Fixed point results for Meir-Keeler-type ϕ-α-contractions on partial metric spaces

Abstract

The purpose of this paper is to study fixed point theorems for a map** satisfying the generalized Meir-Keeler-type ϕ-α-contractions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

MSC:47H10, 54C60, 54H25, 55M20.

1 Introduction and preliminaries

Throughout this paper, by ${\mathbb{R}}^{+}$ we denote the set of all nonnegative real numbers, while ℕ is the set of all natural numbers. In 1994, Mattews [1] introduced the following notion of partial metric spaces.

Definition 1 [1]

A partial metric on a nonempty set X is a function $p:X\times X\to {\mathbb{R}}^{+}$ such that for all $x,y,z\in X$,

($p_1$) $x=y$ if and only if $p(x,x)=p(x,y)=p(y,y)$;

($p_2$) $p(x,x)\le p(x,y)$;

($p_3$) $p(x,y)=p(y,x)$;

($p_4$) $p(x,y)\le p(x,z)+p(z,y)-p(z,z)$.

A partial metric space is a pair $(X,p)$ such that X is a nonempty set and p is a partial metric on X.

Remark 1 It is clear that if $p(x,y)=0$, then from ($p_1$) and ($p_2$), $x=y$. But if $x=y$, $p(x,y)$ may not be 0.

Each partial metric p on X generates a ${\mathcal{T}}_0$ topology ${\tau }_p$ on X which has as a base the family of open p-balls $\{B_p(x,\gamma ):x\in X,\gamma >0\}$, where $B_p(x,\gamma )=\{y\in X:p(x,y)<p(x,x)+\gamma \}$ for all $x\in X$ and $\gamma >0$. If p is a partial metric on X, then the function $d_p:X\times X\to {\mathbb{R}}^{+}$ given by

$$d_p(x,y)=2p(x,y)-p(x,x)-p(y,y)$$

is a metric on X.

We recall some definitions of a partial metric space as follows.

Definition 2 [1]

Let $(X,p)$ be a partial metric space. Then

• (1) a sequence $\{x_n\}$ in a partial metric space $(X,p)$ converges to $x\in X$ if and only if $p(x,x)={lim}_{n\to \mathrm{\infty }}p(x,x_n)$;

• (2) a sequence $\{x_n\}$ in a partial metric space $(X,p)$ is called a Cauchy sequence if and only if ${lim}_{m,n\to \mathrm{\infty }}p(x_m,x_n)$ exists (and is finite);

• (3) a partial metric space $(X,p)$ is said to be complete if every Cauchy sequence $\{x_n\}$ in X converges, with respect to ${\tau }_p$, to a point $x\in X$ such that $p(x,x)={lim}_{m,n\to \mathrm{\infty }}p(x_m,x_n)$;

• (4) a subset A of a partial metric space $(X,p)$ is closed if whenever $\{x_n\}$ is a sequence in A such that $\{x_n\}$ converges to some $x\in X$, then $x\in A$.

Remark 2 The limit in a partial metric space is not unique.

Lemma 1 [1, 2]

• (1) $\{x_n\}$ is a Cauchy sequence in a partial metric space $(X,p)$ if and only if it is a Cauchy sequence in the metric space $(X,d_p)$;

• (2) a partial metric space $(X,p)$ is complete if and only if the metric space $(X,d_p)$ is complete. Furthermore, ${lim}_{n\to \mathrm{\infty }}{d}_p(x_n,x)=0$ if and only if $p(x,x)={lim}_{n\to \mathrm{\infty }}p(x_n,x)={lim}_{n\to \mathrm{\infty }}p(x_n,x_m)$.

In recent years, fixed point theory has developed rapidly on partial metric spaces, see [2–10].

In this study, we also recall the Meir-Keeler-type contraction [11] and α-admissible one [12]. In 1969, Meir and Keeler [11] introduced the following notion of Meir-Keeler-type contraction in a metric space $(X,d)$.

