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Best proximity results for Suzuki and convex type contractions
Fixed Point Theory and Applications volume 2016, Article number: 14 (2016)
Abstract
The aim of the paper is to introduce new Suzuki and convex type contractions and prove new best proximity results for these contractions in the setting of a metric space. As applications, we deduce similar results for such type of contractions in partially ordered metric spaces and derive new Suzuki type fixed point results. An illustrative example is provided here to highlight our findings.
1 Introduction and preliminaries
The background literature on best proximity theory and associated fixed point theory in (ordered) metric spaces, Banach spaces and fuzzy metric spaces is very abundant in the literature; see, for instance, [1–6] and references therein.
For any two nonempty sets A and B in a metric space \((X, d)\), the point \(p \in A\) is called a best proximity point of the mapping \(T : A \to B\) if \(d(p,Tp) = d(A,B)\), where \(d(A,B) = \inf\{d(x,y): x\in A, y \in B\}\). We shall denote the set of best proximity points of T by \(\operatorname {Bpp}(T)\). For more details, we refer the reader to [7–11] and [4–6, 12–31].
We define
Definition 1.1
[20]
For nonempty subsets A, B of metric space \((X,d)\) with \(A_{0}\neq\emptyset\), we say the pair \((A,B)\) satisfy
-
(a)
the P-property if
$$\textstyle\begin{cases} d(x_{1},y_{1})=d(A,B), & \\ d(x_{2},y_{2})=d(A,B), & \end{cases}\displaystyle \quad\Longrightarrow\quad d(x_{1},x_{2})=d(y_{1},y_{2}) $$for all \(x_{1},x_{2}\in A\) and \(y_{1},y_{2}\in B\),
-
(b)
the weak P-property [22, 26] if for any \(x_{1},x_{2} \in A_{0}\) and \(y_{1},y_{2} \in B_{0}\),
$$ d(x_{1},y_{1})=d(A,B) \quad \mbox{and} \quad d(x_{2},y_{2})=d(A,B) \quad \Rightarrow\quad d(x_{1},x_{2})\leq d(y_{1},y_{2}). $$(1.2)
We shall use \(\Psi=\{\psi:[0,+\infty) \to[0,+\infty): \sum_{n=1}^{\infty}\psi^{n}(t)<+\infty\mbox{ for all } t>0\}\), where ψ is nondecreasing function.
Now we introduce new concepts of proximal mappings, for more details see [5].
Definition 1.2
If \(\alpha: A\times A\to[-\infty,\infty)\), then \(T:A\to B\) is called proximal \(\alpha^{+}\)-admissible if
for all \(x_{1},x_{2},u_{1},u_{2}\in A\).
Definition 1.3
The mapping \(T:A\to B\) is called a Suzuki type \(\alpha^{+}\psi\)-proximal contraction, if
for all \(x,y\in A\), where \(d^{*}(x,Tx)=d(x,Tx)-d(A,B)\), \(\alpha: A\times A\to[-\infty,\infty)\), \(\psi\in\Psi\), and
In this manuscript, we propose new types of Suzuki and convex proximal maps to prove best proximity results. We also derive similar results in ordered metric spaces. Several interesting consequences of our obtained results are presented here.
2 Suzuki type \(\alpha^{+}\psi\)-proximal maps
Now we prove our first main result.
Theorem 2.1
Suppose A and B are nonempty closed subsets of a complete metric space X with \(A_{0}\neq\emptyset\). Let \(T:A\to B\) satisfy (1.3) together with the following assertions:
-
(i)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) satisfies the weak P-property,
-
(ii)
T is proximal \(\alpha^{+}\)-admissible,
-
(iii)
there exist \(x_{0}, x_{1} \in A_{0}\) such that
$$d(x_{1},Tx_{0})=d(A,B) \quad \textit{and} \quad \alpha(x_{0},x_{1})\geq0, $$ -
(iv)
T is continuous, or
-
(v)
A is α-regular, that is, if \(\{x_{n}\}\) is a sequence in A such that \(\alpha(x_{n},x_{n+1})\geq0\) and \(x_{n}\to x\in A\) as \(n\to\infty\), then \(\alpha(x_{n}, x)\geq0\) for all \(n\in\mathbb{N}\).
