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Existence and uniqueness of fixed point for mixed monotone ternary operators with application
Fixed Point Theory and Applications volume 2014, Article number: 223 (2014)
Abstract
In this paper, partial order theory is used to study the fixed point of a mixed monotone ternary operator A:P\times P\times P\to P. The existence and uniqueness of a fixed point are obtained without assuming the operators to be compact or continuous. In the end, the application to an integral equation is presented. Our results unify, generalize, and complement various known comparable results from the current literature.
MSC:47H05, 47H10.
1 Introduction
Fixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach fixed point theorem. It is well known that mixed monotone operators were introduced by Guo and Lakshmikantham [1] in 1987. Later, Bhaskar and Lakshmikantham [2] introduced the notion of a coupled fixed point and proved some coupled fixed point results under certain conditions, in a complete metric space endowed with a partial order. Their study has not only important theoretical meaning but also wide applications in engineering, nuclear physics, biological chemistry technology, etc. (see [1–8] and the references therein).
Very recently, Harjani et al. [9] have established the existence results of coupled fixed point for mixed monotone operators, and further obtained their applications to integral equations. Berinde and Borcut [10] have introduced the concept of a triple fixed point and proved some related theorems for contractive type operators in partially ordered metric spaces. Zhai [11] has considered mixed monotone operators with convexity and get the existence and uniqueness of a fixed point (A(u,u)=u type) without assuming the operator to be compact or continuous.
Motivated by the work reported in [9–11], the aim of this paper is to discuss the existence and uniqueness of a fixed point (A(u,u,u)=u type) for mixed monotone ternary operators in the context of ordered metric spaces. Our results unify, generalize, and complement various known comparable results from the current literature.
The rest of the paper is organized as follows. In Section 2, we recall some basic definitions and notations which will be used in the sequel. The existence and uniqueness of a fixed point for mixed monotone ternary operators (without assuming the operators to be compact or continuous) are obtained in Section 3. We also present an application in Section 4 to an integral equation to illustrate our results.
2 Preliminaries
In this section, we recall some standard definitions and notations needed in the following section. For the convenience of the reader, we suggest that one refers to [1, 2, 10–14] for details.
Throughout this paper, unless otherwise specified, suppose that (E,\parallel \cdot \parallel ) is a real Banach space which is partially ordered by a cone P\subset E, i.e., x\le y if and only if yx\in P. If x\le y and x\ne y, then we denote x<y or y>x. By θ we denote the zero element of E. Recall that a nonempty closed convex set P\subset E is a cone if it satisfies (i) x\in P, \lambda \ge 0\Rightarrow \lambda x\in P; (ii) x\in P, x\in P\Rightarrow x=\theta.
Further, P is called normal if there exists a constant N>0 such that, for all x,y\in E, \theta \le x\le y implies \parallel x\parallel \le N\parallel y\parallel; in this case N is called the normality constant of P. If {x}_{1},{x}_{2}\in E, the set [{x}_{1},{x}_{2}]=\{x\in E\mid {x}_{1}\le x\le {x}_{2}\} is called the order interval between {x}_{1} and {x}_{2}.
Definition 2.1 (see [10])
A:P\times P\to P is said to be a mixed monotone operator if A(x,y) is monotone nondecreasing in x and monotone nonincreasing in y, that is, for any x,y\in P,
Definition 2.2 (see [11])
An element x\in P is called a fixed point of A:P\times P\to P if
Definition 2.3 (see [10])
A:P\times P\times P\to P is said to be a mixed monotone operator if A(x,y,z) is monotone nondecreasing in x, z and monotone nonincreasing in y, that is, for any x,y,z\in P
Definition 2.4 An element x\in P is called a fixed point of A:P\times P\times P\to P if
3 Main results
In this section we consider the existence and uniqueness of a fixed point for mixed monotone ternary operators in ordered Banach spaces. Our first main result is the following.
Theorem 3.1 Let E be a real Banach space and let P be a normal cone in E. A:P\times P\times P\to P is a mixed monotone ternary operator which satisfies the following:
(H_{1}) for t\in (0,1), x,y\in P, there exists \alpha (t,x,y)\in (1,+\mathrm{\infty}), such that
(H_{2}) there exist {u}_{0},{v}_{0},{m}_{0}\in P, r\in (0,1), such that
Then A has a unique fixed point {u}^{\ast} in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}]. Moreover, constructing successively the sequences
for any initial values {x}_{0},{y}_{0},{z}_{0}\in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}], we have
as n\to \mathrm{\infty}.
