Abstract
In this paper we give some applications to integral equations as well as homotopy theory via fixed point theorems in partially ordered complete \(S_{b}\)-metric spaces by using generalized contractive conditions. We also furnish an example which supports our main result.
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1 Introduction
Banach contraction principle in metric spaces is one of the most important results in fixed theory and nonlinear analysis in general. Since 1922, when Stefan Banach [1] formulated the concept of contraction and posted a famous theorem, scientists around the world have published new results related to the generalization of a metric space or with contractive mappings (see [1–24]). Banach contraction principle is considered to be the initial result of the study of fixed point theory in metric spaces.
In the year 1989, Bakhtin introduced the concept of b-metric spaces as a generalization of metric spaces [6]. Later several authors proved so many results on b-metric spaces (see [13–16]). Mustafa and Sims defined the concept of a generalized metric space which is called a G-metric space [12]. Sedghi, Shobe and Aliouche gave the notion of an S-metric space and proved some fixed point theorems for a self-mapping on a complete S-metric space [22]. Aghajani, Abbas and Roshan presented a new type of metric which is called \(G_{b}\)-metric and studied some properties of this metric [2].
Recently Sedghi et al. [20] defined \(S_{b}\)-metric spaces using the concept of S-metric spaces [22].
The aim of this paper is to prove some unique fixed point theorems for generalized contractive conditions in complete \(S_{b}\)-metric spaces. Also, we give applications to integral equations as well as homotopy theory. Throughout this paper \(R, R^{+}\) and N denote the sets of all real numbers, non-negative real numbers and positive integers, respectively.
First we recall some definitions, lemmas and examples.
2 Preliminaries
Definition 2.1
[22]
Let X be a non-empty set. An S-metric on X is a function \(S:X^{3} \to[0,+\infty) \) that satisfies the following conditions for each \(x,y,z,a \in X\):
- \((S1)\)::
-
\(0 < S(x, y, z) \) for all \(x,y,z \in X\) with \(x \neq y \neq z \neq x\),
- \((S2)\)::
-
\(S(x, y, z) = 0 \mbox{ if and only if } x = y = z\),
- \((S3)\)::
-
\(S(x,y,z) \leq S(x ,x, a) + S(y, y, a) + S(z, z, a)\) for all \(x,y,z,a \in X\).
Then the pair \((X, S)\) is called an S-metric space.
Definition 2.2
[20]
Let X be a non-empty set and \(b \geq1\) be a given real number. Suppose that a mapping \(S_{b}:X^{3} \to\mathcal{[}0,\infty ) \) is a function satisfying the following properties:
- \((S_{b} 1)\) :
-
\(0 < S_{b}(x,y,z)\) for all \(x,y,z \in X \) with \(x \neq y \neq z \neq x \),
- \((S_{b} 2)\) :
-
\(S_{b}(x, y, z) = 0 \mbox{ if and only if } x = y = z\),
- \((S_{b} 3)\) :
-
\(S_{b}(x,y,z) \leq b(S_{b}(x, x, a) + S_{b}(y, y, a) + S_{b}(z, z, a))\) for all \(x,y,z,a \in X\).
Then the function \(S_{b}\) is called an \(S_{b}\)-metric on X and the pair \((X,S_{b})\) is called an \(S_{b}\)-metric space.
Remark 2.3
[20]
It should be noted that the class of \(S_{b}\)-metric spaces is effectively larger than that of S-metric spaces. Indeed each S-metric space is an \(S_{b}\)-metric space with \(b=1\).
The following example shows that an \(S_{b}\)-metric on X need not be an S-metric on X.
Example 2.4
[20]
Let \((X,S)\) be an S-metric space and \(S_{*}(x,y,z) = S(x,y,z)^{p}\), where \(p>1\) is a real number. Note that \(S_{*}\) is an \(S_{b}\)-metric with \(b = 2^{2(p-1)}\). Also, \((X,S_{*})\) is not necessarily an S-metric space.
