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Fixed point theorems for generalized weakly contractive mappings in metric spaces with applications

Fixed Point Theory and Applications20182018:3

https://doi.org/10.1186/s13663-018-0628-1

  • Received: 15 October 2017
  • Accepted: 9 January 2018
  • Published:

Abstract

In this paper, motivated and inspired by Samet et al., we introduce the notion of generalized weakly contractive mappings in metric spaces and prove the existence and uniqueness of fixed point for such mappings, and we obtain a coupled fixed point theorem in metric spaces. These theorems generalize many previously obtained fixed point results. An example is given to illustrate the main result. Finally, we give applications of our results to fixed point results in partial metric spaces.

Keywords

  • fixed point
  • coupled fixed point
  • generalized weak contractive mapping

MSC

  • 47H10
  • 54H25

1 Introduction and preliminaries

In 1977, Alber et al. [1] generalized Banach’s contraction principle by introducing the concept of weak contraction mappings in Hilbert spaces. Weak contraction principle states that every weak contraction mapping on a complete Hilbert space has a unique fixed point. Rhoades [2] extended weak contraction principle in Hilbert spaces to metric spaces. Since then, many authors (for example, [314]) obtained generalizations and extensions of the weak contraction principle. Khan et al. [15] obtained fixed point theorems in metric spaces by introducing the concept of altering distance functions.

In particular, Choudhury et al. [8] obtained a generalization of the weak contraction principle in metric spaces by using altering distance functions as follows.

Theorem 1.1

([8])

Suppose that a mapping \(g:X\to X\), where X is a metric space with metric d, satisfies the following condition:
$$\begin{aligned} &\psi\bigl(d(gx,gy)\bigr) \\ &\quad \leq \psi\biggl(\max\biggl\{ d(x,y),d(x,gx),d(y,gy), \frac{1}{2}\bigl\{ d(x,gy)+d(y,gx)\bigr\} \biggr\} \biggr) \\ &\qquad {}-\phi\bigl(\max\bigl\{ d(x,y),d(y,gy)\bigr\} \bigr) \end{aligned}$$
(1.1)
for all \(x,y\in X\), where \(\phi:[0,\infty) \to[0,\infty)\) is a continuous function, and \(\psi:[0,\infty) \to[0,\infty)\) is an altering distance function, that is, ψ is a nondecreasing and continuous function, and \(\psi(t)=0\) if and only if \(t=0\).

Then T has a unique fixed point.

Matthews [16] introduced the notion of partial metric spaces, and extended Banach’s contraction principle to partial metric spaces, and then a lot of authors gave fixed point results in partial metric spaces (see [5, 1730]). Also, Aydi et al. [31] extended Ekeland’s variational principle to partial metric spaces, and Aydi et al. [32] extended Caristi’s fixed point theorem to partial metric spaces.

In particular, Abdeljawad [3] extended the result of Choudhury et al. [8] to partial metric spaces.

Samet et al. [33] gave a generalization of Banach’s contraction principle and an application to fixed point results in partial metric spaces.

In this paper, motivated and inspired by Samet et al. [33], we introduce the notion of generalized weakly contractive mappings in metric spaces and prove a fixed point theorem for generalized weakly contractive mappings defined on complete metric spaces, which is a generalization of the results of [810, 33]. Also, we obtain a coupled fixed point theorem in metric spaces by applying our main result, and we give applications to fixed point and coupled fixed point theorems in partial metric spaces.

A function \(f:X\to[0,\infty)\), where X is a metric space, is called lower semicontinuous if, for all \(x \in X\) and \(\{x_{n}\}\subset X\) with \(\lim_{n\to\infty}x_{n}=x\), we have
$$f(x)\leq\lim_{n\to\infty}\inf f(x_{n}). $$
Let
$$\Psi=\bigl\{ \psi: [0,\infty)\to[0,\infty)\mid \psi\text{ is continuous and } \psi(t)=0 \Leftrightarrow t=0\bigr\} . $$
Also, we denote
$$\Phi=\bigl\{ \phi: [0,\infty)\to[0,\infty)\mid \phi\text{ is lower semicontinuous and }\phi(t)=0 \Leftrightarrow t=0\bigr\} . $$

Lemma 1.1

([34])

If a sequence \(\{x_{n}\}\) in X is not Cauchy, then there exist \(\epsilon>0\) and two subsequences \(\{x_{m(k)}\}\) and \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that \(m(k)\) is the smallest index for which \(m(k)>n(k)>k\),
$$ d(x_{m(k)},x_{n(k)})\geq\epsilon, $$
(1.2)
and
$$ d(x_{m(k)-1},x_{n(k)})< \epsilon. $$
(1.3)

Moreover, suppose that \(\lim_{n\to\infty}d(x_{n},x_{n+1})=0\).

Then we have:
  1. (1)

    \(\lim_{n\to\infty}d(x_{m(k)},x_{n(k)})=\epsilon\);

     
  2. (2)

    \(\lim_{n\to\infty}d(x_{m(k)-1},x_{n(k)-1})=\epsilon\);

     
  3. (3)

    \(\lim_{n\to\infty}d(x_{m(k)},x_{n(k)-1})=\epsilon\);

     
  4. (4)

    \(\lim_{n\to\infty}d(x_{m(k)-1},x_{n(k)})=\epsilon\).

     

2 Fixed point results

Let X be a metric space with metric d, let \(T:X \to X\), and let \(\varphi:X \to[0, \infty)\) be a lower semicontinuous function.

Then T is called a generalized weakly contractive mapping if it satisfies the following condition:
$$\begin{aligned} &\psi\bigl(d(Tx,Ty)+\varphi(Tx)+\varphi(Ty)\bigr) \\ &\quad \leq\psi\bigl(m(x,y,d,T,\varphi)\bigr) -\phi\bigl(l(x,y,d,T,\varphi)\bigr) \quad\forall x,y\in X, \end{aligned}$$
(2.1)
where \(\psi\in\Psi, \phi\in\Phi\), and
$$\begin{aligned} m(x,y,d,T,\varphi)={}&\max\biggl\{ d(x,y)+\varphi(x)+\varphi(y), d(x,Tx)+\varphi (x)+\varphi(Tx), \\ &d(y,Ty)+\varphi(y)+\varphi(Ty), {1 \over 2}\bigl\{ d(x,Ty)+ \varphi(x)+\varphi(Ty) \\ &{}+d(y,Tx)+\varphi(y)+\varphi(Tx)\bigr\} \biggr\} , \end{aligned}$$
(2.2)
and
$$\begin{aligned} l(x,y,d,T,\varphi)=\max\bigl\{ d(x,y)+\varphi(x)+\varphi(y) ,d(y,Ty)+\varphi(y)+\varphi(Ty)\bigr\} . \end{aligned}$$
(2.3)

Let X be a metric space with metric d, let \(T:X \to X\), and let \(\varphi:X \to[0, \infty)\) be a lower semicontinuous function.