Proof Let {w}_{0}=r{v}_{0}, \epsilon ={r}^{\alpha (r,{v}_{0},{u}_{0})1}. Then {w}_{0}\ge {u}_{0}, \epsilon \in (0,1), and
Construct successively the sequences
From (3.3)(3.5) and the mixed monotonicity of A, we have
Next we prove that
From (3.2) and (3.3),
Suppose that when n=k, we have
then when n=k+1, note that {u}_{k}\le {w}_{0}=r{v}_{0}\le {v}_{0}, we obtain
By mathematical induction, we know that (3.8) holds. The same procedure may easily be adapted to obtain
On the other hand, from (3.1),
Suppose that when n=k, we have {w}_{k}\le {\epsilon}^{k}{w}_{k}^{\prime}. Then when n=k+1, in view of (3.1), we obtain
By mathematical induction, we have
By (3.6)(3.10) we get
Noting that P is normal and \epsilon \in (0,1), we have
Further,
Here N is the normality constant.
So, we can claim that \{{u}_{n}\}, \{{w}_{n}\}, and \{{m}_{n}\} are Cauchy sequences. Since E is complete, there exist {u}^{\ast},{w}^{\ast},{m}^{\ast}\in P such that
By (3.6), (3.7), respectively, we know that
and then
Further, \parallel {w}^{\ast}{u}^{\ast}\parallel \le N{\epsilon}^{n}\parallel {w}_{0}{u}_{0}\parallel \to 0 (as n\to \mathrm{\infty}), and thus {w}^{\ast}={u}^{\ast}. Similarly, we get \parallel {w}^{\ast}{m}^{\ast}\parallel \le N{\epsilon}^{n}\parallel {w}_{0}{m}_{0}\parallel \to 0 (as n\to \mathrm{\infty}), and thus {w}^{\ast}={m}^{\ast}. Consequently, {w}^{\ast}={u}^{\ast}={m}^{\ast}. Then we obtain
Letting n\to \mathrm{\infty}, then we get
That is, {u}^{\ast} is a fixed point of A in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}].
In the following, we prove that {u}^{\ast} is the unique fixed point of A in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}]. Suppose that there exists {x}^{\ast}\in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}] such that A({x}^{\ast},{x}^{\ast},{x}^{\ast})={x}^{\ast}. Then {u}_{0}\le {x}^{\ast}\le {w}_{0} and {m}_{0}\le {x}^{\ast}\le {w}_{0}. By mathematical induction and the mixed monotonicity of A, we have
Then from the normality of P, we have {x}^{\ast}={u}^{\ast}.
Moreover, constructing successively the sequences
for any initial values {x}_{0},{y}_{0},{z}_{0}\in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}], we have {u}_{n}\le {x}_{n}, {w}_{n}\ge {y}_{n}, {m}_{n}\le {z}_{n}, n=1,2,\dots . Letting n\to \mathrm{\infty} yields {x}_{n}\to {u}^{\ast}, {y}_{n}\to {u}^{\ast}, {z}_{n}\to {u}^{\ast} as n\to \mathrm{\infty}. □
Remark 3.1 It is evident from (3.1) that for t\in (0,1), x,y\in P, there exists \alpha (t,\frac{1}{t}x,y)\in (1,+\mathrm{\infty}), such that
Remark 3.2 Let \alpha (t,x,y) be a constant \alpha \in (1,+\mathrm{\infty}), then Theorem 3.1 also holds.
Corollary 3.2 Let E be a real Banach space and let P be a normal cone in E. A:P\times P\times P\to P is a mixed monotone ternary operator which satisfies (H_{2}) and, for t\in (0,1), x,y\in P, there exists \alpha \in (1,+\mathrm{\infty}), such that A(tx,y,tx)\le {t}^{\alpha}A(x,y,x). Then A has a unique fixed point {u}^{\ast} in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}]. Moreover, constructing successively the sequences
for any initial values {x}_{0},{y}_{0},{z}_{0}\in [{u}_{0},r{v}_{0}]\cap [{m}_{0},r{v}_{0}], we have {u}_{n}\le {x}_{n}, {w}_{n}\ge {y}_{n}, {m}_{n}\le {z}_{n}, n=1,2,\dots . Letting n\to \mathrm{\infty} yields {x}_{n}\to {u}^{\ast}, {y}_{n}\to {u}^{\ast}, {z}_{n}\to {u}^{\ast} as n\to \mathrm{\infty}.
Following the lines of the proof of Theorem 3.1, we obtain an immediate consequence.