Definition 2.5
[20]
Let \((X,S_{b})\) be an \(S_{b}\)-metric space. Then, for \(x \in X\), \(r > 0\), we define the open ball \(B_{S_{b}} (x,r)\) and the closed ball \(B_{S_{b}} [x,r]\) with center x and radius r as follows, respectively:
Lemma 2.6
[20]
In an \(S_{b}\)-metric space, we have
and
Lemma 2.7
[20]
In an \(S_{b}\)-metric space, we have
Definition 2.8
[20]
If \((X,S_{b})\) is an \(S_{b}\)-metric space, a sequence \(\{x_{n}\}\) in X is said to be:
-
(1)
\(S_{b}\)-Cauchy sequence if, for each \(\epsilon> 0\), there exists \(n_{0} \in\mathcal {N}\) such that \(S_{b}(x_{n},x_{n},x_{m}) < \epsilon\) for each \(m,n \geq n_{0}\).
-
(2)
\(S_{b}\)-convergent to a point \(x \in X\) if, for each \(\epsilon> 0\), there exists a positive integer \(n_{0}\) such that \(S_{b}(x_{n},x_{n},x) < \epsilon\) or \(S_{b}(x,x,x_{n}) < \epsilon\) for all \(n \geq n_{0}\), and we denote \(\mathop{\lim} _{n \rightarrow\infty}x_{n} = x\).
Definition 2.9
[20]
An \(S_{b}\)-metric space \((X,S_{b})\) is called complete if every \(S_{b}\)-Cauchy sequence is \(S_{b}\)-convergent in X.
Lemma 2.10
[20]
If \((X,S_{b})\) is an \(S_{b}\)-metric space with \(b\geq1\), and suppose that \(\{x_{n}\}\) is \(S_{b}\)-convergent to x, then we have
and
for all \(y \in X\).
In particular, if \(x=y\), then we have \(\mathop{\lim} _{n \rightarrow \infty} S_{b}(x_{n},x_{n},y) = 0\).
Now we prove our main results.
3 Results and discussions
Definition 3.1
Let \((X, S_{b}, \preceq)\) be a partially ordered complete \(S_{b}\)-metric space which is said to be regular if every two elements of X are comparable,
Definition 3.2
Let \((X, S_{b}, \preceq)\) be a partially ordered complete \(S_{b}\)-metric space which is also regular; let \(f: X \to X \) be a mapping. We say that f satisfies \((\psi, \phi)\)-contraction if there exist \(\psi, \phi : [0, \infty) \to[0, \infty)\) such that
- (3.2.1):
-
f is non-decreasing,
- (3.2.2):
-
ψ is continuous, monotonically non-decreasing and ϕ is lower semi-continuous,
- (3.2.3):
-
\(\psi(t) = 0 = \phi(t)\) if and only if \(t= 0\),
- (3.2.4):
-
\(\psi (4b^{4} {S_{b} ( {fx, fx, fy} )} ) \le \psi ( { M_{f}^{i} ( {x,y} )} ) - \phi ( {M_{f}^{i} ( {x,y} )} )\), \(\forall x, y \in X\), \(x \preceq y\), \(i = 3, 4, 5\) and
$$\begin{aligned} &M_{f}^{5} ( {x,y} ) = \max \left \{ {S_{b}(x, x,y), S_{b}(x, x,fx),S_{b}(y, y, fy),S_{b}(x, x,fy), S_{b}(y, y, fx) } \right \}, \\ &M_{f}^{4} ( {x,y} ) = \max \left \{ {S_{b}(x, x,y), S_{b}(x, x,fx),S_{b}(y, y, fy), \frac{1}{4b^{4}} \bigl[S_{b}(x, x,fy)+S_{b}(y, y, fx) \bigr] } \right \}, \\ &M_{f}^{3} ( {x,y} ) = \max\biggl\{ S_{b}(x, x,y), \frac{1}{4b^{4}} \bigl[S_{b}(x, x,fx)+S_{b}(y, y, fy) \bigr],\\ &\phantom{M_{f}^{3} ( {x,y} ) =}\frac{1}{4b^{4}} \bigl[S_{b}(x, x,fy)+S_{b}(y, y, fx) \bigr] \biggr\} . \end{aligned}$$
Definition 3.3
Suppose that \((X, \preceq)\) is a partially ordered set and f is a mapping of X into itself. We say that f is non-decreasing if for every \(x, y \in X\),
Theorem 3.4
Let \((X,S_{b}, \preceq)\) be an ordered complete \(S_{b}\) metric space, which is also regular, and let \(f : X \to X \) satisfy \((\psi, \phi )\)-contraction with \(i = 5\). If there exists \(x_{0} \in X\) with \(x_{0} \preceq f x_{0}\), then f has a unique fixed point in X.
