 Research
 Open access
 Published:
Discussion of several contractions by Jachymski’s approach
Fixed Point Theory and Applications volume 2016, Article number: 91 (2016)
1 Introduction
The Banach contraction principle [3, 4] is an elegant, forceful tool in nonlinear analysis and has many generalizations. See, e.g., [5–10]. For example, Boyd and Wong in [11] proved the following.
Theorem 1
(Boyd and Wong [11])
Let \((X,d)\) be a complete metric space and let T be a mapping on X. Assume that T is a BoydWong contraction, that is, there exists a function φ from \([0, \infty)\) into itself satisfying the following:

(i)
φ is upper semicontinuous from the right.

(ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\).

(iii)
\(d(Tx,Ty) \leq\varphi\circ d(x,y)\) for any \(x,y \in X\).
Then T has a unique fixed point.
Branciari in [12] introduced contractions of integral type as follows: A mapping T on a metric space \((X,d)\) is a Branciari contraction if there exist \(r \in[0, 1)\) and a locally integrable function f from \([0, \infty)\) into itself such that
for all \(s >0\) and \(x, y \in X\). We have studied contractions of integral type in [13–15].
In this paper, we discuss several contractions of integral type by using Jachymski’s approach. As applications, we give alternative proofs of recent generalizations of the Banach contraction principle due to Ri [1] and Wardowski [2].
2 Preliminaries
Throughout this paper we denote by \(\mathbb {N}\) the set of all positive integers and by \(\mathbb {R}\) the set of all real numbers.
Let f be a function from a subset Q of \(\mathbb {R}\) into \(\mathbb {R}\). Then f is said to satisfy (UR)_{ f } if the following holds:
 (UR)_{ f } :

For any \(t \in Q\), there exist \(\delta> 0\) and \(\varepsilon> 0\) such that \(f(s) \leq t  \varepsilon\) holds for any \(s \in[t,t+\delta) \cap Q\).
We give some lemmas concerning (UR).
Lemma 2
Let f be a function from a subset Q of \(\mathbb {R}\) into \(\mathbb {R}\). Then the following are equivalent:

(i)
f satisfies (UR)_{ f }.

(ii)
\(\limsup[ f(u) : u \to t, u \in Q, t \leq u ] < t \) holds for any \(t \in Q\).

(iii)
\(\limsup[ f(u) : u \to t, u \in Q, t < u ] < t \) and \(f(t) < t \) hold for any \(t \in Q\).
Proof
Obvious. □
Lemma 3
Let f be a function from a subset Q of \(\mathbb {R}\) into \(\mathbb {R}\) such that \(f(t) < t\) for any \(t \in Q\). Assume that f is upper semicontinuous from the right. Then f satisfies (UR)_{ f }.
Proof
Obvious. □
Lemma 4
Let f be a function from a subset Q of \(\mathbb {R}\) into \(\mathbb {R}\) satisfying (UR)_{ f }. Define a function g from Q into \(\mathbb {R}\) by
for \(t \in Q\). Define a mapping L from Q into the power set of \(\mathbb {R}\), a function ℓ from Q into \([\infty,\infty)\) and a function h from Q into \(\mathbb {R}\) by
for \(t \in Q\). Define a function φ from Q into \(\mathbb {R}\) by
for \(t \in Q\). Then the following hold:

(i)
g is upper semicontinuous from the right.

(ii)
h and φ are right continuous.

