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On the existence of fixed points that belong to the zero set of a certain function
Fixed Point Theory and Applications volume 2015, Article number: 152 (2015)
Abstract
Let \(T: X\to X\) be a given operator and \(F_{T}\) be the set of its fixed points. For a certain function \(\varphi: X\to[0,\infty)\), we say that \(F_{T}\) is φ-admissible if \(F_{T}\) is nonempty and \(F_{T}\subseteq Z_{\varphi}\), where \(Z_{\varphi}\) is the zero set of φ. In this paper, we study the φ-admissibility of a new class of operators. As applications, we establish a new homotopy result and we obtain a partial metric version of the Boyd-Wong fixed point theorem.
1 Introduction
Let \((X,d)\) be a metric space. For a given function \(\varphi: X\to[0,\infty)\), we define the set
Let \(T: X\to X\) be a given operator. The set of fixed points of T is denoted by \(F_{T}\), that is,
Definition 1.1
We say that the set \(F_{T}\) is φ-admissible if and only if \(F_{T}\neq\emptyset\) and \(F_{T}\subseteq Z_{\varphi}\).
Let \(\mathcal{F}\) be the set of functions \(F:[0,\infty)^{3}\to[0,\infty )\) satisfying the following conditions:
- (F1):
-
\(\max\{a,b\} \leq F(a,b,c)\), for all \(a, b, c \geq0\);
- (F2):
-
\(F(a,0,0)=a\), for all \(a\geq0\);
- (F3):
-
F is continuous.
As examples, the following functions belong to \(\mathcal{F}\):
-
1.
\(F(a,b,c)=a+b+c\),
-
2.
\(F(a,b,c)=\max\{a,b\}+\ln(c+1)\),
-
3.
\(F(a,b,c)= a+b+c(c+1)\),
-
4.
\(F(a,b,c)=(a+b)e^{c}\),
-
5.
\(F(a,b,c)=(a+b)(c+1)^{n}\), \(n\in\mathbb{N}\).
Let Ψ be the set of functions \(\psi:[0,\infty)\to[0,\infty)\) satisfying the following conditions:
- (Ψ1):
-
ψ is upper semi-continuous from the right;
- (Ψ2):
-
\(\psi(t)< t\), for all \(t>0\).
For given functions \(\varphi:X\to[0,\infty)\), \(F\in\mathcal{F}\), and \(\psi\in\Psi\), we denote by \(\mathcal{T}(\varphi,F,\psi)\) the class of operators \(T: X\to X\) satisfying
The aim of this paper is to study the φ-admissibility of the set \(F_{T}\), where T belongs to the class of operators \(\mathcal{T}(\varphi,F,\psi)\), \((F,\psi)\in\mathcal{F}\times\Psi\). As applications, we obtain an homotopy result and a partial metric version of the Boyd-Wong fixed point theorem.
2 Main result
Our main result is given in the following theorem.
Theorem 2.1
Let \((X,d)\) be a complete metric space and \(T: X\to X\) be a given operator. Suppose that the following conditions hold:
-
(i)
there exist \(\varphi: X\to[0,\infty)\), \(F\in\mathcal{F}\), and \(\psi\in\Psi\) such that \(T\in\mathcal{T}(\varphi,F,\psi)\);
-
(ii)
φ is lower semi-continuous.
Then the set \(F_{T}\) is φ-admissible. Moreover, the operator T has a unique fixed point.