Proof
Since f is a mapping from X into X, there exists a sequence \(\{x_{n} \}\) in X such that
Case (i): If \(x_{n} = x_{n+1}\), then \(x_{n}\) is a fixed point of f.
Case (ii): Suppose \(x_{n} \neq x_{n+1}\ \forall n \).
Since \(x_{0} \preceq fx_{0} = x_{1}\) and f is non-decreasing, it follows that
Now
where
Therefore
By the definition of ψ, we have that
But
From (2) we have that
If \(\frac{1}{2b^{2}} S_{b} ( fx_{0}, fx_{0}, f^{2}x_{0} )\) is maximum, we get a contradiction. Hence
Also
where
Therefore
By the definition of ψ, we have that
But
From (4) we have that
If \(\frac{1}{2b^{2}} S_{b} ( f^{2}x_{0}, f^{2}x_{0}, f^{3}x_{0} )\) is maximum, we get a contradiction. Hence
Continuing this process, we can conclude that
That is,
Now we prove that \(\{ f^{n} x_{0} \}\) is an \(S_{b}\)-Cauchy sequence in \((X, S_{b})\). On the contrary, we suppose that \(\{ f^{n} x_{0} \}\) is not \(S_{b}\)-Cauchy. Then there exist \(\epsilon> 0\) and monotonically increasing sequences of natural numbers \(\{m_{k}\}\) and \(\{ n_{k}\}\) such that \(n_{k} > m_{k}\).
and
So that
Letting \(k \to\infty\) and applying ψ on both sides, we have that
where
But
Also
Therefore
From (8), by the definition of ψ, we have that
which is a contradiction. Hence \(\{ f^{n} x_{0} \}\) is an \(S_{b}\)-Cauchy sequence in complete regular \(S_{b}\)-metric spaces \((X, S_{b}, \preceq)\). By the completeness of \((X, S_{b})\), it follows that the sequence \(\{ f^{n} x_{0} \}\) converges to α in \(( X, S_{b} )\). Thus
Since \(x_{n}, \alpha\in X\) and X is regular, it follows that either \(x_{n} \preceq\alpha\) or \(\alpha\preceq x_{n}\).
Now we have to prove that α is a fixed point of f.
Suppose \(f\alpha\neq\alpha\), by Lemma (2.10), we have that
Now from (3.2.4) and applying ψ on both sides, we have that
Here
Hence from (9) we have that
which is a contradiction. So that α is a fixed point of f.
Suppose that \(\alpha^{\ast}\) is another fixed point of f such that \(\alpha\neq\alpha^{\ast}\).
Consider
which is a contradiction.
Hence α is a unique fixed point of f in \((X, S_{b} )\). □
Example 3.5
Let \(X = [0, 1]\) and \(S : X \times X \times X \to\mathbb {R}^{+}\) by \(S_{b}(x,y,z) = ( \vert y+z-2x \vert + \vert y-z \vert )^{2}\) and ⪯ by \(a \preceq b \iff a\le b\), then \((X, S_{b} , \preceq )\) is a complete ordered \(S_{b}\)-metric space with \(b = 4\). Define \(f: X \rightarrow X\) by \(f(x) = \frac{x}{32\sqrt{2}} \). Also define \(\psi,\phi:\mathbb{R}^{+} \to\mathbb{R}^{+}\) by \(\psi(t) = t \) and \(\phi(t) = \frac{t}{2}\).
where
Hence, all the conditions of Theorem 3.4 are satisfied and 0 is a unique fixed point of f.
Theorem 3.6
Let \((X,S_{b}, \preceq)\) be an ordered complete \(S_{b}\) metric space, and let \(f : X \to X \) satisfy \((\psi, \phi)\)-contraction with \(i = 3\textit{ or }4\). If there exists \(x_{0} \in X\) with \(x_{0} \preceq f x_{0}\), then f has a unique fixed point in X.
Proof
Follows along similar lines of Theorem 3.4 if we take \(M_{f}^{3} ( {x,y} )\) or \(M_{f}^{4} ( {x,y} )\) in place of \(M_{f}^{5} ( {x,y} )\) in Theorem 3.4. □
Theorem 3.7
Let \((X,S_{b}, \preceq)\) be an ordered complete \(S_{b}\) metric space, and let \(f : X \to X \) satisfy
where \(\varphi:[0, \infty) \to[0, \infty) \) and i= 3 or 4 or 5. If there exists \(x_{0} \in X\) with \(x_{0} \preceq f x_{0}\), then f has a unique fixed point in X.
