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Four mappings and generalized contractions
Fixed Point Theory and Applications volume 2016, Article number: 99 (2016)
Abstract
In the paper general theorems on common fixed point for four mappings are presented. The results are compact and they extend and unify the respective part of the fixed point theory.
1 Introduction
The present paper was inspired by the advanced and sophisticated article of Liu et al. [1]. Our assumptions are weaker, as the comparison function is much more general, and we do not assume the spaces under consideration to be metric. In addition the general contraction condition is compact and abstract. Also the proofs are relatively simple.
2 Theorems
Let us recall (see [2]) that Φ is the family of all mappings \(\varphi : [0,\infty)\rightarrow[0,\infty)\) such that \(\varphi(\alpha) < \alpha\), \(\alpha> 0\), and \(\boldsymbol {\Phi _{0}}\) consists of mappings \(\varphi\in\Phi\) such that \(\varphi(0) = 0\). In turn, \(\boldsymbol {\Phi}_{P}\) is the family of all mappings \(\varphi : [0,\infty)\rightarrow[0,\infty)\) for which every sequence \((a_{n})_{n \in\mathbb{N}}\) such that \(a_{n+1}\leq\varphi(a_{n})\), \(n \in\mathbb{N}\), converges to zero. It is well known ([2], Proposition 16) that \(\Phi_{P}\subset\Phi_{0}\).
In turn, \(\boldsymbol {\Psi}_{P}\) (see [3]) consists of all mappings of Φ for which every sequence \((a_{n})_{n \in\mathbb{N}}\) such that \(0 < a_{n+1}\leq\varphi (a_{n})\), \(n \in\mathbb{N}\) converges to zero. It is known ([4], Corollary 2.4) that \(\Psi_{P}\) consists of all mappings \(\varphi\in\Phi\) satisfying
Clearly, \(\Phi_{P}\subset\Psi_{P}\) holds and consequently, all members of \(\Phi_{P}\) satisfy (1).
Lemma 2.1
If a \(\varphi\in\Phi_{0}\) satisfies (1), then \(\varphi\in \Phi_{P}\).
Proof
Let \((a_{n})_{n \in\mathbb{N}}\) be a sequence such that \(a_{n+1}\leq \varphi(a_{n})\), \(n \in\mathbb{N}\) for a \(\varphi\in\Phi_{0}\). Then we have
Therefore, \((a_{n})_{n \in\mathbb{N}}\) is nonincreasing and it converges, say to α. Suppose \(\alpha> 0\). Then from (1) it follows that there exists an interval \((\alpha,\alpha +\epsilon)\) on which \(\varphi(\cdot) \leq\alpha\). For large n all \(a_{n}\) belong to this interval, as
and \(\varphi\in\Phi\) yield \(\alpha\leq\varphi(a_{n+1}) < a_{n+1}\). Now, we have \(\alpha< a_{n+1}\leq\varphi(a_{n}) \leq\alpha\), a contradiction. Consequently, \(\alpha= 0\), i.e. \(\varphi\in\Phi_{P}\). □
Corollary 2.2
\(\Phi_{P}\) consists of all mappings \(\varphi\in\Phi_{0}\) satisfying (1).
The notion of dislocated metric space presented below is due to Hitzler and Seda [5].
Let X be a nonempty set, and \(p : X \times X \rightarrow [0,\infty)\) a mapping satisfying
Then p is called a dislocated metric (briefly a d-metric), and \((X,p)\) is called a dislocated metric space (briefly a d-metric space). The topology of \((X,p)\) is generated by balls.
Many authors applied sophisticated contraction conditions. To present a general idea let us consider a mapping \(h : X^{4} \rightarrow[0,\infty)\) satisfying the following requirements for each \(\alpha> 0\):
In order to present an example let us recall the notion of partial metric due to Matthews ([6], Definition 3.1).
A partial metric is a mapping \(p : X \times X \rightarrow [0,\infty)\) such that
Conditions (4b), (4a) yield (2a) and therefore each partial metric is a d-metric. As regards the respective spaces, the situation is more complicated because their topologies are different.
Example 2.3
Let p be a partial metric on X and let
Then for \(a = d\) we have
Similarly, for \(b = c\) we obtain
Consequently, (3a) holds. From
we obtain (3b), (3c), (3d), and (3e).
