (see, e.g., [7–19]). These fixed-point results have been applied mainly to the existence of solutions of boundary value problems for differential equations, namely [18], has been applied for solving a class of matrix equations.

In [20], Burgić et al. obtained the following global attractivity result for mixed monotone mappings in partially ordered complete metric spaces (see also Gnana Bhaskar and Lakshmikantham [10]).

Theorem 2 (see Burgić et al. [20]). Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝐹∶𝑋×𝑋→𝑋 be a map such that 𝐹(𝑥,𝑦) is nonincreasing in 𝑥 for all 𝑦∈𝑋 and nondecreasing in 𝑦 for all 𝑥∈𝑋. Suppose that the following conditions hold.(i)There exists 𝑘∈[0,1) with 𝑑𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑑(𝑥,𝑢)+𝑑𝑦,𝑣∀𝑥⪯𝑢,𝑦≽𝑣. (4)(ii)There exists 𝑥₀,𝑦₀∈𝑋 such that the following condition holds: 𝑥₀⪯𝐹𝑥₀,𝑦₀,𝑦₀≽𝐹𝑦₀,𝑥₀. (5)(iii)If {𝑥_𝑛}⊂𝑋 is a nondecreasing convergent sequence such that lim_𝑛→∞𝑥_𝑛=𝑥, then 𝑥_𝑛⪯𝑥, for all 𝑛∈ℕ and if {𝑦_𝑛}⊂𝑋 is a nonincreasing convergent sequence such that lim_𝑛→∞𝑦_𝑛=𝑦, then 𝑦_𝑛≽𝑦, for all 𝑛∈ℕ; if 𝑥_𝑛⪯𝑦_𝑛 for every 𝑛, then lim_𝑛→∞𝑥_𝑛⪯lim_𝑛→∞𝑦_𝑛.
Then one has the following.(a)For every initial point 𝑥₀,𝑦₀∈𝑋 such that condition (ii) holds, 𝐹^𝑛(𝑥₀,𝑦₀)→𝑥, 𝐹^𝑛(𝑦₀,𝑥₀)→𝑦, 𝑛→∞, where 𝑥,𝑦∈𝑋 satisfy 𝑥=𝐹𝑥,𝑦,𝑦=𝐹𝑦,𝑥. (6)(b)If 𝑥₀⪯𝑦₀ in condition (ii), then 𝑥=𝑦. If in addition 𝑥=𝑦, then {𝑥_𝑛}, {𝑦_𝑛} converge to the equilibrium of the equation 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑛=1,2,…. (7)(c)In particular, every solution {𝑧_𝑛} of 𝑧_𝑛+1=𝐹𝑧_𝑛,𝑧_𝑛−1,𝑛=2,3,… (8)
such that 𝑥₀⪯𝑧₀, 𝑧₁⪯𝑦₀ converges to the equilibrium of (7).

In this paper, motivated by the results and ideas in a recent work of Berinde and Borcut [9], we extend Theorem 2 to mappings 𝐹∶𝑋×𝑋×𝑋→𝑋. Such extension allows us to study the third-order difference equation 𝑢_𝑛+1=𝐹𝑢_𝑛,𝑢_𝑛−1,𝑢_𝑛−2,𝑛=2,3,…. (9) The presented theorems also extend and generalize the work in [9]. We use our obtained results to build some iterative algorithms to solve a class of matrix equations. A numerical example is also presented to test the validity of the algorithms.

Now we introduce the following concepts.

Definition 3. Let 𝑋 be a nonempty set and 𝐹∶𝑋×𝑋×𝑋→𝑋 a given mapping. One says that (𝑥,𝑦,𝑧)∈𝑋×𝑋×𝑋 is a fixed-point of the third order if 𝐹𝑥,𝑦,𝑥=𝑥,𝐹𝑦,𝑥,𝑦=𝑦,𝐹𝑧,𝑦,𝑧=𝑧. (10)

Definition 4. Let(𝑋,⪯) be a partially ordered set and 𝐹∶𝑋×𝑋×𝑋→𝑋 a given mapping. one says that 𝐹 has the mixed monotone property if 𝐹(𝑥,𝑦,𝑧) is monotonously increasing in 𝑥 and 𝑧 and is monotonously decreasing in 𝑦; that is, for any 𝑥,𝑦,𝑧∈𝑋,(i)𝑥₁,𝑥₂∈𝑋,𝑥₁⪯𝑥₂⇒𝐹(𝑥₁,𝑦,𝑧)⪯𝐹(𝑥₂,𝑦,𝑧),(ii)𝑦₁,𝑦₂∈𝑋,𝑦₁⪯𝑦₂⇒𝐹(𝑥,𝑦₁,𝑧)≽𝐹(𝑥,𝑦₂,𝑧), and(iii)𝑧₁,𝑧₂∈𝑋,𝑧₁⪯𝑧₂⇒𝐹(𝑥,𝑦,𝑧₁)⪯𝐹(𝑥,𝑦,𝑧₂).

Through this paper we will use the following notations.

Let (𝑋,⪯) be a partially ordered set. (i)For 𝑥,𝑦,𝑧∈𝑋×𝑋×𝑋 the notation 𝑥∈[𝑦,𝑧] means that 𝑥≽𝑦 and 𝑥⪯𝑧.(ii)We endow 𝑋×𝑋×𝑋 with the partial order that we denote also by ⪯, defined by ,()⪯()𝑥,𝑦,𝑧𝑢,𝑣,𝑤∈𝑋×𝑋×𝑋,𝑥,𝑦,𝑧𝑢,𝑣,𝑤⟺𝑥⪯𝑢,𝑦≽𝑣,𝑧⪯𝑤. (11)(iii)Let 𝐹∶𝑋×𝑋×𝑋→𝑋 be a given mapping. For all 𝑥,𝑦,𝑧∈𝑋, we denote 𝐹²𝐹,𝐹𝑥,𝑦,𝑧=𝐹𝑥,𝑦,𝑥,𝐹𝑦,𝑥,𝑦,𝐹𝑧,𝑦,𝑧_𝑝+1𝑥,𝑦,𝑧=𝐹𝐹^𝑝𝑥,𝑦,𝑥,𝐹^𝑝𝑦,𝑥,𝑦,𝐹^𝑝𝑧,𝑦,𝑧,∀𝑝≥2. (12)

2. Main Result

Our first result is the following.