Corollary 3.3 (see [11])
Let E be a real Banach space and let P be a normal cone in E. A:P\times P\to P is a mixed monotone operator which satisfies the following:
(H_{3}) for t\in (0,1), x,y\in P, there exists \alpha (t,x,y)\in (1,+\mathrm{\infty}), such that
(H_{4}) there exist {u}_{0},{v}_{0}\in P, r\in (0,1), such that
Then A has a unique fixed point {u}^{\ast} in [{u}_{0},r{v}_{0}]. Moreover, constructing successively the sequences
for any initial values {x}_{0},{y}_{0}\in [{u}_{0},r{v}_{0}], we have
as n\to \mathrm{\infty}.
Theorem 3.4 Let E be a real Banach space and let P be a normal cone in E. A:P\times P\times P\to P is a mixed monotone ternary operator which satisfies (3.1) and
(H_{5}) for R\in (1,+\mathrm{\infty}), x,y,z\in P there exist \alpha (\frac{1}{R},Rx,y,z),\alpha (\frac{1}{R},x,y,Rz)\in (1,+\mathrm{\infty}) such that
(H_{6}) there exist {u}_{0},{v}_{0},{m}_{0}\in P, R\in (1,+\mathrm{\infty}), such that
Then the operator equation A(w,w,w)=bw has a unique solution {w}^{\ast} in [R{u}_{0},{v}_{0}]\cap [R{m}_{0},{v}_{0}], where b=min\{{R}^{\alpha (\frac{1}{R},R{u}_{0},{v}_{0},{m}_{0})1},{R}^{\alpha (\frac{1}{R},{m}_{0},{v}_{0},R{u}_{0})1}\}. Moreover, constructing successively the sequences
for any initial values {x}_{0},{y}_{0},{z}_{0}\in [R{u}_{0},{v}_{0}]\cap [R{m}_{0},{v}_{0}], we have
as n\to \mathrm{\infty}.
Remark 3.3 Two comments with respect to conditions (3.12) and (3.13) are in order:

(a)
A sufficient condition on A for (3.12) to be satisfied is that for t\in (0,1), x,y,z\in P, there exists \alpha (t,x,y,z)\in (1,+\mathrm{\infty}), such that
A(tx,y,z)\le {t}^{\alpha (t,x,y,z)}A(x,y,z). 
(b)
A sufficient condition on A for (3.13) to be satisfied is that for t\in (0,1), x,y,z\in P, there exists \alpha (t,x,y,z)\in (1,+\mathrm{\infty}), such that
A(x,y,tz)\le {t}^{\alpha (t,x,y,z)}A(x,y,z).
Proof of Theorem 3.4 Let {w}_{0}=R{u}_{0}. Then {v}_{0}\ge {w}_{0}. Note that b>1, from (3.12)(3.14),
Set B(x,y,z)={b}^{1}A(x,y,z), x,y,z\in P. Then from the above inequalities, we have
Also, construct successively the sequences
From (3.18) and the mixed monotonicity of A, we have
Next we prove that
By (3.11) and (3.14), we have
From (3.15)(3.17) and (3.22),
Suppose that when n=k, we have
then when n=k+1, recalling (3.16) and (3.22), we obtain
By mathematical induction, we know that (3.21) holds. The same procedure may easily be adapted to obtain
On the other hand, from (3.1),
Suppose that when n=k, we have {v}_{k}\le {(\frac{1}{b})}^{k}{v}_{k}^{\prime}. Then when n=k+1, in view of (3.1), we obtain
By mathematical induction, we have
By (3.19)(3.24) we get
Note that P is normal and b>1, we have
Further,
Here N is the normality constant.
So, we can claim that \{{w}_{n}\}, \{{v}_{n}\}, and \{{m}_{n}\} are Cauchy sequences. Since E is complete, there exist {w}^{\ast},{v}^{\ast},{m}^{\ast}\in P such that
By (3.19), (3.20), respectively, we know that
and then
Further, \parallel {v}^{\ast}{w}^{\ast}\parallel \le N{(\frac{1}{b})}^{n}\parallel {v}_{0}{w}_{0}\parallel \to 0 (as n\to \mathrm{\infty}), and thus {v}^{\ast}={w}^{\ast}. Similarly, we get \parallel {v}^{\ast}{m}^{\ast}\parallel \le N{(\frac{1}{b})}^{n}\parallel {v}_{0}{m}_{0}\parallel \to 0 (as n\to \mathrm{\infty}), and thus {v}^{\ast}={m}^{\ast}. Consequently, {w}^{\ast}={v}^{\ast}={m}^{\ast}. Then we obtain
Letting n\to \mathrm{\infty}, we get
That is, the operator equation A(w,w,w)=bw has a unique solution {w}^{\ast} in [R{u}_{0},{v}_{0}]\cap [R{m}_{0},{v}_{0}].