Then \(\operatorname {Bpp}(T)\) is nonempty.
Proof
Since \(T(A_{0})\subseteq B_{0}\), we have \(x_{2}\in A_{0}\) such that
As T satisfies (iii) and is proximal \(\alpha^{+}\)-admissible, we obtain \(\alpha(x_{1},x_{2})\geq0\). That is,
Again, since \(T(A_{0})\subseteq B_{0}\), there exists \(x_{3}\in A_{0}\) such that
Thus we have
Again since T is proximal \(\alpha^{+}\)-admissible, so \(\alpha(x_{2},x_{3})\geq0\). Hence,
We continue this process, to get
By using the above observations we can write
That is,
Now from (1.3) we get
By a simple calculation we obtain (see for details [2, 5]),
By the weak P-property and (2.1) one obtains
Equations (2.2) and (2.3) imply that
If \(x_{n_{0}}=x_{n_{0}+1}\) for some \(n_{0}\in \mathbb{N}\), from (2.1) one obtains
that is, \(x_{n_{0}}\in \operatorname {Bpp}(T)\). Thus, we suppose that
If, \(\max\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\}=d(x_{n},x_{n+1})\), then (2.4) implies
which is a contradiction. Thus,
Applying the monotonicity of ψ, by induction, it follows from (2.6),
Suppose ϵ is any positive real number. Then there exists \(N\in\mathbb{N}\) such that
If \(m,n\in\mathbb{N}\) with \(m>n\geq N\). We apply the triangle inequality to get
Consequently \(\lim_{m,n,\to+\infty}d(x_{n},x_{m})=0\), which implies \(\{x_{n}\}\) is Cauchy sequence. By completeness of X, \(x_{n}\to z\in X\). If (iv) holds, then \(Tx_{n}\to Tz\) as \(n\to\infty\) and
as required. Next, assume that (v) holds. Then \(\alpha(x_{n},z)\geq0\).
If the following inequalities hold:
for some \(n\in\mathbb{N}\), then by using (2.6) and definition of \(d^{*}\), we obtain the following contradiction:
Consequently,, for any \(n\in\mathbb{N}\), either
holds. Thus, we may pick a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that
for all \(k\in\mathbb{N}\). By (1.3) we get
Notice that
which implies
Further,
which gives
As \(k\to\infty\) in (2.9) we deduce
Therefore from (2.7), (2.8), and (2.10)
Now, if \(d(z,Tz)-d(A,B)>0\), then we get
a contradiction. Hence, \(d(z,Tz)=d(A,B)\) as desired. □
Example 2.1
Suppose \(X=\mathbb{R}^{2}\) is equipped with the metric
for all \((p_{1},p_{2}),(q_{1},q_{2}) \in X\). Let \(A_{1}=\{(p,q) | p=1, 0\leq q\leq\frac{1}{2}\} \), \(A_{2}=\{(p,q) |p=4, q\geq5\}\), \(A_{3}=\{(p,q) p=5, q\geq4\}\), \(A_{4}=\{(p,q) | p=3,q\geq3\}\) and \(A=A_{1}\cup A_{2}\cup A_{3}\cup A_{4}\). Further define \(B_{1}=\{(p,q)| p= \frac{1}{2}, \frac{1}{2}\leq q\leq1\}\), \(B_{2}=\{(p,q)| p=0, q\leq4\}\), \(B_{3}=\{(P,q)| p=4, q\leq0\}\), and \(B=B_{1}\cup B_{2}\cup B_{3}\).