Theorem 5. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝐹∶𝑋×𝑋×𝑋→𝑋 be a mapping having the mixed monotone property on 𝑋. Suppose that the following conditions hold.(i) There exists 𝜆∈[0,1) with 𝑑𝐹()𝑑()()𝐹,𝑑𝐹𝐹,𝑑𝐹𝐹,𝑑𝐹𝑥,𝑦,𝑧,𝐹𝑢,𝑣,𝑤≤𝜆max𝑥,𝑢,𝑑𝑦,𝑣,𝑑𝑧,𝑤,𝑑𝑥,𝑦,𝑥,𝑥𝑥,𝑦,𝑥,𝑢,𝑑𝑦,𝑥,𝑦,𝑦𝑦,𝑥,𝑦,𝑣,𝑑𝑧,𝑦,𝑧,𝑧𝑧,𝑦,𝑧,𝑤 (13) for all (𝑥,𝑦,𝑧)⪯(𝑢,𝑣,𝑤). (ii) There exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑥₀⪯𝐹𝑥₀,𝑦₀,𝑥₀,𝑦₀≽𝐹𝑦₀,𝑥₀,𝑦₀,𝑧₀⪯𝐹𝑧₀,𝑦₀,𝑧₀. (14)(iii) If {𝑥_𝑛}⊂𝑋 is a nondecreasing convergent sequence such that lim_𝑛→∞𝑥_𝑛=𝑥, then 𝑥_𝑛⪯𝑥 for all 𝑛∈ℕ, if {𝑦_𝑛}⊂𝑋 is a nonincreasing convergent sequence such that lim_𝑛→∞𝑦_𝑛=𝑦, then 𝑦_𝑛≽𝑦 for every 𝑛∈ℕ, and if 𝑥_𝑛⪯𝑦_𝑛 for every 𝑛, then lim_𝑛→∞𝑥_𝑛⪯lim_𝑛→∞𝑦_𝑛.
Then one has the following.(a) For every initial point (𝑥₀,𝑦₀,𝑧₀)∈𝑋×𝑋×𝑋 such that condition (14) holds: 𝑥_𝑛=𝐹^𝑛𝑥₀,𝑦₀,𝑥₀⟶𝑥,𝑦_𝑛=𝐹^𝑛𝑦₀,𝑥₀,𝑦₀𝑧⟶𝑦,_𝑛=𝐹^𝑛𝑧₀,𝑦₀,𝑧₀⟶𝑧as𝑛⟶∞, (15) where 𝑥,𝑦,and𝑧 satisfy 𝑥=𝐹𝑥,𝑦,𝑥,𝑦=𝐹𝑦,𝑥,𝑦,𝑧=𝐹𝑧,𝑦,𝑧. (16) If 𝑥₀⪯𝑦₀ and 𝑧₀⪯𝑦₀ in condition (14), then 𝑥=𝑦=𝑧 and {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge to the equilibrium of the equation 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+1=𝐹𝑧_𝑛,𝑦_𝑛,𝑧_𝑛,𝑛=0,1,2…. (17)(b) In particular, every solution {𝑢_𝑛} of 𝑢_𝑛+1=𝐹𝑢_𝑛,𝑢_𝑛−1,𝑢_𝑛−2,𝑛=2,3,…, (18) such that 𝑢₀,𝑢₁,𝑢₂∈[𝑥₀,𝑦₀] (or [𝑧₀,𝑦₀]) converges to the equilibrium of (18).(c) The following estimates hold: 𝑑𝑥,𝐹^𝑛𝑥₀,𝑦₀,𝑥₀≤𝜆^𝑛𝑑𝐹𝑥1−𝜆max₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝑧₀,𝐹𝑧₀,𝑦₀,𝑧₀,𝑑𝑦,𝐹^𝑛𝑦₀,𝑥₀,𝑦₀≤𝜆^𝑛𝑑𝐹𝑥1−𝜆max₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝑧₀,𝐹𝑧₀,𝑦₀,𝑧₀,𝑑𝑧,𝐹^𝑛𝑧₀,𝑦₀,𝑧₀≤𝜆^𝑛𝑑𝐹𝑥1−𝜆max₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝐹𝑧₀,𝑦₀,𝑧₀,𝑧₀. (19)

