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Existence theorems for singlevalued and setvalued mappings with wdistances in metric spaces
Fixed Point Theory and Applications volume 2016, Article number: 38 (2016)
Abstract
In this paper, using the concept of wdistances, and we prove existence theorems for singlevalued mappings and setvalued mappings in a complete metric space which generalize Takahashi, Wong, and Yao’s theorems.
1 Introduction
Let \(\ell^{\infty}\) be the Banach space of bounded sequences with supremum norm and let \((\ell^{\infty})^{*}\) be the dual space of \(\ell^{\infty}\). Let μ be an element of \((\ell^{\infty})^{*}\). We denote by \(\mu(f)\) the value of μ at \(f=\{x_{n}\} \in\ell^{\infty}\). Sometimes, we denote by \(\mu_{n}(x_{n})\) the value \(\mu(f)\). A linear functional μ on \(\ell^{\infty}\) is called a mean if \(\mu(e)=\\mu\=1\), where \(e=\{1, 1, 1, \dots \}\). Hasegawa et al. [1] obtained the following unique fixed point theorem on a complete metric space.
Theorem 1.1
([1])
Let \((X,d)\) be a complete metric space and let S be a mapping of X into itself. Let \(\ell^{\infty}\) be the Banach space of bounded sequences with the supremum norm. Suppose that there exist a real number r with \(0\leq r<1\) and an element \(x\in X\) such that \(\{S^{n} x\}\) is bounded and
for some mean μ on \(l^{\infty}\). Then the following hold:

(1)
S has a unique fixed point \(u\in X\);

(2)
for every \(z\in X\), the sequence \(\{S^{n} z\}\) converges to u.
By using the idea of Caristi’s fixed point theorem [2], Chuang et al. [3] proved a unique fixed point theorem for singlevalued mappings which generalizes Theorem 1.1. Furthermore, they obtained an existence theorem for setvalued mappings in a complete metric space. Using these results, Chuang et al. [3] obtained new and wellknown existence theorems in a complete metric space.
On the other hand, in 1996, Kada et al. [4] introduced the concept of wdistances on a metric space.
Let \((X,d)\) be a metric space. A function \(p:X\times X\to[0, \infty)\) is said to be a wdistance [4] on X if the following are satisfied:

(1)
\(p(x, z)\le p(x, y)+p(y, z)\) for all \(x, y, z\in X\);

(2)
for any \(x\in X\), \(p(x, \cdot):X\to[0, \infty)\) is lower semicontinuous;

(3)
for any \(\varepsilon>0\), there exists \(\delta>0\) such that \(p(z, x)\le\delta\) and \(p(z, y)\le\delta\) imply \(d(x, y)\le\varepsilon\).
Using the concept of wdistances, they improved important results in complete metric spaces. For example, they improved Caristi’s fixed point theorem [2], Ekeland’s variational principle [5] and the nonconvex minimization theorem according to Takahashi [6]. Motivated by Chuang et al. [3], Takahashi et al. [7] improved their unique fixed point theorem for singlevalued mappings by using the concept of wdistances. Furthermore, they extended Chuang et al.’s existence theorem [3] for setvalued mappings to wdistances. However, Takahashi et al. [7] assumed that wdistances are symmetric.
In this paper, without assuming that wdistances are symmetric, we prove Takahashi et al.’s unique fixed point theorems for singlevalued mappings and their existence theorem for setvalued mappings in a complete metric space. Using these results, we obtained new and wellknown existence theorems in a complete metric space. In particular, using this unique fixed point theorem for singlevalued mappings, we obtain a unique fixed point theorem of Caristi’s type [2] with lower semicontinuous functions and wdistances. It seems that the proofs are technical and useful.
2 Preliminaries
Throughout this paper, we denote by \(\mathbb {N}\) and \(\mathbb {R}\) the sets of positive integers and real numbers, respectively. Let X be a metric space with metric d. Then we denote by \(W(X)\) the set of all wdistances on X. A wdistance p on X is called symmetric if \(p(x,y)=p(y, x)\) for all \(x, y\in X\). We denote by \(W_{0}(X)\) the set of all symmetric wdistances on X. Note that the metric d is an element of \(W_{0}(X)\). We also know that there are many important examples of wdistances on X; see [4, 8].
The following lemma was proved by Kada et al. [4]; see also Shioji et al. [9].
Lemma 2.1
([4])
Let \((X,d)\) be a complete metric space and let p be a wdistance on X. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be sequences in X. Let \(\{s_{n}\}\) and \(\{ t_{n}\}\) be sequences in \([0,\infty)\) converging to 0, and let \(x, y, z \in X\). Then the following hold:

