Abstract
The notion of approximate fixed point sequence, emphasized in Chidume (Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. SpringerVerlag London, Ltd., London, 2009), is a very useful tool in proving convergence theorems for fixed point iterative schemes in the class of nonexpansivetype mappings. In the present paper, our aim is to present simple and unified alternative proofs of some classical fixed point theorems emerging from Banach contraction principle, by using a technique based on the concepts of approximate fixed point sequence and graphic contraction.
Introduction
In the monograph [18], Chidume illustrated the role of approximate fixed point sequences in proving convergence theorems for fixed point iterative schemes in the class of nonexpansivetype mappings.
To exemplify this, let K be a nonempty closed convex subset of a real Banach space X and \(T : K\rightarrow K\) a nonexpansive map, i.e., a map satisfying
For arbitrary \(x_0\), \(u\in K\), let \(\{x_n\}\) be the Halperntype iterative sequence defined by
where \(\lambda _n\in [0,1]\) and \(S=(1\delta ) I+\delta T\), for \(\delta \in (0,1)\) (I denotes the identity map).
If X has uniformly Gâteaux differentiable norm and \(\{\lambda _n\}\) satisfies some conditions, then (see [19] and [18], page 214) \(\{x_n\}\) is an approximate fixed point sequence with respect to the averaged map S, that is,
The property of having an approximate fixed point sequence is very important for the class of nonexpansivetype mappings; see for example the very recent paper [64]. So, there are many convergence results for iterative algorithms in such classes of mappings which are proven by using the properties of some approximate fixed point sequence; see [1,2,3, 6, 19, 21, 22, 26, 27, 49, 52, 53, 59, 65, 67] and the references therein.
In this paper our aim is to emphasize, by means of several examples, how one can simplify and unify the proofs of some classical fixed point theorems emerging from Banach contraction principle, such as Kannan fixed point theorem, Chatterjea fixed point theorem, Bianchini fixed point theorem, and Zamfirescu fixed point theorem, using a technique based on the concepts of graphic contractions and approximate fixed point sequence.
Graphic contractions
An important concept that will be useful in this paper is given in the next definition; see for example [5, 48, 56,57,58].
Definition 2.1
Let (X, d) be a metric space. A mapping \(T:X\rightarrow X\) is called a graphic contraction (orbital contraction) if
where \(\alpha \in (0,1)\).
In the following examples, (X, d) is supposed to be a metric space.
Example 2.2
Any Banach contraction, i.e., any mapping \(T:X\rightarrow X\) satisfying the inequality
for some \(a\in [0,1)\), is a graphic contraction with \(\alpha =a\).
Example 2.3
(Kannan [34]) Any Kannan mapping, i.e., any mapping \(T:X\rightarrow X\) satisfying the inequality
for some \(b\in [0,1/2)\), is a graphic contraction with \(\alpha =\dfrac{b}{1b}\).
Example 2.4
(Ćirić [24]; Reich [51]; Rus [55]) Any ĆirićReichRus contraction, i.e., any mapping \(T:X\rightarrow X\) satisfying
where \(a,b\ge 0\) and \(a+2b<1\), is a graphic contraction with \(\alpha =\dfrac{a+b}{1b}\).
Example 2.5
(Bianchini [62]) Any Bianchini mapping, i.e., any mapping \(T:X\rightarrow X\) satisfying
for some \(h\in [0,1)\), is a graphic contraction with \(\alpha =h\).
Example 2.6
(Chatterjea [17]) Any Chatterjea mapping \(T:X\rightarrow X\), i.e., any mapping satisfying
for some \(c\in [0,1/2)\), is a graphic contraction with \(\alpha =\dfrac{c}{1c}\).
Example 2.7
(Zamfirescu [66]) Any Zamfirescu mapping, i.e., any mapping \(T:X\rightarrow X\) for which there exist \(a,b,c\ge 0\) satisfying \(a<1\), \(b<1/2\), \(c<1/2\) such that for each \(x,y\in X\) at least one of the following conditions is true:

(i)
\(d(Tx,Ty)\le a d(x,y)\);

(ii)
\(d(Tx,Ty)\le b \left( d(x,Tx)+d(y,Ty)\right) \);