Note that \(d(A,B)=1\), \(A_{0}=\{(p,q) | p=1, 0\leq q\leq \frac{1}{2}\}\), and \(B_{0}=\{(p,q)| p= \frac{1}{2}, \frac{1}{2}\leq q\leq1\}\). Let, for \(x_{1}=(1, u_{1}), x_{2}=(1, u_{2})\in A_{0}\) and \(y_{1}=(\frac{1}{2}, v_{1}), y_{2}=(\frac{1}{2}, v_{2})\in B_{0}\), us have \(d(x_{1},y_{1})=d(A,B)=1\) and \(d(x_{2},y_{2})=d(A,B)=1\). Then
and
and so \(\vert u_{1}-v_{1}\vert =\frac{1}{2}\) and \(\vert u_{2}-v_{2}\vert =\frac{1}{2}\). Since \(v_{1},v_{2}\geq u_{1}, u_{2}\), we have \(v_{1}=\frac{1}{2}+u_{1}\) and \(v_{2}=\frac{1}{2}+u_{2}\). This shows that \(d(x_{1},x_{2})\leq d(y_{1},y_{2})\). So \((A,B)\) satisfy the weak P-property. Let \(T:A\to B\) be defined by
Notice that \(T(A_{0})\subseteq B_{0}\).
Define the functions \(\psi:[0,+\infty)\rightarrow[0,+\infty)\) and \(\alpha: A\times A\to[-\infty,\infty)\) by
Assume that \(\frac{1}{2}d^{*}(p,Tp)\leq d(p,q)\) and \(\alpha(p,q)\geq0\), for \(p,q\in A\). Then
Since \(d(Tp,Tq)=d(Tq,Tp)\) and \(M(p,q)=M(q,p)\) for all \(p,q\in A\), we can suppose that
Now, we discuss the following cases:
-
(i)
if \((p,q)=((1,0),(4,5))\), then
$$d\bigl(T(1,0),T(4,5)\bigr)=4\leq7=\frac{7}{8} \cdot8=\psi\bigl(d \bigl((1,0),d(4,5)\bigr)\bigr)\leq \psi\bigl(M(p,q)\bigr); $$ -
(ii)
if \((p,q)=((1,0),(5,4))\), then
$$d\bigl(T(1,0),T(5,4)\bigr)=4\leq\frac{7}{8} \cdot8=\psi\bigl(d \bigl((1,0),(5,4)\bigr)\bigr)\leq\psi \bigl(M(p,q)\bigr). $$Consequently, we have
$$ \frac{1}{2}d^{*}(p,Tp)\leq d(p,q)\quad \Rightarrow\quad d(Tp,Tq)\leq \psi \bigl(M(p,q)\bigr). $$Thus all the assumptions of Theorem 2.1 are satisfied and \(\operatorname {Bpp}(T)=\{(1,0)\}\).
The next result can be deduced easily from Theorem 2.1.
Theorem 2.2
Let X, A, \(A_{0}\), and B be as in Theorem 2.1. Assume that \(T:A\to B\) satisfies the assertions (i)-(v) in Theorem 2.1 and
holds for all \(p,q\in A\). Then \(\operatorname {Bpp}(T)\) is nonempty.
If \(\alpha=0\) on A, in Theorem 2.1, we obtain the following new result.
Corollary 2.1
Suppose X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) satisfies the following assumptions:
-
(i)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) satisfies the weak P-property,
-
(ii)
for all \(p,q\in A\) with \(\frac{1}{2}d^{*}(p,Tp)\leq d(p,q)\) we have
$$ d(Tp,Tq)\leq\psi\bigl(M(p,q)\bigr). $$
Then \(\operatorname {Bpp}(T)\) is nonempty.
3 \(\alpha^{+}\) Θ-proximal maps
This section deals with best proximity theorems for Suzuki contractions involving the Θ function which was recently introduced by Jleli and Samet [27].
Let \(\Delta_{\Theta}\) denote the set of all functions \(\Theta: (0,\infty)\rightarrow[1,\infty)\) with the following conditions:
- (\(\Theta_{1}\)):
-
Θ is increasing;
- (\(\Theta_{2}\)):
-
for all sequences \(\{\alpha_{n}\}\subseteq (0,\infty)\), \(\lim_{n\to\infty}\alpha_{n}=0\) if and only if \(\lim_{n\to\infty}\Theta(\alpha_{n})=1\);
- (\(\Theta_{3}\)):
-
there exist \(0< r<1\) and \(\ell\in (0,\infty]\) such that \(\lim_{n\to 0^{+}}\frac{\Theta(t)-1}{t^{r}}=\ell\).