Theorem 2.1

Let X be complete. If T is a generalized weakly contractive mapping, then there exists a unique \(z \in X\) such that \(z=Tz\) and \(\varphi(z)=0\).

Proof

Let \(x_{0}\in X\) be a fixed point, and define a sequence \(\{x_{n}\}\) by \(x_{n+1}=Tx_{n}\) for all \(n=0,1,2,\ldots\) .

If \(x_{n}=x_{n+1}\) for some n, then \(x_{n}=x_{n+1}=Tx_{n}\). So \(x_{n}\) is a fixed point of T, and the proof is finished.

From now on, assume that \(x_{n} \neq x_{n+1}\) for all \(n=0,1,2,\ldots\) .

From (2.2) with \(x=x_{n-1}\) and \(y=x_{n}\) we have
$$\begin{aligned} &m(x_{n-1},x_{n},d,T,\varphi)\\ &\quad =\max\biggl\{ d(x_{n-1},x_{n})+ \varphi (x_{n-1})+\varphi(x_{n}), \\ &\qquad d(x_{n-1},Tx_{n-1})+\varphi (x_{n-1})+ \varphi(Tx_{n-1}), d(x_{n},Tx_{n})+ \varphi(x_{n})+\varphi(Tx_{n}), \\ &\qquad {1 \over 2}\bigl\{ d(x_{n-1},Tx_{n})+ \varphi(x_{n-1})+\varphi(Tx_{n}) +d(x_{n},Tx_{n-1})+ \varphi(x_{n})+\varphi(Tx_{n-1})\bigr\} \biggr\} . \end{aligned}$$
Since
$$\begin{aligned} &{1 \over 2}\bigl\{ d(x_{n-1},Tx_{n})+ \varphi(x_{n-1})+\varphi(Tx_{n}) +d(x_{n},Tx_{n-1})+ \varphi(x_{n})+\varphi(Tx_{n-1})\bigr\} \\ &\quad ={1 \over 2}\bigl\{ d(x_{n-1},x_{n+1})+ \varphi(x_{n-1})+\varphi(x_{n+1}) +d(x_{n},x_{n})+ \varphi(x_{n})+\varphi(x_{n})\bigr\} \\ &\quad \leq{1 \over 2}\bigl\{ d(x_{n-1},x_{n})+ \varphi(x_{n-1})+\varphi(x_{n}) +d(x_{n},x_{n+1})+ \varphi(x_{n})+\varphi(x_{n+1})\bigr\} \\ &\quad \leq\max\bigl\{ d(x_{n-1},x_{n})+\varphi(x_{n-1})+ \varphi(x_{n}), d(x_{n},x_{n+1})+ \varphi(x_{n})+\varphi(x_{n+1})\bigr\} , \end{aligned}$$
we obtain
$$\begin{aligned} &m(x_{n-1},x_{n},d,T,\varphi) \\ &\quad =\max\bigl\{ d(x_{n-1},x_{n})+\varphi(x_{n-1})+ \varphi(x_{n}), d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1})\bigr\} . \end{aligned}$$
(2.4)
Also, we have
$$\begin{aligned} &l(x_{n-1},x_{n},d,T,\varphi) \\ &\quad =\max\bigl\{ d(x_{n-1},x_{n})+\varphi(x_{n-1})+ \varphi(x_{n}) ,d(x_{n},Tx_{n})+ \varphi(x_{n})+\varphi(Tx_{n})\bigr\} \\ &\quad =\max\bigl\{ d(x_{n-1},x_{n})+\varphi(x_{n-1})+ \varphi(x_{n}) ,d(x_{n},x_{n+1})+ \varphi(x_{n})+\varphi(x_{n+1})\bigr\} . \end{aligned}$$
It follows from (2.1) that
$$\begin{aligned} &\psi\bigl(d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1})\bigr) \\ &\quad =\psi\bigl(d(Tx_{n-1},Tx_{n})+\varphi(Tx_{n-1})+ \varphi(Tx_{n})\bigr) \\ &\quad \leq\psi\bigl(m(x_{n-1},x_{n},d,T,\varphi)\bigr)-\phi \bigl(l(x_{n-1},x_{n},d,T,\varphi)\bigr). \end{aligned}$$
(2.5)
If
$$d(x_{n-1},x_{n})+\varphi(x_{n-1})+ \varphi(x_{n}) < d(x_{n},x_{n+1})+ \varphi(x_{n})+\varphi(x_{n+1}) $$
for some positive integer n, then from (2.5) we obtain
$$\begin{aligned} &\psi\bigl(d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1})\bigr) \\ &\quad \leq\psi\bigl(d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1})\bigr) -\phi\bigl(d(x_{n},x_{n+1})+ \varphi(x_{n})+\varphi(x_{n+1})\bigr), \end{aligned}$$
which implies
$$\phi\bigl(d(x_{n+1},x_{n})+\varphi(x_{n+1})+ \varphi(x_{n})\bigr)=0, $$
and so
$$d(x_{n+1},x_{n})+\varphi(x_{n+1})+ \varphi(x_{n})=0. $$
Hence
$$x_{n+1}=x_{n} \quad \text{and}\quad \varphi(x_{n+1})= \varphi(x_{n})=0, $$
which is a contradiction.
Thus we have
$$\begin{aligned} d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1}) \leq d(x_{n-1},x_{n})+ \varphi(x_{n-1})+\varphi(x_{n}) \end{aligned}$$
(2.6)
for all \(n=1,2,3,\dots\), and so
$$m(x_{n-1},x_{n},d,T,\varphi)=d(x_{n-1},x_{n})+ \varphi(x_{n-1})+\varphi(x_{n}) $$
and
$$l(x_{n-1},x_{n},d,T,\varphi)=d(x_{n-1},x_{n})+ \varphi(x_{n-1})+\varphi(x_{n}) $$
for all \(n =1,2,3,\dots\).
It follows from (2.5) that
$$\begin{aligned} &\psi\bigl(d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1})\bigr) \\ &\quad \leq\psi\bigl(d(x_{n-1},x_{n})+\varphi(x_{n-1})+ \varphi(x_{n}) \bigr) \\ &\qquad {}-\phi\bigl(d(x_{n-1},x_{n})+\varphi(x_{n-1})+ \varphi(x_{n})\bigr). \end{aligned}$$
(2.7)
It follows from (2.6) that the sequence \(\{d(x_{n},x_{n+1})+\varphi (x_{n})+\varphi(x_{n+1})\}\) is nonincreasing. Thus we have
$$d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1})\to r \quad \text{as }n\to \infty $$
for some \(r\geq0\).

