 Research
 Open Access
Fixed point theorems of JSquasicontractions
 Zhilong Li^{1, 2} and
 Shujun Jiang^{3}Email author
https://doi.org/10.1186/s1366301605263
© Li and Jiang 2016
 Received: 25 November 2015
 Accepted: 9 March 2016
 Published: 22 March 2016
Abstract
In this paper, we introduce the concept of JSquasicontraction and prove some fixed point results for JSquasicontractions in complete metric spaces under the assumption that the involving function is nondecreasing and continuous. These fixed point results extend and improve many existing results since some assumptions made there are removed or weakened. In addition, we present some examples showing the usability of our results.
Keywords
 fixed point theorem
 JSquasicontraction
MSC
 47H10
 54H25
1 Introductions
Recall the Banach contraction principle [1], which states that each Banach contraction \(T:X\rightarrow X\) (i.e., there exists \(k \in[0,1)\) such that \(d(Tx,Ty)\leq k d(x,y)\) for all \(x,y\in X\)) has a unique fixed point, provided that \((X,d)\) is a complete metric space. According to its importance and simplicity, this principle have been extended and generalized in various directions (see [2–17]). For example, the concepts of Ćirić contraction [5], quasicontraction [6], JScontraction [7], and JSĆirić contraction [8] have been introduced, and many interesting generalizations of the Banach contraction principle are obtained.
 (Ψ1):

\(\psi(t)=1\) if and only if \(t=0\);
 (Ψ2):

for each sequence \(\{t_{n}\}\subset(0,+\infty)\), \(\lim_{n\rightarrow\infty}\psi(t_{n})=1\) if and only if \(\lim_{n\rightarrow\infty}t_{n}=0\);
 (Ψ3):

there exist \(r\in(0,1)\) and \(l\in(0,+\infty]\) such that \(\lim_{t\rightarrow0^{+}}\frac{\psi(t)1}{t^{r}}=l\);
 (Ψ4):

\(\psi(t+s)\leq\psi(t)\psi(s)\) for all \(t,s>0\).
Example 1
Let \(f(t)=e^{te^{t}}\) for \(t\geq0\). Then \(f\in \Phi_{2}\cap\Phi_{3}\), but \(f\notin\Psi\cup\Phi_{1}\cup\Phi_{4}\) since \(\lim_{t\rightarrow0^{+}}\frac{e^{te^{t}}1}{t^{r}}=0\) for each \(r\in (0,1)\) and \(e^{(s+t)e^{s+t}}>e^{se^{s}}e^{te^{t}}\) for all \(s,t>0\).
Example 2
Let \(g(t)=e^{t^{a}}\) for \(t\geq0\), where \(a>0\). When \(a\in(0,1)\), \(g\in\Psi\cap\Phi_{1}\cap\Phi_{2}\cap\Phi_{3}\cap\Phi_{4}\). When \(a=1\), \(g\in \Phi_{2}\cap\Phi_{3}\cap\Phi_{4}\), but \(g\notin\Psi\cup\Phi_{1}\) since \(\lim_{t\rightarrow0^{+}}\frac{e^{t}1}{t^{r}}=0\) for each \(r\in(0,1)\). When \(a>1\), \(g\in\Phi_{2}\cap\Phi_{3}\), but \(g\notin\Psi\cup\Phi_{1}\cup\Phi_{4}\) since \(\lim_{t\rightarrow0^{+}}\frac{e^{t^{a}}1}{t^{r}}=0\) for each \(r\in(0,1)\) and \(e^{(t+s)^{a}}>e^{t^{a}}e^{s^{a}}\) for all \(s,t>0\).
Example 3
Let \(h(t)=1\) for \(t\in[0,a]\) and \(h(t)=e^{ta}\) for \(t>a\), where \(a>0\). Then \(h\in\Phi_{2}\), but \(h\notin\Psi\cup\Phi_{1}\cup\Phi_{3}\cup\Phi_{4}\) since neither (Ψ1) nor (Ψ2) is satisfied.
Example 4
Let \(p(t)=e^{\sqrt{te^{t}}}\) for \(t\geq0\). Then \(p\in\Phi_{1}\cap\Phi_{2}\cap\Phi_{3}\), but \(p\notin\Psi\cup\Phi_{4}\) since \(e^{\sqrt {(t_{0}+s_{0})e^{(t_{0}+s_{0})}}}=e^{\sqrt{2}e}>e^{2\sqrt{e}}=e^{\sqrt{t_{0}e^{t_{0}}}}e^{\sqrt{s_{0}e^{s_{0}}}}\) whenever \(t_{0}=s_{0}=1\).