(1)
If \(p(x_{n},y)\leq s_{n}\) and \(p(x_{n},z)\leq t_{n}\) for all \(n\in \mathbb {N}\), then \(y=z\). In particular, if \(p(x,y)=0\) and \(p(x,z)=0\), then \(y=z\);

(2)
if \(p(x_{n},y_{n})\leq s_{n}\) and \(p(x_{n},z)\leq t_{n}\) for all \(n\in\mathbb {N}\), then the sequence \(\{y_{n}\}\) converges to z;

(3)
if \(p(x_{n},x_{m})\leq s_{n}\) for all \(n, m\in\mathbb {N}\) with \(m>n\), then the sequence \(\{x_{n}\}\) is a Cauchy sequence;

(4)
if \(p(y,x_{n})\leq s_{n}\) for all \(n\in\mathbb {N}\), then \(\{ x_{n}\}\) is a Cauchy sequence.
Let \((X,d)\) be a metric space and let g be a function of X into \((\infty, \infty]=\mathbb {R}\cup\{\infty\}\). Then g is proper if there exists \(x\in X\) such that \(g(x)<\infty \). A function g is lower semicontinuous if for any \(t\in\mathbb {R}\), the set \(\{x\in X: g(x)\leq t\}\) is closed. A function g is bounded below if there exists \(K\in \mathbb {R}\) such that
Kada et al. [4] improved Caristi’s fixed point theorem [2] as follows; see also [8], Theorem 2.2.8.
Theorem 2.2
([4])
Let \((X,d)\) be a complete metric space, \(p\in W(X)\), and let \(\phi:X\to (\infty,\infty]\) be a proper, bounded below, and lower semicontinuous function. Let \(T:X\to X\) be a mapping such that for each \(x\in X\),
Then there exists \(z\in X\) such that \(Tz=z\) and \(p(z,z)=0\).
A mean μ is called a Banach limit on \(\ell^{\infty}\) if \(\mu _{n}(x_{n+1})=\mu_{n}(x_{n})\) for all \(\{x_{n}\}\in\ell^{\infty}\). We know that there exists a Banach limit on \(\ell^{\infty}\). If μ is a Banach limit on \(\ell^{\infty}\), then for \(f=\{x_{n}\} \in\ell^{\infty}\),
In particular, if \(f=\{x_{n}\} \in\ell^{\infty}\) and \(x_{n}\to a\in\mathbb {R}\), then we have \(\mu(f)= \mu_{n} (x_{n}) = a\). For the proof of existence of a Banach limit and its other elementary properties, see [8].
3 Existence theorems for singlevalued mappings
In this section, using means and wdistances, we first prove an existence theorem for mappings in metric spaces which generalizes Takahashi et al. [7].
Theorem 3.1
Let \((X,d)\) be a complete metric space, let \(p\in W(X)\) and let \(\{x_{n}\}\) be a sequence in X such that \(\{p(x_{n}, w)\}\) and \(\{p(w, x_{n})\}\) are bounded for some \(w\in X\). Let μ be a mean on \(\ell^{\infty}\) and let \(\phi:X\to(\infty,\infty]\) be a proper, bounded below, and lower semicontinuous function. Let \(S:X\to X\) be a mapping. Suppose that there exist \(l, m\in\mathbb {N}\cup\{0\}\) such that
for all \(y\in X\). Then there exists \(x_{0}\in X\) such that

(1)
\(x_{0}\) is a unique fixed point of S in \(\{x\in X: \phi(x)< \infty\}\);

(2)
\(x_{0}= \lim_{k\to\infty}S^{k} y\) for all \(y\in X\) with \(\phi (y)< \infty\);