In the following, we prove that {w}^{\ast} is the unique solution of A(w,w,w)=bw in [R{u}_{0},{v}_{0}]\cap [R{m}_{0},{v}_{0}]. Suppose that there exists {x}^{\ast}\in [R{u}_{0},{v}_{0}]\cap [R{m}_{0},{v}_{0}] such that A({x}^{\ast},{x}^{\ast},{x}^{\ast})=b{x}^{\ast}. Then {w}_{0}\le {x}^{\ast}\le {v}_{0} and {m}_{0}\le {x}^{\ast}\le {v}_{0}. By mathematical induction and the mixed monotonicity of A, we have
Then from the normality of P, we have {x}^{\ast}={w}^{\ast}.
Moreover, constructing successively the sequences
for any initial values {x}_{0},{y}_{0},{z}_{0}\in [R{u}_{0},{v}_{0}]\cap [R{m}_{0},{v}_{0}], we have {w}_{n}\le {x}_{n}, {v}_{n}\ge {y}_{n}, {m}_{n}\le {z}_{n}, n=1,2,\dots . Letting n\to \mathrm{\infty} yields {x}_{n}\to {w}^{\ast}, {y}_{n}\to {w}^{\ast}, {z}_{n}\to {w}^{\ast} as n\to \mathrm{\infty}. □
From the proof of Theorem 3.4, we can easily obtain the following conclusion.
Corollary 3.5 (see [11])
Let E be a real Banach space and let P be a normal cone in E. A:P\times P\to P is a mixed monotone operator which satisfies (H_{3}) and
(H_{7}) there exist {u}_{0},{v}_{0}\in P, R\in (1,+\mathrm{\infty}) such that
Then the operator equation A(w,w)=bw has a unique solution {w}^{\ast} in [R{u}_{0},{v}_{0}], where b={R}^{\alpha (\frac{1}{R},R{u}_{0},{v}_{0})1}. Moreover, constructing successively the sequences
for any initial values {x}_{0},{y}_{0}\in [R{u}_{0},{v}_{0}], we have
as n\to \mathrm{\infty}.
4 Application
As application of our results, we investigate the solvability of the following integral equation:
with {\alpha}_{1},{\alpha}_{3}>1, {\alpha}_{2}>0.
Put E=C[0,1] (the space of continuous functions defined on [0,1] endowed with supremum norm). Let P=\{x\in E\mid x(t)\ge 0,\mathrm{\forall}t\in [0,1]\}, then E is a Banach space and P is a normal cone. Suppose that k(\tau ,s):[0,1]\times [0,1]\to {R}^{++} ({R}^{++} denotes the positive real numbers) is continuous and 0<{\int}_{0}^{1}k(\tau ,s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le \frac{1}{2}. In the following, we prove that (4.1) has a unique solution.
Consider the integral operator A:P\times P\times P\to P defined by
with {\alpha}_{1},{\alpha}_{3}>1, {\alpha}_{2}>0.
It is clear that A(u,v,m) is a mixed monotone ternary operator. We shall show that A(u,v,m) satisfies (H_{2}) and for t\in (0,1), x,y\in P, there exists \alpha \in (1,+\mathrm{\infty}), such that A(tx,y,tx)\le {t}^{\alpha}A(x,y,x).
In fact, let {u}_{0}(\tau )\equiv 0, {m}_{0}(\tau )\equiv 0, {v}_{0}(\tau )\equiv 1, then
On the other hand, noting that for any t\in (0,1), letting
we obtain
Hence, all the hypotheses of Corollary 3.2 are satisfied. The operator
has a unique fixed point in [{u}_{0},{v}_{0}], i.e., the integral equation (4.1) has a unique solution in [{u}_{0},{v}_{0}].
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Acknowledgements
This research is supported by the Doctoral Fund of Education Ministry of China (20134219120003), the Natural Science Foundation of Hubei Province (2013CFA131), the National Natural Science Foundation of China (71231007) and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (z201302).
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Bu, C., Feng, Y. & Li, H. Existence and uniqueness of fixed point for mixed monotone ternary operators with application. Fixed Point Theory Appl 2014, 223 (2014). https://doi.org/10.1186/168718122014223
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DOI: https://doi.org/10.1186/168718122014223
Keywords
 fixed point
 mixed monotone ternary operator
 normal cone
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