Proof. Let 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that condition (14) is satisfied. Denote 𝑥₁=𝐹(𝑥₀,𝑦₀,𝑥₀), 𝑦₁=𝐹(𝑦₀,𝑥₀,𝑦₀), and 𝑧₁=𝐹(𝑧₀,𝑦₀,𝑧₀). Since 𝑥₀⪯𝐹(𝑥₀,𝑦₀,𝑥₀)=𝑥₁, 𝑦₀≽𝐹(𝑦₀,𝑥₀,𝑦₀)=𝑦₁, and 𝑧₀⪯𝐹(𝑧₀,𝑦₀,𝑧₀)=𝑦₁, from the mixed monotone property of 𝐹, we get that 𝑥₀⪯𝑥₁=𝐹𝑥₀,𝑦₀,𝑥₀⪯𝑥₂=𝐹𝑥₁,𝑦₁,𝑥₁,𝑦₀≽𝑦₁=𝐹𝑦₀,𝑥₀,𝑦₀≽𝑦₂=𝐹𝑦₁,𝑥₁,𝑦₁,𝑧₀⪯𝑧₁=𝐹𝑧₀,𝑦₀,𝑧₀⪯𝑧₂=𝐹𝑧₁,𝑦₁,𝑧₁. (20) Consider the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} defined by (17). By induction and using the mixed monotone property of 𝐹, we obtain easily that 𝑥₀⪯𝑥₁⪯⋯⪯𝑥_𝑛⪯⋯;𝑦₀≽𝑦₁≽⋯≽𝑦_𝑛𝑧≽⋯;₀⪯𝑧₁⪯⋯⪯𝑧_𝑛⪯⋯. (21) For the sake of clarity, for all 𝑛≥1, denote 𝐷^𝑥_𝑛=𝑑𝑥_𝑛−1,𝑥_𝑛,𝐷^𝑦_𝑛=𝑑𝑦_𝑛−1,𝑦_𝑛,𝐷^𝑧_𝑛=𝑑𝑧_𝑛−1,𝑧_𝑛. (22) We claim that, for all 𝑛≥1, we have 𝐷^𝑥_𝑛+1≤𝜆^𝑛max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁,𝐷^𝑦_𝑛+1≤𝜆^𝑛max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁,𝐷^𝑧_𝑛+1≤𝜆^𝑛max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁. (23) By (13) and (21), we obtain 𝐷^𝑥₂=𝑑𝑥₁,𝑥₂=𝑑𝐹𝑥₀,𝑦₀,𝑥₀,𝐹𝑥₁,𝑦₁,𝑥₁≤𝜆max𝑑𝑥₀,𝑥₁,𝑑𝑦₀,𝑦₁,𝑑𝑥₀,𝑥₁,𝑑𝐹𝑥₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑥₀,𝑦₀,𝑥₀,𝑥₁,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₁,𝑑𝐹𝑥₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑥₀,𝑦₀,𝑥₀,𝑥₁=𝜆max𝑑𝑥₀,𝑥₁,𝑑𝑦₀,𝑦₁,𝐷^𝑦₂=𝑑𝑦₁,𝑦₂=𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝐹𝑦₁,𝑥₁,𝑦₁≤𝜆max𝑑𝑦₀,𝑦₁,𝑑𝑥₀,𝑥₁,𝑑𝑦₀,𝑦₁,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₁,𝑑𝐹𝑥₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑥₀,𝑦₀,𝑥₀,𝑥₁,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₁=𝜆max𝑑𝑦₀,𝑦₁,𝑑𝑥₀,𝑥₁,𝐷^𝑧₂=𝑑𝑧₁,𝑧₂=𝑑𝐹𝑧₀,𝑦₀,𝑧₀,𝐹𝑧₁,𝑦₁,𝑧₁≤𝜆max𝑑𝑧₀,𝑧₁,𝑑𝑦₀,𝑦₁,𝑑𝑧₀,𝑧₁,𝑑𝐹𝑧₀,𝑦₀,𝑧₀,𝑧₀,𝑑𝐹𝑧₀,𝑦₀,𝑧₀,𝑧₁,𝑑𝐹𝑦₀,𝑧₀,𝑦₀,𝑦₀,𝑑𝐹𝑦₀,𝑧₀,𝑦₀,𝑦₁,𝑑𝐹𝑧₀,𝑦₀,𝑧₀,𝑧₀,𝑑𝐹𝑧₀,𝑦₀,𝑧₀,𝑧₁=𝜆max𝑑𝑧₀,𝑧₁,𝑑𝑦₀,𝑦₁. (24) Thus we get that 𝐷^𝑥₂≤𝜆max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁,𝐷^𝑦₂≤𝜆max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁,𝐷^𝑧₂≤𝜆max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁. (25) Then our claim holds for 𝑛=1. Suppose now that (23) holds for some fixed 𝑛≥1. Similarly, by (13) and (21), we obtain 𝐷^𝑥_𝑛+2=𝑑𝑥_𝑛+1,𝑥_𝑛+2=𝑑𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝐹𝑥_𝑛+1,𝑦_𝑛+1,𝑥_𝑛+1≤𝜆max𝐷^𝑥_𝑛+1,𝐷^𝑦_𝑛+1≤𝜆^𝑛+1max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑦₁. (26) Similarly, one can show that 𝐷^𝑦_𝑛+2≤𝜆^𝑛+1max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁,𝐷^𝑧_𝑛+2≤𝜆^𝑛+1max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁. (27) Then by the induction principle, (23) holds for all 𝑛≥1.
Now we will prove that {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} are Cauchy sequences in the metric space (𝑋,𝑑). Using (23) and the triangular inequality, for 𝑚>𝑛, we have 𝑑𝑥_𝑚,𝑥_𝑛≤^𝑚−1_𝑖=𝑛𝑑𝑥_𝑖,𝑥_𝑖+1=^𝑚−1_𝑖=𝑛𝐷^𝑥_𝑖+1≤^𝑚−1_𝑖=𝑛𝜆^𝑖max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁≤^𝑚−1_𝑖=𝑛𝜆^𝑖max𝐷^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁≤𝜆^𝑛𝐷1−𝜆max^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁⟶0as𝑛⟶∞. (28) This implies that {𝑥_𝑛} is a Cauchy sequence. Similarly, one can prove that {𝑦_𝑛} and {𝑧_𝑛} are Cauchy sequences.
Since (𝑋,𝑑) is complete, there exist 𝑥,𝑦,𝑧∈𝑋 such that 𝑥_𝑛⟶𝑥,𝑦_𝑛𝑧⟶𝑦,_𝑛⟶𝑧as𝑛⟶∞. (29) From condition (iii) and (21), we get that 𝑥_𝑛⪯𝑥,𝑦_𝑛≽𝑦,𝑧_𝑛⪯𝑧,∀𝑛. (30) We claim that 𝑥=𝐹(𝑥,𝑦,𝑥). Indeed, we have 𝑑𝑥,𝐹𝑥,𝑦,𝑥≤𝑑𝑥,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝐹𝑥,𝑦,𝑥=𝑑𝑥,𝑥_𝑛+1+𝑑𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝐹𝑥,𝑦,𝑥≤𝑑𝑥,𝑥_𝑛+1+𝜆max𝑑𝑥_𝑛,𝑥,𝑑𝑦_𝑛,𝑑𝑥,𝑦_𝑛,𝑥,𝑑𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝑥_𝑛,𝑑𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝑥,𝑑𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝑦_𝑛,𝑑𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝑦,𝑑𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝑥_𝑛,𝑑𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝑥=𝑑𝑥,𝑥_𝑛+1+𝜆max𝑑𝑥_𝑛,𝑥,𝑑𝑦_𝑛().,𝑦⟶0as𝑛⟶∞by29 (31)
Then, our claim holds; that is, 𝑥=𝐹(𝑥,𝑦,𝑥). Similarly, one can show that 𝑦=𝐹(𝑦,𝑥,𝑦) and 𝑧=𝐹(𝑧,𝑦,𝑧). Thus we proved (16).
On the other hand, from (28), for 𝑚>𝑛 with 𝑛 being fixed, we have 𝑑𝑥_𝑚,𝑥_𝑛=𝑑𝑥_𝑚,𝐹^𝑛𝑥₀,𝑦₀,𝑥₀≤𝜆^𝑛𝐷1−𝜆max^𝑥₁,𝐷^𝑦₁,𝐷^𝑧₁. (32) Letting 𝑚→∞ in the above inequality, we obtain 𝑑𝑥,𝐹^𝑛𝑥₀,𝑦₀,𝑥₀≤𝜆^𝑛𝑑𝑥1−𝜆max₀,𝑥₁,𝑑𝑦₀,𝑦₁,𝑑𝑧₀,𝑧₁; (33) that is, 𝑑𝑥,𝐹^𝑛𝑥₀,𝑦₀,𝑥₀≤𝜆^𝑛𝑑𝑥1−𝜆max₀,𝐹𝑥₀,𝑦₀,𝑥₀,𝑑𝑦₀,𝐹𝑦₀,𝑥₀,𝑦₀,𝑑𝑧₀,𝐹𝑧₀,𝑦₀,𝑧₀. (34) Similarly, one can show that 𝑑𝑦,𝐹^𝑛𝑦₀,𝑥₀,𝑦₀≤𝜆^𝑛𝑑𝑥1−𝜆max₀,𝐹𝑥₀,𝑦₀,𝑥₀,𝑑𝑦₀,𝐹𝑦₀,𝑥₀,𝑦₀,𝑑𝑧₀,𝐹𝑧₀,𝑦₀,𝑧₀,𝑑𝑧,𝐹^𝑛𝑧₀,𝑦₀,𝑧₀≤𝜆^𝑛𝑑𝑥1−𝜆max₀,𝐹𝑥₀,𝑦₀,𝑥₀,𝑑𝑦₀,𝐹𝑦₀,𝑥₀,𝑦₀,𝑑𝑧₀,𝐹𝑧₀,𝑦₀,𝑧₀. (35) Thus we proved (19).
Now, if 𝑥₀⪯𝑦₀ and 𝑧₀⪯𝑦₀, we claim that, for all 𝑛∈ℕ,𝑥_𝑛⪯𝑦_𝑛 and 𝑧_𝑛⪯𝑦_𝑛. Indeed, by the mixed monotone property of 𝐹, 𝑥₁=𝐹𝑥₀,𝑦₀,𝑥₀⪯𝐹𝑦₀,𝑥₀,𝑦₀=𝑦₁,𝑧₁=𝐹𝑧₀,𝑦₀,𝑧₀⪯𝐹𝑦₀,𝑥₀,𝑦₀=𝑦₁. (36) Assume that 𝑥_𝑛⪯𝑦_𝑛 and 𝑧_𝑛⪯𝑦_𝑛 for some 𝑛. Then, 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛⪯𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛=𝑦_𝑛+1, (37) and similarly for 𝑦_𝑛 and 𝑧_𝑛. Thus we proved that 𝑥_𝑛⪯𝑦_𝑛,𝑧_𝑛⪯𝑦_𝑛,∀𝑛∈ℕ. (38)
Next, from (38), we have 𝑑𝑥,𝑦≤𝑑𝑥,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑦≤𝑑𝑥,𝑥_𝑛+1+𝑑𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛+𝑑𝑦_𝑛+1,𝑦≤𝑑𝑥,𝑥_𝑛+1+𝜆𝑑𝑥_𝑛,𝑧_𝑛+𝑑𝑦_𝑛+1⋮,𝑦≤𝑑𝑥,𝑥_𝑛+1+𝜆^𝑛𝑑𝑥₀,𝑦₀+𝑑𝑦_𝑛+1,𝑦⟶0as𝑛⟶∞, (39) which implies that 𝑥=𝑦. Similarly, we obtain that 𝑑(𝑦,𝑧)=0, that is, 𝑦=𝑧. Then 𝑥=𝑦=𝑧.
Now, assume that 𝑥₀⪯𝑦₀. Then, in view of the monotonicity of 𝐹, we have 𝑥₁=𝐹𝑥₀,𝑦₀,𝑥₀⪯𝑥₂=𝐹𝑥₁,𝑦₁,𝑥₁⪯𝐹𝑦₁,𝑥₁,𝑦₁=𝑦₂⪯𝐹𝑦₀,𝑥₀,𝑦₀=𝑦₁. (40) Continuing this process, one can show that 𝑥_𝑛⪯𝑥_𝑛+1⪯𝑦_𝑛+1⪯𝑦_𝑛,∀𝑛. (41) If we assume that 𝑢₀,𝑢₁,𝑢₂∈[𝑥₀,𝑦₀], in view of the monotonicity of 𝐹, we have 𝑥₁=𝐹𝑥₀,𝑦₀,𝑥₀⪯𝐹𝑢₂,𝑢₁,𝑢₀=𝑢₃⪯𝐹𝑦₀,𝑥₀,𝑦₀=𝑦₁,𝑥₁=𝐹𝑥₀,𝑦₀,𝑥₀⪯𝐹𝑢₃,𝑢₂,𝑢₁=𝑢₄⪯𝐹𝑦₀,𝑥₀,𝑦₀=𝑦₁,𝑥₁=𝐹𝑥₀,𝑦₀,𝑥₀⪯𝐹𝑢₄,𝑢₃,𝑢₂=𝑢₅⪯𝐹𝑦₀,𝑥₀,𝑦₀=𝑦₁. (42) Continuing in a similar way, we can prove that 𝑥_𝑘⪯𝑢_3𝑘,𝑢_3𝑘+1,𝑢_3𝑘+2⪯𝑦_𝑘,∀𝑘≥1. (43) Letting 𝑘→∞ and using (iii) and (29), we get that 𝑢_𝑛→𝑥 as 𝑛→∞, where 𝑥 is the equilibrium of (17). Similarly, if 𝑧₀⪯𝑦₀ and 𝑢₀,𝑢₁,𝑢₂∈[𝑧₀,𝑦₀], we obtain also the same result.