(3)
\(\phi(x_{0})=\inf_{v\in X}\phi(v)\).
Proof
Since \(\{p(x_{n}, w)\}\) is bounded for some \(w\in X\), we have, for any \(y\in X\), \(\{p(x_{n},y)\}\) is bounded. In fact, we have, for any \(n\in\mathbb {N}\),
Furthermore, since \(\{p(w, x_{n})\}\) is bounded, we see that \(\{p(z, x_{n})\}\) is bounded for all \(z\in X\). In fact, we have, for any \(n\in\mathbb {N}\),
We have from (3.1)
for all \(y\in X\). For \(y\in X\) with \(\phi(y)<\infty\), we have from (3.2) \(\phi(S^{k}y)< \infty\) for all \(k\in \mathbb {N}\cup\{0\}\) and hence
and
Then we see that \(\{\phi(S^{k}y)\}\) is a decreasing sequence which is bounded below. Hence \(\lim_{k\to\infty}\phi(S^{k}y)\) exists. Put \(s=\lim_{k\to\infty}\phi(S^{k}y)\). Since
and
for all \(k\in\mathbb {N}\), we have
Then we have
We have, for any \(k, n\in\mathbb {N}\),
Since μ is a mean on \(\ell^{\infty}\), we have from (3.3) and (3.4), for any \(k\in\mathbb {N}\),
We have from (3.6), for any \(h,k \in\mathbb {N}\) with \(k> h\),
where \(\alpha_{h}=\phi(S^{l+h}y)\) and \(\beta_{h}=\phi(S^{m+h}y)\). Since \(\alpha_{h}s +\beta_{h}s \to0\) as \(h\to\infty\), we see from Lemma 2.1 that \(\{S^{l+m+k}y\}\) is a Cauchy sequence in X. Since X is complete, there exists \(y_{0}\in X\) such that \(\lim_{k\to\infty}S^{l+m+k} y=y_{0}\). We know from the definition of p that, for any \(n\in\mathbb {N}\), \(y\mapsto p(x_{n},y)\) is lower semicontinuous. Using this and following the technique of [7], we have, for any \(n\in\mathbb {N}\),
and hence
On the other hand, we have from (3.7), for any \(h,k, n \in \mathbb {N}\) with \(k> h\),
and hence
Applying μ to both sides of the inequality, we have
Letting \(h\to\infty\), we get from (3.5) that
Then we have from (3.8) and (3.9)
This implies that
Similarly, for another \(u\in X\) with \(\phi(u)<\infty\), there exists \(u_{0}\in X\) such that \(\lim_{k\to\infty}S^{l+m+k}u=u_{0}\) and \(\mu_{n} p(x_{n},u_{0})=0\). We also have, for \(k,n\in\mathbb {N}\),
and hence
Furthermore, we have, for \(k,n\in\mathbb {N}\),
and hence
We know that \(\mu_{n}p(S^{l+m+k}y, x_{n})\to0\) as \(k\to\infty\). Thus, we have from (3.11), (3.12), and Lemma 2.1 \(y_{0}=u_{0}\). Therefore we have \(x_{0}=\lim_{k\to\infty}S^{k} z\) for all \(z\in X\) with \(\phi(z)<\infty\). Since ϕ is lower semicontinuous and \(\lim_{k\to\infty}S^{k} z=x_{0}\) for all \(z\in X\) with \(\phi(z)<\infty\), we have
This implies that
We finally prove that \(x_{0}\) is a unique fixed point of S in \(\{x\in X: \phi(x)< \infty\}\). Since, from (3.13),
we have \(\mu_{n} p(x_{n},S^{l}x_{0})=0\). We also know \(\mu_{n} p(x_{n},x_{0})=0\). For \(k,n \in\mathbb {N}\), we have
and
Then, as in the above argument, we have
and
We also know from (3.5) that \(\mu_{n}p(S^{m+k}y, x_{n})\to0\) as \(k\to\infty\). Therefore, from (3.14), (3.15), and Lemma 2.1 \(S^{l}x_{0}=x_{0}\). Using \(S^{l}x_{0}=x_{0}\), we have from (3.13)
and hence \(\mu_{n} p(x_{n},Sx_{0})=0\). Since, for \(k,n \in\mathbb {N}\),
we have
We have from (3.15), (3.16), and Lemma 2.1 \(Sx_{0}=x_{0}\). We show that \(x_{0}\) is a unique fixed point of S in \(\{x\in X: \phi (x)< \infty\}\). Indeed, if \(z_{0}\) is a fixed point of S with \(\phi(z_{0})<\infty\), then
and hence \(\mu_{n}p(x_{n}, z_{0})=0\). Since, for \(k,n \in\mathbb {N}\),
we have
Since \(\mu_{n}p(S^{m+k}y, x_{n})\to0\) as \(k\to\infty\), from (3.15), (3.17), and Lemma 2.1, we have \(z_{0}=x_{0}\). Therefore \(x_{0}\) is a unique fixed point of S in \(\{y\in X: \phi(y)<\infty\}\). This completes the proof. □
Using Theorem 3.1, we can obtain the following result proved by Takahashi et al. [7].
Theorem 3.2
([7])
Let \((X,d)\) be a complete metric space, let \(p\in W_{0}(X)\) and let \(\{x_{n}\}\) be a sequence in X such that \(\{p(x_{n}, x)\}\) is bounded for some \(x\in X\). Let μ be a mean on \(\ell^{\infty}\) and let \(\psi:X\to(\infty,\infty]\) be a proper, bounded below, and lower semicontinuous function. Let \(T:X\to X\) be a mapping. Suppose that there exists \(m\in\mathbb {N}\cup \{0\}\) such that
Then there exists \(\bar{x}\in X\) such that