Assume that \(r>0\).

Letting \(n\to\infty\) in (2.7), by the continuity of ψ and the lower semicontinuity of ϕ it follows that
$$\begin{aligned} \psi(r) \leq&\psi(r)-\lim_{n\to\infty}\inf\phi \bigl(d(x_{n-1},x_{n})+ \varphi(x_{n-1}) +\varphi(x_{n})\bigr) \\ \leq& \psi(r)-\phi(r). \end{aligned}$$
Since \(r>0\), \(\phi(r)>0\). Hence
$$\psi(r)\leq \psi(r)-\phi(r)< \psi(r), $$
a contradiction.
Hence
$$\lim_{n\to\infty}\bigl\{ d(x_{n},x_{n+1})+ \varphi(x_{n})+\varphi(x_{n+1})\bigr\} =0, $$
which implies
$$\begin{aligned} \lim_{n\to\infty}d(x_{n},x_{n+1})=0 \end{aligned}$$
(2.8)
and
$$\begin{aligned} \lim_{n\to\infty}\varphi(x_{n})=0. \end{aligned}$$
(2.9)

Now, we prove that the sequence \(\{x_{n}\}\) is Cauchy.

If \(\{x_{n}\}\) is not Cauchy, then by Lemma 1.1 there exist \(\epsilon >0\)and subsequences \(\{x_{m(k)}\}\) and \(\{x_{n(k)}\}\) of \(\{x_{n}\}\) such that (1.2) and (1.3) hold.

From (2.2) we have
$$\begin{aligned} &m(x_{n(k)},x_{m(k)},d,T,\varphi) \\ &\quad =\max\biggl\{ d(x_{n(k)},x_{m(k)})+ \varphi (x_{n(k)})+\varphi(x_{m(k)}), \\ &\qquad d(x_{n(k)},Tx_{n(k)})+\varphi (x_{n(k)})+ \varphi(Tx_{n(k)}), d(x_{m(k)},Tx_{m(k)})+ \varphi(x_{m(k)})+\varphi(Tx_{m(k)}), \\ &\qquad {1 \over 2}\bigl\{ d(x_{n(k)},Tx_{m(k)})+ \varphi(x_{n(k)})+\varphi(Tx_{m(k)}) +d(x_{m(k)},Tx_{n(k)})+ \varphi(x_{m(k)})+\varphi(Tx_{n(k)})\bigr\} \biggr\} \\ &\quad =\max\biggl\{ d(x_{n(k)},x_{m(k)})+\varphi(x_{n(k)})+ \varphi(x_{m(k)}), d(x_{n(k)},x_{n(k)+1})+ \varphi(x_{n(k)})+\varphi(x_{n(k)+1}), \\ &\qquad d(x_{m(k)},x_{m(k)+1})+\varphi(x_{m(k)})+ \varphi(x_{m(k)+1}), {1 \over 2}\bigl\{ d(x_{n(k)},x_{m(k)+1})+ \varphi(x_{n(k)})+\varphi (x_{m(k)+1}) \\ &\qquad {}+d(x_{m(k)},x_{n(k)+1})+\varphi(x_{m(k)})+ \varphi(x_{n(k)+1})\bigr\} \biggr\} . \end{aligned}$$
(2.10)
Letting \(k\to\infty\) in (2.10) and applying Lemma 1.1, (2.8), and (2.9), it follows that
$$\begin{aligned} \lim_{k\to\infty}m(x_{n(k)},x_{m(k)},d,T, \varphi)=\epsilon. \end{aligned}$$
(2.11)
Also, it follows from (2.3) that
$$\begin{aligned} &l(x_{n(k)},x_{m(k)},d,T,\varphi) \\ &\quad =\max\bigl\{ d(x_{n(k)},x_{m(k)})+\varphi(x_{n(k)})+\varphi (x_{m(k)}),d(x_{m(k)},Tx_{m(k)})+\varphi(x_{m(k)})+ \varphi(Tx_{m(k)})\bigr\} \\ &\quad =\max\bigl\{ d(x_{n(k)},x_{m(k)})+\varphi(x_{n(k)})+\varphi (x_{m(k)}),d(x_{m(k)},x_{m(k)+1})+\varphi(x_{m(k)})+ \varphi (x_{m(k)+1})\bigr\} . \end{aligned}$$
Hence
$$\begin{aligned} \lim_{n\to\infty}l(x_{n(k)},x_{m(k)},d,T, \varphi)=\epsilon. \end{aligned}$$
(2.12)
From (2.1) we have
$$\begin{aligned} &\psi\bigl(d(x_{n(k)+1},x_{m(k)+1}) +\varphi(x_{n(k)+1})+ \varphi(x_{m(k)+1})\bigr) \\ &\quad \leq\psi\bigl(m(x_{n(k)},x_{m(k)},d,T,\varphi)\bigr) -\phi \bigl(l(x_{n(k)},x_{m(k)},d,T,\varphi)\bigr). \end{aligned}$$
Letting \(k \to\infty\) in this inequality, by Lemma 1.1, the continuity of ψ, the lower semicontinuity of ϕ, and by (2.9), (2.11), and (2.12) we have
$$\psi(\epsilon)\leq\psi(\epsilon)-\phi(\epsilon), $$
which is a contradiction because \(\phi(\epsilon)>0\).
Hence the sequence \(\{x_{n}\}\) is Cauchy, and hence
$$\lim_{n\to\infty}x_{n}=z \in X \text{ exists } $$
because X is complete. Since φ is lower semicontinuous,
$$\varphi(z)\leq\lim_{n \to\infty}\inf\varphi(x_{n})\leq\lim _{n \to \infty} \varphi(x_{n})=0, $$
which implies
$$\begin{aligned} \varphi(z)=0. \end{aligned}$$
(2.13)
It follows from (2.2) that
$$\begin{aligned} m(x_{n},z,d,T,\varphi)={}&\max\biggl\{ d(x_{n},z)+ \varphi(x_{n})+\varphi(z), \\ &{}d(x_{n},Tx_{n})+\varphi(x_{n})+ \varphi(Tx_{n}), d(z,Tz)+\varphi(z)+\varphi(Tz), \\ &{}{1 \over 2}\bigl\{ d(x_{n},Tz)+\varphi(x_{n})+ \varphi(Tz) +d(z,Tx_{n})+\varphi(z)+\varphi(Tx_{n})\bigr\} \biggr\} \\ ={}&\max\biggl\{ d(x_{n},z)+\varphi(x_{n})+\varphi(z), \\ &{}d(x_{n},x_{n+1})+\varphi(x_{n})+ \varphi(x_{n+1}), d(z,Tz)+\varphi(z)+\varphi(Tz), \\ &{}{1 \over 2}\bigl\{ d(x_{n},Tz)+\varphi(x_{n})+ \varphi(Tz) +d(z,x_{n+1})+\varphi(z)+\varphi(x_{n+1})\bigr\} \biggr\} . \end{aligned}$$
So we have
$$\begin{aligned} &\lim_{n\to\infty}m(x_{n},z,d,T,\varphi) =d(z,Tz)+\varphi(z)+\varphi(Tz)=d(z,Tz)+\varphi(Tz). \end{aligned}$$
(2.14)
Also, we have
$$\begin{aligned} \lim_{n\to\infty}l(x_{n},z,d,T,\varphi) &=\lim_{n\to\infty}\max\bigl\{ d(x_{n},z)+ \varphi(x_{n})+\varphi(z), d(z,Tz)+\varphi(z)+\varphi(Tz)\bigr\} \\ &=d(z,Tz)+\varphi(z)+\varphi(Tz)=d(z,Tz)+\varphi(Tz). \end{aligned}$$
(2.15)
It follows from (2.1) that
$$\begin{aligned} \psi\bigl(d(x_{n+1},Tz)+\varphi(x_{n+1})+\varphi(Tz)\bigr) &=\psi\bigl(d(Tx_{n},Tz)+\varphi(Tx_{n})+\varphi(Tz)\bigr) \\ &\leq\psi\bigl(m(x_{n},z,d,T,\varphi)\bigr) -\phi\bigl(l(x_{n},z,d,T, \varphi)\bigr). \end{aligned}$$
(2.16)
By taking the limit as \(n\to\infty\) in (2.16) and by applying the continuity of ψ, the lower semicontinuity of ϕ, (2.14), and (2.15) we have
$$\psi\bigl(d(z,Tz)+\varphi(Tz)\bigr)\leq\psi\bigl(d(z,Tz)+\varphi(Tz)\bigr) -\phi \bigl(d(z,Tz)+\varphi(Tz)\bigr). $$
Hence \(d(z,Tz)+\varphi(Tz)=0\), and hence \(z=Tz\) and \(\varphi(Tz)=0\).