(iii)
\(f(t) \leq g(t) \leq h(t) < \varphi(t) < t\) holds for any \(t \in Q\).
Proof
Since f satisfies (UR)_{ f }, we have \(f(t) \leq g(t) < t\) for any \(t \in Q\). In order to show (i), we fix \(t \in Q\) and let \(\{ t_{n} \}\) be a strictly decreasing sequence in Q converging to t. Fix \(\varepsilon> 0\). Then for every \(n \in \mathbb {N}\), there exists \(s_{n} \in Q\) satisfying \(t_{n} \leq s_{n} \leq t_{n} + 1/n\) and \(g(t_{n}) \leq f(s_{n}) + \varepsilon\). Since \(\{ s_{n} \}\) converges to t, we have
Since \(\varepsilon> 0\) is arbitrary, we obtain \(\limsup_{n} g(t_{n}) \leq g(t) \). Therefore we have shown (i). We shall show \(h(t) < t\) for any \(t \in Q\). Arguing by contradiction, we assume \(h(t) \geq t\) for some \(t \in Q\). Then since \(g(t) < t\), there exists a strictly increasing sequence \(\{ s_{n} \}\) such that \(\lim_{n} s_{n} = t\) and \(\lim_{n} g(s_{n}) = h(t)\). Since \(g(s_{n}) < s_{n}\) for \(n \in \mathbb {N}\), we have \(h(t) = t\). Therefore \(t \in L(t)\), which implies \(h(t) = g(t) < t\). This is a contradiction. So \(h(t) < t\) holds. It is obvious that \(h(t) < \varphi(t) < t\) for any \(t \in Q\). Therefore we have shown (iii). In order to show (ii), we fix \(t \in Q\) and \(\varepsilon> 0\) with \(h(t) + \varepsilon< t\). From (i), there exists \(\delta> 0\) such that
for \(s \in(t,t+\delta) \cap Q\). Let \(\{ t_{n} \}\) be a strictly decreasing sequence \(\{ t_{n} \}\) in Q such that \(t_{1} < t + \delta\) and \(\{ t_{n} \}\) converges to t. Then we note \(\ell(t) = \ell(t_{n})\) for \(n \in \mathbb {N}\). So we have
for \(n \in \mathbb {N}\). Hence
Since \(\varepsilon> 0\) is arbitrary, we obtain \(\lim_{n} h(t_{n}) = h(t)\). Thus, h is right continuous. It is obvious that φ is also right continuous. We have shown (ii). □
Remark
See Theorem 2 in [7]. Note that the domain of h is Q. We cannot extend the domain of h to \(\bigcup [ [t,\infty) : t \in Q ]\), considering the function f from \((\infty,0) \cup(0,\infty)\) into \(\mathbb {R}\) defined by
3 Definitions
We list the following notation in order to simplify the statement of the results of this paper:

(A1)
Let D be a subset of \((0,\infty)^{2}\).

(A2)
Let θ be a function from \((0,\infty)\) into \(\mathbb {R}\). Put \(\Theta= \theta ( (0,\infty) )\) and
$$\Theta_{\leq}= \bigcup \bigl[ [t,\infty) : t \in\Theta \bigr] . $$
Jachymski in [8] discussed several contractions by using subsets of \([0,\infty)^{2}\). Since this approach seems to be very reasonable for considering future studies, we use an approach similar to Jachymski’s.
Definition 5
Assume (A1).

(1)
D is said to be contractive (Cont for short) [3, 4] if there exists \(r \in(0,1)\) such that \(u \leq r t\) holds for any \((t,u) \in D\).

(2)
D is said to be a Browder (Bro, for short) [16] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:

(2i)
φ is nondecreasing and right continuous.

(2ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\).

(2iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in D\).

(2i)

(3)
D is said to be BoydWong (BW for short) [11] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:

(3i)
φ is upper semicontinuous from the right.

(3ii)
\(\varphi(t) < t\) holds for any \(t \in(0, \infty)\).

(3iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in D\).

(3i)

(4)
D is said to be MeirKeeler (MK for short) [17] if for any \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(u < \varepsilon\) holds for any \((t,u) \in D\) with \(t < \varepsilon+ \delta\); see also [18–20].

(5)
D is said to be Matkowski (Mat for short) [21] if there exists a function φ from \((0, \infty)\) into itself satisfying the following:

(5i)
φ is nondecreasing.

(5ii)
\(\lim_{n} \varphi^{n}(t) = 0\) for every \(t \in(0, \infty)\).

(5iii)
\(u \leq\varphi(t)\) holds for any \((t,u) \in D\).

(5i)

(6)
D is said to be CJM [6, 22–24] if the following hold:

(6i)
For any \(\varepsilon> 0\), there exists \(\delta> 0\) satisfying \(u \leq\varepsilon\) holds for any \((t,u) \in D\) with \(t < \varepsilon+ \delta\).

(6ii)
\(u < t\) holds for any \((t,u) \in D\).

(6i)
Remark
We know the following implications; see, e.g., [5, 7, 10].