Definition 3.1
A mapping \(T:A\to B\) is called a Suzuki type \(\alpha^{+}\Theta\)-proximal contraction, if for all \(x,y\in A\) with \(\frac{1}{2}d^{*}(x,Tx)\leq d(x,y)\) and \(d(Tx,Ty)>0\),
where \(\alpha: A\times A\to[-\infty,\infty)\), \(0\leq k<1\), and \(\Theta\in\Delta_{\Theta}\).
Theorem 3.1
Assume that X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) satisfy (3.1) and the assertions (i)-(v) in Theorem 2.1. Then \(\operatorname {Bpp}(T)\) is nonempty.
Proof
As in the proof of Theorem 2.1, we can construct a sequence \(\{x_{n}\}\) satisfying
and
Now (3.1) implies
In Theorem 2.1 we obtain
and
Therefore from (3.3) and (3.4) we get
Now if \(\max\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\}=d(x_{n},x_{n+1})\), then from (3.5) we get
which is a contradiction. Hence,
Therefore,
Taking the limit as \(n\to\infty\) in (3.7) we have
and since \(\Theta\in\Delta_{\Theta}\) we obtain
Again since \(\Theta\in\Delta_{\Theta}\), there exist \(0< r<1\) and \(0<\ell\leq\infty\) with
Assume that \(\ell<\infty\). Let \(C=\frac{\ell}{2}\). Thus there exists \(n_{0}\in\mathbb{N}\) such that
hence
and so
where \(D=\frac{1}{C}\). If \(\ell=\infty\), then there exists \(n_{0}\in \mathbb{N}\),
which implies
where \(D=\frac{1}{C}\). Hence, in all cases there exist \(D>0\) and \(n_{0}\in\mathbb{N}\) such that
Now (3.7) implies
and on letting \(n\to\infty\) we obtain
It follows from (3.10) that there is \(n_{1}\in\mathbb{N}\) with
for all \(n>n_{1}\). This implies
for all \(n>n_{1}\). If \(m>n>n_{1}\), then
Since \(0< r<1\), \(\sum_{i=n}^{\infty}\frac{1}{i^{1/r}}\) is convergent. Thus, \(d(x_{n},x_{m})\to0\) as \(m,n\to\infty\), which shows that \(\{x_{n}\}\) is a Cauchy sequence. Thus there is \(z\in X\) such that \(x_{n}\to z\) as \(n\to\infty\). Assume that (iv) holds. Thus \(Tx_{n}\to Tz\) as \(n\to\infty\), which implies
as required. Next, assume that (v) holds. As in the proof of Theorem 2.1 we can deduce there is a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) satisfying
for all \(k\in\mathbb{N}\). By (3.1) we get
which implies
As in Theorem 2.1 we obtain
and
therefore,
which is a contradiction when \(d(z,Tz)>d(A,B)\). So, \(d(z,Tz)=d(A,B)\), that is, \(\operatorname {Bpp}(T)\) is nonempty. □
Corollary 3.1
Suppose X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) satisfies the assertions (i)-(v) in Theorem 3.1. If
holds for all \(p,q\in A\) where \(\alpha: A\times A\to [-\infty,\infty)\) and \(\Theta\in\Delta_{\Theta}\), then \(\operatorname {Bpp}(T)\) is nonempty.
Corollary 3.2
Suppose X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) satisfies the following assertions:
-
(i)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) satisfies the weak P-property;
-
(ii)
for all \(p,q\in A\) with \(\frac{1}{2}d^{*}(p,Tp)\leq d(p,q)\) we have
$$ \Theta\bigl(d(Tp,Tq)\bigr)\leq\bigl[\Theta\bigl(M(p,q)\bigr) \bigr]^{k}, $$where \(\Theta\in\Delta_{\Theta}\).