Remark 6. If we replace the condition, if {𝑥_𝑛}⊂𝑋 is a nondecreasing convergent sequence such that lim_𝑛→∞𝑥_𝑛=𝑥, then 𝑥_𝑛⪯𝑥 for all 𝑛∈ℕ and if {𝑦_𝑛}⊂𝑋 is a nonincreasing convergent sequence such that lim_𝑛→∞𝑦_𝑛=𝑦, then 𝑦_𝑛≽𝑦 for every 𝑛∈ℕ, by the continuity of 𝐹, we can check easily that the result of Theorem 5 holds also in this case.

Theorem 7. In addition to the hypotheses of Theorem 5, suppose that for every (𝑥,𝑦,𝑧), (𝑥^′,𝑦^′,𝑧^′)∈𝑋×𝑋×𝑋, there exists (𝑢,𝑣,𝑤) such that (𝑥,𝑦,𝑧)⪯(𝑢,𝑣,𝑤) and (𝑥^′,𝑦^′,𝑧^′)⪯(𝑢,𝑣,𝑤). Then one obtains the uniqueness of the fixed point of the third order.

Proof. From (a) of Theorem 5, we know that 𝐹 admits a fixed point of the third order (𝑥,𝑦,𝑧)∈𝑋×𝑋×𝑋; that is, 𝑥=𝐹𝑥,𝑦,𝑥,𝑦=𝐹𝑦,𝑥,𝑦,𝑧=𝐹𝑧,𝑦,𝑧, (44) where 𝑥_𝑛=𝐹^𝑛𝑥₀,𝑦₀,𝑥₀⟶𝑥,𝑦_𝑛=𝐹^𝑛𝑦₀,𝑥₀,𝑦₀𝑧⟶𝑦,_𝑛=𝐹^𝑛𝑧₀,𝑦₀,𝑧₀⟶𝑧as𝑛⟶∞. (45) Suppose that (𝑥^∗,𝑦^∗,𝑧^∗)∈𝑋×𝑋×𝑋 is another fixed point of the third order of 𝐹; that is, 𝑥^∗=𝐹𝑥^∗,𝑦^∗,𝑥^∗,𝑦^∗=𝐹𝑦^∗,𝑥^∗,𝑦^∗,𝑧^∗=𝐹𝑧^∗,𝑦^∗,𝑧^∗. (46) We will prove that 𝜂𝑥,𝑦,𝑧,𝑥^∗,𝑦^∗,𝑧^∗=0, (47) where 𝜂𝑥,𝑦,𝑧,𝑥^∗,𝑦^∗,𝑧^∗=max𝑑𝑥,𝑥^∗,𝑑𝑦,𝑦^∗,𝑑𝑧,𝑧^∗. (48) From the hypothesis of Theorem 7, there exists (𝑢,𝑣,𝑤) such that 𝑥,𝑦,𝑧⪯(𝑢,𝑣,𝑤),𝑥^∗,𝑦^∗,𝑧^∗⪯(𝑢,𝑣,𝑤). (49) Since 𝐹 is a mixed monotone operator, we have 𝐹^𝑛𝑥,𝑦,𝑥,𝐹^𝑛𝑦,𝑥,𝑦,𝐹^𝑛=⪯𝐹𝑧,𝑦,𝑧𝑥,𝑦,𝑧^𝑛(𝑢,𝑣,𝑤),𝐹^𝑛(𝑣,𝑢,𝑣),𝐹^𝑛(),𝐹𝑤,𝑣,𝑤_𝑛𝑥^∗,𝑦^∗,𝑥^∗,𝐹^𝑛𝑦^∗,𝑥^∗,𝑦^∗,𝐹^𝑛𝑧^∗,𝑦^∗,𝑧^∗=𝑥^∗,𝑦^∗,𝑧^∗⪯𝐹^𝑛(𝑢,𝑣,𝑤),𝐹^𝑛(),𝐹𝑣,𝑢,𝑣_𝑛(𝑤,𝑣,𝑤). (50) We have 𝜂𝑥,𝑦,𝑧,𝑥^∗,𝑦^∗,𝑧^∗=𝜂𝐹^𝑛𝑥,𝑦,𝑥,𝐹^𝑛𝑦,𝑥,𝑦,𝐹^𝑛,𝐹𝑧,𝑦,𝑧_𝑛𝑥^∗,𝑦^∗,𝑥^∗,𝐹^𝑛𝑦^∗,𝑥^∗,𝑦^∗,𝐹^𝑛𝑧^∗,𝑦^∗,𝑧^∗≤𝜂𝐹^𝑛𝑥,𝑦,𝑥,𝐹^𝑛𝑦,𝑥,𝑦,𝐹^𝑛,𝐹𝑧,𝑦,𝑧_𝑛(𝑢,𝑣,𝑢),𝐹^𝑛(𝑣,𝑢,𝑣),𝐹^𝑛()𝐹𝑤,𝑣,𝑤+𝜂^𝑛(𝑢,𝑣,𝑢),𝐹^𝑛(𝑣,𝑢,𝑣),𝐹^𝑛(),𝐹𝑤,𝑣,𝑤_𝑛𝑥^∗,𝑦^∗,𝑥^∗,𝐹^𝑛𝑦^∗,𝑥^∗,𝑦^∗,𝐹^𝑛𝑧^∗,𝑦^∗,𝑧^∗≤𝜆^𝑛max𝑑(𝑥,𝑢),𝑑𝑦,𝑣,𝑑(𝑧,𝑤)+𝜆^𝑛max𝑑𝑢,𝑥^∗,𝑑𝑣,𝑦^∗,𝑑𝑤,𝑧^∗⟶0as𝑛⟶∞, (51) which implies that 𝜂𝑥,𝑦,𝑧,𝑥^∗,𝑦^∗,𝑧^∗=0; (52) that is, 𝑥=𝑥^∗,𝑦=𝑦^∗,𝑧=𝑧^∗. (53)
Example 8. Let 𝑋=[0,1] be an ordered set with the natural ordering of real numbers and 𝑑 a usual metric on 𝑋. Let 𝐹∶𝑋×𝑋×𝑋→𝑋 be defined by 𝐹𝑥,𝑦,𝑧=1+𝑥−𝑦+2𝑧8∀𝑥,𝑦,𝑧∈𝑋. (54) It is easy to check that 𝐹 has the mixed monotone property on 𝑋. For 𝑥,𝑦,𝑧,𝑢,𝑣,𝑤∈𝑋 with (𝑥,𝑦,𝑧)⪯(𝑢,𝑣,𝑤), we have 𝑑𝐹()=1𝑥,𝑦,𝑧,𝐹𝑢,𝑣,𝑤8(≤1𝑥−𝑢−𝑦−𝑣+2𝑧−𝑤)8|+|≤3|𝑥−𝑢𝑦−𝑣+2𝑧−𝑤|4|,,|=3max|𝑥−𝑢𝑦−𝑣𝑧−𝑤|4𝑑()(≤3max𝑥,𝑢,𝑑𝑦,𝑣,𝑑𝑧,𝑤)4𝑑()(),𝑑𝐹𝐹,𝑑𝐹𝐹,𝑑𝐹𝐹.max𝑥,𝑢,𝑑𝑦,𝑣,𝑑𝑧,𝑤𝑥,𝑦,𝑥,𝑥,𝑑𝑥,𝑦,𝑥,𝑢𝑦,𝑥,𝑦,𝑦,𝑑𝑦,𝑥,𝑦,𝑣𝑧,𝑦,𝑧,𝑧,𝑑𝑧,𝑦,𝑧,𝑤 (55) Thus (13) is satisfied for 𝜆=3/4∈[0,1). Thus all the conditions of Theorems 5 and 7 are satisfied. Moreover, there exists a unique (𝑥,𝑦,𝑧)=(1/6,1/6,1/6) in 𝑋×𝑋×𝑋 such that 𝐹𝑥,𝑦,𝑥=𝑥,𝐹𝑦,𝑥,𝑦=𝑦,𝐹𝑧,𝑦,𝑧=𝑧. (56)