Cont ⇒ Bro ⇒ BW ⇒ MK ⇒ CJM;

Cont ⇒ Bro ⇒ Mat ⇒ CJM.
We give one proposition on the concept of BoydWong. Note that we can easily obtain similar results on the other concepts.
Proposition 6
Let T be a mapping on a metric space \((X,d)\) and define a subset D of \((0,\infty)^{2}\) by
Then T is a BoydWong contraction iff D is BoydWong.
Proof
We first note
because \(Tx \neq Ty\) implies \(x \neq y\). We assume that D is BoydWong. Then there exists φ satisfying (3i)(3iii) in Definition 5. Define a function η from \([0,\infty)\) into itself by \(\eta(0) = 0\) and \(\eta(t) = \varphi(t)\) for \(t \in(0,\infty)\). Then we have \((\mathrm{i})_{\eta}\) and \((\mathrm{ii})_{\eta}\) in Theorem 1. If either \(x=y\) or \(Tx=Ty\) holds, then \(d(Tx,Ty) \leq\eta\circ d(x,y) \) obviously holds. Considering this fact, we have \((\mathrm{iii})_{\eta}\) in Theorem 1. Therefore T is a BoydWong contraction. Conversely, we next assume that T is a BoydWong contraction. Then there exists η satisfying \((\mathrm{i})_{\eta}\)\((\mathrm{iii})_{\eta}\) in Theorem 1. Define a function φ from \((0,\infty)\) into itself by
for any \(t \in(0,\infty)\). Then φ satisfies (3i) and (3ii) in Definition 5. We also have
for any \(x,y \in X\) with \(Tx \neq Ty\). So (3iii) holds. Therefore D is BoydWong. □
The following are variants of Corollaries 9 and 14 in [14].
Proposition 7
([14])
Assume (A1), (A2) and the following:

(i)
θ is nondecreasing and continuous.

(ii)
There exists an upper semicontinuous function ψ from Θ into \(\mathbb {R}\) satisfying \(\psi(\tau) < \tau\) for any \(\tau\in\Theta\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).
Then D is Browder.
Proposition 8
([14])
Assume (A1), (A2), and the following:

(i)
θ is nondecreasing.

(ii)
There exists an upper semicontinuous function ψ from \(\Theta_{\leq}\) into \(\mathbb {R}\) satisfying \(\psi(\tau) < \tau\) for any \(\tau\in\Theta_{\leq}\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).
Then D is CJM.
Remark
From the proof in [14], we can weaken (ii) of Proposition 8 to the following:
 (ii)′:

There exists a function ψ from \(\Theta_{\leq}\) into \(\mathbb {R}\) such that ψ is upper semicontinuous from the right, \(\psi(\tau) < \tau\) for any \(\tau\in\Theta_{\leq}\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).
4 Main results
In this section, we prove our main results. We begin with BoydWong.
Proposition 9
Assume (A1), (A2), and the following:

(i)
θ is nondecreasing and continuous.

(ii)
There exists a function ψ from Θ into \(\mathbb {R}\) satisfying \((\mathrm{UR})_{\psi}\) and \(\theta(u) \leq\psi\circ\theta(t) \) for any \((t,u) \in D\).
Then D is BoydWong.
Proof
Define a function \(\theta_{+}^{1}\) from \(\mathbb {R}\) into \([0, \infty]\) by
We also define a function η from \((0,\infty)\) into \([0,\infty)\) by \(\eta= \theta_{+}^{1} \circ\psi\circ\theta\). We note
Since \(\psi(\tau) < \tau\) for any \(\tau\in\Theta\), we have \(\psi\circ\theta(t) < \theta(t) \leq\theta(s)\) for any \(t, s \in(0,\infty)\) with \(t \leq s\). Hence \(\eta(t) \leq t\) holds for any \(t \in(0,\infty)\). Arguing by contradiction, we assume that \((\mathrm{UR})_{\eta}\) does not hold. Then there exist \(t \in(0,\infty)\) and a sequence \(\{ t_{n} \}\) in \([t,\infty)\) such that \(\{ t_{n} \}\) converges to t and
holds for \(n \in \mathbb {N}\). Since \(\eta(t_{n}) > 0\),
holds. Hence there exists a sequence \(\{ u_{n} \}\) in \((0,\infty)\) satisfying
for \(n \in \mathbb {N}\). Since θ is nondecreasing, \(u_{n} < t_{n}\) holds for any \(n \in \mathbb {N}\). Thus \(\{ u_{n} \}\) also converges to t. Hence by the continuity of θ,
This contradicts \((\mathrm{UR})_{\psi}\). Therefore \((\mathrm{UR})_{\eta}\) holds. For any \((t,u) \in D\), since \(\theta(u) \leq\psi\circ\theta(t)\), we have
By Lemma 4, there exists a right continuous function φ from \((0,\infty)\) into itself satisfying \(\eta(t) < \varphi(t) < t \). It is obvious that \(u \leq\eta(t) < \varphi(t) \) for any \((t,u) \in D\). Therefore D is BoydWong. □
Remark
There appears \(\theta_{+}^{1}\) in Proposition 2.1 in [15].
We next discuss MeirKeeler.
Proposition 10
Assume (A1), (A2), and the following:

(i)
θ is nondecreasing and right continuous.

(ii)
For any \(\varepsilon\in\Theta\), there exists \(\delta> 0\) such that \(\theta(t) < \varepsilon+ \delta\) implies \(\theta(u) < \varepsilon\) for any \((t,u) \in D\).
Then D is MeirKeeler.
Proof
Fix \(\varepsilon> 0\). Then from (ii), there exists \(\alpha> 0\) such that
for any \((t,u) \in D\). From the right continuity of θ, there exists \(\delta> 0\) such that \(\theta(\varepsilon+ \delta) < \theta(\varepsilon) + \alpha\). Fix \((t,u) \in D\) with \(t < \varepsilon+ \delta\). Then we have
and hence \(\theta(u) < \theta(\varepsilon) \). Therefore \(u < \varepsilon\) holds. So D is MeirKeeler. □
We obtain the following, which is a generalization of Corollary 17 in [14].
Corollary 11
Assume (A1), (A2), (i) of Proposition 10, and (ii) of Proposition 9. Then D is MeirKeeler.
Let us discuss Matkowski.
Proposition 12
Assume (A1), (A2), and the following:

(i)
θ is nondecreasing and left continuous.

(ii)
minΘ does not exist.

(iii)
There exist a subset Q of \(\mathbb {R}\) and a nondecreasing function ψ from Q into Q satisfying \(\Theta\subset Q \subset\Theta_{\leq}\),
$$\lim_{n \to\infty} \psi^{n} (\tau) = \inf\Theta $$for any \(\tau\in Q\) and \(\theta(u) \leq\psi\circ\theta(t)\) for any \((t,u) \in D\).
Then D is Matkowski.
Proof
We first note that \(\inf\Theta= \inf Q = \inf\Theta_{\leq}\) holds and neither minΘ, minQ nor \(\min\Theta_{\leq}\) does exist. So, from (ii) and (iii), \(\psi(\tau) < \tau\) holds for any \(\tau\in Q\). Define a function \(\theta_{+}^{1}\) from Q into \((0,\infty]\) by
Since θ is left continuous, we have \(\tau< \theta(t)\) implies \(\theta_{+}^{1}(\tau) < t\). We also have
provided \(\tau< \sup\Theta\). Hence \(\theta\circ\theta_{+}^{1}(\tau) \leq\tau\) provided \(\tau< \sup\Theta\). It is obvious that \(\theta_{+}^{1}\) is nondecreasing. Define a function φ from \((0,\infty)\) into itself by \(\varphi= \theta_{+}^{1} \circ\psi\circ\theta\). Then for any \(t \in(0, \infty)\), since \(\psi\circ\theta(t) < \theta(t)\), we have \(\varphi(t) < t \). Since θ, ψ, and \(\theta_{+}^{1}\) are nondecreasing, φ is also nondecreasing. Noting \(\psi\circ\theta(t) < \theta(t) \leq\sup\Theta\), we have
Continuing this argument, we can prove \(\varphi^{n} (t) \leq\theta_{+}^{1} \circ\psi^{n} \circ\theta(t) \) by induction. Since \(\lim_{n} \psi^{n} \circ\theta(t) = \inf\Theta\), we have \(\lim_{n} \theta_{+}^{1} \circ\psi^{n} \circ\theta(t) = 0\) from (ii). Therefore we obtain
for any \(t \in(0, \infty)\). Since \(u \leq\theta_{+}^{1} \circ\theta(u) \leq\theta_{+}^{1} \circ\psi\circ\theta(t)\), we obtain \(u \leq\varphi(t) \) for any \((t,u) \in D\). Therefore D is Matkowski. □
5 Counterexamples
In this section, we give counterexamples connected with the results in Section 4.
Example 13
(Example 2.3 in [15], Example 10 in [14])
Define a complete metric space \((X, d)\) by
