The objective of this paper is to establish a theorem involving a pair of weakly compatible mappings fulfilling a contractive condition of rational type in the context of dislocated quasi metric space. Besides we proved the existence and uniqueness of coupled coincidence and coupled common fixed point for such mappings. This work offers extension as well as considerable improvement of some results in the existing literature. Lastly, an illustrative example is given to validate our newly proved results.
The concept of dislocated metric space was introduced by Hitzler [1] in an effort to generalize the well known Banach contraction principle. Later his work was generalized by Zeyada [2] and many papers covering fixed point results for a single and a pair of mappings satisfying various types of contraction conditions are also published, see [2,3,4]. Similarly, Bhaskar and Lakshmikantham [5] introduced the concept of coupled fixed point for non-linear contractions in partially ordered metric spaces. After wards, Lakshmikantham and Ciric [6] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in a complete partially ordered metric space. This area of research has attracted the interest of many researchers and a number of works has been published in different spaces, see [7,8,9,10]. Most recently, Mohammad et al., [11] has obtained coupled fixed point finding in the context of dislocated quasi metric space. In this paper, we have established and proved existence and uniqueness of coupled coincidence and coupled common fixed points for a pair of maps in the context of dislocated quasi metric spaces.
Definition 1. [1] Let \(X\) be a non-empty set and let \(d : X\times X\rightarrow \Re^{+}\cup \{0\}\) be a function satisfying the conditions
Definition 2. [2] A sequence \(\{x_n\}\) in a dislocated quasi metric space \((X, d)\) is said to converge to a point \(x \in X\) if and only if \(\lim\limits_{n\to\infty} d(x_{n}, x)=\lim\limits_{n\to\infty} d(x,x_{n}) = 0.\)
Definition 3. [2] A sequence \(\{x_n\}\) in a dislocated quasi metric space \((X, d)\) is called a Cauchy sequence if for every \(\epsilon>0\), there exists a positive integer \(n_{0}\) such that for \(m,n>n_{0}\), we have \(d(x_{n},x_{m})< \epsilon\). That is, \(\lim\limits_{n,m\to\infty} d(x_{n}, x_{m})=0.\)
Definition 4. [2] A dislocated quasi metric space is called complete if every Cauchy sequence converges to an element in the same metric space.
Definition 5. [12] Let \((X,d)\) be a metric space and \(T : X \rightarrow X\) be a self-map, then \(T\) is said to be a contraction mapping if there exists a constant \(k \in [0, 1)\) called a contraction factor, such that \(d(Tx,Ty) \leq kd(x,y)\)for all \(x,y \in X.\)
Definition 6. [12] Let X be a nonempty set and \(T : X \rightarrow X\) a self-map. We say that x is a fixed point of T if Tx = x.
Theorem 7. [12] Suppose \((X,d)\) be a complete metric space and \(T : X\rightarrow X\) be a contraction, then \(T\) has a unique fixed point.
Definition 8.[5] An element \((x,y) \in X\times X\) , where \(X\) is any non-empty set, is called a coupled fixed point of the mapping \(F:X\times X \rightarrow X\) if \(F(x,y)=x\) and \(F(y,x)=y.\)
Definition 9. [6] An element \((x, y) \in X \times X\) is called a coupled coincidence point of the mappings \(F:X \times X \rightarrow X\) and \(g : X \rightarrow X\) if F(x, y) = g(x) and F(y, x) = g(y), and \((gx,gy)\) is called coupled point of coincidence.
Definition 10. [6] An element \((x,y) \in X\times X\), where \(X\) is any non-empty set, is called a coupled common fixed point of the mappings \(F:X\times X \rightarrow X\) and and \(g : X \rightarrow X\) if \(F(x,y)=g(x)=x\) and \(F(y,x)=g(y)=y\).
Definition 11. [6] The mappings \(F : X \times X \rightarrow X\) and \(g : X \rightarrow X\) are called commutative if \(g(F(x, y)) = F(gx, gy)\) for all \(x,y \in X\).
Definition 12. [6] The mappings \(F : X \times X \rightarrow X\) and \(g : X \rightarrow X\) are called w-Compatible if \(g(F(x, y)) = F(gx, gy)\) and \(g(F(y, x)) = F(gy, gx)\) whenever \(gx = F(x, y)\) and \(gy = F(y, x)\).