Corollary 9. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝐹∶𝑋×𝑋×𝑋→𝑋 be a mapping having the mixed monotone property on 𝑋. Suppose that the following conditions hold. (1) There exists 𝑎_𝑖's≥0 for 𝑖=1,2,…,9 and ∑⁹_𝑖=1𝑎_𝑖<1 with 𝑑𝐹𝑥,𝑦,𝑧,𝐹(𝑢,𝑣,𝑤)≤𝑎₁𝑑(𝑥,𝑢)+𝑎₂𝑑𝑦,𝑣+𝑎₃𝑑(𝑧,𝑤)+𝑎₄𝑑𝐹𝑥,𝑦,𝑥,𝑥+𝑎₅𝑑𝐹𝑥,𝑦,𝑥,𝑢+𝑎₆𝑑𝐹𝑦,𝑥,𝑦,𝑦+𝑎₇𝑑𝐹𝑦,𝑥,𝑦,𝑣+𝑎₈𝑑𝐹𝑧,𝑦,𝑧,𝑧+𝑎₉𝑑𝐹𝑧,𝑦,𝑧,𝑤 (57) for all (𝑥,𝑦,𝑧)⪯(𝑢,𝑣,𝑤). (2) There exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑥₀⪯𝐹𝑥₀,𝑦₀,𝑥₀,𝑦₀≽𝐹𝑦₀,𝑥₀,𝑦₀,𝑧₀⪯𝐹𝑧₀,𝑦₀,𝑧₀. (58)(3) If {𝑥_𝑛}⊂𝑋 is a nondecreasing convergent sequence such that lim_𝑛→∞𝑥_𝑛=𝑥, then 𝑥_𝑛⪯𝑥 for all 𝑛∈ℕ, if {𝑦_𝑛}⊂𝑋 is a nonincreasing convergent sequence such that lim_𝑛→∞𝑦_𝑛=𝑦, then 𝑦_𝑛≽𝑦 for every 𝑛∈ℕ, and if 𝑥_𝑛⪯𝑦_𝑛 for every 𝑛∈ℕ, then lim_𝑛→∞𝑥_𝑛⪯lim_𝑛→∞𝑦_𝑛.
Then one has the following.(a) For every initial point (𝑥₀,𝑦₀,𝑧₀)∈𝑋×𝑋×𝑋 such that condition (14) holds, 𝑥_𝑛=𝐹^𝑛𝑥₀,𝑦₀,𝑥₀⟶𝑥,𝑦_𝑛=𝐹^𝑛𝑦₀,𝑥₀,𝑦₀𝑧⟶𝑦,_𝑛=𝐹^𝑛𝑧₀,𝑦₀,𝑧₀⟶𝑧as𝑛⟶∞, (59) where 𝑥,𝑦,and𝑧 satisfy 𝑥=𝐹𝑥,𝑦,𝑥,𝑦=𝐹𝑦,𝑥,𝑦,𝑧=𝐹𝑧,𝑦,𝑧. (60) If 𝑥₀⪯𝑦₀ and 𝑧₀⪯𝑦₀ in condition (14), then 𝑥=𝑦=𝑧 and {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge to the equilibrium of the equation 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+1=𝐹𝑧_𝑛,𝑦_𝑛,𝑧_𝑛,𝑛=0,1,2…. (61)(b) In particular, every solution {𝑢_𝑛} of 𝑢_𝑛+1=𝐹𝑢_𝑛,𝑢_𝑛−1,𝑢_𝑛−2,𝑛=2,3,… (62) such that 𝑢₀,𝑢₁,𝑢₂∈[𝑥₀,𝑦₀] (or [𝑧₀,𝑦₀]) converges to the equilibrium of (18).

Proof. Note that (14) implies that 𝑑𝐹()𝑑()(),𝑑𝐹𝐹,𝑑𝐹𝐹,𝑑𝐹𝐹𝑥,𝑦,𝑧,𝐹𝑢,𝑣,𝑤≤ℎmax𝑥,𝑢,𝑑𝑦,𝑣,𝑑𝑧,𝑤𝑥,𝑦,𝑥,𝑥,𝑑𝑥,𝑦,𝑥,𝑢𝑦,𝑥,𝑦,𝑦,𝑑𝑦,𝑥,𝑦,𝑣𝑧,𝑦,𝑧,𝑧,𝑑𝑧,𝑦,𝑧,𝑤 (63) for all (𝑥,𝑦,𝑧)⪯(𝑢,𝑣,𝑤), where ℎ=∑⁹_𝑖=1𝑎_𝑖<1 and the result follows from Theorem 5.

