1 Introduction

In 1981, Bhakta and Basu [1] proved the following common fixed point theorem.

Theorem BB

Let \((X,d)\) be a complete metric space and \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Suppose that the following conditions are satisfied:

  1. (BB)

    \(d(Sx,Ty)\le\varphi(x)-\varphi(Sx)+\psi(y)-\psi(Ty)\), for all \(x,y\in X\).

  2. (oc)

    S and T are orbitally continuous.

Then S and T have a unique common fixed point.

Recall that \(T:X\to X\) is orbitally continuous if for every \(x,w\in X\), the following implication holds:

$$T^{n_{j}}x\to w\quad\implies\quad T\bigl(T^{n_{j}}x\bigr)\to Tw. $$

Let \((X,d)\) be a metric space. A function \(p:X\times X\to[0,\infty)\) is called a w-distance on X if the following conditions are satisfied:

  1. (w1)

    \(p(x,y)\le p(x,z)+p(z,y)\), for all \(x,y,z\in X\).

  2. (w2)

    For each \(x\in X\), the function \(y\mapsto p(x,y)\) is lower semicontinuous.

  3. (w3)

    For each \(\varepsilon>0\) there exists \(\delta>0\) such that \(d(y,z)\le\varepsilon\) whenever \(p(x,y)\le\delta\) and \(p(x,z)\le\delta\).

Using this notion, the following interesting result was proved.

Theorem KST

([2])

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(T:X\to X\) be a mapping. Let \(\psi:X\to[0,\infty)\) be any function. Suppose that the following conditions hold:

  1. (KST)

    \(p(x,Tx)\le\psi(x)-\psi(Tx)\), for all \(x\in X\).

  2. (lsc)

    ψ is lower semicontinuous.

Then there exists \(u\in X\) such that \(u=Tu\) and \(p(u,u)=0\).

It is clear that Theorem KST includes Caristi’s theorem as a special case.

Obama and Kuroiwa [3] used the concept of w-distance to prove the following two theorems. The first one is a generalization of Theorem BB.

Theorem OK1

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to [0,\infty)\) be any functions. Suppose that the following conditions are satisfied:

  1. (OK1)

    \(\max\{p(Sx,Ty),p(Ty,Sx)\}\le\varphi(x)-\varphi(Sx)+\psi (y)-\psi(Ty)\), for all \(x,y\in X\).

  2. (oc)

    S and T are orbitally continuous.

Then S and T have a unique common fixed point.

Theorem OK2

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to [0,\infty)\) be any functions. Suppose that the following conditions are satisfied:

  1. (OK2)

    \(\max\{p(x,y),p(y,x)\}+p(x,Sx)+p(y,Ty)\le\varphi(x)-\varphi (Sx)+\psi(y)-\psi(Ty)\), for all \(x,y\in X\).

  2. (oc)

    S and T are orbitally continuous.

Then S and T have a unique common fixed point.

The main tool of our result via the w-distance is based on the following lemma.

Lemma KST

([2])

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences in X, let \(\{\alpha_{n}\} \) and \(\{\beta_{n}\}\) be sequences in \([0,\infty)\) converging to 0, and let \(x,y,z\in X\). Then the following statements hold:

  1. (1)

    If \(p(x_{n},y_{n})\leq\alpha_{n}\) and \(p(x_{n},z)\leq\beta_{n}\), for all \(n\in\mathbb{N}\), then \(\{y_{n}\}\) converges to z. In particular, if \(p(x,y)=0\) and \(p(x,z)=0\), then \(y=z\).

  2. (2)

    If \(p(x_{n},x_{m})\leq\alpha_{n}\), for all \(n,m\in\mathbb{N}\) with \(m>n\), then \(\{x_{n}\}\) is a Cauchy sequence.

2 Discussions on Theorems OK1 and OK2

In the appearance of the condition (OK1) or (SS) of Lemma 1 below, the problem of finding a common fixed point of S and T reduces to that of finding a fixed point of each mapping individually.

Lemma 1

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Let \(x_{0},y_{0}\in X\). Suppose that one of the following conditions is satisfied:

  1. (OK1)

    \(\max\{p(Sx,Ty),p(Ty,Sx)\}\leq\varphi(x)-\varphi(Sx)+\psi (y)-\psi(Ty)\), for all \(x,y\in X\).

  2. (SS)

    \(m(x,y)+p(x,Sx)+p(y,Ty)\leq\varphi(x)-\varphi(Sx)+\psi (y)-\psi(Ty)\), for all \(x,y\in X\) where

    $$m(x,y):=\min \left \{ \begin{array}{@{}c@{}} p(x,y), p(y,x), p(x,Ty), p(Ty,x),\\ p(y,Sx), p(Sx,y), p(Sx,Ty), p(Ty,Sx) \end{array} \right \}. $$

If there are \(\widehat{x},\widehat{y}\in X\) such that \(\widehat {x}=S\widehat{x}\) and \(\widehat{y}=T\widehat{y}\), then \(\widehat {x}=\widehat{y}\) and \(\widehat{x}\) is a unique common fixed point of S and T. Moreover, \(p(\widehat{x},\widehat{x})=0\).

Proof

Let \(\widehat{x},\widehat{y}\in X\) be such that \(\widehat {x}=S\widehat{x}\) and \(\widehat{y}=T\widehat{y}\).

Suppose that (OK1) holds. Then

$$\begin{aligned} \max\bigl\{ p(\widehat{x},\widehat{y}),p(\widehat{y},\widehat {x})\bigr\} &=\max \bigl\{ p(S\widehat{x},T\widehat{y}),p(T\widehat{y},S\widehat {x})\bigr\} \\ &\le\varphi(\widehat{x})-\varphi(S\widehat{x})+\psi(\widehat{y})-\psi (T \widehat{y})=0. \end{aligned}$$

It follows that \(p(\widehat{x},\widehat{y})=p(\widehat{y},\widehat {x})=0\) and so

$$p(\widehat{x},\widehat{x})\le p(\widehat{x},\widehat{y})+p(\widehat {y}, \widehat{x})=0. $$

Hence \(\widehat{x}=\widehat{y}\), and we have \(\widehat{x}=S\widehat {x}=T\widehat{x}\). The uniqueness is obvious.