Suppose that u is another fixed point of T.

Then
$$u=Tu\quad \text{and}\quad \varphi(u)=0. $$
By applying (2.1) with \(x=z\) and \(y=u\) we have
$$\begin{aligned} \psi\bigl(d(z,u)\bigr) &=\psi\bigl(d(Tz,Tu)\bigr) \\ &=\psi\bigl(d(Tz,Tu)+\varphi(Tz)+\varphi(Tu)\bigr) \\ &\leq \psi\bigl(m(z,u,d,T,\varphi)\bigr) -\phi\bigl(l(z,u,d,T,\varphi)\bigr) \\ &=\psi\bigl(d(z,u)\bigr) -\phi\bigl(d(z,u)\bigr), \end{aligned}$$
which implies \(z=u\). □

The following example illustrates Theorem 2.1 and shows that it is a real generalization of Theorem 3.1 in [8].

Example 2.1

Let \(X=[0,\infty)\) and \(d(x,y)= \vert x-y \vert \) for \(x,y\in X\), let \(\psi(t)=\frac {3}{2}t\) for \(t\geq0\), and let
$$\varphi(t)= \textstyle\begin{cases} \frac{1}{2}t&(0\leq t\leq1), \\\frac{1}{2}t+\frac {1}{2}&(1< t\leq2), \\ t & (t>2). \end{cases} $$

Then \(\psi\in\Psi\), φ is lower semicontinuous, and \(\frac {1}{2}t \leq\varphi(t) \leq t\), \(t\geq0\).

Define the map \(T: X\to X\) by
$$Tx=\frac{x^{2}}{2(1+x)}. $$
Assume that a function \(\phi: [0,\infty)\to[0,\infty)\) is defined by
$$\phi(t)=\frac{3t}{4+2t}. $$

Then \(\phi\in\Phi\).

We now show that (2.1) holds.

Without loss of generality, suppose that \(x\geq y\).

Then we have
$$\begin{aligned} &{1 \over 2}\bigl\{ d(x,Ty)+\varphi(x)+\varphi(Ty) +d(y,Tx)+ \varphi(y)+\varphi(Tx)\bigr\} \\ &\quad \geq{1 \over 2}\biggl\{ d(x,Ty)+\frac{1}{2}x+ \frac{1}{2}Ty +d(y,Tx)+\frac{1}{2}y+\frac{1}{2}Tx\biggr\} \\ &\quad \geq {1 \over 2}\biggl\{ {1 \over 2}\bigl\{ d(x,Ty)+x+Ty +d(y,Tx)+y+Tx\bigr\} \biggr\} \\ &\quad = \textstyle\begin{cases}{1 \over 2}\{x+ \frac{x^{2}}{1+x}\}&(y\leq\frac {x^{2}}{2(1+x)} ), \\ {1 \over 2}\{x+y\}& \text{otherwise} \end{cases}\displaystyle \\ &\quad >\frac{1}{2}x . \end{aligned}$$
Thus we have
$$\begin{aligned} &m(x,y,d,T,\varphi) \\ &\quad =\max\biggl\{ d(x,y)+\varphi(x)+\varphi(y), d(x,Tx)+\varphi(x)+\varphi(Tx), \\ &\qquad d(y,Ty)+\varphi(y)+\varphi(Ty), {1 \over 2}\bigl\{ d(x,Ty)+ \varphi(x)+\varphi(Ty) +d(y,Tx)+\varphi(y)+\varphi(Tx)\bigr\} \biggr\} \\ &\quad \geq\frac{1}{2}\max\biggl\{ d(x,y)+x+y, d(x,Tx)+x+Tx, \\ &\qquad d(y,Ty)+y+Ty, {1 \over 2}\bigl\{ d(x,Ty)+x+Ty +d(y,Tx)+y+Tx\bigr\} \biggr\} \\ &\quad =\frac{1}{2}\max\biggl\{ 2x,2x,2y,\frac{1}{2}x\biggr\} \\ &\quad =x \end{aligned}$$
and
$$\begin{aligned} l(x,y,d,T,\varphi) &=\max\bigl\{ d(x,y)+\varphi(x)+\varphi(y) ,d(y,Ty)+\varphi(y)+\varphi(Ty)\bigr\} \\ &\leq\max\bigl\{ d(x,y)+x+y ,d(y,Ty)+y+Ty\bigr\} \\ &=\max\{2x, 2y\} \\ & =2x. \end{aligned}$$
Also, we have
$$\begin{aligned} \psi\bigl(d(Tx,Ty)+\varphi(Tx)+\varphi(Ty)\bigr) &\leq\psi\bigl(d(Tx,Ty)+Tx+Ty\bigr) \\ &=\frac{3}{2}\biggl( \biggl\vert \frac{x^{2}}{2(1+x)}-\frac{y^{2}}{2(1+y)} \biggr\vert +\frac {x^{2}}{2(1+x)}+\frac{y^{2}}{2(1+y)}\biggr) \\ &=\frac{3}{2}\cdot\frac{ 2x^{2}}{2(1+x)} \\ &= \frac{3}{2}\cdot\frac{ x^{2}}{1+x}. \end{aligned}$$
Hence
$$\begin{aligned} \psi\bigl(m(x,y,d,T,\varphi)\bigr)-\phi\bigl(l(x,y,d,T,\varphi)\bigr) &\geq \frac{3}{2}x- \frac{3x/2}{1+x} \\ &=\frac{3}{2}\cdot\frac{ x^{2}}{1+ x} \\ &\geq \psi\bigl(d(Tx,Ty)+\varphi(Tx)+\varphi(Ty)\bigr), \end{aligned}$$
where the equality is satisfied when \(x=0\).