Then \(\operatorname {Bpp}(T)\) is nonempty.
Remark 3.1
-
(a)
The results proved in the above sections generalize the corresponding results of Zhang et al. [26], Suzuki [22], Hussain et al. [2, 3] and many others.
-
(b)
Several more best proximity point theorems can be obtained using more choices for the function Θ, and some other concrete choices of α and \(\psi\in\Psi\) in the results of the above sections.
4 Best proximity results for convex type contractions
We discuss two new and general types of proximal convex contractions and establish corresponding best proximity results (see also [8]).
Definition 4.1
Suppose \(T:A\to B\) be a mapping where A and B are two nonempty subsets of a metric space X. Then T is an
-
(1)
\(\alpha^{+}\)-convex proximal contractive map of the first type if, for \(x,y,u,u^{*}, v\in A\),
$$ \left . \textstyle\begin{array}{l} \alpha(x,y)\geq0,\\ d(u,Tx)=d(A,B),\\ d(u^{*},Tu)=d(A,B),\\ d(v,Ty)=d(A,B),\\ d(v^{*},Tv)=d(A,B) \end{array}\displaystyle \right \} \quad \Longrightarrow\quad d\bigl(u^{*},v^{*}\bigr)\leq r_{1}d(u,v)+r_{2}d(x,y) $$(4.1)holds where \(r_{1},r_{2}\geq0\), \(r_{1}+r_{2}<1\);
-
(2)
\(\alpha^{+}\)-convex proximal contractive map of second type if for \(x,y,u,u^{*}, v\in A\),
$$\begin{aligned} &\left . \textstyle\begin{array}{l} \alpha(x,y)\geq0,\\ d(u,Tx)=d(A,B),\\ d(u^{*},Tu)=d(A,B),\\ d(v,Ty)=d(A,B),\\ d(v^{*},Tv)=d(A,B) \end{array}\displaystyle \right \} \\ &\quad \Longrightarrow\quad d\bigl(u^{*},v^{*}\bigr)\leq r_{1}d(x,u)+r_{2}d \bigl(u,u^{*}\bigr)+r_{3}d(y,v)+r_{4}d\bigl(v,v^{*}\bigr) \end{aligned}$$(4.2)holds where \(r_{1},r_{2},r_{3},r_{4}\geq0\), \(r_{1}+r_{2}+r_{3}+r_{4}<1\).
Theorem 4.1
Suppose X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) satisfy (4.1) with \(T(A_{0})\subseteq B_{0}\) and the conditions (ii)-(iv) in Theorem 2.1. Then \(\operatorname {Bpp}(T)\) is nonempty. Moreover, \(\operatorname {Bpp}(T)\) is a singleton if \(\alpha(x,y)\geq0\) for all \(x,y\in \operatorname {Bpp}(T)\).
Proof
Following the technique of the proof in Theorem 2.1, one can find a sequence \(\{x_{n}\}\) such that
For
equation (4.1) implies
By taking \(\vartheta=d(x_{2},x_{1})+d(x_{1},x_{0})\) and \(r=r_{1}+r_{2}\) we have
where \(m=2l\) or \(m=2l+1\). Let \(m=2l\). Then for \(n=2p\) with \(p>2\) and \(l\geq1\) and \(m< n\) we deduce
Similarly, for \(m=2l\) and \(n=2p+1\) with \(p\geq1\) and \(l\geq1\) and \(m< n\) we get
Now, assume that \(m=2l+1\). Then for \(n=2p\) with \(p\geq2\) and \(l\geq 1\) and \(m< n\) we have
Similarly, for \(m=2l+1\) and \(n=2p+1\) with \(p\geq1\) and \(l\geq1\) and \(m< n\) we deduce
Hence, for all \(m,n\in\mathbb{N}\) with \(m< n\) we have
which letting \(l\to\infty\) implies \(d(x_{m},x_{n})\to0\). That is, \(\{x_{n}\}\) is a Cauchy sequence and hence there is \(z\in X\) such that \(x_{n}\to z\) as \(n\to\infty\). Continuity of T implies \(Tx_{n}\to Tz\) as \(n\to\infty\). Hence,
Let \(w,z\in \operatorname {Bpp}(T)\) with \(w\neq z\). Then \(\alpha(w,z)\geq0\). Now with
(4.1) implies
which is a contradiction and hence \(d(w,z)=0\). i.e., \(w=z\). Thus \(\operatorname {Bpp}(T)\) is a singleton. □
By taking, \(\alpha(x,y)=0\), in the above theorem we deduce the following result.