Suppose that (SS) holds. Then

$$\begin{aligned} m(\widehat{x},\widehat{y})+p(\widehat{x},\widehat {x})+p(\widehat{y}, \widehat{y}) &=m(\widehat{x},\widehat{y})+p(\widehat{x},S\widehat{x})+p(\widehat {y},T\widehat{y}) \\ &\le\varphi(\widehat{x})-\varphi(S\widehat{x})+\psi(\widehat{y})-\psi (T \widehat{y})=0. \end{aligned}$$

If \(m(\widehat{x},\widehat{y})=p(\widehat{x},\widehat{y})\), then

$$p(\widehat{x},\widehat{x})=p(\widehat{x},\widehat{y})=0. $$

So \(\widehat{x}=\widehat{y}\) and hence \(\widehat{x}=S\widehat {x}=T\widehat{x}\). The same conclusion holds if \(m(\widehat{x},\widehat {y})=p(\widehat{y},\widehat{x})\). The uniqueness is obvious. □

Remark 2

It is clear that (OK2) ⇒ (SS).

2.1 On Theorem OK1

Let \((X,d)\) be a metric space and \(S:X\to X\) be a mapping. Recall that \(G:X\to\mathbb{R}\) is S-orbitally lower semicontinuous at \(x_{0}\in X\) if \(G(\widehat{x})\le\liminf_{n\to\infty}G(x_{n})\) whenever \(\{x_{n}\}\) is a sequence in \(O(x_{0},S):=\{x_{0},Sx_{0},S^{2}x_{0},\ldots\}\) and \(\lim_{n\to\infty }x_{n}=\widehat{x}\).

The following result is motivated by the one proved by Bollenbacher and Hicks [4].

Theorem 3

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Let \(x_{0},y_{0}\in X\). Suppose that the following conditions are satisfied:

  1. (BH)

    \(\max\{p(Sx,Ty),p(Ty,Sx)\}\leq\varphi(x)-\varphi(Sx)+\psi (y)-\psi(Ty)\), for all \(x\in O(x_{0},S)\) and \(y\in O(y_{0},T)\).

  2. (cc)

    Every Cauchy sequence in \(O(x_{0},S)\) converges to a point in X and every Cauchy sequence in \(O(y_{0},T)\) converges to a point in X.

Then the following statements are true:

  1. (i)

    There exists \(z\in X\) such that \(\lim_{n\to\infty}S^{n}x_{0}=\lim_{n\to\infty}T^{n}y_{0}=z\). Moreover,

    $$\lim_{n\to\infty}p\bigl(S^{n}x_{0},z\bigr)=\lim _{n\to\infty}p\bigl(T^{n}y_{0},z\bigr)=0. $$
  2. (ii)

    \(p(S^{n+1}x_{0},z)\leq\varphi(S^{n}x_{0})+\varphi (S^{n+1}x_{0})+2\psi(T^{n}y_{0})\) and \(p(T^{n+1}y_{0},z)\leq2\varphi(S^{n+1}x_{0})+\psi(T^{n}y_{0})+\psi (T^{n+1}y_{0})\), for all \(n\ge0\).

  3. (iii)

    Define \(G(x):=p(x,Sx)\), for all \(x\in X\), and \(H(y):=p(y,Ty)\), for all \(y\in X\). Then the following statements are equivalent:

    1. (a)

      \(z=Sz=Tz\) and \(p(z,z)=0\).

    2. (b)

      G is S-orbitally lower semicontinuous at \(x_{0}\) and H is T-orbitally lower semicontinuous at \(y_{0}\).

Proof

Let \(x_{0},y_{0}\in X\). Define \(x_{n}=S^{n}x_{0}\) and \(y_{n}=T^{n}y_{0}\), for all \(n\in\mathbb{N}\). Then, for all \(i\in\mathbb{N}\), we have

$$p(x_{i},y_{i})\leq\varphi(x_{i-1})- \varphi(x_{i})+\psi(y_{i-1})-\psi(y_{i}) $$

and

$$p(y_{i},x_{i+1})\leq\varphi(x_{i})- \varphi(x_{i+1})+\psi(y_{i-1})-\psi(y_{i}). $$

For all integers \(n\ge1\) and \(k\ge0\), we have

$$\begin{aligned} \sum_{i=n}^{n+k}p(x_{i},y_{i}) \leq \varphi(x_{n-1})-\varphi(x_{n+k})+\psi(y_{n-1})- \psi(y_{n+k}) \end{aligned}$$
(1)

and

$$\begin{aligned} \sum_{i=n}^{n+k}p(y_{i},x_{i+1}) \leq\varphi(x_{n})-\varphi(x_{n+k+1})+\psi (y_{n-1})- \psi(y_{n+k}). \end{aligned}$$
(2)

In particular,

$$\sum_{n=1}^{\infty}p(x_{n},y_{n}) \leq\varphi(x_{0})+\psi(y_{0})<\infty\quad\mbox{and}\quad\sum _{n=1}^{\infty}p(y_{n},x_{n+1}) \leq\varphi(x_{1})+\psi(y_{0})<\infty. $$

Thus \(\sum_{n=0}^{\infty} p(x_{n},x_{n+1})<\infty\). It follows that

$$\begin{aligned} p(x_{n},x_{n+k+1})&\leq\sum_{i=n}^{n+k}p(x_{i},x_{i+1}) \to 0\quad\mbox{as }n\to\infty. \end{aligned}$$

Therefore, \(\{x_{n}\}\) is a Cauchy sequence in \(O(x_{0},S)\) and hence \(x_{n}\to z\) for some \(z\in X\). So

$$p(x_{n},z)\leq\liminf_{k\to\infty}p(x_{n},x_{n+k}) \leq\sum_{i=n}^{\infty }p(x_{i},x_{i+1}) \to0\quad\mbox{as }n\to\infty. $$

From \(\sum_{n=1}^{\infty}p(x_{n},y_{n})<\infty\), we have \(p(x_{n},y_{n})\to0\). It follows from Lemma KST that \(y_{n}\to z\). It is not difficult to show that \(p(y_{n},z)\to0\). Hence (i) holds.