(iii)
\(d(Tx,Ty)\le c\left( d(x,Ty)+d(y,Tx)\right) \),
is a graphic contraction with
Example 2.8
(Ćirić [24]) Any strong Ćirić quasi contraction, i.e., any mapping \(T:X\rightarrow X\) satisfying, for all \(x,y \in X\),
for some \(h\in [0,1)\), is a graphic contraction with \(\alpha =h\).
Example 2.9
(Hardy and Rogers [32]) Any Hardy and Rogers contraction, i.e., any mapping \(T:X\rightarrow X\) satisfying, for all \(x,y \in X\),
for \(a_1,a_2,a_3,a_4,a_5\ge 0\) and \(a_1+a_2+a_3+a_4+a_5<1\), is a graphic contraction with
Example 2.10
(Berinde [5]) Any almost contraction, that is, any mapping \(T:X\rightarrow X\) satisfying
where \(a\in [0,1)\) and \(L\ge 0\), is a graphic contraction with \(\alpha =a\).
The following notion is related to that of graphic contraction, as it is shown by Lemma 2.12. According to MathScinet, the first papers that consider explicitly this concept are by Lin [41] and Khamsi [37].
Definition 2.11
s Let (X, d) be a metric space and \(T:X\rightarrow X\) a selfmapping. A sequence \(\{x_n\}\subset X\) is called an approximate fixed point sequence with respect to T if
The next lemma will be extremely useful in proving some classical fixed results in metric fixed point theory.
Lemma 2.12
Let (X, d) be a metric space. Any graphic contraction \(T:X\rightarrow X\) admits an approximate fixed point sequence.
Proof
Denote
and
Obviously, \(\delta \ge 0\).
Asume \(\delta >0\). Then, since \(D_1\subseteq D\), by using (2) we get
a contradiction. So, \(\delta =0\), i.e.,
which, by the definition of infimum, shows that there exists a sequence \(\{x_n\}\subset X\) such that
\(\square \)
Remark 2.13

1.
Note that the Picard iteration associated with a graphic contraction T, i.e., the sequence \(\{x_n\}\) defined by \(x_{n+1}=T x_n\), \(n\ge 0\), for some \(x_0\in X\), is an approximate fixed point sequence with respect to T.
However, the approximate fixed point sequence \(\{x_n\}\) ensured by Lemma 2.12 is not necessarily the Picard iteration associated with T.