Corollary 4.1
Suppose X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) is a continuous convex proximal contractive mapping of the first type satisfying \(T(A_{0})\subseteq B_{0}\). Then \(\operatorname {Bpp}(T)\) is nonempty.
Theorem 4.2
Suppose X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) is an \(\alpha^{+}\)-convex proximal contractive map of second type with \(T(A_{0})\subseteq B_{0}\) and satisfying conditions (ii)-(iv) in Theorem 2.1. Then \(\operatorname {Bpp}(T)\) is nonempty. Moreover, \(\operatorname {Bpp}(T)\) is a singleton if \(\alpha(x,y)\geq0\) for all \(x,y\in \operatorname {Bpp}(T)\).
Proof
As in Theorem 2.1, one may find a sequence \(\{x_{n}\}\) such that
For
with \(r=r_{1}+r_{2}+r_{3}\), \(\eta=1-r_{4}\), and \(\vartheta=d(x_{2},x_{1})+d(x_{1},x_{0})\), (4.2) implies
Now if \(n=2\), then
which implies \((1-r_{4})d(x_{2},x_{3})\leq r\vartheta\). That is, \(d(x_{2},x_{3})\leq \frac{r}{\eta}\vartheta\). Again by taking \(n=3\) in (4.5) we get
which implies \(d(x_{3},x_{4})\leq \frac{r}{\eta}\vartheta\). Similarly, \(d(x_{4},x_{5})\leq (\frac{r}{\eta})^{2}\vartheta\) and \(d(x_{5},x_{6})\leq (\frac{r}{\eta})^{2}\vartheta\). By continuing this process, we get \(d(x_{m},x_{m+1})\leq (\frac{r}{\eta})^{l}\vartheta\) when \(m=2l\) or \(m=2l+1\). Let \(m=2l\). Then for \(n=2p\) with \(p>2\) and \(l\geq1\) and \(m< n\) we deduce
Similarly, for \(m=2l\) and \(n=2p+1\) with \(p\geq1\) and \(l\geq1\) and \(m< n\) we get
Now, assume that \(m=2l+1\). Then for \(n=2p\) with \(p\geq2\) and \(l\geq 1\) and \(m< n\) we have
Similarly, for \(m=2l+1\) and \(n=2p+1\) with \(p\geq1\) and \(l\geq1\) and \(m< n\) we deduce
Hence, for all \(m,n\in\mathbb{N}\) with \(m< n\) we have
Letting \(l\to\infty\), we get \(d(x_{m},x_{n})\to0\). That is, \(\{x_{n}\}\) is a Cauchy sequence and so there is \(z\in X\) such that \(x_{n}\to z\) as \(n\to\infty\). BY continuity of T, \(Tx_{n}\to Tz\) as \(n\to\infty\). Hence,
The proof that \(\operatorname {Bpp}(T)\) is a singleton is similar to the above theorem and so is omitted. □
By taking, \(\alpha(x,y)=0\), in the above theorem, we deduce the following result.
Corollary 4.2
Suppose X, A, \(A_{0}\), and B are as in Theorem 2.1 and \(T:A\to B\) is a continuous convex proximal contractive mapping of the second type satisfying \(T(A_{0})\subseteq B_{0}\). Then \(\operatorname {Bpp}(T)\) is a singleton.
5 Results in partially ordered sets
In this section, we deduce best proximity theorems for Suzuki and convex proximal maps in partially ordered sets.