To see (ii), from (1) and (2), we have

$$\begin{aligned} \sum_{i=n}^{n+k}p(x_{i},x_{i+1}) \leq\varphi(x_{n-1})-\varphi (x_{n+k})+2\psi(y_{n-1})-2 \psi(y_{n+k}) +\varphi(x_{n})-\varphi(x_{n+k+1}). \end{aligned}$$

It follows that

$$\begin{aligned} &p(x_{n},x_{n+k+1}) \\ &\quad\leq\sum_{i=n}^{n+k}p(x_{i},x_{i+1}) \\ &\quad\leq\varphi(x_{n-1})-\varphi(x_{n+k})+2 \psi(y_{n-1})-2\psi (y_{n+k})+\varphi(x_{n})- \psi(y_{n+k+1}) \\ &\quad\leq\varphi(x_{n-1})+\varphi(x_{n})+2 \psi(y_{n-1}). \end{aligned}$$

Thus

$$p(x_{n},z)\leq\liminf_{k\to\infty}p(x_{n},x_{n+k+1}) \leq\varphi (x_{n-1})+\varphi(x_{n})+2\psi(y_{n-1}). $$

Hence, \(p(S^{n+1}x_{0},z)\leq\varphi(S^{n}x_{0})+\varphi(S^{n}x_{0})+2\psi(T^{n}y_{0})\). Similarly, we have

$$p\bigl(T^{n+1}y_{0},z\bigr)\leq2\varphi\bigl(S^{n+1}x_{0} \bigr)+\psi\bigl(T^{n}y_{0}\bigr)+\psi\bigl(T^{n+1}y_{0} \bigr). $$

This proves (ii).

Finally, we prove (iii). (a) ⇒ (b) Assume that \(z=Sz=Tz\) and \(p(z,z)=0\). Let \(\{z_{n}\}\) be a sequence in \(O(x_{0},S)\). We may assume that \(z_{n}\to z\). It follows that \(G(z)=0\leq\liminf_{n\to\infty }G(z_{n})\). Hence G is S-orbitally lower semicontinuous at \(x_{0}\). Similarly, H is T-orbitally lower semicontinuous at \(y_{0}\).

(b) ⇒ (a) Assume that G is S-orbitally lower semicontinuous at \(x_{0}\) and H is T-orbitally lower semicontinuous at \(y_{0}\). By the proof of (i), we have \(x_{n}=S^{n}x_{0}\to z\), \(y_{n}=T^{n}y_{0}\to z\), \(\lim_{n\to\infty}G(x_{n})=\lim_{n\to\infty }p(S^{n}x_{0},S^{n+1}x_{0})=0\), and \(\lim_{n\to\infty}H(y_{n})=\lim_{n\to\infty }p(T^{n}y_{0}, T^{n+1}y_{0})=0\). By the assumption, we have \(G(z)=H(z)=0\), that is,

$$p(z,Sz)=p(z,Tz)=0. $$

Hence \(Sz=Tz\). Thus

$$p\bigl(S^{n}x_{0},Sz\bigr)\leq p\bigl(S^{n}x_{0},z \bigr)+p(z,Sz)\to0 \quad\mbox{as }n\to\infty. $$

By (i) and Lemma KST, we have \(z=Sz\). Hence \(z=Sz=Tz\) and \(p(z,z)=0\). □

Remark 4

Our Theorem 3 improves Theorem OK1 in the following ways:

  1. (1)

    We replace the completeness of X with the weaker assumption (cc).

  2. (2)

    A necessary and sufficient condition for the existence of a common fixed point of S and T is given in terms of the orbitally lower semicontinuities of \(G(x):=p(x,Sx)\) and \(H(y):=p(y,Ty)\).

  3. (3)

    We obtain some estimating expression for the iterative sequences.

  4. (4)

    In the presence of the conditions (BH) and (cc) of Theorem 3, it is clear that if S and T are orbitally continuous, then \(G(x):=p(x,Sx)\) is S-orbitally lower semicontinuous at \(x_{0}\) and \(H(y):=p(y,Ty)\) is T-orbitally lower semicontinuous at \(y_{0}\).

2.2 On Theorem OK2

We can follow the proof of Bollenbacher and Hicks’ result [4] and the proof of Theorem 3 to obtain the following result in terms of a w-distance. This result is related to Theorem KST.

Theorem 5

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(T:X\to X\) be a given mapping and \(\Phi:X\to[0,\infty)\) be any function. Let \(x_{0}\in X\). Suppose that the following conditions are satisfied:

  1. (BH)

    \(p(x,Tx)\leq\Phi(x)-\Phi(Tx)\), for all \(x\in O(x_{0},T)\).

  2. (cc)

    Every Cauchy sequence in \(O(x_{0},T)\) converges to a point in X.

Then the following statements are true:

  1. (i)

    There exists \(z\in X\) such that \(\lim_{n\to\infty }d(T^{n}x_{0},z)=\lim_{n\to\infty}p(T^{n}x_{0},z)=0\).