Remark 1
 (i)
Clearly, \(\Psi\subseteq\Phi_{1}\) and \(\Phi _{4}\subseteq\Phi_{3}\subseteq\Phi_{2}\). Moreover, from Examples 24 it follows that \(\Psi\subset\Phi_{1}\) and \(\Phi_{4}\subset\Phi_{3}\subset\Phi_{2}\).
 (ii)
From Examples 14 we can conclude that \(\Phi_{2}\not\subset\Phi _{1}\), \(\Phi_{4}\not\subset\Psi\), \(\Phi_{1}\cap\Phi_{2}\neq\varnothing\), and \(\Psi\cap\Phi_{4}\neq\varnothing\).
Definition 1
 (i)a Ćirić contraction [5] if there exist nonnegative numbers q, r, s, t with \(q+r+s+2t<1\) such that$$d(Tx, Ty)\leq qd(x,y)+rd(x, Tx) + sd(y, Ty)+t\bigl[d(x, Ty) + d(y, Tx)\bigr], \quad \forall x,y\in X; $$
 (ii)a quasicontraction [6] if there exists \(\lambda\in[0,1)\) such thatwhere \(M_{d}(x,y)=\max\{d(x,y),d(x,Tx),d(y,Ty),\frac {d(x,Ty)+d(y,Tx)}{2}\}\);$$d(Tx, Ty)\leq\lambda M_{d}(x,y),\quad \forall x,y\in X, $$
 (iii)a JScontraction [7] if there exist \(\psi\in\Phi_{1}\) and \(\lambda \in[0,1)\) such that$$ \psi\bigl(d(Tx, Ty)\bigr)\leq\psi\bigl(d(x,y)\bigr)^{\lambda}, \quad\forall x,y\in X \mbox{ with } Tx\neq Ty; $$(1)
 (iv)a JSĆirić contraction [8] if there exist \(\psi\in\Psi\) and nonnegative numbers q, r, s, t with \(q+r+s+2t<1\) such that$$\begin{aligned} &\psi\bigl(d(Tx, Ty)\bigr)\leq\psi\bigl(d(x,y)\bigr)^{q}\psi\bigl(d(x, Tx)\bigr)^{r} \psi\bigl(d(y, Ty)\bigr)^{s}\psi\bigl(d(x, Ty) + d(y, Tx)\bigr)^{t}, \\ &\quad \forall x,y\in X. \end{aligned}$$(2)
In the 1970s, Ćirić [5, 6] established the following two wellknown generalizations of the Banach contraction principle.
Theorem 1
(see [5])
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\) a Ćirić contraction. Then T has a unique fixed point in X.
Theorem 2
(see [6])
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\) a quasicontraction. Then T has a unique fixed point in X.
Recently, Jleli and Samet [7] proved the following fixed point result of JScontractions, which is a real generalization of the Banach contraction principle.
Theorem 3
(see [7], Corollary 2.1)
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\) a JScontraction with \(\psi\in\Phi_{1}\). Then T has a unique fixed point in X.
Remark 2
Recently, Hussain et al. [8] presented the following extension of Theorem 1 and Theorem 3.
Theorem 4
(see [8], Theorem 2.3)
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\) a continuous JSĆirić contraction with \(\psi\in\Psi\). Then T has a unique fixed point in X.
Remark 3
In this paper, we generalize and improve Theorems 14 and remove or weaken the assumptions made on ψ appearing in [7, 8].
2 Main results
Definition 2
Remark 4
(i) Each quasicontraction is a JSquasicontraction with \(\psi(t)=e^{t}\).
(ii) Each JScontraction is a JSquasicontraction whenever ψ is nondecreasing.
(iv) Let \(T:X\rightarrow X\) and \(\psi:[0,+\infty)\rightarrow[1,+\infty )\) be such that (2) is satisfied. Suppose that ψ is a nondecreasing function such that (Ψ4) is satisfied. Then, \(\psi(d(x,Ty)+d(y,Tx))^{t} \leq\psi(\frac{d(x,Ty)+d(y,Tx)}{2})^{2t}\) for all \(x,y\in X\), and so (4) holds. Moreover, if (Ψ1) is satisfied, then it follows from (iii) that T is a JSquasicontraction with \(\lambda=p+r+s+2t\). Therefore, a JSĆirić contraction with \(\psi\in\Phi_{4}\) or \(\psi \in\Psi\) is certainly a JSquasicontraction.
Theorem 5
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\) a JSquasicontraction with \(\psi\in\Phi_{2}\). Then T has a unique fixed point in X.
Proof
Fix \(x_{0}\in X\) and let \(x_{n}=T^{n}x_{0}\) for each n.
Remark 5
In view of Example 2 and (i) of Remark 4, Theorem 2 is a particular case of Theorem 5 with \(\psi(t)=e^{t}\in\Phi_{2}\). The following example shows that Theorem 5 is a real generalization of Theorem 2.