Theorem 13. [11] Let \((X,d)\) be a complete dislocated quasi-metric space and \(T : X \rightarrow X\) be a continuous mapping satisfying the following rational type contractive condition \begin{align*} &d[T(x,y),T(u,v)] \leq a_{1}\left[d(x,u)+d(y,v)\right]+a_{2}\left[d\left(x,T(x,y)\right)+d(u,T(u,v))\right]+a_{3}\left[d\left(x,T(u,v)\right)+d\left(u,T(x,y)\right)\right]\\ &\ +a_{4}\left[\frac{d\left(x,T(x,y)\right)d\left(u,T(u,v)\right)}{d(x,u)+d(y,v)}\right] +a_{5}\left[\frac{\left(d(x,u)+d(y,v)\right)\times\left(d\left(x,T(x,y)\right)+d\left(u,T(u,v)\right)\right)}{1+d(x,u)+d(y,v)}\right]\\ &\ +a_{6}\left[\frac{d\left(x,T(x,y)\right)+d\left(x,T(u,v)\right)}{1+d\left(u,T(u,v)\right)d\left(u,T(x,y)\right)}\right] \end{align*} for all \(x,y,u,v \in X\) and \(a_{1}, a_{2}, a_{3},a_{4}, a_{5}\), and \(a_{6}\) are non-negative constants with \(2(a_{1}+a_{2}+a_{5})+4(a_{3}+a_{6})+a_{4}< 1\), then \(T\) has a unique coupled fixed point in \(X\times X\).
Theorem 14. Let \((X,d)\) be a dislocated quasi-metric space and \(T : X\times X \rightarrow X\) and \(g : X \rightarrow X\) be a continuous and commutative mappings satisfying the following rational type contractive condition
Proof. Let \(x_{0}\) and \(y_{0} \in X\) and set \(gx_{1}=T(x_{0},y_{0}) \mbox{ and }gy_{1}=T(y_{0},x_{0}).\) This is possible since \(T(X\times X) \subseteq g(X)\). Proceeding this way, we can construct two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in \(X\) such that \(gx_{n+1}=T(x_{n},y_{n}) \mbox{ and } gy_{n+1}=T(y_{n},x_{n}).\) Consider \(d(gx_{n},gx_{n+1})=d\bigl[T(x_{n-1},y_{n-1}),T(x_{n},y_{n})\bigr].\) This is in order to show that \(\{gx_{n}\}\) and \(\{gy_{n}\}\) are Cauchy sequences in \(g(X)\). Now applying (1), we get \begin{eqnarray*} && d(gx_{n},gx_{n+1}) \leq a_{1} \left[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\right] +a_{2}\left[d(gx_{n-1},T(x_{n-1},y_{n-1}))+d(gx_{n},T(x_{n},y_{n}))\right]\nonumber\\&&{} +a_{3}\left[d(gx_{n-1},T(x_{n},y_{n}))+d(gx_{n},T(x_{n-1},y_{n-1}))\right]+a_{4}\left[\frac{d(gx_{n-1},T(x_{n-1},y_{n-1}))d(gx_{n},T(x_{n},y_{n}))}{d(gx_{n-1},gx_{n}) +d(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{5}\left[\frac{\left(d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\right)\times \left(d\bigl(gx_{n-1},T(x_{n-1},y_{n-1})) +d\left(gx_{n},T(x_{n},y_{n})\right)\right)}{1+d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{6}\left[\frac{d(gx_{n-1},T(x_{n-1},y_{n-1}))+d(gx_{n-1},T(x_{n},y_{n}))}{1+d(gx_{n},T(x_{n},y_{n})) d(gx_{n},T(x_{n-1},y_{n-1}))}\right] +a_{7}\left[\frac{d(gx_{n-1},T(x_{n-1},y_{n-1}))d(gx_{n},T(x_{n},y_{n}))}{1+d(gx_{n-1},gx_{n}) +d(gx_{n},T(x_{n},y_{n}))}\right]. \end{eqnarray*} At this point, we are going to make use of the definitions of the sequences \(\{gx_{n}\}\) and \(\{gy_{n}\}\) to get \begin{eqnarray*} && d(gx_{n},gx_{n+1}) \leq a_{1} \left[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\right]+a_{2}\left[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\right]\nonumber\\&&{} +a_{3}\left[d(gx_{n-1},gx_{n+1})+d(gx_{n},gx_{n})\right] +a_{4}\left[\frac{d(gx_{n-1},gx_{n})d(gx_{n},gx_{n+1})}{d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{5}\left[\frac{(d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n}))\times(d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1}))} {1+d(gx_{n-1},gx_{n})+(gy_{n-1},gy_{n})}\right]\nonumber\\&&{} +a_{6}\left[\frac{d(gx_{n-1},gx_{n})+d(gx_{n-1},gx_{n+1})}{1+d(gx_{n},gx_{n+1})d(gx_{n},gx_{n})}\right] +a_{7}\left[\frac{d(gx_{n-1},gx_{n})d(gx_{n},gx_{n+1})}{1+d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})}\right]. \end{eqnarray*} Applying