  2. (ii)

    \(p(T^{n}x_{0},z)\leq\Phi(T^{n}x_{0})\), for all \(n\ge1\).

  3. (iii)

    \(z=Tz\) and \(p(z,z)=0\) if and only if \(H(x):=p(x,Tx)\) is T-orbitally lower semicontinuous at \(x_{0}\).

In this subsection, we give another proof of Theorem OK2 via Theorem 5 and the following lemmas. We obtain the same conclusion as Theorem 3.

Lemma 6

Let \((X,d)\) be a metric space and \(p:X\times X\to[0,\infty)\) be a mapping. Let \(S,T:X\to X\) be two given mappings and \(\boldsymbol{X}:=X\times X\). Then the following statements are true:

  1. (i)

    Define \(\boldsymbol{d}:\boldsymbol{X}\times\boldsymbol{X}\to [0,\infty)\) by

    $$\boldsymbol {d}\bigl((x,y),(z,w)\bigr)=d(x,z)+d(y,w), $$

    for all \(x,y,z,w\in X\). Then d is a metric on X.

  2. (ii)

    Define \(\boldsymbol{p}:\boldsymbol{X}\times\boldsymbol{X}\to [0,\infty)\) by

    $$\boldsymbol{p}\bigl((x,y),(z,w)\bigr)=p(x,z)+p(y,w), $$

    for all \(x,y,z,w\in X\). If p is a w-distance (w.r.t. d), then p is a w-distance (w.r.t. d).

Lemma 7

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let X, d, and p be the same as in Lemma  6. Let \(\boldsymbol{x}_{0}=(x_{0},y_{0})\in\boldsymbol {X}\). Define \(\boldsymbol{T}:\boldsymbol{X}\to\boldsymbol{X}\) by

$$\boldsymbol{T}(\boldsymbol{x})=(Sx,Ty), $$

for all \(\boldsymbol{x}=(x,y)\in\boldsymbol{X}\). If \(G(x):=p(x,Sx)\) is S-orbitally lower semicontinuous at \(x_{0}\) and \(H(y):=p(y,Ty)\) is T-orbitally lower semicontinuous at \(y_{0}\), then \(\boldsymbol {H}(\boldsymbol{z}):=\boldsymbol{p}(\boldsymbol{z},\boldsymbol {T}\boldsymbol{z})\) is T-orbitally lower semicontinuous at \(\boldsymbol{x}_{0}\).

Proof

Let \(\{\boldsymbol{z}_{n}\}\) be a sequence in \(O(\boldsymbol {x}_{0},\boldsymbol{T})\) (\(:=\{\boldsymbol{x}_{0},\boldsymbol{T}\boldsymbol {x}_{0},\boldsymbol{T}^{2}\boldsymbol{x}_{0},\ldots\}\)) such that \(\boldsymbol {z}_{n}\to\boldsymbol{z}\) for some \(\boldsymbol{z}=(z,w)\in\boldsymbol {X}\). We write \(\boldsymbol{z}_{n}=(x_{n},y_{n})\) where \(x_{n},y_{n}\in X\), for all \(n\in\mathbb{N}\). Then \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences in \(O(x_{0},S)\) and \(O(y_{0},T)\), respectively. Moreover, \(x_{n}\to z\) and \(y_{n}\to w\). By the assumption, we have

$$p(z,Sz)=G(z)\le\liminf_{n\to\infty}G(x_{n})=\liminf _{n\to\infty}p(x_{n},Sx_{n}) $$

and

$$p(w,Tw)=H(w)\le\liminf_{n\to\infty}H(y_{n})=\liminf _{n\to\infty}p(y_{n},Ty_{n}). $$

It follows that

$$\begin{aligned} \boldsymbol{H}(\boldsymbol{z})&=\boldsymbol{p}(\boldsymbol {z},\boldsymbol{T} \boldsymbol{z}) \\ &=p(z,Sz)+p(w,Tw) \\ &\le\liminf_{n\to\infty}p(x_{n},Sx_{n})+\liminf _{n\to\infty}p(y_{n},Ty_{n}) \\ &\le\liminf_{n\to\infty}\boldsymbol{H}(\boldsymbol{z}_{n}). \end{aligned}$$

Hence H is T-orbitally lower semicontinuous at \(\boldsymbol{x}_{0}\). □

The following result follows directly from Theorem 5, Lemmas 6, and 7.

Lemma 8

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Let \(x_{0},y_{0}\in X\). Suppose that the following conditions are satisfied:

  1. (OK2*)

    \(p(x,Sx)+p(y,Ty)\leq\varphi(x)-\varphi(Sx)+\psi(y)-\psi (Ty)\), for all \(x\in O(x_{0},S)\) and \(y\in O(y_{0},T)\).

  2. (cc)

    Every Cauchy sequence in \(O(x_{0},S)\) converges to a point in X and every Cauchy sequence in \(O(y_{0},T)\) converges to a point in X.

Then the following statements are true:

  1. (i)

    There exist \(z,w\in X\) such that \(\lim_{n\to\infty}S^{n}x_{0}=z\) and \(\lim_{n\to\infty}T^{n}y_{0}=w\). Moreover,

    $$\lim_{n\to\infty}p\bigl(S^{n}x_{0},z\bigr)=\lim _{n\to\infty}p\bigl(T^{n}y_{0},w\bigr)=0. $$
  2. (ii)

    \(p(S^{n}x_{0},z)+p(T^{n}y_{0},w)\leq\varphi(S^{n}x_{0})+\psi(T^{n}y_{0})\), for all \(n\ge1\).

  3. (iii)

    Define \(G(x):=p(x,Sx)\), for all \(x\in X\), and \(H(y):=p(y,Ty)\), for all \(y\in X\). Then the following statements are equivalent:

    1. (a)

      \(z=Sz\) and \(w=Tw\) and \(p(z,z)=p(w,w)=0\).