Definition 3 Let $(X,d)$ be a metric space, $f:X\to X$. Then f is called a Meir-Keeler-type contraction whenever, for each $\eta >0$, there exists $\gamma >0$ such that

$$\eta \le d(x,y)<\eta +\gamma ⟹d(fx,fy)<\eta .$$

The following definition was introduced in [12].

Definition 4 Let $f:X\to X$ be a self-map** of a set X and $\alpha :X\times X\to {\mathbb{R}}^{+}$. Then f is called α-admissible if

$$x,y\in X,\alpha (x,y)\ge 1⟹\alpha (fx,fy)\ge 1.$$

The purpose of this paper is to study fixed point theorems for a map** satisfying the generalized Meir-Keeler-type ϕ-α-contractions in complete partial metric spaces. Our results generalize or improve many recent fixed point theorems in the literature.

2 Main results

In the article, we denote by Φ the class of functions $\varphi :{{\mathbb{R}}^{+}}^4\to {\mathbb{R}}^{+}$ satisfying the following conditions:

(${\varphi }_1$) ϕ is an increasing and continuous function in each coordinate;

(${\varphi }_2$) for $t\in {\mathbb{R}}^{+}\mathrm{\setminus }\{0\}$, $\varphi (t,t,t,t)\le t$, $\varphi (t,0,0,t)\le t$, $\varphi (0,0,t,\frac{t}{2})\le t$; and $\varphi (t_1,t_2,t_3,t_4)=0$ iff $t_1=t_2=t_3=t_4=0$.

We now state the new notions of generalized Meir-Keeler-type ϕ-contractions and generalized Meir-Keeler-type ϕ-α-contractions in partial metric spaces as follows.

Definition 5 Let $(X,p)$ be a partial metric space, $f:X\to X$ and $\varphi \in \mathrm{\Phi }$. Then f is called a generalized Meir-Keeler-type ϕ-contraction whenever, for each $\eta >0$, there exists $\delta >0$ such that

$$\begin{array}{r}\eta \le \varphi (p(x,y),p(x,fx),p(y,fy),\frac{1}{2}[p(x,fy)+p(y,fx)])<\eta +\delta \\ ⟹p(fx,fy)<\eta .\end{array}$$

Definition 6 Let $(X,p)$ be a partial metric space, $f:X\to X$, $\varphi \in \mathrm{\Phi }$ and $\alpha :X\times X\to {\mathbb{R}}^{+}$. Then f is called a generalized Meir-Keeler-type ϕ-α-contraction if the following conditions hold:

• (1) f is α-admissible;

• (2) for each $\eta >0$, there exists $\delta >0$ such that

  $$\begin{array}{r}\eta \le \varphi (p(x,y),p(x,fx),p(y,fy),\frac{1}{2}[p(x,fy)+p(y,fx)])<\eta +\delta \\ ⟹\alpha (x,x)\alpha (y,y)p(fx,fy)<\eta .\end{array}$$

  (2.1)

Remark 3 Note that if f is a generalized Meir-Keeler-type ϕ-α-contraction, then we have that for all $x,y\in X$,

$$\begin{array}{r}\alpha (x,x)\alpha (y,y)p(fx,fy)\\ \le \varphi (p(x,y),p(x,fx),p(y,fy),\frac{1}{2}[p(x,fy)+p(y,fx)]).\end{array}$$

Further, if $\varphi (p(x,y),p(x,fx),p(y,fy),\frac{1}{2}[p(x,fy)+p(y,fx)])=0$, then $p(fx,fy)=0$. On the other hand, if $\varphi (p(x,y),p(x,fx),p(y,fy),\frac{1}{2}[p(x,fy)+p(y,fx)])>0$, then $\alpha (x,x)\alpha (y,y)p(fx,fy)<\varphi (p(x,y),p(x,fx),p(y,fy),\frac{1}{2}[p(x,fy)+p(y,fx)])$.

We now state our main result for the generalized Meir-Keeler-type ϕ-α-contraction as follows.