Corollary 10. In addition to the hypotheses of Corollary 9, suppose that for every (𝑥,𝑦,𝑧), (𝑥^′,𝑦^′,𝑧^′)∈𝑋×𝑋×𝑋, there exists (𝑢,𝑣,𝑤) such that (𝑥,𝑦,𝑧)⪯(𝑢,𝑣,𝑤) and (𝑥^′,𝑦^′,𝑧^′)⪯(𝑢,𝑣,𝑤). Then one obtains the uniqueness of the fixed point of the third order.

Corollary 11. Let (𝑋,⪯) be a partially ordered set and suppose there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝐹∶𝑋×𝑋×𝑋→𝑋 be a mapping having the mixed monotone property on 𝑋. Suppose that the following conditions hold. (1)There exists 𝜆∈[0,1) with 𝑑𝐹()𝑑()(∀⪯().𝑥,𝑦,𝑧,𝐹𝑢,𝑣,𝑤≤𝜆max𝑥,𝑢,𝑑𝑦,𝑣,𝑑𝑧,𝑤)𝑥,𝑦,𝑧𝑢,𝑣,𝑤 (64)(2)There exist 𝑥₀,𝑦₀,𝑧₀∈𝑋 such that 𝑥₀⪯𝐹𝑥₀,𝑦₀,𝑥₀,𝑦₀≽𝐹𝑦₀,𝑥₀,𝑦₀,𝑧₀⪯𝐹𝑧₀,𝑦₀,𝑧₀. (65)(3)If {𝑥_𝑛}⊂𝑋 is a nondecreasing convergent sequence such that lim_𝑛→∞𝑥_𝑛=𝑥, then 𝑥_𝑛⪯𝑥 for all 𝑛∈ℕ; if {𝑦_𝑛}⊂𝑋 is a nonincreasing convergent sequence such that lim_𝑛→∞𝑦_𝑛=𝑦, then 𝑦_𝑛≽𝑦 for every 𝑛∈ℕ, and if 𝑥_𝑛⪯𝑦_𝑛 for every 𝑛, then lim_𝑛→∞𝑥_𝑛⪯lim_𝑛→∞𝑦_𝑛.
Then one has the following.(a) For every initial point (𝑥₀,𝑦₀,𝑧₀)∈𝑋×𝑋×𝑋 such that condition (14) holds, 𝑥_𝑛=𝐹^𝑛𝑥₀,𝑦₀,𝑥₀⟶𝑥,𝑦_𝑛=𝐹^𝑛𝑦₀,𝑥₀,𝑦₀𝑧⟶𝑦,_𝑛=𝐹^𝑛𝑧₀,𝑦₀,𝑧₀⟶𝑧as𝑛⟶∞, (66) where 𝑥,𝑦,and𝑧 satisfy 𝑥=𝐹𝑥,𝑦,𝑥,𝑦=𝐹𝑦,𝑥,𝑦,𝑧=𝐹𝑧,𝑦,𝑧. (67) If 𝑥₀⪯𝑦₀ and 𝑧₀⪯𝑦₀ in condition (14), then 𝑥=𝑦=𝑧 and {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converge to the equilibrium of the equation 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛,𝑧_𝑛+1=𝐹𝑧_𝑛,𝑦_𝑛,𝑧_𝑛,𝑛=0,1,2…. (68)(b) In particular, every solution {𝑢_𝑛} of 𝑢_𝑛+1=𝐹𝑢_𝑛,𝑢_𝑛−1,𝑢_𝑛−2,𝑛=2,3,… (69) such that 𝑢₀,𝑢₁,𝑢₂∈[𝑥₀,𝑦₀] (or [𝑧₀,𝑦₀]) converges to the equilibrium of (18).(c) The following estimates hold: 𝑑𝑥,𝐹^𝑛𝑥₀,𝑦₀,𝑥₀≤𝜆^𝑛𝑑𝐹𝑥1−𝜆max₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝑧₀,𝐹𝑧₀,𝑦₀,𝑧₀,𝑑𝑦,𝐹^𝑛𝑦₀,𝑥₀,𝑦₀≤𝜆^𝑛𝑑𝐹𝑥1−𝜆max₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝑧₀,𝐹𝑧₀,𝑦₀,𝑧₀,𝑑𝑧,𝐹^𝑛𝑧₀,𝑦₀,𝑧₀≤𝜆^𝑛𝑑𝐹𝑥1−𝜆max₀,𝑦₀,𝑥₀,𝑥₀,𝑑𝐹𝑦₀,𝑥₀,𝑦₀,𝑦₀,𝑑𝐹𝑧₀,𝑦₀,𝑧₀,𝑧₀. (70)

Corollary 12. In addition to the hypotheses of Corollary 11, suppose that for every (𝑥,𝑦,𝑧), (𝑥^′,𝑦^′,𝑧^′)∈𝑋×𝑋×𝑋, there exists (𝑢,𝑣,𝑤) such that (𝑥,𝑦,𝑧)⪯(𝑢,𝑣,𝑤) and (𝑥^′,𝑦^′,𝑧^′)⪯(𝑢,𝑣,𝑤). Then one obtains the uniqueness of the fixed point of the third order.

3. Application: Solving a Class of Third-Order Difference Matrix Equations

In this section, we apply our main results to the study of a class of third-order difference matrix equations. At first, we start by fixing some notations and recalling some preliminaries.

We will use the symbol 𝐻(𝑁) for the set of all 𝑁×𝑁 Hermitian matrices. We denote by 𝑃(𝑁) the set of all 𝑁×𝑁 Hermitian positive definite matrices. Instead of 𝑋∈𝑃(𝑁) we will also write 𝑋≻0. Furthermore, 𝑋≽0 means that 𝑋 is positive semidefinite. As a different notation for 𝑋−𝑌≽0,𝑋−𝑍≽0 and 𝑋⪯𝑍⪯𝑌 we will use, respectively, 𝑋≽𝑌,𝑋≽𝑍, and 𝑍∈[𝑋,𝑌]. The symbol ‖⋅‖ denotes the spectral norm, that is, ‖𝐴‖=𝜆^1/2_max(𝐴^∗𝐴), the largest eigenvalue of 𝐴^∗𝐴. We denote by ‖⋅‖₁ the Ky Fan norm defined by ‖𝐴‖₁=∑^𝑁_𝑗=1𝑠_𝑗(𝐴), where 𝑠_𝑗(𝐴),𝑗=1,…,𝑁 are the singular values of 𝐴. For a given 𝑄∈𝑃(𝑁), we define the modified norm ‖⋅‖_1,𝑄 given by ‖𝐴‖_1,𝑄=‖𝑄^1/2𝐴𝑄^1/2‖₁. The set 𝐻(𝑁) endowed with this norm is a complete metric space for any positive definite matrix 𝑄. For any 𝑁×𝑁 matrix 𝐴, we denote by tr(𝐴) the trace of the matrix 𝐴.

The following lemmas will be useful later.

Lemma 13 (see [18]). Let 𝐴≽0 and 𝐵≽0 be 𝑁×𝑁 matrices, then 0≤tr(𝐴𝐵)≤‖𝐴‖tr(𝐵). (71)

Lemma 14 (See [21]). Let 𝐴∈𝐻(𝑁) satisfy −𝐼≺𝐴≺𝐼, then ‖𝐴‖<1.

Finally, we recall the well-known Schauder fixed-point theorem.

Theorem 15 (The Schauder fixed-point theorem). Let 𝑆 be a nonempty, compact, and convex subset of a normed vector space. Every continuous function 𝑓∶𝑆→𝑆 mapping 𝑆 into itself has a fixed point.