(a)
\(\bar{x}= \lim_{k\to\infty}T^{k} y\) for all \(y\in X\) with \(\psi (y)< \infty\);

(b)
\(\psi(\bar{x})=\inf_{u\in X}\psi(u)\);

(c)
x̄ is a unique fixed point of T in \(\{x\in X: \psi (x)< \infty\}\).
Proof
Since \(\{x_{n}\}\) is a bounded sequence in X such that \(\{p(x_{n}, x)\}\) is bounded for some \(x\in X\), we see from \(p\in W_{0}(X)\) that \(\{p(x, x_{n})\}\) is bounded. Putting \(S=T\), \(l=m\), and \(\phi=2\psi\) in Theorem 3.1, we have
and hence
Thus we have the desired result from Theorem 3.1. □
Using Theorem 3.1 and the generalized Caristi’s fixed point theorem (Theorem 2.2), we also have a unique fixed point theorem of Caristi’s type [2] with lower semicontinuous functions and wdistances.
Theorem 3.3
Let \((X,d)\) be a complete metric space and let \(p\in W(X)\) such that \(p(x,x)=0\) for all \(x\in X\). Let \(\phi:X\to(\infty, \infty]\) be a proper, bounded below, and lower semicontinuous function. Let \(S:X\to X\) be a mapping. Suppose that there exists \(\alpha\in\mathbb{R}\) such that
Then there exists \(x_{0}\in X\) such that

(1)
\(x_{0}\) is a unique fixed point of S in \(\{x\in X: \phi(x)< \infty\}\);

(2)
\(x_{0}= \lim_{k\to\infty}S^{k} y\) for all \(y\in X\) with \(\phi (y)< \infty\);