Theorem 1 Let $(X,p)$ be a complete partial metric space, and $\varphi \in \mathrm{\Phi }$. If $\alpha :X\times X\to {\mathbb{R}}^{+}$ satisfies the following conditions:

(${\alpha }_1$) there exists $x_0\in X$ such that $\alpha (x_0,x_0)\ge 1$;

(${\alpha }_2$) if $\alpha (x_n,x_n)\ge 1$ for all $n\in \mathbb{N}$, then ${lim}_{n\to \mathrm{\infty }}\alpha (x_n,x_n)\ge 1$;

(${\alpha }_3$) $\alpha :X\times X\to {\mathbb{R}}^{+}$ is a continuous function in each coordinate.

Suppose that $f:X\to X$ is a generalized Meir-Keeler-type ϕ-α-contraction. Then f has a fixed point in X.

Proof Let $x_0$ and let $x_{n+1}=f{x}_n=f^n{x}_0$ for $n=0,1,2,\dots$ . Since f is α-admissible and $\alpha (x_0,x_0)\ge 1$, we have

$$\alpha (f{x}_0,f{x}_0)=\alpha (x_1,x_1)\ge 1.$$

By continuing this process, we get

$$\alpha (x_n,x_n)\ge 1\text{for all }n\in \mathbb{N}\cup \{0\}.$$

(2.2)

If there exists $n_0\in \mathbb{N}$ such that $x_{n_0+1}=x_{n_0}$, then we finished the proof. Suppose that $x_{n+1}\ne x_n$ for any $n=0,1,2,\dots$ . By the definition of the function ϕ, we have $\varphi (p(x_n,x_{n+1}),p(x_n,f{x}_n),p(x_{n+1},f{x}_{n+1}),\frac{1}{2}[p(x_n,f{x}_{n+1})+p(x_{n+1},f{x}_n)])>0$ for all $n\in \mathbb{N}\cup \{0\}$.

Step 1. We shall prove that

$$\underset{n\to \mathrm{\infty }}{lim}p(x_n,x_{n+1})=0,\text{that is}\underset{n\to \mathrm{\infty }}{lim}{d}_p(x_n,x_{n+1})=0.$$

By Remark 3 and ($p_4$), using (2.2), we have

$$\begin{array}{r}p(x_{n+1},x_{n+2})\\ =p(f{x}_n,f{x}_{n+1})\\ \le \alpha (x_n,x_n)\alpha (x_{n+1},x_{n+1})p(f{x}_n,f{x}_{n+1})\\ <\varphi (p(x_n,x_{n+1}),p(x_n,f{x}_n),p(x_{n+1},f{x}_{n+1}),\frac{1}{2}[p(x_n,f{x}_{n+1})+p(x_{n+1},f{x}_n)])\\ =\varphi (p(x_n,x_{n+1}),p(x_n,x_{n+1}),p(x_{n+1},x_{n+2}),\frac{1}{2}[p(x_n,x_{n+2})+p(x_{n+1},x_{n+1})])\\ \le \varphi (p(x_n,x_{n+1}),p(x_n,x_{n+1}),p(x_{n+1},x_{n+2}),\frac{1}{2}[p(x_n,x_{n+1})+p(x_{n+1},x_{n+2})]).\end{array}$$

(2.3)

If $p(x_n,x_{n+1})\le p(x_{n+1},x_{n+2})$, then

$$\begin{array}{rl}p(x_{n+1},x_{n+2})& =p(f{x}_n,f{x}_{n+1})\\ <\varphi (p(x_{n+1},x_{n+2}),p(x_{n+1},x_{n+2}),p(x_{n+1},x_{n+2}),p(x_{n+1},x_{n+2}))\\ \le p(x_{n+1},x_{n+2}),\end{array}$$

which implies a contradiction, and hence $p(x_n,x_{n+1})<p(x_{n-1},x_n)$. From the argument above, we also have that for each $n\in \mathbb{N}$,

$$\begin{array}{rl}p(x_{n+1},x_{n+2})& =p(f{x}_n,f{x}_{n+1})\\ <\varphi (p(x_n,x_{n+1}),p(x_n,x_{n+1}),p(x_n,x_{n+1}),p(x_n,x_{n+1}))\\ \le p(x_n,x_{n+1}).\end{array}$$