2.
The main idea behind Lemma 2.12 is taken from Joseph and Kwack [33].

3.
There exist mappings which are not graphic contractions but they admit an approximate fixed point sequence. Indeed, let \(X=[0,1]\) with the usual metric and \(T:X\rightarrow X\) be given by \(Tx=\dfrac{7}{8}\), if \(0\le x<1\) and \(T1=\dfrac{1}{4}\). Then T has an approximate fixed point sequence \(\{x_n\}\) (see [2], Example 2.1), T is asymptotically regular on X but T is not a graphic contraction (just take \(x=1\) in (2) to get \(\alpha \ge 5\), a contradiction).
To shorten the statements of the fixed point theorems presented in this paper, we also need the following concepts.
Let \(T: X\rightarrow X\) be a mapping. Denote by
the set of all fixed points of T. The map T is called a weakly Picard operator, see for example [58], if
(p1) \(Fix\,(T)\ne \emptyset \);
(p2) the Picard iteration \(\{x_n\}^\infty _{n=0}\) defined by
converges to some \(p\in Fix\,(T)\), for any \(x_0\in X\).
If T is a weakly Picard operator and \(Fix\,(T)=\{p\}\), then T is called a Picard operator.
Our first main result in this section is an alternative proof of the wellknown ĆirićReichRus fixed point theorem, from which are then obtained as particular cases the classical fixed point theorems due to Banach [4] and Kannan [34].
The innovation brought by Lemma 2.12 is that the Cauchyness is established for an arbitrary approximate fixed point sequence and not necessarily for the Picard iteration.
Theorem 2.14
(Ćirić [24]; Reich [51]; Rus [55]) Let (X, d) be a complete metric space and \(T:X\rightarrow X\) be a Ćirić–Reich–Rus contraction. Then T is a Picard operator.
Proof
By Example 2.4, T is a graphic contraction with \(\alpha =\dfrac{a+b}{1b}<1\).
Hence, by Lemma 2.12, there exists an approximate fixed point sequence \(\{x_n\}\) with respect to T, that is, a sequence \(\{x_n\}\subset X\) with the property
Now, for n, m positive integers, by the contraction condition (5) we have
which, by virtue of (14), shows that \(\{x_n\}\) is a Cauchy sequence. Let
By using once again the Ćirić–Reich–Rus condition (5), we obtain
which, by (14) and (15), proves that \(T p=p\), i.e., \(Fix\,(T)\ne \emptyset \).
Assume that \(q\ne p\) is another fixed point of T. Then, by (5)
a contradiction. This proves that \(Fix\,(T)=\{p\}\).
Now, let \(\{y_n\}\subset X\) be the Picard iteration defined by \(y_0\in X\) and
Then, by (5) one obtains
which, by induction, yields
This proves that \(\{y_n\}\) converges to p as \(n\rightarrow \infty \). So, T is a Picard operator. \(\square \)
Corollary 2.15
(Banach [4, 16]) Let (X, d) be a complete metric space and \(T:X\rightarrow X\) a Banach contraction. Then T is a Picard operator.
Proof
Any Banach contraction is a Ćirić–Reich–Rus contraction with the constant \(b=0\).
We apply Theorem 2.14 and get the conclusion. \(\square \)
Corollary 2.16
(Kannan [34]) Let (X, d) be a complete metric space and \(T:X\rightarrow X\) a Kannan mapping. Then T is a Picard operator.
Proof
Any Kannan mapping is a Ćirić–Reich–Rus contraction with the coefficient \(a=0\).
The conclusion follows by applying Theorem 2.14. \(\square \)
Theorem 2.17
(Bianchini [62]) Let (X, d) be a complete metric space and \(T:X\rightarrow X\) a Bianchini mapping. Then T is a Picard operator.
Proof
By Example 2.5, T is a graphic contraction with \(\alpha =h\). Hence, by Lemma 2.12, there exists an approximate fixed point sequence \(\{x_n\}\) with respect to T, i.e., a sequence \(\{x_n\}\subset X\) such that
Now, for n, m positive integers, by the Bianchini contraction condition (6) we have
which, by (19), shows that \(\{x_n\}\) is a Cauchy sequence. Let
Again by the Bianchini contraction condition (6), we get
Now, if \(\max \{d(x_n,T x_n),d(p,T p)\}=d(x_n,T x_n)\), then it follows that
which, by (19) and (20), proves that \(Tp=p\).
If \(\max \{d(x_n,T x_n),d(p,T p)\}=d(p,T p)\), then
which, by (19) and (20), also proves that \(Tp=p\), i.e., \(Fix\,(T)\ne \emptyset \).
Assume that \(q\ne p\) is another fixed point of T. Then, by (6)
a contradiction. This proves that \(Fix\,(T)=\{p\}\).
Now, let \(\{y_n\}\subset X\) be defined by \(y_0\in X\) and
Then, by Example 2.5, one obtains
which, by induction, yields
and this proves that \(\{y_n\}\) converges to p, for any starting point \(y_0\in X\).
\(\square \)
Theorem 2.18
(Chatterjea [17]) Let (X, d) be a complete metric space and \(T:X\rightarrow X\) a Chatterjea mapping. Then T is a Picard operator.
Proof
By Example 2.6, T is a graphic contraction with \(\alpha =\dfrac{c}{1c}<1\). Hence, by Lemma 2.12, there exists an approximate fixed point sequence \(\{x_n\}\) with respect to T, i.e., a sequence \(\{x_n\}\subset X\) such that
By Chatterjea contraction condition (7) and for n, m positive integers, we get
which, by (23), shows that \(\{x_n\}\) is a Cauchy sequence. Let
Again, by the Chatterjea contraction condition (7) we get
which, by (23) and (24), proves that \(Tp=p\), i.e., \(Fix\,(T)\ne \emptyset \).
Assume that \(q\ne p\) is another fixed point of T. Then, by (7)
a contradiction. This proves that \(Fix\,(T)=\{p\}\).
Now, let \(\{y_n\}\subset X\) be defined by \(y_0\in X\) and
Then, by Example 2.6 one obtains
which, by induction, yields
and this proves that \(\{y_n\}\) converges to p, for any starting point \(y_0\in X\).
\(\square \)
Theorem 2.19
(Zamfirescu [66]) Let (X, d) be a complete metric space and \(T:X\rightarrow X\) a Zamfirescu mapping. Then T is a Picard operator.
Proof
By Example 2.7, T is a graphic contraction with
Hence, by Lemma 2.12, there exists an approximate fixed point sequence \(\{x_n\}\) with respect to T, i.e., a sequence \(\{x_n\}\subset X\) such that
Now, if for \(x_n,x_m\in X\) and T we have condition (i) in Example 2.7 satisfied, then
If for \(x_n,x_m\in X\) and T we have condition (ii) in Example 2.7 satisfied, then
while, if for \(x_n,x_m\in X\) and T we have condition (iii) in Example 2.7 satisfied, then by the proof of Theorem 2.18 we have
By (27), (28), (29) and (30), we obtain that \(\{x_n\}\) is a Cauchy sequence. Let
On the other hand, we have
which, by (27) and (31), proves that \(Tp=p\), i.e., \(Fix\,(T)\ne \emptyset \).
Assume that \(q\ne p\) is another fixed point of T. Then, by considering separately each of the cases (i), (ii) and (iii), we obtain the contradiction
which proves that \(Fix\,(T)=\{p\}\).
Now, let \(\{y_n\}\subset X\) be defined by \(y_0\in X\) and
Then, by Example 2.7 one obtains
which, by induction, yields
and this proves that \(\{y_n\}\) converges to p, for any starting point \(y_0\in X\).
\(\square \)
Remark 2.20
Note that to prove the fixed point theorem corresponding to almost contractions (Example 2.10), which are weakly Picard operators, we have to use Picard iteration as approximate fixed point sequence and not an arbitrary approximate fixed point sequence as above; see the complete proof in [5].
Maia fixed point theorems
One of the most interesting generalizations of the contraction mapping principle is the socalled Maia fixed point theorem, see [43], which was obtained by splitting the assumptions in the contraction mapping principle among two metrics defined on the same set. We provide an alternate proof to this result by using the concept of approximate fixed point sequence.
Theorem 3.1
(Maia [43]) Let X be a nonempty set, d and \(\rho \) two metrics on X and \(T : X \rightarrow X\) a mapping. Suppose that