The notion of a 0-complete d-metric space (or a set) was presented in [3], Definition 2.3 (condition (2.5)). Let us note that if \(\lim_{n \rightarrow\infty}p(y,x_{n}) = \lim_{n \rightarrow\infty}p(x,x_{n})= 0\) holds, then from \(p(x,y) \leq p(x,x_{n}) + p(y,x_{n})\) it follows that \(x = y\). In addition, \(p(x,x)\leq2p(x,x_{n})\) means that \(p(x,x)= 0\). Therefore, condition (2.5) of [3] is equivalent to
The idea of 0-completeness for partial metric spaces is due to Romaguera ([7], Definition 2.1). A partial metric space \((X,p)\) is called 0-complete if any 0-Cauchy sequence \((x_{n})_{n \in\mathbb{N}}\) in X (i.e. such that \(\lim_{m,n \rightarrow\infty}p(x_{n},x_{m})=0\)), converges (in the topology of \((X,p)\)) to a point \(x \in X\) for which \(p(x,x)= 0\).
It is well known (e.g. [6]) that \(x \in\lim_{n \rightarrow\infty}x_{n}\) in a partial metric space \((X,p)\) iff \(\lim_{n \rightarrow\infty}p(x,x_{n})= p(x,x)\). In addition, as it was noticed before, each partial metric is a d-metric. Hence we obtain the following.
Corollary 2.4
Any partial metric space \((X,p)\) is 0-complete iff \((X,p)\) treated as a d-metric space is 0-complete, and iff (6) is satisfied.
Proposition 2.5
Let \((X,p)\) be a d-metric space and let \((x_{n})_{n \in\mathbb{N}}\) be a sequence of points of X such that
holds for a \(\varphi\in\Phi_{P}\). Then we have
Proof
Let us adopt \(a_{n}= p(x_{n+1},x_{n})\). Clearly, \(a_{n+1}\leq\varphi(a_{n})\), \(n \in\mathbb{N}\) holds, and \(\lim_{n \rightarrow\infty}a_{n}= 0\), as \(\varphi\in\Phi_{P}\). Now, \(p(x_{n},x_{n}) \leq2p(x_{n+1},x_{n}) = 2a_{n}\) completes (7). □
In the following \(f(X)\) is replaced by fX, \(f(x)\) is replaced by fx, etc. The precise order in \(\max\{p(a,b),p(c,a),p(d,b)\}\) informs on the variables of the mapping h, and they are not shown in the proofs.
Lemma 2.6
Let \((X,p)\) be a d-metric space and let \(f,g,i,j\) be self mappings in X satisfying the following conditions:
for a \(h : X^{4} \rightarrow[0,\infty)\) such that (3a), (3b) hold, and a \(\varphi\in\Phi_{P}\) (see Corollary 2.2). Then there exist sequences \((x_{n})_{n \in \mathbb{N}}\), \((y_{n})_{n \in\mathbb{N}}\) in X such that
and \((x_{n})_{n \in\mathbb{N}}\) converges to a point x such that \(p(x,x)= 0\), and
Proof
From (8), (9), and (3a) (for \(b = c\)) it follows that
and (for \(a = d\))
If e.g. \(p(fx,ix) \leq p(gy,fx)\) holds, then \(p(gy,fx) \leq \varphi(p(gy,fx))\) means that \(p(gy,fx) = 0\), as \(\varphi\in\Phi_{0}\). Consequently, (8) and (9) yield
and
For an \(x_{0}\in X\) let us adopt \(x_{1}= gx_{0}= iy_{1}\), \(x_{2} = fy_{1}\), where \(p(x_{2},x_{1}) \leq\varphi(p(x_{1},jx_{0}))\) (see (13)). Now, we define \(x_{3} = gy_{2}\) for \(y_{2}\) such that \(x_{2} = fy_{1}= jy_{2}\) and \(p(x_{3},x_{2}) \leq \varphi (p(x_{2},x_{1}))\) (see (12)). In turn \(x_{4} = fy_{3}\) for \(y_{3}\) such that \(x_{3} = gy_{2} = iy_{3}\) and \(p(x_{4},x_{3}) \leq\varphi(p(x_{3},x_{2}))\) (see (13)). By induction we define two sequences \((x_{n})_{n \in\mathbb{N}}\), \((y_{n})_{n \in\mathbb{N}}\) satisfying (10) and
In view of Proposition 2.5 we have \(\lim_{n \rightarrow \infty}p(x_{n+1},x_{n}) = 0\).