Proof
The proof follows from Theorems 3.4 and 3.6 by taking \(\psi(t) = t\) and \(\phi(t) = \varphi(t)\). □
Theorem 3.8
Let \((X,S_{b}, \preceq)\) be an ordered complete \(S_{b}\) metric space, and let \(f : X \to X \) satisfy
where \(\lambda\in [ {0, \frac{1}{4b^{4}}} )\) and \(i = 3, 4, 5\). If there exists \(x_{0} \in X\) with \(x_{0} \preceq f x_{0}\), then f has a unique fixed point in X.
3.1 Application to integral equations
In this section, we study the existence of a unique solution to an initial value problem as an application to Theorem 3.4.
Theorem 3.9
Consider the initial value problem
where \(T: I \times [ {\frac{{x_{0} }}{4},\infty} ) \to [ {\frac{{x_{0} }}{4},\infty} )\) and \(x_{0} \in\mathbb{R}\). Then there exists a unique solution in \(C (I, [ {\frac{{x_{0} }}{4},\infty} ) )\) for initial value problem (10).
Proof
The integral equation corresponding to initial value problem (10) is
Let \(X =C (I, [ {\frac{{x_{0} }}{4},\infty} ) )\) and \(S_{b}(x,y,z) = ( \vert y+z-2x \vert + \vert y-z \vert )^{2} for x, y \in X\). Define \(\psi, \phi: [0, \infty) \to[0, \infty)\) by \(\psi(t) = t \), \(\phi(t)= \frac{5t}{9} \). Define \(f: X \to X\) by
Now
where
It follows from Theorem 3.4 that f has a unique fixed point in X. □
3.2 Application to homotopy
In this section, we study the existence of a unique solution to homotopy theory.
Theorem 3.10
Let \((X, S_{b})\) be a complete \(S_{b}\)-metric space, U be an open subset of X and U̅ be a closed subset of X such that \(U \subseteq\overline{U}\). Suppose that \(H : \overline{U} \times[0, 1] \to X \) is an operator such that the following conditions are satisfied:
-
(i)
\(x \neq H(x, \lambda)\) for each \(x \in\partial{U}\) and \(\lambda \in[0, 1]\) (here ∂U denotes the boundary of U in X),
-
(ii)
\(\psi(4b^{4} S_{b}(H(x, \lambda),H(x, \lambda), H(y, \lambda) )) \leq\psi( S_{b}(x, x, y)) - \phi( S_{b}(x, x, y))\) \(\forall x, y \in\overline{U}\) and \(\lambda\in[0, 1]\), where \(\psi :[0,\infty) \to[0,\infty)\) is continuous, non-decreasing and \(\phi :[0,\infty) \to[0,\infty)\) is lower semi-continuous with \(\phi(t)>0\) for \(t>0\),
-
(iii)
there exists \(M\geq0\) such that
$$S_{b} \bigl(H(x, \lambda),H(x, \lambda), H(x, \mu) \bigr) \leq M \vert \lambda - \mu \vert $$for every \(x \in\overline{U}\) and \(\lambda, \mu\in[0, 1]\).
Then \(H(\cdot, 0)\) has a fixed point if and only if \(H(\cdot, 1)\) has a fixed point.
Proof
Consider the set
Since \(H(\cdot, 0)\) has a fixed point in U, we have that \(0 \in A\). So that A is a non-empty set.
We will show that A is both open and closed in \([0, 1]\), and so, by the connectedness, we have that \(A = [0, 1]\). As a result, \(H(\cdot, 1)\) has a fixed point in U. First we show that A is closed in \([0, 1]\). To see this, let \(\{ { \lambda_{n}} \}_{n = 1}^{\infty }\subseteq A\) with \(\lambda_{n} \to\lambda\in[0, 1]\) as \(n \to\infty \).
We must show that \(\lambda\in A\). Since \(\lambda_{n} \in A\) for \(n = 1, 2, 3, \ldots \) , there exists \(x_{n} \in U\) with \(x_{n} = H(x_{n}, \lambda _{n})\).
Consider
Letting \(n \to\infty\), we get
Since ψ is continuous and non-decreasing, we obtain
By the definition of ψ, it follows that
So that
Now we prove that \(\{x_{n}\}\) is an \(S_{b}\)-Cauchy sequence in \((X, d_{p})\). On the contrary, suppose that \(\{x_{n} \}\) is not \(S_{b}\)-Cauchy.