Definition 5.1
[18]
Let \((X,d, \preceq)\) be a partially ordered metric space. A map \(T:A\to B\) is called proximally order-preserving if, for all \(x_{1},x_{2},u_{1},u_{2}\in A\),
Definition 5.2
A mapping \(T:A\to B\) is said to be Suzuki type ordered ψ-proximal contraction, if for \(x,y\in A\)
Similarly, we can define order versions of other maps discussed in above sections.
Theorem 5.1
Let A and B be nonempty closed subsets of a complete partially ordered metric space \((X,d, \preceq)\) such that \(A_{0}\) is nonempty and \(T:A\to B\) be a Suzuki type ordered ψ-proximal map satisfying the following assertions:
-
(i)
\(T(A_{0})\subseteq B_{0}\) and \((A,B)\) satisfies the weak P-property,
-
(ii)
T is proximally ordered-preserving,
-
(iii)
there are \(x_{0}\) and \(x_{1}\) in \(A_{0}\) such that
$$d(x_{1},Tx_{0})=d(A,B)\quad \textit{and}\quad x_{0}\preceq x_{1}, $$ -
(iv)
T is continuous, or
-
(v)
if \(\{x_{n}\}\) is a increasing sequence in A with \(x_{n}\to x\in A\) as \(n\to\infty\), then \(x_{n}\preceq x\) for all \(n\in\mathbb{N}\).
Then \(\operatorname {Bpp}(T)\) is nonempty.
Proof
Define \(\alpha:A\times A \to[-\infty,+\infty)\) by
T is proximal \(\alpha^{+}\)-admissible mapping as follows.
implies
Since T is proximally ordered-preserving, \(u\preceq v\), that is, \(\alpha(u,v)\geq0\). Further, by (ii) we have
Note that, if \(x\preceq y\), then \(\alpha(x,y)=0\) and otherwise, \(\alpha(x,y)=-\infty\). Since T is a Suzuki type ordered ψ-proximal map, we have the following inequality:
Further, let \(\{x_{n}\}\) be a sequence, such that \(\alpha(x_{n},x_{n+1})\geq0\) for all \(n\in\mathbb{N}\cup\{0\}\) with \(x_{n}\to x\) as \(n\to\infty\), then \(x_{n}\preceq x_{n+1}\) for all \(n\in\mathbb{N}\cup\{0\}\) with \(x_{n}\to x\) as \(n\to\infty\). That is, \(\{x_{n}\}\) is an increasing sequence, with \(x_{n}\to x\) as \(n\to\infty\). So from (v) we have \(x_{n}\preceq x\) for all \(n\in \mathbb{N}\cup\{0\}\). That is, \(\alpha(x_{n},x_{n})\geq0\) for all \(n\in\mathbb{N}\cup\{0\}\). Thus all the assumptions of Theorem 2.1 hold and \(\operatorname {Bpp}(T)\) is nonempty. □
Similarly we can prove the following theorems.
Theorem 5.2
Suppose X, A, \(A_{0}\), and B are as in Theorem 5.1 and \(T:A\to B\) is a Suzuki type ordered Θ-proximal contraction where we have the assumptions (i)-(v) of Theorem 5.1. Then \(\operatorname {Bpp}(T)\) is nonempty.
Theorem 5.3
Suppose X, A, \(A_{0}\), and B are as in Theorem 5.1 and \(T:A\to B\) is an ordered convex proximal contractive mapping of the first type (or the second type) satisfying \(T(A_{0})\subseteq B_{0}\) and the conditions (ii)-(iv) of Theorem 5.1. Then \(\operatorname {Bpp}(T)\) is nonempty. Moreover, \(\operatorname {Bpp}(T)\) is singleton if \(\alpha(x,y)\geq0\) for all \(x,y\in \operatorname {Bpp}(T)\).
6 Application to fixed point theory
Here we deduce certain new and general fixed point results for Suzuki and convex contractions. Our results contain properly the main theorem due to Suzuki [24] and many of its extensions [23] (see also [28]).
If \(A=B=X\), then definition (1.2) reduces to the following.
Definition 6.1
A map \(T:X\to X\), is called \(\alpha^{+}\)-admissible if
for all \(x,y\in X\).