Example 5
Let \(X=\{\tau_{n}\}\) and \(d(x,y)=xy\), where \(\tau_{n}=\frac{n(n+1)(n+2)}{3}\) for each n. Clearly, \((X,d)\) is a complete metric space. Define the mapping \(T:X\rightarrow X\) by \(T\tau_{1}=\tau_{1}\) and \(T\tau_{n}=\tau_{n1}\) for each \(n\geq2\).
On the other hand, it is not hard to check that there exists \(\lambda \in(0,1)\) (resp. nonnegative numbers q, r, s, t with \(q + r + s + 2t < 1\)) such that (1) (resp. (2)) is satisfied with \(\psi(t)=e^{te^{t}}\). But neither Theorem 3 nor Theorem 4 is applicable in this situation since \(e^{te^{t}}\notin\Psi\cup\Phi_{1}\) by Example 1.
Example 6
Let \(X=\{1,2,3\}\) and \(d(x,y)=xy\). Clearly, \((X,d)\) is a complete metric space. Define the mapping \(T:X\rightarrow X\) by \(T1=T2=1\) and \(T3=2\).
When \(x=2\) and \(y=3\), we have \(d(T2,T3)=d(2,3)=1\) and hence \(\frac {d(T2,T3)e^{d(T2,T3)d(2,3)}}{d(2,3)}=1\), which implies that T is not a JScontraction with \(\psi(t)=e^{\sqrt{te^{t}}}\). Therefore, Theorem 3 is not applicable here.
In addition, it is not hard to check that there exist nonnegative numbers q, r, s, t with \(q + r + s + 2t < 1\) such that (2) is satisfied with \(\psi(t)=e^{\sqrt {te^{t}}}\). However, Theorem 4 is not applicable here since \(e^{\sqrt{te^{t}}}\notin\Psi\) by Example 4.
Theorem 6
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\). Assume that there exist \(\psi\in\Phi_{3}\) and nonnegative numbers q, r, s, t with \(q+r+s+2t<1\) such that (4) is satisfied. Then T has a unique fixed point in X.
Proof
In view of (iii) of Remark 4, T is a JSquasicontraction with \(\lambda=q+r+s+2t\). In the case where \(q+r+s+2t=0\), by (3) we have \(\psi(d(Tx,Ty))=1\) for all \(x,y\in X\). Moreover, by (Ψ1) we get \(d(Tx,Ty)=0\) for all \(x,y\in X\). This shows that \(y=Tx\) is a fixed point of T. Let z be another fixed point of T. Then \(d(y,z)=d(Ty,Tz)=0\), and hence \(y=z\), that is, T has a unique fixed point. In the case where \(0< q+r+s+2t<1\), the conclusion immediately follows from Theorem 5. The proof is completed. □
Remark 6
Theorem 2 and Theorem 1 are respectively particular cases of Theorem 5 and Theorem 6 with \(\psi(t)=e^{t}\), whereas they are not particular cases of Theorem 3 and Theorem 4 with \(\psi(t)=e^{t}\) since \(e^{t}\in\Phi_{2}\cap\Phi_{3}\) but \(e^{t}\notin\Psi\cup\Phi_{1}\). Hence, Theorem 5 and Theorem 6 are new generalizations of Theorem 2 and Theorem 1.
In view of (ii) and (iv) of Remark 4, we have the following two corollaries of Theorem 5 and Theorem 6.
Corollary 1
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\) a JScontraction with \(\psi\in\Phi_{2}\). Then T has a unique fixed point in X.
Corollary 2
Let \((X,d)\) be a complete metric space, and \(T:X\rightarrow X\) a JSĆirić contraction with \(\psi\in\Phi_{4}\). Then T has a unique fixed point in X.
Remark 7
Conditions (Ψ2) and (Ψ3) assumed in Theorems 3 and 4 are removed from Corollaries 1 and 2 at the expense that ψ is continuous. Thus, Corollaries 1 and 2 partially improve Theorems 3 and 4.
Taking \(\psi(t)=e^{t^{a}}\) (\(a>0\)) in Theorem 6, we have the following new generalization of Theorem 1.
Corollary 3
Corollary 4
(see [8], Theorem 2.4 and Corollary 2.9)
Proof
For each \(a\in(0,1]\), we have \((d(x,Ty)+d(y,Tx))^{a}\leq 2(\frac{d(x,Ty)+d(y,Tx)}{2})^{a}\), and so (23) immediately follows from (24). Thus, by Corollary 3, T has a unique fixed point. The proof is completed. □
Remark 9
Declarations
Acknowledgements
The work was supported by the Natural Science Foundation of China (11161022, 11561026, 71462015), the Natural Science Foundation of Jiangxi Province (20142BCB23013, 20143ACB21012, 20151BAB201003, 20151BAB201023), the Natural Science Foundation of Jiangxi Provincial Education Department (KJLD14034, GJJ150479).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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