There exists \(\epsilon> 0\) and monotone increasing sequences of natural numbers \(\{ m_{k}\}\) and \(\{ n_{k}\}\) such that \(n_{k} > m_{k}\),
and
Letting \(k \to\infty\) and applying ψ on both sides, we have that
But
It follows that
Thus
Hence from (15) and the definition of ψ, we have that
which is a contradiction.
Hence \(\{x_{n} \}\) is an \(S_{b}\)-Cauchy sequence in \((X, S_{b})\) and, by the completeness of \((X, S_{b})\), there exists \(\alpha\in U\) with
It follows that \(\alpha= H(\alpha, \lambda)\).
Thus \(\lambda\in A\). Hence A is closed in \([0, 1]\).
Let \(\lambda_{0} \in A\). Then there exists \(x_{0} \in U\) with \(x_{0} = H(x_{0}, \lambda_{0})\).
Since U is open, there exists \(r > 0\) such that \(B_{S_{b}}(x_{0}, r) \subseteq U\).
Choose \(\lambda\in(\lambda_{0} - \epsilon, \lambda_{0} + \epsilon)\) such that \(\vert \lambda- \lambda_{0} \vert \leq\frac{1}{M^{n}} < \epsilon\).
Then, for \(x \in\overline{B_{p} (x_{0}, r)} = \{x \in X / S_{b}(x, x, x_{0}) \leq r + b^{2} S_{b}(x_{0}, x_{0}, x_{0}) \}\),
Letting \(n \to\infty\), we obtain
Since ψ is continuous and non-decreasing, we have
Since ψ is non-decreasing, we have
Thus, for each fixed \(\lambda\in(\lambda_{0} - \epsilon, \lambda_{0} + \epsilon)\), \(H(\cdot, \lambda): \overline{B_{p} (x_{0}, r)} \to\overline {B_{p} (x_{0}, r)}\).
Since also (ii) holds and ψ is continuous and non-decreasing and ϕ is continuous with \(\phi(t)> 0\) for \(t > 0\), then all the conditions of Theorem (3.10) are satisfied.
Thus we deduce that \(H(\cdot, \lambda)\) has a fixed point in U̅. But this fixed point must be in U since (i) holds.
Thus \(\lambda\in A\) for any \(\lambda\in(\lambda_{0} - \epsilon, \lambda _{0} + \epsilon)\).
Hence \((\lambda_{0} - \epsilon, \lambda_{0} + \epsilon) \subseteq A\) and therefore A is open in [0, 1].
For the reverse implication, we use the same strategy. □
Corollary 3.11
Let \((X, p)\) be a complete partial metric space, U be an open subset of X and \(H : \overline{U} \times[0, 1] \to X \) with the following properties:
-
(1)
\(x \neq H(x, t)\) for each \(x \in\partial U\) and each \(\lambda \in[0, 1]\) (here ∂U denotes the boundary of U in X),
-
(2)
there exist \(x, y \in\overline{U} \) and \(\lambda\in[0, 1], L \in [0, \frac{1}{4b^{4}} )\) such that
$$S_{b} \bigl(H(x, \lambda),H(x, \lambda), H(y, \mu) \bigr) \leq L S_{b}(x, x, y), $$ -
(3)
there exists \(M \geq0\) such that
$$S_{b} \bigl(H(x, \lambda), H(x, \lambda), H(x, \mu) \bigr) \leq M \vert \lambda- \mu \vert $$for all \(x \in\overline{U} \) and \(\lambda, \mu\in[0, 1]\).
If \(H(\cdot, 0)\) has a fixed point in U, then \(H(\cdot, 1)\) has a fixed point in U.
Proof
Proof follows by taking \(\psi(x) = x, \phi(x) = x - Lx \mbox{ with } L \in [0, \frac{1}{4b^{4}} )\) in Theorem (3.10). □
4 Conclusions
In this paper we conclude some applications to homotopy theory and integral equations by using fixed point theorems in partially ordered \(S_{b}\)-metric spaces.
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Kishore, G., Rao, K., Panthi, D. et al. Some applications via fixed point results in partially ordered \(S_{b}\)-metric spaces. Fixed Point Theory Appl 2017, 10 (2016). https://doi.org/10.1186/s13663-017-0603-2
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DOI: https://doi.org/10.1186/s13663-017-0603-2