Proof
Let ξ be an arbitrary element of the set \(F_{T}\). Take \(x=y=\xi\) in (1.1), and we get
If \(F(0,\varphi(\xi),\varphi(\xi))\neq0\), from (Ψ2), we get
which is impossible from (2.1). Consequently, we have
Using the above equality and (F1), we obtain
which yields
Consequently, we have
Now, we have to prove that \(F_{T}\) is a nonempty set. Let \(x_{0}\) be an arbitrary element of X. Consider the Picard sequence \(\{x_{n}\}\subset X\) defined by
where \(T^{n}\) is the nth iterate of T. If for some \(N\in \mathbb{N}\) we have \(x_{N}=x_{N+1}\), then \(x_{N}\) will be an element of \(F_{T}\). As a result we can suppose that
Using (1.1), we have
here \(\mathbb{N}^{*}=\{1,2,\ldots\}\). If for some \(N\in\mathbb{N}^{*}\), we have
then property (F1) yields
which is a contradiction with (2.3). Thus
Using (2.4), (2.5), the definition of the sequence \(\{x_{n}\}\), and (Ψ2), we have
It follows immediately from (2.6) that there exists some \(c\geq 0\) such that
Suppose now that \(c>0\). Using the properties (Ψ1)-(Ψ2), we deduce from (2.7) that
which is a contradiction. As a consequence, we have \(c=0\), that is,
Using (F1) and (2.8), we get
Next, we show that \(\{x_{n}\}\) is a Cauchy sequence in the metric space \((X,d)\). Suppose that \(\{x_{n}\}\) is not a Cauchy sequence. Then there exists \(\varepsilon>0\) for which we can find two sequences of positive integers \(\{m(k)\}\) and \(\{n(k)\}\) such that, for all \(k\in\mathbb{N}\),
Using (2.10), for all \(k\in\mathbb{N}\) we have
which yields
Letting \(k\to\infty\) in the above inequality and using (2.9), we obtain
Using the properties (F2)-(F3), (2.9), and (2.12), we get
Using the above limit and (Ψ1), we obtain
On the other hand, using (1.1) and (F1), for all \(k\in\mathbb {N}\) we have
Passing to the limit superior as \(k\to\infty\), using (2.9), (2.13), and (Ψ2), we obtain
which is a contradiction. As a consequence, \(\{x_{n}\}\) is a Cauchy sequence. Since \((X,d)\) is a complete metric space, there is a \(z\in X\) such that
Since φ is lower semi-continuous, it follows from (2.14) and (2.9) that
which yields
Now we show that \(z\in F_{T}\). Using (1.1), (F1), and (2.15), we have
Also using the continuity of F, (F2), (2.14), and (2.9), we have
Note that from (Ψ2), we have
Then
Now, passing \(n\to\infty\) in (2.16) and using (2.17), we get
The uniqueness of the limit yields \(z=Tz\). Thus \(F_{T}\) is a nonempty set, and the φ-admissibility of \(F_{T}\) is proved. Finally, in order to prove the uniqueness of the fixed point, let us assume that \(w\in F_{T}\) with \(d(z,w)>0\). Since \(F_{T}\) is φ-admissible, we know that \(z,w\in Z_{\varphi}\). Now, applying (1.1) with \((x,y)=(z,w)\), we obtain
Using the properties (F2) and (Ψ2), we get
which is a contradiction. Thus T has a unique fixed point. □
Remark 2.2
Take \(\varphi\equiv0\) and \(F(a,b,c)=a+b+c\) in Theorem 2.1, and we recover the Boyd-Wong fixed point theorem [1].
Now, we give some examples to illustrate our main result given by Theorem 2.1.
Example 2.3
We endow the set \(X=[0,\infty)\) with the standard metric
Let \(T: X\to X\) be the mapping defined by
Observe that T is not continuous in X. So, there is no \(\psi\in\Psi \) such that
Then the Boyd-Wong fixed point theorem cannot be applied in this case. Let \(\varphi: X\to[0,\infty)\) be the function defined by
Let \(F: [0,\infty)^{3}\to[0,\infty)\) be the function defined by
Let \(\psi:[0,\infty)\to[0,\infty)\) be the function defined by
Observe that F belongs to the set \(\mathcal{F}\) and ψ belongs to the set Ψ. We claim that \(T\in\mathcal{T}(\varphi,F,\psi)\), that is,
In order to prove our claim, we distinguish three cases.
Case 1. \((x,y)\in[0,1]\times[0,1]\).
In this case, we have
Case 2. \((x,y)\in[0,1]\times(1,\infty)\).