Suppose that there exists an infinite set \(\mathbb{K}\subset \mathbb{N}\) such that for each \(k \in\mathbb{K}\) there exists an \(n \in\mathbb{N}\) for which \(0 < \alpha< p(x_{2k},x_{2n+1+2k})\) holds. Let \(n = n(k) > 0\) be the smallest numbers satisfying this inequality for \(k \in\mathbb{K}\). We have (see (2c))
for \(n = n(k)\), and therefore (see (7)),
Now, (9), (10), (7), and the above equality yield
for large k. In turn, (3b) yields
and hence we obtain \(\max\{p(x_{2k-1},x_{2n+2k}),h(\cdot)\} < \alpha+ \epsilon\) for large k. From \(\varphi(\beta) < \alpha\) for \(\beta\leq\alpha\), and (1) we get
for large k, a contradiction. Therefore
holds, and (7) with the triangle inequality (2c) yield \(\lim_{m,n \rightarrow\infty}p(x_{n},x_{m}) = 0\). Consequently (see (6)), there exists an \(x \in X\) such that \(\lim_{n \rightarrow\infty}p(x,x_{n})= p(x,x)= 0\), and (11) holds. □
Let us recall (see [8]) that a pair \((f,i)\) of mappings \(f,i : X \rightarrow X\) is called weakly compatible if
Now, we are ready to prove our theorem.
Theorem 2.7
Let \((X,p)\) be a d-metric space such that (4b) holds. Assume that f, g, i, j are self mappings in X satisfying
(8), and (9) for a \(\varphi\in\Phi_{P}\) (see Corollary 2.2) and a mapping \(h : X^{4} \rightarrow[0,\infty)\) for which the system of conditions (3a)-(3e) holds. Then f, g, i, j have a unique common fixed point, say x, and \(p(x,x)= 0\).
Proof
Let \((x_{n})_{n \in\mathbb{N}}\), \((y_{n})_{n \in\mathbb{N}}\), and x be as in Lemma 2.6. Assume e.g. that \(x \in jX\) (for 0-complete jX or fX). Then there exists a v such that \(x = jv\). Let us prove that \(gv = x\). We have
Suppose \(p(gv,x) > 0\). Then we obtain
for large k (see (9), (3c). Hence,
yields \(p(gv,x) = 0\) (\(\varphi\in\Phi_{0}\)), and \(gv = x\). Thus we have \(x = gv = jv\).
From \(gX \subset iX\) it follows that there exists a w such that \(x = iw\). Let us show that \(fw = x\). We have
Suppose \(p(fw,x) > 0\). Then we have
for large k (see (9)), (3d)). Hence, we obtain
and \(p(fw,x) = 0\), i.e. \(x = fw = iw\).
Now, (14) yields
Let us show that \(fx = gx\). We have (see (9), (3e), (4b))
and therefore \(p(fx,gx) = 0\), i.e. \(fx = gx\) (let us note that \(p(fx,fx) = 0\) not necessarily holds if p is a d-metric).
Now, it is clear that \(fx = gx = ix = jx\) holds.
Let us prove that \(gx = x\). We have (see (9), (3e), (4b))
and consequently, \(p(x,gx) = 0\), i.e. \(gx = x\).
We have proved that \(x = fx = gx = ix = jx\).
If y is a common fixed point of our mappings, then (see (9), (3e), (4b))
yields \(p(x,y) = 0\), i.e. \(x = y\). □
The authors of [1] consider (in metric spaces) formulas \(M_{1}\), \(M_{2}\), and \(M_{3}\) instead of ‘max…’ from our condition (9). It would be rather exhausting to cite \(M_{1}\) or \(M_{2}\), nevertheless all three formulas are applied in conditions that are particular cases of (9) for h satisfying (3a)-(3e).
The next theorem is a consequence of Corollary 2.4, and Theorem 2.7 (see also Example 2.3).
Theorem 2.8
Let \((X,p)\) be a partial metric space. Assume that f, g, i, j are self mappings in X satisfying (14), (8), and (9) for a \(\varphi\in\Phi_{P}\) (see Corollary 2.2) and a mapping \(h : X^{4} \rightarrow[0,\infty)\) for which the system of conditions (3a)-(3e) holds. Then f, g, i, j have a unique common fixed point, say x, and \(p(x,x)= 0\).
The above theorem is a far extension of [9], Theorem 2.8 in its part concerning a unique common fixed point; our mapping φ need not be continuous (see also Example 2.3).
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Acknowledgements
The work has been supported by the Polish Ministry of Science and Higher Education.
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Pasicki, L. Four mappings and generalized contractions. Fixed Point Theory Appl 2016, 99 (2016). https://doi.org/10.1186/s13663-016-0591-7
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DOI: https://doi.org/10.1186/s13663-016-0591-7