Thus (2.1) is satisfied.

By Theorem 2.1, T has a unique fixed point \(z=0\), and \(\varphi(z)=0\).

However, (1.1) is not satisfied. In fact, let \(x=3\), \(y=1\), and \(\varphi (t)=0\), \(t\geq0\).

Then
$$\begin{aligned} &\psi\bigl(m\bigl((x,y,d,T,\varphi)\bigr)\bigr)=\frac{45}{2}, \\ &\phi\bigl(l\bigl((x,y,d,T,\varphi)\bigr)\bigr)=\frac{3}{4}, \\ &\psi\bigl(d(Tx,Ty)\bigr)=\frac{51}{2}, \end{aligned}$$
and so
$$\psi\bigl(d(Tx,Ty)\bigr)=\frac{204}{8}>\frac{147}{8} =\psi\bigl(m \bigl((x,y,d,T,\varphi)\bigr)\bigr)-\phi\bigl(l\bigl((x,y,d,T,\varphi)\bigr) \bigr). $$

The proofs of the following Corollary 2.2 and Corollary 2.3 are similar to that of Theorem 2.1. So, here the proofs are omitted.

Corollary 2.2

Let X be complete. Suppose that T satisfies the following condition:
$$\begin{aligned} &\psi\bigl(d(Tx,Ty)+\varphi(Tx)+\varphi(Ty)\bigr) \\ &\quad \leq\psi\bigl(m(x,y,d,T,\varphi)\bigr) -\phi\bigl(m(x,y,d,T,\varphi)\bigr) \\ &\qquad \forall x,y\in X, \textit{ where }\psi\in\Psi\textit{ and }\phi\in\Phi. \end{aligned}$$

Then there exists a unique \(z \in X\) such that \(z=Tz\) and \(\varphi(z)=0\).

Corollary 2.3

Let \((X,d)\) be complete. Suppose that T satisfies the following condition:
$$\begin{aligned} &\psi\bigl(d(Tx,Ty)+\varphi(Tx)+\varphi(Ty)\bigr) \\ &\quad \leq\psi\bigl(d(x,y)+\varphi(x)+\varphi(y)\bigr) -\phi\bigl(d(x,y)+\varphi(x)+ \varphi(y)\bigr) \\ &\qquad \forall x,y\in X, \textit{ where }\psi\in\Psi\textit{ and }\phi\in\Phi. \end{aligned}$$

Then there exists a unique \(z \in X\) such that \(z=Tz\) and \(\varphi(z)=0\).

Corollary 2.4

Let X be complete. Suppose that T satisfies the following condition:
$$\begin{aligned} &\psi\bigl(d\bigl(T^{k}x,T^{k}y\bigr)+\varphi \bigl(T^{k}x\bigr)+\varphi\bigl(T^{k}y\bigr)\bigr) \\ &\quad \leq\psi\bigl(m\bigl(x,y,d,T^{k},\varphi\bigr)\bigr) -\phi\bigl(l \bigl(x,y,d,T^{k},\varphi\bigr)\bigr) \\ &\qquad \forall x,y\in X, \textit{ where }\psi\in\Psi, \phi\in\Phi, \textit{ and } k \textit{ is a positive integer.} \end{aligned}$$

Then there exists a unique \(z \in X\) such that \(z=Tz\) and \(\varphi(z)=0\).

Proof

Let \(S=T^{k}\). Then by Theorem 2.1 S has a unique fixed point, say z.

Then \(T^{k}z=Sz=z\) and
$$\varphi(z)=\varphi(Sz)=\varphi\bigl(T^{k}z\bigr)=0. $$
Since \(T^{k+1}z=Tz\),
$$STz=T^{k}(Tz)=T^{k+1}z=Tz, $$
and so Tz is a fixed point of S. By the uniqueness of a fixed point of S, \(Tz=z\). □

Remark 2.1

If we have \(\varphi=0\), then ψ is nondecreasing and continuous, and ϕ is continuous in Theorem 2.1 (resp. Corollary 2.3, Corollary 2.4), then we obtain Theorem 3.1 of [8] (resp. Theorem 2.1 of [10], Corollary 3.1 of [8]).

Remark 2.2

If \(\varphi=0\) and if ψ and ϕ are nondecreasing and continuous in Corollary 2.3, then we obtain Theorem 2.1 of [10].

Remark 2.3

If \(\varphi=0\) and ψ is nondecreasing and continuous in Corollary 2.2, then we obtain Theorem 2.2 of [9].

3 Discussion

In this section, we obtain a new coupled fixed point result from Theorem 2.1.

Let X be a nonempty set.

We say that \((x,y)\in X \times X\) is a coupled fixed point [35] of a mapping \(G: X \times X \to X\) if
$$G(x,y)=x \quad \text{and}\quad G(y,x)=y. $$

Lemma 3.1

([36])

Let X be a nonempty set, \((x,y) \in X \times X\), and let \(G: X \times X \to X\).