In this case, we have
while
Then we have to prove that
Observe that the function \(h: [0,1]\to\mathbb{R}\) defined by
has a global minimum at \(x_{n}= (\frac{1}{n} )^{\frac{1}{n-1}}\) which is equal to \(x_{n} (\frac{1-n}{n} )\geq\frac{1-n}{n}\). So, we have just to check that
Since \(y>1\), we have
Then our claim holds in this case.
Case 3. \(x,y>1\).
In this case, we have
and
Obviously, we have
Finally, in all cases our claim (2.18) holds, which yields \(T\in \mathcal{T}(\varphi,F,\psi)\). By Theorem 2.1, the set \(F_{T}\) is φ-admissible and T has a unique fixed point. In this example, \(F_{T}=\{0\}\) and \(\varphi(0)=0\).
Example 2.4
We endow the set \(X=[\sqrt{2},\infty)\) with the standard metric
Let \(T: X\to X\) be the mapping defined by
As in the previous example, the Boyd-Wong fixed point theorem cannot be applied in this case. Let \(\varphi: X\to[0,\infty)\) be the function defined by
Let \(F: [0,\infty)^{3}\to[0,\infty)\) be the function defined by
Let \(\psi:[0,\infty)\to[0,\infty)\) be the function defined by
We distinguish three cases.
Case 1. \((x,y)\in[\sqrt{2}, 2\sqrt{2}]\times [\sqrt{2}, 2\sqrt{2}]\).
In this case, we have
Case 2. \((x,y)\in[\sqrt{2}, 2\sqrt{2}]\times (2\sqrt{2},\infty)\).
In this case, we have
while
Clearly, we have
Case 3. \((x,y)\in(2\sqrt{2},\infty)\).
In this case, we have
while
Also, we have
As a consequence, the mapping T belongs to \(\mathcal{T}(\varphi,F,\psi )\). By Theorem 2.1, the set \(F_{T}\) is φ-admissible and T has a unique fixed point. In this example, \(F_{T}=\{\sqrt{2}\}\) and \(\varphi(\sqrt{2})=0\).
Example 2.5
Let \((X,d)\) be the metric space considered in Example 2.4. We take the functions φ, F, and ψ defined in Example 2.4. Let \(T: X\to X\) be the mapping defined by
Similarly, we have \(T\in\mathcal{T}(\varphi,F,\psi)\). By Theorem 2.1, the set \(F_{T}\) is φ-admissible and T has a unique fixed point. In this example, \(F_{T}=\{\sqrt{2}\}\) and \(\varphi(\sqrt{2})=0\).
3 Applications
3.1 An homotopy result
Let us denote by \(\mathcal{F}^{*}\) the set of functions \(F\in \mathcal{F}\) satisfying the following property:
- (F4):
-
for all \(a,b,c,d\geq0\),
$$a\leq c+d \quad\Longrightarrow\quad F(a,b,0)\leq F(c,b,0)+d. $$
As examples, the following functions belong to \(\mathcal{F}^{*}\):
-
1.
\(F(a,b,c)=(a+b)e^{c}\),
-
2.
\(F(a,b,c)=(a+b)(c+1)^{n}\), \(n\in\mathbb{N}\).
Observe that \(\mathcal{F}^{*}\subsetneq\mathcal{F}\). To see this, let us consider the function
It is not difficult to check that \(F\in\mathcal{F}\) but \(F\notin \mathcal{F}^{*}\).
We have the following homotopy result.