(3)
\(\phi(x_{0})=\inf_{v\in X}\phi(v)\).
Proof
Let us first consider \(\alpha>0\). Putting \(y=x\) in (3.19), we have from \(p(x,x)=0\)
and hence
By Theorem 2.2, there exists \(u_{0}\in X\) such that \(Su_{0}=u_{0}\). Putting \(x=u_{0}\) in (3.19) again, we have, for any \(y\in X\),
Since \(Su_{0}=u_{0}\), we have, for any \(y\in X\),
By Theorem 3.1, we see that \(x_{0}\) is a unique fixed point of S in \(\{x\in X: \phi(x)<\infty\}\) such that \(\phi(x_{0})=\inf_{u\in X}\phi(u)\) and \(x_{0}=\lim_{k\to\infty}S^{k} z\) for all \(z\in X\) with \(\phi(z)< \infty\).
Next let us consider the case of \(\alpha=0\). Then we have
Replacing x and y by Sx and x in (3.20), respectively, we have
and hence
We also see from Theorem 2.2 that there exists \(u_{0}\in X\) such that \(Su_{0}=u_{0}\). Putting \(x=u_{0}\) in (3.19), we have also
By Theorem 3.1, we see that \(x_{0}\) is a unique fixed point of S in \(\{x\in X: \phi(x)<\infty\}\) such that \(\phi(x_{0})=\inf_{u\in X}\phi(u)\) and \(x_{0}=\lim_{k\to\infty}S^{k} z\) for all \(z\in X\) with \(\phi(z)< \infty\).
In the case of \(\alpha<0\), we have \(1\alpha>0\). Furthermore, replacing y by Sx in (3.19), we have from \(p(Sx, Sx)=0\)
and hence
Take \(x\in X\) with \(\phi(x)<\infty\). Then we have, for any \(n\in \mathbb {N}\),
Adding these inequalities, we have
Since \(\{\phi(S^{n}x)\}\) is a decreasing sequence and bounded below, we see that there exists \(s= \lim_{n\to\infty}\phi(S^{n}x)\). Thus we have, for any \(n\in\mathbb {N}\),
Then \(\{p(x,S^{n}x)\}\) is bounded. Furthermore, from (3.21) we have
As in the above argument, we have, for any \(n\in\mathbb {N}\),
Then \(\{p(S^{n}x,x)\}\) is bounded. Replacing x by \(S^{n}x\) in (3.19), we have, for any \(n\in \mathbb {N}\),
Applying a Banach limit μ to the both sides of this inequality, we have
Since \(\mu_{n} p(S^{n+1}x,y)+\mu_{n} p(y,S^{n+1}x) = \mu_{n} p(S^{n}x,y)+\mu _{n} p(y,S^{n}x)\), we get
By Theorem 3.1, S has a unique fixed point \(x_{0}\) in \(\{x\in X: \phi(x)<\infty\}\) such that \(\phi(x_{0})=\inf_{u\in X}\phi(u)\) and \(x_{0}=\lim_{k\to\infty}S^{k} z\) for all \(z\in X\) with \(\phi(z)< \infty\). □
4 Existence theorems for setvalued mappings
Using wdistances, we have the following existence theorem for setvalued mappings in a complete metric space. Let \((X,d)\) be a metric space and let \(P(X)\) be the class of all nonempty subsets of X. A mapping of X into \(P(X)\) is called a setvalued mapping, or a multivalued mapping.
Theorem 4.1
Let \((X,d)\) be a complete metric space, let \(p\in W(X)\), and let \(\{x_{n}\}\) be a sequence in X such that \(\{p(x_{n}, w)\}\) and \(\{p(w, x_{n})\}\) are bounded for some \(w\in X\). Let μ be a mean on \(\ell^{\infty}\) and let \(\phi:X\to(\infty, \infty]\) be a proper, bounded below, and lower semicontinuous function. Let \(S:X\to P(X)\) be a setvalued mapping such that for each \(x\in X\), there exists \(y\in Sx\) satisfying
Then there exists \(x_{0}\in X\) such that

(1)
\(x_{0}\in Sx_{0}\);

(2)
\(\phi(x_{0})=\inf_{y\in X}\phi(y)\);