    2. (b)

      G is S-orbitally lower semicontinuous at \(x_{0}\) and H is T-orbitally lower semicontinuous at \(y_{0}\).

Proof

Let X, d and p be the same as in Lemma 6 and T be the same as in Lemma 7. Define \(\boldsymbol{\Phi}:\boldsymbol{X}\to [0,\infty)\) by

$$\boldsymbol{\Phi}(\boldsymbol{x})=\varphi(x)+\psi(y), $$

for all \(\boldsymbol{x}=(x,y)\in\boldsymbol{X}\). Note that the condition (OK2*) is equivalent to

$$\boldsymbol{p}(\boldsymbol{x},\boldsymbol{T}\boldsymbol{x})\leq \boldsymbol{ \Phi}(\boldsymbol{x})-\boldsymbol{\Phi}(\boldsymbol {T}\boldsymbol{x}), $$

for all \(\boldsymbol{x}\in O(\boldsymbol{x}_{0},\boldsymbol{T})\) and \(\boldsymbol{x}_{0}=(x_{0},y_{0})\in\boldsymbol{X}\). Define \(\boldsymbol {x}_{n}=\boldsymbol{T}^{n}\boldsymbol{x}_{0}\), for all \(n\in\mathbb{N}\). Then by Theorem 5, we obtain the statements (i) and (ii).

We finally prove the statement (iii).

(b) ⇒ (a) It follows from Theorem 5 and Lemma 7.

(a) ⇒ (b) It is similar to the proof of (iii) of Theorem 3 so we omit the proof. □

We now obtain our result related to Theorem OK2.

Theorem 9

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Let \(x_{0},y_{0}\in X\). Suppose that the following conditions are satisfied:

  1. (OK2)

    \(\max\{p(x,y),p(y,x)\}+p(x,Sx)+p(y,Ty)\leq\varphi (x)-\varphi(Sx)+\psi(y)-\psi(Ty)\), for all \(x\in O(x_{0},S)\) and \(y\in O(y_{0},T)\).

  2. (cc)

    Every Cauchy sequence in \(O(x_{0},S)\) converges to a point in X and every Cauchy sequence in \(O(y_{0},T)\) converges to a point in X.

Then the following statements are true:

  1. (i)

    There exists \(z\in X\) such that \(\lim_{n\to\infty}S^{n}x_{0}=\lim_{n\to\infty}T^{n}y_{0}=z\). Moreover,

    $$\lim_{n\to\infty}p\bigl(S^{n}x_{0},z\bigr)=\lim _{n\to\infty}p\bigl(T^{n}y_{0},z\bigr)=0. $$
  2. (ii)

    \(p(S^{n}x_{0},z)+p(T^{n}y_{0},z)\leq\varphi(S^{n}x_{0})+\psi(T^{n}y_{0})\), for all \(n\ge1\).

  3. (iii)

    Define \(G(x):=p(x,Sx)\), for all \(x\in X\) and \(H(y):=p(y,Ty)\), for all \(y\in X\). Then the following statements are equivalent:

    1. (a)

      \(z=Sz=Tz\) and \(p(z,z)=0\).

    2. (b)

      G is S-orbitally lower semicontinuous at \(x_{0}\) and H is T-orbitally lower semicontinuous at \(y_{0}\).

Proof

Let \(x_{0},y_{0}\in X\). Define \(x_{n}=S^{n}x_{0}\) and \(y_{n}=T^{n}y_{0}\), for all \(n\ge 1\). Note that (OK2) ⇒ (OK2*). It follows from Lemma 8(i) that there exist \(z,w\in X\) such that \(\lim_{n\to\infty }x_{n}=z\), \(\lim_{n\to\infty}y_{n}=w\), and \(\lim_{n\to\infty}p(x_{n},z)=\lim_{n\to\infty}p(y_{n},w)=0\). By the condition (OK2), we have

$$\begin{aligned} p(x_{n},y_{n})&\le\max\bigl\{ p(x_{n},y_{n}),p(y_{n},x_{n}) \bigr\} +p(x_{n},x_{n+1})+p(y_{n},y_{n+1}) \\ &\le\varphi(x_{n})-\varphi(x_{n+1})+\psi(y_{n})- \psi(y_{n+1}), \end{aligned}$$

for all \(n\ge0\). So \(\sum_{n=0}^{\infty}p(x_{n},y_{n})\le\varphi(x_{0})+\psi (y_{0})<\infty\), that is, \(p(x_{n},y_{n})\to0\). Hence \(y_{n}\to z\), that is, \(z=w\). The statements (ii) and (iii) follow trivially. □

Remark 10

Our Theorem 9 improves Theorem OK2 in the following ways:

  1. (1)

    We replace the completeness of X with the weaker assumption (cc).

  2. (2)

    A necessary and sufficient condition for the existence of a common fixed point of S and T is given in terms of the orbitally lower semicontinuities of \(G(x):=p(x,Sx)\) and \(H(y):=p(y,Ty)\).

  3. (3)

    We obtain some estimating expression for the iterative sequences.

  4. (4)

    In the presence of the conditions (OK2) and (cc) of Theorem 9, it is clear that if S and T are orbitally continuous, then \(G(x):=p(x,Sx)\) is S-orbitally lower semicontinuous at \(x_{0}\) and \(H(y):=p(y,Ty)\) is T-orbitally lower semicontinuous at \(y_{0}\).