Define a mapping T on X and functions θ and ψ from \((0, \infty)\) into itself by
and \(\psi(t) = t / 2\). Define D by (1). Then all the assumptions of Propositions 9 and 12 except the left continuity of θ are satisfied. However, D is neither BoydWong nor Matkowski.
Remark
By Corollary 11, D is MeirKeeler. We define E by
Then \(E \subset \{ 2 \} \times (1/4,1)\) holds. Hence E is contractive.
Proof
We have
Hence D is neither BoydWong nor Matkowski. □
Example 14
(Example 2.6 in [13], Example 11 in [14])
Define a complete metric space \((X, d)\) by \(X = [0, \infty)\) and \(d(x,y) = x + y\) for \(x, y \in X\) with \(x \neq y\). Define a mapping T on X and functions θ and ψ from \((0, \infty)\) into itself by
and \(\psi(t) = t / 2\). Define D by (1). Then all the assumptions of Proposition 10 except the right continuity of θ are satisfied. However, D is not MeirKeeler. Therefore D is not BoydWong.
Remark
By Proposition 12, D is Matkowski. We define E by (2). Then \(E = \{ (2,1) \} \) holds. Hence E is contractive.
Proof
We have
Hence D is not MeirKeeler. □
Example 15
Define a complete metric space \((X, d)\) by \(X = \{ 0, 1 \} \) and \(d(0,1) = 1\). Define a mapping T on X and functions θ and ψ from \((0, \infty)\) into itself by
Define D by (1). Then all the assumptions of Proposition 12 except (ii) are satisfied. However, D is not Matkowski.
Proof
Obvious. □
6 Applications
In this section, as applications, we give alternative proofs of some recent generalizations of the Banach contraction principle. Ri in [1] proved the following fixed point theorem.
Theorem 16
(Ri [1])
Let \((X,d)\) be a complete metric space and let T be a mapping on X. Assume there exists a function ψ from \([0,\infty)\) into itself satisfying the following:

(R1)
\(\psi(t) < t\) for any \(t \in(0,\infty)\).

(R2)
\(\limsup_{s \to t+0} \psi(s) < t\) for any \(t \in(0,\infty)\).

(R3)
\(d(Tx, Ty) \leq\psi ( d(x, y) )\) for any \(x, y \in X\).
Then T has a unique fixed point.
We give an alternative proof of Theorem 16 by showing that a mapping T in Theorem 16 is a BoydWong contraction.
Proof of Theorem 16
By Lemma 2, the restriction ψ to \((0,\infty)\) satisfies \((\mathrm{UR})_{\psi}\). Then by Lemma 4, there exists a right continuous function φ from \((0,\infty)\) into itself satisfying \(\psi(t) < \varphi(t) < t\) for \(t \in(0,\infty)\). Thus T is a BoydWong contraction. So T has a unique fixed point. □
Wardowski in [2] proved a fixed point theorem on Fcontraction.
Theorem 17
(Wardowski [2])
Let \((X,d)\) be a complete metric space and let T be a Fcontraction on X, that is, there exist a function F from \((0,\infty)\) into \(\mathbb {R}\) and real numbers \(\eta\in(0,\infty)\) and \(k \in(0,1)\) satisfying the following:

(F1)
F is strictly increasing.

(F2)
For any sequence \(\{ \alpha_{n} \}\) of positive numbers, \(\lim_{n} \alpha_{n} = 0\) iff \(\lim_{n} F(\alpha_{n})=\infty\).

(F3)
\(\lim_{t \to+0} t^{k} F(t) = 0\) holds.

