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A fixed point problem under two constraint inequalities
 Mohamed Jleli^{1} and
 Bessem Samet^{1}Email author
https://doi.org/10.1186/s1366301605049
© Jleli and Samet 2016
 Received: 3 December 2015
 Accepted: 25 January 2016
 Published: 3 February 2016
Abstract
Let \((X,d)\) be a metric space. Suppose that the set X is equipped with two partial orders ⪯_{1} and ⪯_{2}. Let \(T,A,B,C,D: X\to X\) be given operators. We provide sufficient conditions for the existence of a fixed point of T satisfying the two constraint inequalities: \(Ax\preceq_{1} Bx\) and \(Cx\preceq_{2} Dx\).
Keywords
 fixed point
 constraint inequalities
 partial order
1 Introduction and basic definitions
Recently there have been many developments concerning the existence of fixed points for operators defined in a metric space equipped with a partial order. This trend was initiated by Turinici [1]. Next, Ran and Reurings [2] extended the Banach contraction principle to continuous monotone operators defined in a partially ordered metric space. They also presented some applications to the existence of positive solutions to certain classes of nonlinear matrix equations. The result obtained in [2] was extended and generalized by many authors in different directions (see [3–11] and the references therein).
The following definitions will be used throughout the paper.
Definition 1.1
 (i)
For every \(x\in X\), we have \(x\preceq x\).
 (ii)For every \(x,y,z\in X\), we have$$x\preceq y, \quad y\preceq z\quad \Longrightarrow\quad x\preceq z. $$
 (iii)For every \(x,y\in X\), we have$$x\preceq y,\quad y\preceq x\quad \Longrightarrow \quad x=y. $$
Definition 1.2
Example 1.3
Definition 1.4
Example 1.5
Example 1.6
Now, we are ready to state and prove our main result.
2 Main result
 (\(\Phi_{1}\)):

φ is a lower semicontinuous function.
 (\(\Phi_{2}\)):

\(\varphi^{1}(\{0\})=\{0\}\).
Our main result in this paper is giving by the following theorem.
Theorem 2.1
 (i)
\(\preceq_{i}\) is dregular, \(i=1,2\).
 (ii)
A, B, C, and D are continuous.
 (iii)There exists \(x_{0}\in X\) such that$$Ax_{0}\preceq_{1} Bx_{0}. $$
 (iv)
T is \((A,B,C,D,\preceq_{1},\preceq_{2})\)stable.
 (v)
T is \((C,D,A,B,\preceq_{2},\preceq_{1})\)stable.
 (vi)There exists \(\varphi\in\Phi\) such that$$Ax\preceq_{1} Bx,\quad Cy\preceq_{2} Dy\quad \Longrightarrow \quad d(Tx,Ty)\leq d(x,y)\varphi\bigl(d(x,y)\bigr). $$
 (I)The sequence \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\) satisfying$$Ax^{*}\preceq_{1} Bx^{*} \quad \textit{and} \quad Cx^{*} \preceq_{2} Dx^{*}. $$
 (II)
The point \(x^{*}\in X\) is a solution to (1.1).
Proof
• Case 1. \(\Lambda=\infty\).
• Case 2. \(\Lambda<\infty\).
In the next section, we present some consequences following from Theorem 2.1.
3 Some consequences
3.1 A fixed point problem under one constraint equality
Corollary 3.1
 (i)
⪯ is dregular.
 (ii)
A and B are continuous.
 (iii)There exists \(x_{0}\in X\) such that$$Ax_{0}\preceq Bx_{0}. $$
 (iv)For all \(x\in X\), we have$$Ax\preceq Bx\quad \Longrightarrow \quad BTx\preceq ATx. $$
 (v)For all \(x\in X\), we have$$Bx\preceq Ax\quad \Longrightarrow \quad ATx\preceq BTx. $$
 (vi)There exists \(\varphi\in\Phi\) such that$$Ax\preceq Bx, \quad By\preceq Ay \quad \Longrightarrow \quad d(Tx,Ty)\leq d(x,y)\varphi \bigl(d(x,y)\bigr). $$
 (I)
The sequence \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\) satisfying \(Ax^{*}=Bx^{*}\).
 (II)
The point \(x^{*}\in X\) is a solution to (3.1).
3.2 A common fixed point problem
Corollary 3.2
 (i)
⪯ is dregular.
 (ii)
B is continuous.
 (iii)There exists \(x_{0}\in X\) such that$$x_{0}\preceq Bx_{0}. $$
 (iv)For all \(x\in X\), we have$$x\preceq Bx \quad \Longrightarrow \quad BTx\preceq Tx. $$
 (v)For all \(x\in X\), we have$$Bx\preceq x\quad \Longrightarrow \quad Tx\preceq BTx. $$
 (vi)There exists \(\varphi\in\Phi\) such that$$x\preceq Bx, \quad By\preceq y \quad \Longrightarrow \quad d(Tx,Ty)\leq d(x,y) \varphi\bigl(d(x,y)\bigr). $$
 (I)
The sequence \(\{T^{n}x_{0}\}\) converges to some \(x^{*}\in X\) satisfying \(x^{*}=Bx^{*}\).
 (II)
The point \(x^{*}\in X\) is a solution to (3.2).
Next, we present an example that illustrates the above result.
Example 3.3
Taking \(B=T\) in Corollary 3.2, we obtain the following fixed point result.
Corollary 3.4
 (i)
⪯ is dregular.
 (ii)
T is continuous.
 (iii)There exists \(x_{0}\in X\) such that$$x_{0}\preceq Tx_{0}. $$
 (iv)For all \(x\in X\), we have$$x\preceq Tx \quad \Longrightarrow \quad T^{2}x\preceq Tx. $$
 (v)For all \(x\in X\), we have$$Tx\preceq x\quad \Longrightarrow \quad Tx\preceq T^{2}x. $$
 (vi)There exists \(\varphi\in\Phi\) such that$$x\preceq Tx,\quad Ty\preceq y \quad \Longrightarrow \quad d(Tx,Ty)\leq d(x,y) \varphi\bigl(d(x,y)\bigr). $$
4 Conclusion
In this paper, we obtained sufficient conditions for the existence of a fixed point of a certain operator under two constraint inequalities with respect to two partial orders. The used technique can also be adapted for any finite number of constraint inequalities and other contractive conditions. An interesting question is the existence of a best proximity point of a certain operator under constraint inequalities. Such a question will be studied in a future work.
Declarations
Acknowledgements
The authors extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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