(3)
for any \(z\in X\) with \(\phi(z)<\infty\), there exists a sequence \(\{z_{m}\}\subset X\) such that \(z_{m+1}\in Sz_{m}\), \(m\in\mathbb {N} \cup\{0\}\) and \(z_{m}\to x_{0}\) as \(m\to\infty\).
Proof
For each \(z_{1}=z\in X\) with \(\phi(z)<\infty\), there exists \(z_{2}\in Sz_{1}\) such that
Repeating this process, we get a sequence \(\{z_{m}\}\) in X such that \(z_{m+1}\in Sz_{m}\) and
for each \(m\in\mathbb{N}\). Clearly, \(\{\phi(z_{m})\}\) is a decreasing sequence which is bounded below. Hence \(\lim_{m\to\infty}\phi(z_{m})\) exists. Put \(s=\lim_{m\to\infty}\phi(z_{m})\). We have from (4.2)
We have, for any \(m, n\in\mathbb {N}\),
Since μ is a mean on \(\ell^{\infty}\), we have, for any \(m\in\mathbb {N}\),
We have from (4.4), for any \(l,m \in\mathbb {N}\) with \(m> l\),
and \(2\phi(z_{l})2s \to0\) as \(l\to\infty\). We see from Lemma 2.1 that \(\{z_{m}\}\) is a Cauchy sequence in X. Since X is complete, there exists a point \(x_{0}\in X\) such that \(\lim_{m\to\infty}z_{m}=x_{0}\). We know from the definition of p that, for any \(n\in\mathbb {N}\), \(y\mapsto p(x_{n},y)\) is lower semicontinuous. Using this and following the technique of [7], we have, for any \(n\in\mathbb {N}\),
and hence
On the other hand, we have from (4.5), for any \(l,k, n \in \mathbb {N}\) with \(m> l\),
and hence
Applying μ to both sides of the inequality, we have
Letting \(l\to\infty\), we get
We have from (4.3), (4.6), and (4.7)
This implies that
Doing the same argument as above for each \(y_{1}=y\in X\) with \(\phi (y)<\infty\), we can construct a sequence \(\{y_{m}\}\) in X such that \(\{\phi(y_{m})\}\) is a decreasing sequence, \(\lim_{m\to\infty}y_{m}=y_{0}\) for some \(y_{0}\in X\), and \(\mu_{n} p(x_{n},y_{0})=0\). We show that \(x_{0}=y_{0}\). We have, for any \(m,n \in\mathbb {N}\),
Then, we have
Furthermore, we have, for any \(m,n \in\mathbb {N}\),
and hence
We know from (4.3) that \(\mu_{n}p(z_{m}, x_{n})\to0\) as \(m\to\infty\). Therefore, from (4.9), (4.10), and Lemma 2.1 \(x_{0}=y_{0}\). Since ϕ is lower semicontinuous,
Since \(y_{1}\) is any point of X with \(\phi(y_{1})<\infty\), we have
Using (4.1), we have \(u_{0}\in X\) such that \(u_{0}\in Sx_{0}\) and
Furthermore, repeating this process, we have \(v_{0}\in X\) such that \(v_{0}\in Su_{0}\) and
Using (4.11), we have
Then we have from (4.12) and (4.13)
This implies that
Since \(p(z_{m}, u_{0})\leq p(z_{m}, x_{n})+p(x_{n},u_{0} )\) for \(m,n\in\mathbb {N}\), we have
We know from (4.3) that \(\mu_{n}p(z_{m}, x_{n})\to0\) as \(m\to\infty\). Therefore, from (4.9), (4.14), and Lemma 2.1 \(x_{0}=u_{0}\). Since \(u_{0}\in Sx_{0}\), we have \(x_{0}\in Sx_{0}\). This completes the proof. □
Let \((X,d)\) be a metric space. Then \(S:X\to P(X)\) is called a multivalued weakly Picard operator [10] if for each \(x\in X\) and each \(y\in Sx\), there exists a sequence \(\{x_{n}\}\) in X such that

(1)
\(x_{0}=x\), \(x_{1}=y\);

(2)
\(x_{n+1}\in Sx_{n}\), \(n\in\mathbb {N} \cup\{0\}\);

(3)
\(\{x_{n}\}\) is convergent and its limit is a fixed point of S.
Using Theorem 4.1, we can get the following result proved by Takahashi et al. [7].
Theorem 4.2
([7])
Let \((X,d)\) be a complete metric space, let \(p\in W_{0}(X)\) and let \(\{x_{n}\}\) be a sequence in X such that \(\{p(x_{n}, x)\}\) is bounded for some \(x\in X\). Let μ be a mean on \(\ell^{\infty}\) and let \(\psi:X\to(\infty, \infty)\) be a bounded below and lower semicontinuous function. Let \(T:X\to P(X)\) be a setvalued mapping such that for each \(u\in X\), there exists \(v\in Tu\) satisfying
Then T is a multivalued weakly Picard operator.
Proof
Putting \(S=T\) and \(\phi=2\psi\) in Theorem 4.1, we see that, for each \(x\in X\), there exists \(y\in Tx\) such that
and hence
For each \(x\in X\) and each \(y\in Tx\), put \(u_{0}=x\) and \(u_{1}=y\). Then we can take \(u_{2}\in Tu_{1}\) such that
Repeating this process, we get a sequence \(\{u_{m}\}\) in X such that \(u_{m+1}\in Tu_{m}\) and
for each \(m\in\mathbb{N}\cup\{0\}\). Thus we have the desired result from Theorem 4.1. □
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Acknowledgements
The second author was partially supported by GrantinAid for Scientific Research No. 15K04906 from Japan Society for the Promotion of Science. The fourth author was partially supported by the grant MOST 1032923E039001MY3.
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Kaneko, S., Takahashi, W., Wen, CF. et al. Existence theorems for singlevalued and setvalued mappings with wdistances in metric spaces. Fixed Point Theory Appl 2016, 38 (2016). https://doi.org/10.1186/s1366301605272
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DOI: https://doi.org/10.1186/s1366301605272
MSC
 47H10
 37C25
 58J20
Keywords
 complete metric space
 contractive mapping
 fixed point theorem
 generalized hybrid mapping
 wdistance