  5. (5)

    It is easy to see that the condition (OK2) can be replaced by the weaker condition (SS):

    $$m(x,y)+p(x,Sx)+p(y,Ty)\le\varphi(x)-\varphi(Sx)+\psi(y)-\psi(Ty), $$

    for all \(x\in O(x_{0},S)\) and \(y\in O(y_{0},T)\). In fact, if \(m(x_{n},y_{n})\to 0\), then there is a strictly increasing sequence \(\{n_{k}\}\) on ℕ such that one of the following sequences converges to zero:

    $$\begin{aligned}& \bigl\{ p(x_{n_{k}},y_{n_{k}})\bigr\} , \qquad\bigl\{ p(y_{n_{k}},x_{n_{k}})\bigr\} , \qquad\bigl\{ p(x_{n_{k}},y_{{n_{k}}+1}) \bigr\} , \qquad\bigl\{ p(y_{{n_{k}}+1},x_{n_{k}})\bigr\} , \\& \bigl\{ p(x_{{n_{k}}+1},y_{n_{k}})\bigr\} ,\qquad \bigl\{ p(y_{n_{k}},x_{{n_{k}}+1})\bigr\} ,\qquad \bigl\{ p(x_{{n_{k}}+1},y_{{n_{k}}+1}) \bigr\} ,\qquad \bigl\{ p(y_{{n_{k}}+1},x_{{n_{k}}+1})\bigr\} . \end{aligned}$$

Before moving to the next section, we give an example which satisfies our conditions in Theorems 3 and 9 but cannot be concluded from Theorems OK1 and OK2.

Example 11

Let \(X=[0,1/3)\) be equipped with the usual metric d. Define \(S,T:X\to X\) by \(Sx=0\) and \(Tx=x^{2}\), for all \(x\in X\). Also, define \(\varphi,\psi:X\to[0,\infty)\) by \(\varphi(x)=2x\) and \(\psi (x)=5x/2\), for all \(x\in X\). It is clear that X is not complete. Moreover, it is not hard to see that

$$d(Sx,Ty)\le d(x,y)+d(x,Sx)+d(y,Ty)\le\varphi(x)-\varphi(Sx)+\psi(y)-\psi(Ty), $$

for all \(x,y\in X\).

3 Results on Theorems OK1 and OK2 with the lower semicontinuities of φ and ψ

As studied in Theorem KST, it is more practical to put an assumption on the dominated function ψ than to put one on the mapping T itself. In this section, we discuss Theorems OK1 and OK2 where the functions φ and ψ are assumed to be lower semicontinuous.

3.1 Results related to Theorem OK1

The following observation is obvious.

Lemma 12

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(T:X\to X\) be a given mapping. Let \(\varphi :X\to[0,\infty)\) be any function. Let \(\widehat{y}\in X\). Define \(\widehat{p}:X\times X\to[0,\infty)\) and \(\widehat{\varphi}:X\to [0,\infty)\) by

$$\widehat{p}(x,y):=\frac{1}{2}p(T\widehat{y},x)+\frac{1}{2}p(T \widehat {y},y),\quad \textit{for all }x,y\in X, $$

and

$$\widehat{\varphi}(x):=\varphi(x)+\frac{1}{2}p(T\widehat{y},x),\quad \textit{for all }x\in X. $$

Then the following statements hold:

  1. (i)

    \(\widehat{p}\) is a w-distance on X.

  2. (ii)

    If φ is lower semicontinuous, then so is \(\widehat {\varphi}\).

In the setting of Theorem OK1 with the appearance of the lower semicontinuities of φ and ψ in place of the orbital continuities of S and T, we get a partial result with some additional assumption.

Theorem 13

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to [0,\infty)\) be any functions. Suppose that the following conditions are satisfied:

  1. (OK1)

    \(\max\{p(Sx,Ty),p(Ty,Sx)\}\le\varphi(x)-\varphi(Sx)+\psi (y)-\psi(Ty)\), for all \(x,y\in X\).

  2. (lsc)

    φ and ψ are lower semicontinuous.

Then the following statements hold:

  • S has a fixed point if and only if there exists an element \(\widehat{y}\in X\) such that \(\psi(\widehat{y})\le\psi(T\widehat{y})\).

  • T has a fixed point if and only if there exists an element \(\widehat{x}\in X\) such that \(\varphi(\widehat{x})\le\varphi(S\widehat{x})\).

In each case above, we find that S and T have a unique common fixed point.

Proof

We first prove the following two statements.

  1. (i)

    If there exists an element \(\widehat{y}\in X\) such that \(\psi (\widehat{y})\le\psi(T\widehat{y})\), then S has a fixed point.

  2. (ii)

    If there exists an element \(\widehat{x}\in X\) such that \(\varphi(\widehat{x})\le\varphi(S\widehat{x})\), then T has a fixed point.

Since the proof of (ii) is similar to that of (i), we prove only (i). Assume that there exists an element \(\widehat{y}\in X\) such that \(\psi (\widehat{y})\le\psi(T\widehat{y})\). Then we have

$$\begin{aligned} \max\bigl\{ p(Sx,T\widehat{y}),p(T\widehat {y},Sx)\bigr\} \le \varphi(x)-\varphi(Sx), \quad\mbox{for all }x\in X. \end{aligned}$$
(3)

Let \(\widehat{p}\) and \(\widehat{\varphi}\) be the same as in Lemma 12. Then it follows from (3) that

$$\begin{aligned} \widehat{p}(x,Sx)&=\frac{1}{2}p(T\widehat{y},x)+\frac {1}{2}p(T \widehat{y},Sx) \\ &=\frac{1}{2}p(T\widehat{y},x)+p(T\widehat{y},Sx)-\frac{1}{2}p(T \widehat {y},Sx) \\ &\le\frac{1}{2}p(T\widehat{y},x)+\max\bigl\{ p(Sx,T\widehat{y}),p(T\widehat {y},Sx)\bigr\} -\frac{1}{2}p(T\widehat{y},Sx) \\ &\le\frac{1}{2}p(T\widehat{y},x)+\varphi(x)-\varphi(Sx)- \frac {1}{2}p(T\widehat{y},Sx) \\ &=\widehat{\varphi}(x)-\widehat{\varphi}(Sx), \end{aligned}$$

for all \(x\in X\). By Lemma 12 and Theorem KST, there exists \(\widehat{x}\in X\) such that \(\widehat{x}=S\widehat{x}\).