Then the following are equivalent.
  1. (1)

    \(G(x,y)=x\) and \(G(y,x)=y\);

     
  2. (2)

    \(H(x,y)=(x,y)\), that is, H has a fixed point \((x,y)\),

    where \(H: X \times X \to X \times X\) is a mapping defined by
    $$\begin{aligned} H(x,y)=\bigl(G(x,y), G(y,x)\bigr). \end{aligned}$$
    (3.1)
     

Lemma 3.2

([37])

Let \((X,d)\) be a complete metric space (resp. complete partial metric space).

Define \(\rho:X \times X \to[0,\infty)\) by
$$\begin{aligned} \rho\bigl((x,y),(u,v)\bigr)=\max\bigl\{ d(x,u),d(y,v)\bigr\} . \end{aligned}$$
(3.2)

Then \((X\times X,\rho)\) is a complete metric space (resp. complete partial metric space).

Let \((X,d)\) be a metric space (or partial metric space), \(G: X \times X \to X\), and let \(H: (X \times X, \rho)\to(X \times X, \rho)\) be a mapping defined as in (3.1).

Let
$$\begin{aligned} &M\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr) \\ &\quad = \max\bigg\{ \max\bigl\{ d(x,u),d(y,v)\bigr\} +\varphi^{*}(x,y)+ \varphi^{*}(u,v), \\ &\qquad \max\bigl\{ d\bigl(x,G(x,y)\bigr),d\bigl(y,G(y,x)\bigr)\bigr\} + \varphi^{*}(x,y)+\varphi ^{*}\bigl(G(x,y),G(y,x)\bigr), \\ &\qquad \max\bigl\{ d\bigl(u,G(u,v)\bigr),d\bigl(v,G(v,u)\bigr)\bigr\} + \varphi^{*}(u,v)+\varphi ^{*}\bigl(G(u,v),G(v,u)\bigr), \\ &\qquad {1 \over 2}\bigl[\max\{d\bigl(x,G(u,v)\bigr),d\bigl(y,G(v,u)\bigr)+ \varphi^{*}(x,y)+\varphi ^{*}\bigl(G(u,v),G(v,u)\bigr) \\ &\qquad {}+\max\bigl\{ d\bigl(u,G(x,y)\bigr),d\bigl(v,G(y,x)\bigr)\bigr\} + \varphi^{*}(u,v)+\varphi ^{*}\bigl(G(x,y),G(y,x)\bigr)\bigr]\bigg\} , \end{aligned}$$
and let
$$\begin{aligned} &L\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr) \\ &\quad = \max\bigl\{ \max\bigl\{ d(x,u),d(y,v)\bigr\} +\varphi^{*}(x,y)+ \varphi^{*}(u,v), \\ &\qquad \max\bigl\{ d\bigl(u,G(x,y)\bigr),d\bigl(v,G(y,x)\bigr)\bigr\} + \varphi^{*}(u,v)+\varphi ^{*}\bigl(G(x,y),G(y,x)\bigr)\bigr\} , \end{aligned}$$
where \(\varphi^{*}:X \times X \to[0,\infty)\).
Then we have
$$\begin{aligned} &M\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr) \\ &\quad =\max\biggl\{ \rho\bigl((x,y),(u,v)\bigr) +\varphi^{*}(x,y)+ \varphi^{*}(u,v), \\ &\qquad \rho\bigl((x,y), H(x,y)\bigr)+\varphi^{*}(x,y)+\varphi^{*} \bigl(H(x,y)\bigr), \\ &\qquad \rho\bigl((u,v), H(u,v)\bigr)+\varphi^{*}(u,v)+\varphi^{*} \bigl(H(u,v)\bigr), \\ &\qquad {1 \over 2}\bigl[\rho\bigl((x,y),H(u,v)\bigr)+\varphi^{*}(x,y)+ \varphi^{*}\bigl(H(u,v)\bigr) \\ &\qquad {}+\rho\bigl((u,v),H(x,y)\bigr)+\varphi^{*}(u,v)+\varphi^{*} \bigl(H(x,y)\bigr)\bigr]\biggr\} \\ &\quad =m\bigl((x,y),(u,v),\rho,H,\varphi^{*}\bigr). \end{aligned}$$
(3.3)
Also, we have
$$\begin{aligned} L\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr)=l \bigl((x,y),(u,v),\rho,H,\varphi^{*}\bigr). \end{aligned}$$
(3.4)

Theorem 3.1

Let X be complete. Suppose that \(G:X \times X \to X\) is a mapping such that
$$\begin{aligned} &\psi\bigl(d\bigl(G(x,y),G(u,v)\bigr)+\varphi\bigl(G(x,y),G(y,x)\bigr)+\varphi \bigl(G(u,v),G(v,u)\bigr) \bigr) \\ &\quad \leq\psi\bigl(M\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr)\bigr)-\phi \bigl(L\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr)\bigr), \end{aligned}$$
(3.5)
for all \((x,y),(u,v)\in X \times X\), where \(\psi\in\Psi\), \(\phi\in\Phi \), and \(\varphi^{*}: X \times X \to[0,\infty)\) is lower semicontinuous.

Then G has a unique coupled fixed point \((x^{*},y^{*})\in X\times X\), and \(\varphi^{*}(x^{*},y^{*})=0\).

Proof

Let ρ be the metric on \(X \times X\) defined as (3.2), and let \(H: (X \times X, \rho)\to(X \times X, \rho)\) be a mapping defined as in (3.1) for \((x,y),(u,v)\in X\times X\).

It follows from (3.3), (3.4), and (3.5) that
$$\begin{aligned} &\rho\bigl(H(x,y),H(u,v)\bigr) \leq\psi(m\bigl((x,y),(u,v),\rho,H,\varphi^{*}\bigr)-\phi\bigl(l \bigl((x,y),(u,v),\rho ,H,\varphi^{*}\bigr)\bigr) \end{aligned}$$
for \((x,y),(u,v)\in X \times X\).

By Theorem 2.1, H has a unique fixed point, and so by Lemma 3.1 G has a unique coupled fixed point. □

Corollary 3.2

Let X be complete. Suppose that \(G:X \times X \to X\) is a mapping such that
$$\begin{aligned} &\psi\bigl(d\bigl(G(x,y),G(u,v)\bigr)+\varphi\bigl(G(x,y),G(y,x)\bigr)+\varphi \bigl(G(u,v),G(v,u)\bigr) \bigr) \\ &\quad \leq\psi\bigl(M\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr)\bigr)-\phi \bigl(M\bigl((x,y),(u,v),d,G,\varphi^{*}\bigr)\bigr) \end{aligned}$$
for all \((x,y),(u,v)\in X \times X\), where \(\psi\in\Psi\), \(\phi\in\Phi \), and \(\varphi^{*}: X \times X \to[0,\infty)\) is lower semicontinuous.