Theorem 3.1
Let \((X,d)\) be a complete metric space, U be an open subset of X, and V be a closed subset of X with \(U\subset V\). Suppose that \(H: V\times[0,1]\to X\) has the following properties:
- (C1):
-
\(x\neq H(x,\lambda)\) for every \(x\in V\backslash U\) and \(\lambda\in[0,1]\);
- (C2):
-
there exist a continuous function \(\varphi: X\to[0,\infty )\), \(L\in(0,1)\), and \(F\in\mathcal{F}^{*}\) such that for all \(x,y\in V\) and \(\lambda\in[0,1]\),
$$F \bigl(d\bigl(H(x,\lambda),H(y,\lambda)\bigr),\varphi\bigl(H(x,\lambda)\bigr), \varphi \bigl(H(y,\lambda) \bigr)\bigr)\leq L F \bigl(d(x,y),\varphi(x), \varphi(y) \bigr); $$ - (C3):
-
there exists a continuous function \(\eta:[0,1]\to\mathbb {R}\) such that for all \(x\in V\) and \(\lambda,\mu\in[0,1]\),
$$F \bigl(d\bigl(H(x,\lambda),H(x,\mu)\bigr),\varphi\bigl(H(x,\lambda)\bigr), \varphi \bigl(H(x,\mu ) \bigr)\bigr)\leq \bigl|\eta(\lambda)-\eta(\mu)\bigr|. $$
Then \(H(\cdot,0)\) has a fixed point if and only if \(H(\cdot,1)\) has a fixed point.
Proof
Suppose that \(H(\cdot,0)\) has a fixed point. Consider the set
From (C1), clearly 0 is an element of Q, so Q is a nonempty set. We will show that Q is both closed and open in \([0,1]\), and so by the connectedness of \([0,1]\), we are finished since \(Q=[0,1]\). First, let us prove that Q is open in \([0,1]\). Let \(t_{0}\in Q\) and \(x_{0}\in U\) with \(x_{0}= H(x_{0},t_{0})\). Using (C2) with \(x=y=x_{0}\) and \(\lambda=t_{0}\), we obtain
which implies since \(L\in(0,1)\) that
Then (F1) yields
Moreover, observe that, for all \(t\in[0,1]\), if \(x\in U\) is a fixed point of \(H(\cdot,t)\), then \(\varphi(x)=0\). On the other hand, since U is open in \((X,d)\), there exists \(r>0\) such that \(B(x_{0},r) \subseteq U\), where
Consider the set
Clearly \(\Lambda(x_{0},\varphi)\) is nonempty (since \(x_{0}\in \Lambda (x_{0},\varphi)\)) and \(\Lambda(x_{0},\varphi)\subseteq B(x_{0},r)\). Let \(\varepsilon= (1-L)r>0\). Since η is continuous on \(t_{0}\), there exists \(\alpha(\varepsilon)>0\) such that
Let \(t\in(t_{0}-\alpha(\varepsilon),t_{0}+\alpha(\varepsilon))\cap[0,1]\). For \(x\in\overline{\Lambda(x_{0},\varphi)}\) (the closure of \(\Lambda(x_{0},\varphi)\)), we have
Also since
using the properties (F1), (F4) we get
Thus we proved that, for all \(t\in(t_{0}-\alpha(\varepsilon),t_{0}+\alpha (\varepsilon))\cap[0,1]\), the operator
is well defined. Now, using (C2) and Theorem 2.1, we deduce that, for all \(t\in (t_{0}-\alpha(\varepsilon), t_{0}+\alpha(\varepsilon))\cap[0,1]\), the operator \(H(\cdot,t)\) has a fixed point in V. However, such a fixed point should be in U from (C1). As a consequence,
which proves that Q is open in \([0,1]\). Next, we show that Q is closed in \([0,1]\). To see this, let \(\{t_{n}\}\) be a sequence in Q with \(t_{n}\to t\in[0,1]\) as \(n\to\infty\). We have to prove that \(t\in Q\). From the definition of Q, for all \(n\in\mathbb{N}\), there exists \(x_{n}\in U\) with
Also for all \(m,n\in\mathbb{N}\), we have
which yields
Letting \(m,n\to\infty\) in the above inequality and using the continuity of η, we get \(d(x_{n},x_{m})\to0\) as \(m,n\to\infty\), which implies that \(\{x_{n}\}\) is a Cauchy sequence in the complete metric space \((X,d)\). Then there is some \(z\in V\) (since V is closed) such that
since φ is lower semi-continuous. Now, for all \(n\in\mathbb {N}\) we have
Letting \(n\to\infty\) in the above inequality, we obtain
The uniqueness of the limit yields \(z=H(z,t)\). Using (C1), we deduce that \(z\in U\) and \(t\in Q\). Thus Q is closed in \([0,1]\).