Next, we prove the following statement:

  1. (iii)

    If S has a fixed point, then there exists an element \(\widehat{y}\in X\) such that \(\psi(\widehat{y})\le\psi(T\widehat{y})\).

In fact, if \(\widehat{x}=S\widehat{x}\), then \(\varphi(\widehat {x})=\varphi(S\widehat{x})\). It follows from (ii) that T has a fixed point, that is, there exists \(\widehat{y}\in X\) such that \(\widehat {y}=T\widehat{y}\), so we obtain (iii). The uniqueness follows immediately from Lemma 1. □

Remark 14

In (i) of the proof of Theorem 13, we also have \(ST\widehat{y}=T\widehat{y}\). To see this, we note that \(p(\widehat{x},\widehat{x})=0\) and \(p(\widehat{x},\widehat{y})\le\psi (\widehat{y})-\psi(T\widehat{y})\le0\). This gives \(\widehat{x}=\widehat {y}\) and hence \(ST\widehat{y}=S\widehat{x}=\widehat{x}=T\widehat{y}\).

Since d is a w-distance, we immediately obtain this corollary which is related to Theorem BB where the condition (oc) is replaced by the condition (lsc).

Corollary 15

Let \((X,d)\) be a complete metric space and \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Suppose that the following conditions are satisfied:

  1. (BB)

    \(d(Sx,Ty)\le\varphi(x)-\varphi(Sx)+\psi(y)-\psi(Ty)\), for all \(x,y\in X\).

  2. (lsc)

    φ and ψ are lower semicontinuous.

Then the following statements hold:

  • S has a fixed point if and only if there exists an element \(\widehat{y}\in X\) such that \(\psi(\widehat{y})\le\psi(T\widehat{y})\).

  • T has a fixed point if and only if there exists an element \(\widehat{x}\in X\) such that \(\varphi(\widehat{x})\le\varphi(S\widehat{x})\).

In each case above, S and T have a unique common fixed point.

3.2 Results related to Theorem OK2

Lemma 16

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Suppose that the following conditions hold:

  1. (OK2*)

    \(p(x,Sx)+p(y,Ty)\leq\varphi(x)-\varphi(Sx)+\psi(y)-\psi (Ty)\), for all \(x,y\in X\).

  2. (lsc)

    φ and ψ are lower semicontinuous.

Then there exists \((\widehat{x},\widehat{y})\in X\times X\) such that \(\widehat{x}=S\widehat{x}\) and \(\widehat{y}=T\widehat{y}\).

Proof

Let X, d, and p be the same as in Lemma 6 and T be the same as in Lemma 7. It is clear that \((\boldsymbol{X},\boldsymbol{d})\) is complete. Define \(\boldsymbol{\Phi}:\boldsymbol{X}\to[0,\infty)\) by

$$\boldsymbol{\Phi}(\boldsymbol{x})=\varphi(x)+\psi(y), $$

for all \(\boldsymbol{x}=(x,y)\in\boldsymbol{X}\). Since φ and ψ are lower semicontinuous, we conclude that Φ is lower semicontinuous. Note that (OK2*) is equivalent to

$$\boldsymbol{p}(\boldsymbol{x},\boldsymbol{T}\boldsymbol{x})\leq \boldsymbol{ \Phi}(\boldsymbol{x})-\boldsymbol{\Phi}(\boldsymbol {T}\boldsymbol{x}), \quad \mbox{for all } \boldsymbol{x}\in\boldsymbol{X}. $$

By Theorem KST, there exists \(\boldsymbol{\widehat{x}}=(\widehat {x},\widehat{y})\in\boldsymbol{X}\) such that \(\boldsymbol{\widehat {x}}=\boldsymbol{T}\boldsymbol{\widehat{x}}\), that is, \(\widehat{x}=S\widehat{x}\) and \(\widehat{y}=T\widehat{y}\). □

We now obtain a result related to Theorem OK2 where the condition (oc) is replaced by the condition (lsc).

Theorem 17

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Suppose that the following conditions hold:

  1. (OK2)

    \(\max\{p(x,y),p(y,x)\}+p(x,Sx)+p(y,Ty)\leq\varphi (x)-\varphi(Sx)+\psi(y)-\psi(Ty)\), for all \(x,y\in X\).

  2. (lsc)

    φ and ψ are lower semicontinuous.

Then S and T have a unique common fixed point.

Proof

It follows immediately from Lemmas 1 and 16. □

Remark 18

It is easy to see that the condition (OK2) can be replaced by the weaker condition (SS):

$$m(x,y)+p(x,Sx)+p(y,Ty)\le\varphi(x)-\varphi(Sx)+\psi(y)-\psi(Ty), $$

for all \(x,y\in X\).

With a slight modification of the condition (OK2*), we can conclude a common fixed point even if we assume that either φ or ψ is lower semicontinuous. However, the uniqueness is not guaranteed.

Theorem 19

Let \((X,d)\) be a complete metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to [0,\infty)\) be any functions. Suppose that the following conditions are satisfied:

  1. (SS*)

    \(p(x,Sx)+p(y,Ty)\le\varphi(x)-\psi(Sx)+\psi(y)-\varphi (Ty)\), for all \(x,y\in X\).

  2. (lsc*)

    Either φ or ψ is lower semicontinuous.

Then S and T have a common fixed point.