(F4)
If \(Tx \neq Ty\), then
$$F \bigl( d(Tx,Ty) \bigr) \leq F \bigl( d(x,y) \bigr)  \eta $$holds.
Then T has a unique fixed point.
Remark
By (F1), we note that (F2) is equivalent to the following:
 (F2)′:

\(\lim_{t \to+0} F(t) =  \infty\) holds.
We give an alternative proof of Theorem 17 by showing that mappings satisfying (F1) and (F4) are CJM contractions.
Proof of Theorem 17
Define a subset D of \((0,\infty)^{2}\) by (1). Define θ and ψ by \(\theta= F\) and \(\psi(\tau) = \tau \eta\). Then all the assumptions of Proposition 8 hold. So, by Proposition 8, D is CJM. Therefore T has a unique fixed point. □
Remark
We assume (F4) and that F is nondecreasing instead of (F1)(F4). Then D defined by (1) is CJM. Moreover, the following hold:
References
Ri, SI: A new fixed point theorem in the fractal space. Indag. Math. 27, 8593 (2016)
Wardowski, D: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, Article ID 94 (2012)
Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133181 (1922)
Caccioppoli, R: Un teorema generale sull’esistenza di elementi uniti in una transformazione funzionale. Rend. Accad. Naz. Lincei 11, 794799 (1930)
Jachymski, J: A generalization of the theorem by Rhoades and Watson for contractive type mappings. Math. Jpn. 38, 10951102 (1993)
Jachymski, J: Equivalent conditions and the MeirKeeler type theorems. J. Math. Anal. Appl. 194, 293303 (1995)
Jachymski, J: Equivalence of some contractivity properties over metrical structures. Proc. Am. Math. Soc. 125, 23272335 (1997)
Jachymski, J: Remarks on contractive conditions of integral type. Nonlinear Anal. 71, 10731081 (2009)
Kirk, WA: Contraction mappings and extensions. In: Kirk, WA, Sims, B (eds.) Handbook of Metric Fixed Point Theory, pp. 134. Kluwer Academic, Dordrecht (2001)
Suzuki, T, Alamri, B: A sufficient and necessary condition for the convergence of the sequence of successive approximations to a unique fixed point II. Fixed Point Theory Appl. 2015, Article ID 59 (2015)
Boyd, DW, Wong, JSW: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458464 (1969)
Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29, 531536 (2002)
Suzuki, T: MeirKeeler contractions of integral type are still MeirKeeler contractions. Int. J. Math. Math. Sci. 2007, Article ID 39281 (2007)
Suzuki, T: Comments on some recent generalization of the Banach contraction principle. J. Inequal. Appl. 2016, Article ID 111 (2016)
Suzuki, T, Vetro, C: Three existence theorems for weak contractions of Matkowski type. Int. J. Math. Stat. 6, 110120 (2010)
Browder, FE: On the convergence of successive approximations for nonlinear functional equations. Proc. K. Ned. Akad. Wet., Ser. A, Indag. Math. 30, 2735 (1968)
Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326329 (1969)
Lim, TC: On characterizations of MeirKeeler contractive maps. Nonlinear Anal. 46, 113120 (2001)
Proinov, PD: Fixed point theorems in metric spaces. Nonlinear Anal. 64, 546557 (2006)
Suzuki, T: Fixed point theorem for asymptotic contractions of MeirKeeler type in complete metric spaces. Nonlinear Anal. 64, 971978 (2006)
Matkowski, J: Integrable solutions of functional equations. Diss. Math. 127, 168 (1975)
Ćirić, LB: A new fixedpoint theorem for contractive mappings. Publ. Inst. Math. (Belgr.) 30, 2527 (1981)
Kuczma, M, Choczewski, B, Ger, R: Iterative Functional Equations. Encyclopedia of Mathematics and Its Applications, vol. 32. Cambridge University Press, Cambridge (1990)
Matkowski, J: Fixed point theorems for contractive mappings in metric spaces. Čas. Pěst. Mat. 105, 341344 (1980)
Acknowledgements
The author is supported in part by JSPS KAKENHI Grant Number 16K05207 from Japan Society for the Promotion of Science.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
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.
About this article
Cite this article
Suzuki, T. Discussion of several contractions by Jachymski’s approach. Fixed Point Theory Appl 2016, 91 (2016). https://doi.org/10.1186/s1366301605819
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366301605819