(2.4)

Since the sequence $\{p(x_n,x_{n+1})\}$ is decreasing, it must converge to some $\eta \ge 0$, that is,

$$\underset{n\to \mathrm{\infty }}{lim}p(x_n,x_{n+1})=\eta .$$

(2.5)

It follows from (2.4) and (2.5) that

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (p(x_n,x_{n+1}),p(x_n,x_{n+1}),p(x_n,x_{n+1}),p(x_n,x_{n+1}))=\eta .$$

(2.6)

Notice that $\eta =inf\{p(x_n,x_{n+1}):n\in \mathbb{N}\}$. We claim that $\eta =0$. Suppose, to the contrary, that $\eta >0$. Since f is a generalized Meir-Keeler-type ϕ-contraction, corresponding to η use, and taking into account the above inequality (2.6), there exist $\delta >0$ and a natural number k such that

$$\begin{array}{r}\eta \le \varphi (p(x_k,x_{k+1}),p(x_k,x_{k+1}),p(x_k,x_{k+1}),p(x_k,x_{k+1}))<\eta +\delta \\ ⟹\alpha (x_k,x_k)\alpha (x_{k+1},x_{k+1})p(f{x}_k,f{x}_{k+1})<\eta ,\end{array}$$

which implies

$$p(x_{k+1},x_{k+2})=p(f{x}_k,f{x}_{k+1})\le \alpha (x_k,x_k)\alpha (x_{k+1},x_{k+1})p(f{x}_k,f{x}_{k+1})<\eta .$$

So, we get a contradiction since $\eta =inf\{p(x_n,x_{n+1}):n\in \mathbb{N}\}$. Thus we have that

$$\underset{n\to \mathrm{\infty }}{lim}p(x_n,x_{n+1})=0.$$

(2.7)

By ($p_2$), we also have

$$\underset{n\to \mathrm{\infty }}{lim}p(x_n,x_n)=0.$$

(2.8)

Since $d_p(x,y)=2p(x,y)-p(x,x)-p(y,y)$ for all $x,y\in X$, using (2.7) and (2.8), we obtain that

$$\underset{n\to \mathrm{\infty }}{lim}{d}_p(x_n,x_{n+1})=0.$$

(2.9)

Step 2. We show that $\{x_n\}$ is a Cauchy sequence in the partial metric space $(X,p)$, that is, it is sufficient to show that $\{x_n\}$ is a Cauchy sequence in the metric space $(X,d_p)$.

Suppose that the above statement is false. Then there exists $ϵ>0$ such that for any $k\in \mathbb{N}$, there are $n_k,m_k\in \mathbb{N}$ with $n_k>m_k\ge k$ satisfying

$$d_p(x_{m_k},x_{n_k})\ge ϵ.$$

(2.10)

Further, corresponding to $m_k\ge k$, we can choose $n_k$ in such a way that it is the smallest integer with $n_k>m_k\ge k$ and $d(x_{2m_k},x_{2n_k})\ge ϵ$. Therefore

$$d_p(x_{m_k},x_{n_k-2})<ϵ.$$

(2.11)

Now we have that for all $k\in \mathbb{N}$,

$$\begin{array}{rl}ϵ& \le d_p(x_{m_k},x_{n_k})\\ \le d_p(x_{m_k},x_{n_k-2})+d_p(x_{n_k-2},x_{n_k-1})+d_p(x_{n_k-1},x_{n_k})\\ <ϵ+d_p(x_{n_k-2},x_{n_k-1})+d_p(x_{n_k-1},x_{n_k}).\end{array}$$

(2.12)