Then G has a unique coupled fixed point \((x^{*},y^{*})\in X\times X\), and \(\varphi^{*}(x^{*},y^{*})=0\).

Corollary 3.3

Let X be complete. Suppose that \(G:X \times X \to X\) is a mapping such that
$$\begin{aligned} &\psi\bigl(d\bigl(G(x,y),G(u,v)\bigr)+\varphi\bigl(G(x,y),G(y,x)\bigr)+\varphi \bigl(G(u,v),G(v,u)\bigr) \bigr) \\ &\quad \leq\psi\bigl(\max\bigl\{ d(x,u),d(y,v)\bigr\} +\varphi^{*}(x,y)+ \varphi^{*}(u,v)\bigr) \\ &\qquad {}-\phi\bigl(\max\bigl\{ d(x,u),d(y,v)\bigr\} +\varphi^{*}(x,y)+ \varphi^{*}(u,v)\bigr) \end{aligned}$$
for all \((x,y),(u,v)\in X \times X\), where \(\psi\in\Psi\), \(\phi\in\Phi \), and \(\varphi^{*}: X \times X \to[0,\infty)\) is lower semicontinuous.

Then G has a unique coupled fixed point \((x^{*},y^{*})\in X\times X\), and \(\varphi^{*}(x^{*},y^{*})=0\).

Taking \(\varphi^{*}=0\) in Theorem 3.1, we have the following corollary.

Corollary 3.4

Let X be complete. Suppose that \(G:X \times X \to X\) is a mapping such that
$$\begin{aligned} &\psi\bigl(d\bigl(G(x,y),G(u,v)\bigr) \bigr) \leq \psi(M\bigl((x,y),(u,v),d,G,0\bigr)-\phi\bigl(L\bigl((x,y),(u,v),d,G,0 \bigr)\bigr) \end{aligned}$$
for all \((x,y),(u,v)\in X \times X\), where \(\psi\in\Psi\) and \(\phi\in \Phi \).

Then G has a unique coupled fixed point.

Corollary 3.5

Let X be complete. Suppose that \(G:X \times X \to X\) is a mapping such that
$$\begin{aligned} &\psi\bigl(d\bigl(G(x,y),G(u,v)\bigr) \bigr) \\ &\quad \leq\psi(M\bigl((x,y),(u,v),d,G,0\bigr)-\phi\bigl(M\bigl((x,y),(u,v),d,G,0 \bigr)\bigr) \end{aligned}$$
for all \((x,y),(u,v)\in X \times X\), where \(\psi\in\Psi\) and \(\phi\in \Phi \).

Then G has a unique coupled fixed point.

Corollary 3.6

Let X be complete. Suppose that \(G:X \times X \to X\) is a mapping such that
$$\begin{aligned} &\psi\bigl(d\bigl(G(x,y),G(u,v)\bigr)\bigr) \\ &\quad \leq\psi(\max\bigl\{ d(x,u),d(y,v)\bigr\} -\phi\bigl(\max\bigl\{ d(x,u),d(y,v) \bigr\} \bigr) \end{aligned}$$
for all \((x,y),(u,v)\in X \times X\), where \(\psi\in\Psi\) and \(\phi\in \Phi \).

Then G has a unique coupled fixed point.

4 Applications

In this section, we give application to fixed point theorems in partial metric spaces.

Recall some definitions and basic results in partial metric spaces. For more details, we refer to [16].

Let Z be a nonempty set. A function \(p:Z\times Z \to[0,\infty)\) is called a partial metric on Z if, for all \(x,y,z\in Z\), the following are satisfied:
  1. (1)

    \(p(x,x)=p(y,y)=p(x,y) \Leftrightarrow x=y\);

     
  2. (2)

    \(p(x,x)\leq p(x,y)\);

     
  3. (3)

    \(p(x,y)=p(y,x)\);

     
  4. (4)

    \(p(x,z)\leq p(x,y)+p(y,z)-p(y,y)\).

     

The pair \((Z,p)\) is called a partial metric space.

Note that if \(p(x,y)=0\), then \(x=y\).

An example of a partial metric defined on \([0,\infty)\) is \(p(x,y)=\max\{ x,y\}\), \(x,y\geq0\). For more examples of partial metrics, we refer to [16].

It is well known that each partial metric p on a nonempty set Z generates a \(T_{0}\) topology on Z and that \(\{B(x,\epsilon):\epsilon >0,x\in Z\}\) is a base for the topology, where \(B(x,\epsilon)=\{y\in Z:p(x,y)< p(x,x)+\epsilon\}\) for all \(x\in Z\) and \(\epsilon>0\).

Also, it is known that the function \(p_{s}:Z \times Z \to[0,\infty)\) defined by
$$\begin{aligned} p_{s}(x,y)=2p(x,y)-p(x,x)-p(y,y) \end{aligned}$$
(4.1)
is a metric on Z.
Let Z be a partial metric space with partial metric p, let \(\{ x_{n}\} (\subset Z)\) be a sequence, and let \(x\in Z\). Then we say that
  1. (1)

    \(\{x_{n}\}\) is convergent to x if \(\lim_{n\to\infty}p(x,x_{n})=p(x,x)\);

     
  2. (2)

    \(\{x_{n}\}\) is called a Cauchy sequence if there exists \(\lim_{n,m \to\infty}p(x_{n},x_{m})\) such that it is finite;

     
  3. (3)
    Z is complete if every Cauchy sequence in Z is convergent to a point \(z\in Z\) such that
    $$\lim_{n,m\to\infty}p(x_{n},x_{m})=p(z,z). $$
     

Remark 4.1

A partial metric space Z is complete if and only if for every Cauchy sequence \(\{x_{n}\}\) in Z, there exists \(z\in Z\) such that
$$\lim_{n,m\to\infty}p(x_{n},x_{m})=\lim _{n,m\to\infty}p(x_{n},z)=p(z,z). $$

Remark 4.2

Let \(\{x_{n}\} (\subset Z)\) be a sequence, and let \(x\in Z\). If the sequence \(\{x_{n}\}\) is convergent to x in \((Z,p_{s})\), then it is convergent to x in \((Z,p)\), and the converse is not true (see [16]).