For the reverse implication, we use the same technique. □
3.2 A partial metric version of Boyd-Wong fixed point theorem
In this part, using Theorem 2.1, we obtain a partial metric version of the Boyd-Wong fixed point theorem.
We start by recalling some basic definitions and properties of partial metric spaces. For more details of such spaces, we refer the reader to [2–20].
A partial metric on a nonempty set X is a function \(p: X\to X \to[0, \infty)\) such that for all \(x,y,z\in X\), we have
-
(i)
\(p(x,x)=p(y,y)=p(x,y)\Longleftrightarrow x=y\);
-
(ii)
\(p(x,x)\leq p(x,y)\);
-
(iii)
\(p(x,y)=p(y,x)\);
-
(iv)
\(p(x,y)\leq p(x,z)+p(z,y)-p(z,z)\).
A partial metric space is a pair \((X,p)\) such that X is a nonempty set and p is a partial metric on X. It is clear that, if \(p(x,y)=0\), then from (i)-(ii), \(x=y\); but if \(x=y\), \(p(x,y)\) may not be 0. A basic example of a partial metric space is the pair \(([0, \infty),p)\), where \(p(x,y)= \max\{x,y\}\).
Each partial metric p on X generates a \(T_{0}\) topology \(\tau_{p}\) on X which has as a base the family of open p-balls \(\{ B_{p}(x,\varepsilon): x\in X, \varepsilon>0\}\), where
Let \((X,p)\) be a partial metric space. A sequence \(\{x_{n}\}\subset X\) converges to some \(x\in X\) with respect to p if and only if
A sequence \(\{x_{n}\}\subset X\) is said to be a Cauchy sequence if and only if \(\lim_{m,n\to\infty}p(x_{n},x_{m})\) exists and is finite. The partial metric space \((X,p)\) is said to be complete if and only if every Cauchy sequence \(\{x_{n}\}\) in X converges to some \(x\in X\) such that \(\lim_{n,m\to\infty}p(x_{n},x_{m})=p(x,x)\).
If p is a partial metric on X, then the function \(d_{p}: X\to X\to [0,\infty)\) defined by
is a metric on X.
Lemma 3.2
Let \((X,p)\) be a partial metric space. Then:
-
(i)
\(\{x_{n}\}\) is a Cauchy sequence in \((X,p)\) if and only if \(\{x_{n}\}\) is a Cauchy sequence in the metric space \((X,d_{p})\);
-
(ii)
the partial metric space \((X,p)\) is complete if and only if the metric space \((X,d_{p})\) is complete. Furthermore, \(\lim_{n\to\infty} d_{p}(x_{n}, x)=0\) if and only if
$$\lim_{n\to\infty}p(x_{n},x)=p(x,x)=\lim _{m,n\to\infty}p(x_{n},x_{m}). $$
We have the following result.
Corollary 3.3
Let \((X,p)\) be a complete partial metric space and let \(T: X\to X\) be an operator such that
where \(\psi\in\Psi\). We have the following results:
-
(i)
if \(z\in X\) is a fixed point of T then \(p(z,z)=0\);
-
(ii)
T has a unique fixed point.
Proof
Let \(d_{p}\) be the metric on X defined by (3.1). We have
where
Then (3.2) yields
where
From (ii) Lemma 3.2, the metric space \((X,d)\) is complete and the function φ is continuous with respect to the topology of d. Finally the desired result follows from Theorem 2.1. □
Remark 3.4
Take in Corollary 3.3, \(\psi(t)= kt\) with \(k\in(0,1)\), and we recover Matthews fixed point theorem [9].
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Acknowledgements
The third author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project No. IRG14-04.
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Karapinar, E., O’Regan, D. & Samet, B. On the existence of fixed points that belong to the zero set of a certain function. Fixed Point Theory Appl 2015, 152 (2015). https://doi.org/10.1186/s13663-015-0401-7
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DOI: https://doi.org/10.1186/s13663-015-0401-7