Now, we consider the class of third-order difference matrix equations: 𝑈_𝑛+1=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑈_𝑛𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑈_𝑛−1𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑈_𝑛−2𝐶_𝑖,𝑛=2,3,… (72) for given 𝑈₀,𝑈₁,𝑈₂≻0, where 𝑄≻0 and 𝐴_𝑖, 𝐵_𝑖, and 𝐶_𝑖 are 𝑁×𝑁 Hermitian matrices. This type of difference equations often arises from many areas such as ladder networks [22, 23], dynamic programming [24, 25], and control theory [26, 27].

3.1. A Convergence Result

Theorem 16. Suppose that ^𝑚_𝑖=1𝐴_𝑖𝑄𝐴_𝑖≺𝑄4,^𝑚_𝑖=1𝐵_𝑖𝑄𝐵_𝑖≺𝑄2_𝑚𝑖=1,𝐶_𝑖𝑄𝐶_𝑖≺𝑄4. (73) Then, one has the following.(i) Equation (72) has one and only one equilibrium point 𝑋>0.(ii)𝑋∈[𝐹(0,𝑄,0),𝐹(2𝑄,0,2𝑄)], where 𝐹(0,2𝑄,0)=𝑄−2^𝑚_𝑖=1𝐵_𝑖𝑄𝐵_𝑖,𝐹(2𝑄,0,2𝑄)=𝑄+2^𝑚_𝑖=1𝐴_𝑖𝑄𝐴_𝑖+2^𝑚_𝑖=1𝐶_𝑖𝑄𝐶_𝑖. (74)(iii) The sequences (𝑋_𝑘) and (𝑌_𝑘) defined by 𝑋₀=0=𝑍₀, 𝑌₀=2𝑄, and 𝑋_𝑘+1=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑋_𝑘𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑌_𝑘𝐵_𝑖+^𝑚_𝑖=1𝐴_𝑖𝑋_𝑘𝐴_𝑖,𝑌_𝑘+1=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑌_𝑘𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑋_𝑘𝐵_𝑖+^𝑚_𝑖=1𝐴_𝑖𝑌_𝑘𝐴_𝑖,𝑍_𝑘+1=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑍_𝑘𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑌_𝑘𝐵_𝑖+^𝑚_𝑖=1𝐴_𝑖𝑍_𝑘𝐴_𝑖,𝑘=0,1,2,… (75) converge to 𝑋, and the error estimation is given by max𝑋_𝑘+1−𝑋_1,𝑄,𝑌_𝑘+1−𝑋_1,𝑄,𝑍_𝑘+1−𝑍_1,𝑄≤𝜆^𝑘()𝑋1−𝑘max₁−𝑋_01,𝑄,𝑌₁−𝑌_01,𝑄,𝑍₁−𝑍_01,𝑄, (76) for all 𝑘=0,1,…, where 𝜆 is a certain constant in [0,1).(iv) For every 𝑈₀,𝑈₁,𝑈₂∈[0,2𝑄], every solution {𝑈_𝑛} of (72) converges to 𝑋.

Proof. In order to make the proof easy, we divide it into several steps.
Step 1. We claim that there exists a unique (𝑋,𝑌,𝑍)∈𝐻(𝑁)×𝐻(𝑁)×𝐻(𝑁) solution to the system 𝑋=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑋𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑌𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑋𝐶_𝑖,𝑌=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑌𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑋𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑌𝐶_𝑖,𝑍=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑍𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑌𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑍𝐶_𝑖. (77) Consider the mapping 𝐹∶𝐻(𝑁)×𝐻(𝑁)×𝐻(𝑁)→𝐻(𝑁) defined by 𝐹(𝑋,𝑌,𝑍)=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑋𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑌𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑍𝐶_𝑖, (78) for all 𝑋,𝑌,𝑍∈𝐻(𝑁). It is clear that 𝐹 is a mapping having the mixed monotone property with respect to the partial order ⪯. Let 𝑋,𝑌,𝑍,𝑈,𝑉,𝑊∈𝐻(𝑁) such that 𝑋≽𝑈, 𝑌⪯𝑉, and 𝑍≽𝑊. Using Lemma 13, we have ‖𝐹(𝑋,𝑌,𝑍)−𝐹(𝑈,𝑉,𝑊)‖_1,𝑄=tr𝑄^1/2(𝐹(𝑋,𝑌,𝑍)−𝐹(𝑈,𝑉,𝑊))𝑄^1/2=tr𝑄^1/2𝑚_𝑖=1𝐴_𝑖𝑋𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑌𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑍𝐶_𝑖−^𝑚_𝑖=1𝐴_𝑖𝑈𝐴_𝑖+^𝑚_𝑖=1𝐵_𝑖𝑉𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑊𝐶_𝑖𝑄^1/2=^𝑚_𝑖=1tr𝑄^1/2𝐴_𝑖(𝑋−𝑈)𝐴_𝑖+𝐵_𝑖(𝑉−𝑌)𝐵_𝑖+𝐶_𝑖(𝑍−𝑊)𝐶_𝑖𝑄^1/2=^𝑚_𝑖=1tr𝑄^1/2𝐴_𝑖(𝑋−𝑈)𝐴_𝑖𝑄^1/2+tr𝑄^1/2𝐵_𝑖(𝑉−𝑌)𝐵_𝑖𝑄^1/2+tr𝑄^1/2𝐶_𝑖(𝑍−𝑊)𝐶_𝑖𝑄^1/2=^𝑚_𝑖=1tr𝐴_𝑖𝑄𝐴_𝑖(𝑋−𝑈)+tr𝐵_𝑖𝑄𝐵_𝑖()𝐶𝑉−𝑌+tr_𝑖𝑄𝐶_𝑖()=𝑍−𝑊_𝑚𝑖=1tr𝐴_𝑖𝑄𝐴_𝑖𝑄^−1/2𝑄^1/2(𝑋−𝑈)𝑄^1/2𝑄^−1/2+tr𝐵_𝑖𝑄𝐵_𝑖𝑄^−1/2𝑄^1/2(𝑉−𝑌)𝑄^1/2𝑄^−1/2+tr𝐶_𝑖𝑄𝐶_𝑖𝑄^−1/2𝑄^1/2(𝑍−𝑊)𝑄^1/2𝑄^−1/2=^𝑚_𝑖=1tr𝑄^−1/2𝐴_𝑖𝑄𝐴_𝑖𝑄^−1/2𝑄^1/2(𝑋−𝑈)𝑄^1/2+tr𝑄^−1/2𝐵_𝑖𝑄𝐵_𝑖𝑄^−1/2𝑄^1/2(𝑉−𝑌)𝑄^1/2+tr𝑄^−1/2𝐶_𝑖𝑄𝐶_𝑖𝑄^−1/2𝑄^1/2(𝑍−𝑊)𝑄^1/2=tr^𝑚_𝑖=1𝑄^−1/2𝐴_𝑖𝑄𝐴_𝑖𝑄^−1/2𝑄^1/2(𝑋−𝑈)𝑄^1/2+tr^𝑚_𝑖=1𝑄^−1/2𝐵_𝑖𝑄𝐵_𝑖𝑄^−1/2𝑄^1/2(𝑉−𝑌)𝑄^1/2+tr^𝑚_𝑖=1𝑄^−1/2𝐶_𝑖𝑄𝐶_𝑖𝑄^−1/2𝑄^1/2(𝑍−𝑊)𝑄^1/2≤^𝑚_𝑖=1𝑄^−1/2𝐴_𝑖𝑄𝐴_𝑖𝑄^−1/2‖𝑋−𝑈‖_1,𝑄+^𝑚_𝑖=1𝑄^−1/2𝐵_𝑖𝑄𝐵_𝑖𝑄^−1/2‖𝑉−𝑌‖_1,𝑄+^𝑚_𝑖=1𝑄^−1/2𝐶_𝑖𝑄𝐶_𝑖𝑄^−1/2‖𝑍−𝑊‖_1,𝑄≤𝛿max‖𝑋−𝑈‖_1,𝑄,‖𝑉−𝑌‖_1,𝑄,‖𝑋−𝑈‖_1,𝑄, (79) where 𝛿=max^𝑚_𝑖=1𝑄^−1/2𝐴_𝑖𝑄𝐴_𝑖𝑄^−1/2,_𝑚𝑖=1𝑄^−1/2𝐵_𝑖𝑄𝐵_𝑖𝑄^−1/2,^𝑚_𝑖=1𝑄^−1/2𝐶_𝑖𝑄𝐶_𝑖𝑄^−1/2. (80) From (73) and Lemma 14, we have 𝛿<1. Thus, the contractive condition of Theorem 5 is satisfied for all 𝑋,𝑌,𝑍,𝑈,𝑉,𝑊∈𝐻(𝑁) with 𝑋≽𝑈, 𝑌⪯𝑉, and 𝑍≽𝑊. Moreover, from (73), we have 𝑋₀=𝑍₀=0≺𝐹(0,2𝑄,0) and 𝐹(2𝑄,0,2𝑄)≺2𝑄=𝑌₀.
Now, all the hypotheses of Theorem 5 are satisfied. Consequently, there exists (𝑋,𝑌,𝑍)∈𝐻(𝑁)×𝐻(𝑁)×𝐻(𝑁) solution to (77). Since for every 𝑋,𝑌,𝑍∈𝐻(𝑁) there is a greatest lower bound and a least upper bound with respect to the partial order ⪯, we deduce from Theorem 7 the uniqueness of the solution to (77). Then, our claim holds.
Step 2. We claim that 𝑋=𝑌=𝑍=𝑋.
Since 𝑋₀=𝑍₀=0≺𝑌₀=2𝑄 (note that 𝑄 is a positive definite matrix), applying Theorem 5, we obtain the equality 𝑋=𝑌=𝑍. This proves our claim.
From Steps 1 and 2, we know that (72) has a unique equilibrium point 𝑋∈𝐻(𝑁). Now, we need to prove that 𝑋∈𝑃(𝑁). This is the goal of the next step.
Step 3 (proof of (i)–(iii)). Define the mapping 𝐺∶[𝐹(0,2𝑄,0),𝐹(2𝑄,0,2𝑄)]→𝐻(𝑁) by 𝐺𝑋=𝑄+^𝑚_𝑖=1𝐴_𝑖𝑋𝐴_𝑖−^𝑚_𝑖=1𝐵_𝑖𝑋𝐵_𝑖+^𝑚_𝑖=1𝐶_𝑖𝑋𝐶_𝑖, (81) for all 𝑋∈[𝐹(0,2𝑄,0),𝐹(2𝑄,0,2𝑄)].
From (73), one can show that 𝐺([𝐹(0,2𝑄,0),𝐹(2𝑄,0,2𝑄)])⊆[𝐹(0,2𝑄,0),𝐹(2𝑄,0,2𝑄)]. Now, applying the Schauder fixed-point theorem (see Theorem 15), we deduce the existence of a fixed point of 𝐺 in [𝐹(0,2𝑄,0),𝐹(2𝑄,0,2𝑄)]. But a fixed point of 𝐺 is an equilibrium point of (72), and from Step 2, we know that (72) has a unique equilibrium point in 𝐻(𝑁). Consequently, 0≺𝐹(0,2𝑄,0)⪯𝑋⪯𝐹(2𝑄,0,2𝑄). This proves (i) and (ii). For the proof of (iii), we have only to apply the inequalities (19).
Step 4. Let 𝑈₀=0, 𝑈₁=𝑄, and 𝑈₂=2𝑄. Then, we have 𝑈₀,𝑈₁,𝑈₂∈[0,2𝑄], thus by (b) Theorem 5, every solution {𝑈_𝑛} to (72) converges to 𝑋.