Definition 6.2
A mapping \(T:X\to X\) is called a Suzuki type \(\alpha^{+}\psi\)-contraction, if
for all \(x,y\in X\).
Definition 6.3
A mapping \(T:X\to X\) is called a Suzuki type \(\alpha^{+}\Theta\)-contraction, if
for all \(x,y\in X\), \(\alpha: X\times X\to[-\infty,\infty)\) and \(\Theta\in\Delta_{\Theta}\).
Now from Theorems 2.1, 2.2 and 3.1, we derive the following new fixed point theorems.
Theorem 6.1
Assume that X is a complete metric space and \(T:X\to X\) is a Suzuki type \(\alpha^{+}\psi\)-contraction with the following assertions:
-
(i)
T is \(\alpha^{+}\)-admissible,
-
(ii)
there is \(x_{0}\) with \(\alpha(x_{0},Tx_{0})\geq0\),
-
(iii)
T is continuous or,
-
(iv)
X is α-regular.
Then \(F(T)\) is nonempty.
Theorem 6.2
Assume that X is a complete metric space and \(T:X\to X\) is a Suzuki type \(\alpha^{+}\Theta\)-contraction satisfying the conditions (i)-(iv) in Theorem 6.1. Then \(F(T)\) is nonempty.
Theorem 6.3
Suppose X is a complete metric space and \(T:X\to X\) is an \(\alpha^{+}\)-convex contractive mapping of the first (or the second) type with the following assertions:
-
(i)
T is \(\alpha^{+}\)-admissible,
-
(ii)
there exists \(x_{0}\) such \(\alpha(x_{0},Tx_{0})\geq0\),
-
(iii)
T is continuous.
Then \(F(T)\) is nonempty.
By taking \(\alpha(x,y)=0\) for all \(x,y\in X\) in the above theorem, we obtain the main results of Istrǎţescu [29] as corollaries.
Definition 6.4
A mapping \(T:X\to X\) is called a Suzuki type ordered ψ-contraction, if
for \(x,y\in X\), \(\psi\in\Psi\).
Definition 6.5
A mapping \(T:X\to X\) is called a Suzuki type ordered Θ-contraction, if
for \(x,y\in X\) and \(\Theta\in\Delta_{\Theta}\).
Theorem 6.4
Suppose \((X,d, \preceq)\) is a complete partially ordered metric space and \(T:X\to X\) is a Suzuki type ordered ψ-contraction with the following assertions:
-
(i)
T is an increasing mapping,
-
(ii)
there is \(x_{0}\in X\) such that \(x_{0}\preceq Tx_{0}\),
-
(iii)
T is continuous or,
-
(iv)
X is regular.
Then \(F(T)\) is nonempty.
Theorem 6.5
Suppose \((X,d, \preceq)\) is a complete partially ordered metric space and \(T:X\to X\) is a Suzuki type ordered Θ-contraction satisfying the conditions (i)-(iv) in Theorem 6.4. Then \(F(T)\) is nonempty.
Theorem 6.6
Assume that \((X,d, \preceq)\) is a complete partially ordered metric space and \(T:X\to X\) is an ordered convex contractive mapping of the first (or the second) type with the following assertions:
-
(i)
T is increasing,
-
(ii)
there is \(x_{0}\) such \(x_{0}\preceq Tx_{o}\),
-
(iii)
T is continuous.
Then \(F(T)\) is singleton.
Remark 6.1
Several more fixed point theorems can be obtained using more choices for function Θ, and/or some other concrete choices of α and \(\psi\in\Psi\).
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This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, KAU for financial support.
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Hussain, N., Hezarjaribi, M., Kutbi, M. et al. Best proximity results for Suzuki and convex type contractions. Fixed Point Theory Appl 2016, 14 (2016). https://doi.org/10.1186/s13663-016-0499-2
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DOI: https://doi.org/10.1186/s13663-016-0499-2
Keywords
- best proximity point
- proximal α-admissible mapping
- Suzuki type proximal contractions