4.1 Fixed points on partial metric spaces

Theorem 4.1

Let Z be complete with partial metric p. Suppose that \(T:Z \to Z\) is a mapping such that
$$\begin{aligned} \psi\bigl(p(Tx,Ty)\bigr)\leq{}&\psi\biggl(\max\biggl\{ p(x,y),p(x,Tx),p(y,Ty), \frac{1}{2}\bigl\{ p(x,Ty)+p(y,Tx)\bigr\} \biggr\} \biggr) \\ &{}-\phi\bigl(\max\bigl\{ p(x,y),p(y,Ty)\bigr\} \bigr) \end{aligned}$$
(4.2)
for all \(x,y\in Z\), where \(\psi\in\Psi\) and \(\phi\in\Phi \).

Then there exists a unique \(z\in Z\) such that \(z=Tz\) and \(p(z,z)=0\).

Proof

From (4.1) we have
$$p(x,y)=\frac{p_{s}(x,y)+p(x,x)+p(y,y)}{2} $$
for all \(x,y\in Z\).

Let \(d(x,y)=\frac{p_{s}(x,y)}{2}\) and \(\varphi(x)=\frac{p(x,x)}{2}\) for all \(x,y\in Z\).

Then Z is a complete metric space with metric d, and \(\varphi:Z \to [0,\infty)\) is a lower semicontinuous function. Also, (4.2) reduces to (2.1). By Theorem 2.1 there exists a unique \(z\in Z\) such that \(z=Tz\) and \(p(z,z)=0\). □

Remark 4.3

Theorem 4.1 is a generalization of Theorem 8 of [3]. In fact, let ϕ and ψ be nondecreasing and continuous functions.

Then Theorem 4.1 reduces to Theorem 8 of [3].

Corollary 4.2

Let Z be complete with partial metric p. Suppose that \(T:Z \to Z\) is a mapping such that
$$\begin{aligned} \psi\bigl(p(Tx,Ty)\bigr)\leq{}&\psi\biggl(\max\biggl\{ p(x,y),p(x,Tx),p(y,Ty), \frac{1}{2}\bigl\{ p(x,Ty)+p(y,Tx)\bigr\} \biggr\} \biggr) \\ &{}-\phi\biggl(\max\biggl\{ p(x,y),p(x,Tx),p(y,Ty), \frac{1}{2}\bigl\{ p(x,Ty)+p(y,Tx)\bigr\} \biggr\} \biggr) \end{aligned}$$
for all \(x,y\in Z\), where \(\psi\in\Psi\) and \(\phi\in\Phi\).

Then there exists a unique \(z\in Z\) such that \(z=Tz\) and \(p(z,z)=0\).

Remark 4.4

If ϕ is continuous in Corollary 4.2, then we obtain Theorem 2.5 of [6].

Corollary 4.3

Let Z be complete with partial metric p. Suppose that \(T:Z \to Z\) is a mapping such that
$$\begin{aligned} \psi\bigl(p(Tx,Ty)\bigr)\leq\psi\bigl(p(x,y)\bigr)-\phi\bigl(p(x,y)\bigr) \end{aligned}$$
for all \(x,y\in Z\), where \(\psi\in\Psi\) and \(\phi\in\Phi \).

Then there exists a unique \(z\in Z\) such that \(z=Tz\) and \(p(z,z)=0\).

4.2 Coupled fixed points on partial metric spaces

Theorem 4.4

Let Z be complete with partial metric p. Suppose that \(G:Z \times Z \to Z\) is a mapping such that
$$\begin{aligned} &\psi\bigl(p\bigl(G(x,y),G(u,v)\bigr)\bigr) \\ &\quad \leq\psi\bigl(M\bigl((x,y),(u,v),p,G,0\bigr)\bigr)-\phi\bigl(L \bigl((x,y),(u,v),p,G,0\bigr)\bigr) \end{aligned}$$
(4.3)
for all \((x,y),(u,v)\in Z \times Z\), where \(\psi\in\Psi\) and \(\phi\in \Phi \).

Then G has a unique coupled fixed point.

Proof

Let ρ be the partial metric on \(Z\times Z\) defined as in (3.2), and let \(H: (Z \times Z, \rho)\to(Z \times Z, \rho)\) be a mapping defined as in (3.1).

It follows from (4.3), (3.3), and (3.4) with \(\varphi^{*}=0\) that
$$\begin{aligned} &\rho\bigl(H(x,y),H(u,v)\bigr) \leq\psi(m\bigl((x,y),(u,v),\rho,H,0\bigr)-\phi\bigl(l\bigl((x,y),(u,v), \rho,H,0\bigr)\bigr) \end{aligned}$$
for all \((x,y),(u,v)\in Z \times Z\).

By Theorem 4.1, H has a unique fixed point, and so by Lemma 3.1 G has a unique coupled fixed point. □

Corollary 4.5

Let Z be complete with partial metric p. Suppose that \(G:Z \times Z \to Z\) is a mapping such that
$$\begin{aligned} &\psi\bigl(p\bigl(G(x,y),G(u,v)\bigr)\bigr) \\ &\quad \leq\psi\bigl(M\bigl((x,y),(u,v),p,G,0\bigr)\bigr)-\phi\bigl(M \bigl((x,y),(u,v),p,G,0\bigr)\bigr) \end{aligned}$$
for all \((x,y),(u,v)\in Z \times Z\), where \(\psi\in\Psi\) and \(\phi\in \Phi \).

Then G has a unique coupled fixed point.

Corollary 4.6

Let Z be complete with partial metric p. Suppose that \(G:Z \times Z \to Z\) is a mapping such that
$$\begin{aligned} &\psi\bigl(p\bigl(G(x,y),G(u,v)\bigr)\bigr) \\ &\quad \leq\psi\bigl(\max\bigl\{ p(x,y),p(u,v)\bigr\} \bigr)-\phi\bigl(\max\bigl\{ p(x,y),p(u,v)\bigr\} \bigr) \end{aligned}$$
for all \((x,y),(u,v)\in Z \times Z\), where \(\psi\in\Psi\) and \(\phi\in \Phi \).

Then G has a unique coupled fixed point.

5 Conclusions

Motivated by the result of Samet et al. [33], we introduce the notion of generalized weakly contractive mappings and prove the existence and uniqueness of fixed points for such mappings. We give applications to the existence of fixed points in partial metric spaces.

This investigation can be extended to a quasi-metric spaces with applications to studies of fixed points in quasi-partial metric spaces.

Declarations

Acknowledgements

The author would like to thank the anonymous reviewers and editor for their valuable comments.

Availability of data and materials

“Not applicable”

Funding

“Not applicable”

Authors’ contributions

The author read and approved the final manuscript.

Competing interests

The author declares that he has no competing interests.

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

(1)
Department of Mathematics, Hanseo University, Seosan, South Korea

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