3.2. Numerical Example

Consider a third-order difference matrix equation 𝑈_𝑛+1=𝑄+𝐴𝑈_𝑛𝐴−𝐵𝑈_𝑛−1𝐵+𝐶𝑈_𝑛−2𝐶, (82) where 𝑄, 𝐴, 𝐵, 𝐶, 𝑈₀, 𝑈₁, and 𝑈₂ are given by ,,𝑈𝑄=1.50.30.30.31.50.30.30.31.5,𝐴=0.30.050.050.050.30.050.050.050.3𝐵=0.10.20.10.20.10.20.10.20.1,𝐶=0.10.10.10.10.10.10.10.10.1₀=0,𝑈₁=𝑄,𝑈₂=2𝑄, (83)

We are interested to approximate the unique positive equilibrium to (82).

We use Algorithm (iii) of Theorem 16 with 𝑋₀=0=𝑍₀ and 𝑌₀=2𝑄.

After 35 iterations, we get the unique solution 𝑋 given by 𝑋=𝑋₃₅=𝑌₃₅=𝑍₃₅=,𝑈1.6156434875710.318746203393550.335643487570950.318746203393551.53742213234620.318746203393550.335643487570950.318746203393551.615643487571₃₅=.1.61564348758170.318746203405830.335643487581710.318746203405831.53742213236040.318746203405830.335643487581710.318746203405831.6156434875817 (84)

The residual errors are 𝑅(𝑋)₃₅=𝑋₃₅−𝑄+𝐴₁𝑋₃₅𝐴₁+𝐴₂𝑋₃₅𝐴₂−𝐵₁𝑋₃₅𝐵₁−𝐵₂𝑋₃₅𝐵₂=3.792×10⁻¹⁶,𝑅(𝑌)₃₅=𝑌₃₅−𝑄+𝐴₁𝑌₃₅𝐴₁+𝐴₂𝑌₃₅𝐴₂−𝐵₁𝑌₃₅𝐵₁−𝐵₂𝑌₃₅𝐵₂=1.796×10⁻¹⁶,𝑅(𝑍)₃₅=𝑍₃₅−𝑄+𝐴₁𝑍₃₅𝐴₁+𝐴₂𝑍₃₅𝐴₂−𝐵₁𝑍₃₅𝐵₁−𝐵₂𝑍₃₅𝐵₂=2.351×10⁻¹⁶,𝑅(𝑈)₃₅=𝑈₃₅−𝑄+𝐴₁𝑈₃₅𝐴₁+𝐴₂𝑈₃₅𝐴₂−𝐵₁𝑈₃₅𝐵₁−𝐵₂𝑈₃₅𝐵₂=3.352×10⁻¹¹. (85)

The convergence history is given by Figure 1, where (a) corresponds to ‖𝑋_𝑘−𝐹(𝑋_𝑘,𝑋_𝑘,𝑋_𝑘)‖, (b) corresponds to ‖𝑌_𝑘−𝐹(𝑌_𝑘,𝑌_𝑘,𝑌_𝑘)‖, (c) corresponds to ‖𝑍_𝑘−𝐹(𝑍_𝑘,𝑍_𝑘,𝑍_𝑘)‖, and (d) corresponds to ‖𝑈_𝑘−𝐹(𝑈_𝑘,𝑈_𝑘,𝑈_𝑘)‖.

(a)

(b)

(c)

(d)

Figure 1 

Convergence history for (82).