Letting $k\to \mathrm{\infty }$ in the above inequality and using (2.12), we get

$$\underset{n\to \mathrm{\infty }}{lim}{d}_p(x_{m_k},x_{n_k})=ϵ.$$

(2.13)

On the other hand, we have

$$\begin{array}{rl}ϵ& \le d_p(x_{m_k},x_{n_k})\\ \le d_p(x_{m_k},x_{m_{k+1}})+d_p(x_{m_{k+1}},x_{n_{k+1}})+d_p(x_{n_{k+1}},x_{n_k})\\ \le d_p(x_{m_k},x_{m_{k+1}})+d_p(x_{m_{k+1}},x_{m_k})+d_p(x_{m_k},x_{n_k})+d_p(x_{n_k},x_{n_{k+1}})+d_p(x_{n_{k+1}},x_{n_k}).\end{array}$$

Letting $n\to \mathrm{\infty }$, we obtain that

$$\underset{n\to \mathrm{\infty }}{lim}{d}_p(x_{m_{k+1}},x_{n_{k+1}})=ϵ.$$

(2.14)

Since $d_p(x,y)=2p(x,y)-p(x,x)-p(y,y)$ and using (2.13) and (2.14), we have that

$$\underset{n\to \mathrm{\infty }}{lim}p(x_{m_k},x_{n_k})=\frac{ϵ}{2}$$

(2.15)

and

$$\underset{n\to \mathrm{\infty }}{lim}p(x_{m_{k+1}},x_{n_{k+1}})=\frac{ϵ}{2}$$

(2.16)

By Remark 3 and ($p_4$), we have

$$\begin{array}{r}p(x_{m_{k+1}},x_{n_{k+1}})\\ =p(f{x}_{m_k},f{x}_{n_k})\\ \le \alpha (x_{m_k},x_{m_k})\alpha (x_{n_k},x_{n_k})p(f{x}_{m_k},f{x}_{n_k})\\ <\varphi (p(x_{m_k},x_{n_k}),p(x_{m_k},f{x}_{m_k}),p(x_{n_k},f{x}_{n_k}),\frac{1}{2}[p(x_{m_k},f{x}_{n_k})+p(x_{n_k},f{x}_{m_k})])\\ =\varphi (p(x_{m_k},x_{n_k}),p(x_{m_k},x_{m_k+1}),p(x_{n_k},x_{n_{k+1}}),\\ \frac{1}{2}[p(x_{m_k},x_{n_k+1})+p(x_{n_k},x_{m_k+1})]).\end{array}$$

(2.17)

Since

$$p(x_{m_k},x_{n_k+1})\le p(x_{m_k},x_{m_k+1})+p(x_{m_k+1},x_{n_k+1})-p(x_{m_k+1},x_{m_k+1})$$

(2.18)

and

$$p(x_{n_k},x_{m_k+1})\le p(x_{n_k},x_{n_k+1})+p(x_{n_k+1},x_{m_k+1})-p(x_{n_k+1},x_{n_k+1}).$$

(2.19)

Taking into account the above inequalities (2.8), (2.17), (2.18) and (2.19), letting $k\to \mathrm{\infty }$, we have

$$\frac{ϵ}{2}<\varphi (\frac{ϵ}{2},0,0,\frac{ϵ}{2})\le \frac{ϵ}{2},$$

which implies a contradiction. Thus, $\{x_n\}$ is a Cauchy sequence in the metric space $(X,d_p)$.

Step 3. We show that f has a fixed point ν in ${\bigcap }_{i=1}^m{A}_i$.