the triangle inequality and the fact that \(d(x,y) \geq 0\), we obtain \begin{eqnarray*} && d(gx_{n},gx_{n+1}) \leq a_{1} \bigl[d(gx_{n-1},gx_{n})+d(gy_{n-1},gy_{n})\bigr]+a_{2}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\bigr]\nonumber\\ &&{}+a_{3}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})+d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\bigr]+a_{4}d(gx_{n},gx_{n+1})\nonumber\\ &&{}+a_{5}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})\bigr]+a_{6}\bigl[d(gx_{n-1},gx_{n})+d(gx_{n},gx_{n+1})+d(gx_{n-1},gx_{n})\nonumber\\ &&{}+d(gx_{n},gx_{n+1})\bigr]+a_{7}d(gx_{n-1},gx_{n}). \end{eqnarray*} Simplification yields \[ \alpha d(gx_{n},gx_{n+1}) \leq \beta d(gx_{n-1},gx_{n})+a_{1} d(gy_{n-1},gy_{n}) \] where \(\alpha = 1-(a_{2}+2a_{3}+a_{4}+a_{5}+2a_{6}),\) and \(\beta = a_{1}+a_{2}+2a_{3}+a_{5}+2a_{6}+a_{7}.\) It follows that
Similarly, we can show that
Similarly, we can show that \(g(y)=T(y,x)\). Hence, \((gx,gy)\) is coupled point of coincidence of \(T\) and \(g\). Now, we claim that \((gx,gy)\) is the unique coupled point of coincidence of \(T\) and \(g\). Suppose, we have another coupled point of coincidence say \((gx_{1},gy_{1})\) where \((x_{1},y_{1}) \in X^{2}\) with \(gx_{1}=T(x_{1},y_{1})\) and \(gy_{1}=T(y_{1},x_{1})\).
Using (1), we have \begin{align*} &d(gx,gx) = d\left[T(x,y),T(x,y)\right] \leq a_{1}\left[d(gx,gx)+d(gy,gy)\right]+a_{2}\left[d(gx,gx)+d(gx,gx)\right] +a_{3}\left[d(gx,gx)+d(gx,gx)\right]\\&{} +a_{4}\left[\frac{d(gx,gx)d(gx,gx)}{d(gx,gx)+d(gy,gy)}\right]+a_{5}\left[\frac{[d(gx,gx)+d(gy,gy)][d(gx,gx)+d(gx,gx)]}{1+d(gx,gx)+d(gy,gy)}\right]+a_{6}\left[\frac{d(gx,gx)+d(gx,gx)} {1+d(gx,gx)d(gx,gx)}\right]\\&{} +a_{7}\left[\frac{d(gx,gx)d(gx,gx)}{1+d(gx,gx)+d(gx,gx)}\right]. \end{align*} Now, we have \begin{align*} & d(gx,gx)\leq a_{1}\bigl[d(gx,gx)+d(gy,gy)\bigr] +a_{2}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{3}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{4}d(gx,gx)\\&{}+a_{5}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{6}\bigl[d(gx,gx)+d(gx,gx)\bigr]+a_{7}d(gx,gx). \end{align*} It follows thatNext, we show that \(gx=gy\).
\begin{align*} & d(gx,gy)= d\bigl[T(x,y),T(y,x)\bigr] \leq a_{1}\bigl[d(gx,gy)+d(gy,gx)\bigr] +a_{2}\Bigl[d\bigl(gx,T(x,y)\bigr)+d\bigl(gy,T(y,x)\bigr)\Bigr]\nonumber\\&{}+a_{3}\Bigl[d\bigl(gx,T(y,x)\bigr)+d\bigl(gy,T(x,y)\bigr)\Bigr] +a_{4}\left[\frac{d\bigl(gx,T(x,y)\bigr)d\bigl(gy,T(y,x)\bigr)}{d(gx,gy)+d(gy,gx)}\right]\nonumber\\ &{}+a_{5}\Biggl[\frac{\Bigl[d(gx,gy)+d(gy,gx)\Bigr]\Bigl[d(gx,T(x,y))+d(gy,T(y,x))\Bigr]}{1+d(gx,gy)+d(gy,gx)}\Biggr]\nonumber\\ &{}+a_{6}\left[\frac{d\bigl(gx,T(x,y)\bigr)+d\bigl(gx,T(y,x)\bigr)}{1+d\bigl(gy,T(y,x)\bigr)d\bigl(gy,T(y,x)\bigr)}\right]+a_{7}\left[\frac{d\bigl(gx,T(x,y)\bigr)d\bigl(gy,T(y,x)\bigr)}{1+d(gx,gy)+d\bigl(gy,T(y,x)\bigr)}\right].\nonumber \end{align*} Using (1) and the fact that \(gx=T(x,y)\) and \(gy=T(y,x)\), we have \begin{align*} &d(gx,gy)\leq a_{1}\bigl[d(gx,gy)+d(gy,gx)\bigr] +a_{2}\bigl[d(gx,gx)+d(gy,gy)\bigr]+a_{3}\bigl[d(gx,gy)+d(gy,gx)\bigr]\\ &{}+a_{4}\left[\frac{d(gx,gx)d(gy,gy)}{d(gx,gy)+d(gy,gx)}\right] +a_{5}\left[\frac{\Bigl[d(gx,gy)+d(gy,gx)\Bigr]\Bigl[d(gx,gx)+d(gy,gy)\Bigr]}{1+d(gx,gy)+d(gy,gx)}\right]\\ &{}+a_{6}\left[\frac{d(gx,gx)+d(gx,gy)}{1+d(gy,gy)d(gy,gy)}\right] +a_{7}\left[\frac{d(gx,gx)d(gy,gy)}{1+d(gx,gy)+d(gy,gy)}\right].\nonumber \end{align*} Thus, we haveRemark 1. If we take \(g=I\) (the identity map) and \(a_{7}=0\) in Theorem 14, we get Theorem 13 of [11].