Proof

We may assume that φ is lower semicontinuous. Let \(y=Sx\) in the condition (SS*) and we have

$$\begin{aligned} p(x,TSx)&\le p(x,Sx)+p(Sx,TSx) \\ &\le\varphi(x)-\psi(Sx)+\psi(Sx)-\varphi(TSx) \\ &\le\varphi(x)-\varphi(TSx). \end{aligned}$$

By Theorem KST, there exists \(\widehat{x}\in X\) such that \(\widehat {x}=TS\widehat{x}\). It follows that

$$p(\widehat{x},\widehat{x})\le p(\widehat{x},S\widehat{x})+p(S\widehat {x},TS \widehat{x})\le\varphi(\widehat{x})-\psi(S\widehat{x})+\psi (S\widehat{x})- \varphi(TS\widehat{x})=0. $$

Thus \(p(\widehat{x},\widehat{x})=p(\widehat{x},S\widehat{x})=p(S\widehat {x},TS\widehat{x})=0\). It follows from the first equality that \(\widehat {x}=S\widehat{x}\). Then \(p(\widehat{x},T\widehat{x})=0\) and hence \(\widehat{x}=T\widehat{x}\). This completes the proof. □

Remark 20

The condition (SS*) in Theorem 19 is motivated by Lemma 2.2 of [5]. More precisely, let \((X,d)\) be a metric space and let \(S,T:X\to X\) be two mappings with two nonnegative real numbers λ and μ such that \(\lambda+\mu<1\) and

$$d(Sx,TSx)\le\frac{\lambda}{1-\mu}d(x,Sx) $$

and

$$d(Tx,STx)\le\frac{\lambda}{1-\mu}d(x,Tx), $$

for all \(x\in X\). Set \(\varphi(x)=\frac{1-\mu}{1-\lambda-\mu}d(x,Sx)\) and \(\psi(x)=\frac{1-\mu}{1-\lambda-\mu}d(x,Tx)\). It follows then that

$$d(x,Sx)\le\varphi(x)-\psi(Sx) $$

and

$$d(x,Tx)\le\psi(x)-\varphi(Tx) $$

for all \(x\in X\). In particular, S and T satisfy the condition (SS*).

3.3 The existence of a common fixed point of S and T is equivalent to their orbital continuities

First, let us start with the following easy observation.

Lemma 21

Let \((X,d)\) be a metric space and p be a w-distance. Let \(\varphi:X\to[0,\infty)\) be any function. Suppose that \(S:X\to X\) is a mapping satisfying the following condition:

  • There exists \(z\in X\) such that

    $$p(z,Sx)\le\varphi(x)-\varphi(Sx) $$

    for all \(x\in X\).

Then the following statements are true:

  1. (i)

    \(\lim_{n\to\infty}p(z,S^{n}x)=0\) for all \(x\in X\).

  2. (ii)

    \(z=Sz\) if and only if S is orbitally continuous and \(p(z,z)=0\).

Proof

(i) Let \(x\in X\) be given. It follows that

$$0\le p\bigl(z,S^{n}x\bigr)\le\varphi\bigl(S^{n}x\bigr)- \varphi\bigl(S^{n+1}x\bigr) $$

for all integers \(n\ge0\). Then the sequence \(\{\varphi(S^{n}x)\}\) is nonincreasing and hence it is convergent. In particular, \(\lim_{n\to \infty}p(z,S^{n}x)=0\).

(ii) (⇒) Assume that \(z=Sz\). In particular, \(p(z,z)=p(z,Sz)\le\varphi(z)-\varphi(Sz)=\varphi(z)-\varphi(z)=0\). To show that S is orbitally continuous, let \(x\in X\) be such that \(S^{n_{j}}x\to w\) for some \(w\in X\). It follows from (i) and Lemma KST that \(S^{n}x\to z\). So, we have \(w=z\). Now, \(S(S^{n_{j}}x)\to z=Sz\). (⇐) Assume that S is orbitally continuous and \(p(z,z)=0\). It follows from (i) and Lemma KST that \(S^{n}z\to z\). Since S is orbitally continuous, we have \(S(S^{n}z)\to Sz\). It follows then that \(z=Sz\). □

The following result shows that the condition (oc) is not only sufficient but also necessary for the existence of a common fixed point in Theorems OK1 and OK2.

Theorem 22

Let \((X,d)\) be a metric space and p be a w-distance on X. Let \(S,T:X\to X\) be two given mappings. Let \(\varphi,\psi:X\to[0,\infty)\) be any functions. Suppose that one of the following conditions holds:

  1. (OK1)

    \(\max\{p(Sx,Ty),p(Ty,Sx)\}\le\varphi(x)-\varphi(Sx)+\psi (y)-\psi(Ty)\) for all \(x,y\in X\).

  2. (OK2)

    \(\max\{p(x,y),p(y,x)\}+p(x,Sx)+p(y,Ty)\leq\varphi (x)-\varphi(Sx)+\psi(y)-\psi(Ty)\) for all \(x,y\in X\).

If S and T have a (unique) common fixed point, then:

  1. (oc)

    S and T are orbitally continuous.

Proof

We assume that S, T satisfy the condition (OK1) and \(z=Sz=Tz\). Letting \(y=z\) in the condition (OK1) gives \(p(z,Sx)\le\varphi(x)-\varphi(Sx)\) for all \(x\in X\). It follows from the preceding lemma that S is orbitally continuous. Similarly, interchanging the role of S and T ensures that T is orbitally continuous as well.

We assume that S, T satisfy the condition (OK2) and \(z=Sz=Tz\). It is clear that \(p(z,z)=0\). Because of this, letting \(y=z\) in the condition (OK2) gives \(p(z,Sx)\le p(z,x)+p(x,Sx)+p(z,Tz) \le\varphi(x)-\varphi (Sx)+\psi(z)-\psi(Tz)=\varphi(x)-\varphi(Sx)\) for all \(x\in X\). As proved in the first part, we conclude that S and T are orbitally continuous. □