Since $(X,p)$ is complete, then from Lemma 1, we have that $(X,d_p)$ is complete. Thus, there exists $\nu \in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}{d}_p(x_n,\nu )=0.$$

Moreover, it follows from Lemma 1 that

$$p(\nu ,\nu )=\underset{n\to \mathrm{\infty }}{lim}p(x_n,\nu )=\underset{n,m\to \mathrm{\infty }}{lim}p(x_n,x_m).$$

(2.20)

On the other hand, since the sequence $\{x_n\}$ is a Cauchy sequence in the metric space $(X,d_p)$, we also have

$$\underset{n\to \mathrm{\infty }}{lim}{d}_p(x_n,x_m)=0.$$

Since $d_p(x,y)=2p(x,y)-p(x,x)-p(y,y)$, we can deduce that

$$\underset{n\to \mathrm{\infty }}{lim}p(x_n,x_m)=0.$$

(2.21)

Using (2.20) and (2.21), we have

$$p(\nu ,\nu )=\underset{n\to \mathrm{\infty }}{lim}p(x_n,\nu )=\underset{n\to \mathrm{\infty }}{lim}p(x_{n_k},\nu )=0.$$

Again, by Remark 3, ($p_4$), and the conditions of the map** α, we have

$$\begin{array}{rl}p(x_{n+1},f\nu )& =p(f{x}_n,f\nu )\\ \le \alpha (x_n,x_n)\alpha (\nu ,\nu )p(f{x}_n,f\nu )\\ <\varphi (p(x_n,\nu ),p(x_n,f{x}_n),p(\nu ,f\nu ),\frac{1}{2}[p(x_n,f\nu )+p(\nu ,f{x}_n)])\\ =\varphi (p(x_n,\nu ),p(x_n,x_{n+1}),p(\nu ,f\nu ),\frac{1}{2}[p(x_n,f\nu )+p(\nu ,x_{n+1})]).\end{array}$$

(2.22)

Letting $n\to \mathrm{\infty }$ in (2.22), we get

$$p(\nu ,f\nu )<\varphi (0,0,p(\nu ,f\nu ),\frac{1}{2}p(\nu ,f\nu ))\le p(\nu ,f\nu ),$$

a contradiction. So, we have $p(\nu ,f\nu )=0$, that is, $f\nu =\nu$. □

We give the following example to illustrate Theorem 2.

Example 1 Let $X=[0,1]$. We define the partial metric p on X by

$$p(x,y)=max\{x,y\}.$$

Let $\alpha :[0,1]\times [0,1]\to {\mathbb{R}}^{+}$ be defined as

$$\alpha (x,y)=1+x+y,$$

let $f:X\to X$ be defined as

$$f(x)=\frac{1}{16}{x}^2,$$

and, let $\varphi :{{\mathbb{R}}^{+}}^4\to {\mathbb{R}}^{+}$ denote

$$\psi (t_1,t_2,t_3,t_4)=\frac{1}{2}\cdot max\{t_1,t_2,t_3,\frac{1}{2}{t}_4\}.$$

Then f is α-admissible.

Without loss of generality, we assume that $x>y$ and verify the inequality (2.1). For all $x,y\in [0,1]$ with $x>y$, we have

$$\begin{array}{c}\alpha (x,x)\alpha (y,y)p(fx,fy)\ge \frac{1}{16}{x}^2,\hfill \\ p(x,y)=x,p(x,fx)=x,p(y,fy)=y\text{and}\hfill \\ \begin{array}{rl}\frac{1}{2}[p(x,fy)+p(y,fx)]& =\frac{1}{2}[max\{x,y^2\}+max\{y,x^2\}]\\ \le \frac{1}{2}[max\{x,y\}+max\{y,x\}]\\ <x,\end{array}\hfill \end{array}$$

and hence $\varphi (p(x,y),p(x,fx),p(y,fy),\frac{1}{2}[p(x,fy)+p(y,fx)])=\frac{1}{2}x$. Therefore, all the conditions of Theorem 1 are satisfied, and we obtained that 0 is a fixed point of f.

If we let

$$\alpha (x,y)=1\text{for }x,y\in X,$$

then it is easy to get the following theorem.

Theorem 2 Let $(X,p)$ be a complete partial metric space and $\varphi \in \mathrm{\Phi }$. Suppose that $f:X\to X$ is a generalized Meir-Keeler-type ϕ-contraction. Then f has a fixed point in X.