The following example supports our main theorem.
Example 1. Let \(X=[0,1)\) and \(d : X\times X\rightarrow \Re^{+}\) be defined by \(d(x,y)=|x-y|+|y|\) for all \(x,y \in X\). Then \((X,d)\) is \(dq\)-metric space. We define the functions \(T : X\times X\rightarrow X\) and \(g : X\rightarrow X\) by \[gx=\begin{cases} \frac{1}{3}x &\quad \text{ if \(0\leq x< \frac{9}{10}\),} \\ \frac{3}{10} &\quad \text{ if \(\frac{9}{10}\leq x < 1,\)} \end{cases} \] and \[T(x,y)=\begin{cases} \frac{x+y}{27} &\quad \text{ if \(0\leq x,y< \frac{9}{10},\)} \\ \frac{1}{30}y &\quad \text{ if \(\frac{9}{10}\leq x< 1\) and \(0 \leq y < \frac{9}{10},\)}\\ \frac{1}{30}x &\quad \text{ if \(\frac{9}{10}< y< 1\) and \(0\leq x< \frac{9}{10},\)}\\ \frac{1}{15} &\quad \text{ if \(\frac{9}{10}\leq x< 1\) and \(\frac{9}{10}\leq y < 1.\)} \end{cases} \] Clearly \(T\) and \(g\) are continuous, \(T(X\times X) \subseteq g(X)\), and \(g(X)\) is complete. Following four cases will arise for \(x\), \(u\), \(v\), and \(y\);
Case 1: For \(0\leq x, u, y, v< \frac{9}{10}\), we have \begin{eqnarray*} d[T(x,y),T(u,v)]&=& d\left(\frac{x+y}{27},\frac{u+v}{27}\right) \nonumber\\&=&\left|\frac{x+y}{27}-\frac{u+v}{27}\right|+\left|\frac{u+v}{27}\right|\nonumber\\ &=& \left|\frac{x}{27}+\frac{y}{27}-\frac{u}{27}-\frac{v}{27}\right|+\left|\frac{u}{27}+\frac{v}{27}\right|\nonumber\\ &\leq & \left|\frac{x}{27}-\frac{u}{27}\right|+\left|\frac{y}{27}-\frac{v}{27}\right|+\left|\frac{u}{27}\right|+\left|\frac{v}{27}\right|\nonumber\\ &=&\frac{1}{9}\left[\left(\left|\frac{x}{3}-\frac{u}{3}\right|+\left|\frac{u}{3}\right|\right)+\left(\left|\frac{y}{3}-\frac{v}{3}\right|+\left|\frac{v}{3}\right|\right)\right]\nonumber\\ &\leq & \frac{1}{9}\bigl[d(gx,gu)+d(gy,gv)\bigr]\nonumber\\ &\leq & \frac{2}{9}\bigl[d(gx,gu)+d(gy,gv)\bigr]. \end{eqnarray*} Similarly, for Cases (2) to (4), we obtain \begin{equation*} d\bigl[T(x,y),T(u,v)\bigr]\leq \frac{2}{9}\bigl[d(gx,gu)+d(gy,gv)\bigr]. \end{equation*} Hence all the conditions of Theorem 13 are satisfied with \(a_{1}=\frac{2}{9}, a_{2}=\frac{1}{120}, a_{3}=\frac{1}{64}, a_{4}=\frac{1}{80}, a_{5}=\frac{1}{100}, a_{6}=\frac{1}{128}\), and \(a_{7}=\frac{1}{32}\). Therefore, \(T\) and \(g\) have unique coupled point of coincidence and unique coupled common fixed point which are \((g0,g0)\) and \((0,0)\) respectively. This is due to the fact that \(g T(0,0)=T(g0,g0)=T(0,0)=0.\)