(i)
\(d(x,y)\le \rho (x,y) \), for each \(x,y \in X\);

(ii)
(X, d) is a complete metric space;

(iii)
\(T : X \rightarrow X\) is continuous with respect to the metric d;

(iv)
T is a contraction mapping with respect to the metric \(\rho \), with contraction coefficient \(a\in [0,1)\).
Then T is a Picard operator.
Proof
By assumption (iv), T is a graphic contraction with respect to the metric \(\rho \), with \(\alpha = a\). Then, by Lemma 2.12, there exists an approximate fixed point sequence \(\{x_n\}\) with respect to T, i.e., a sequence \(\{x_n\}\subset X\) such that
For this sequence, by the contraction condition
we obtain
which, by virtue of (34), shows that \(\{x_n\}\) is a Cauchy sequence in the metric space \((X,\rho )\).
By (i), \(\{x_n\}\) is a Cauchy sequence in the metric space (X, d), too, and by (ii) it follows that it converges with respect to the metric d. Let
Now, by (iii) we obtain that \(p\in Fix\,(T)\) and by (iv) that \(Fix\,(T)=\{p\}\). \(\square \)
Remark 3.2
Theorem 3.1 remains valid if one replaces assumption (i) by the following more general condition:
\((i')\) There exists \(C>0\) such that \(d(x,y)\le C\cdot \rho (x,y) \), for each \(x,y \in X\).
If we have \(\rho \equiv d\), from Theorem 3.1 one obtains the Picard–Banach fixed point principle (Corollary 2.15).
A more general Maiatype result, which generalizes Theorem 2.14, is given by the following:
Theorem 3.3
Let X be a nonempty set, d and \(\rho \) two metrics on X and \(T : X \rightarrow X\) a mapping. Suppose that

(i)
there exists \(C>0\) such that \(d(x,y)\le C\cdot \rho (x,y) \), for each \(x,y \in X\);

(ii)
(X, d) is a complete metric space;

(iii)
\(T : X \rightarrow X\) is continuous with respect to the metric d;

(iv)
T is a Ćirić–Reich–Rus contraction with respect to the metric \(\rho \), with contraction coefficients \(a,b\in [0,1)\).
Then T is a Picard operator.
Proof
Based on the same arguments like in the proof of Theorem 2.14 and using assumption (iv), we can easily deduce that T is a graphic contraction with respect to the metric \(\rho \), with \(\alpha =\dfrac{a+b}{1b}<1\).
Then, by Lemma 2.12, there exists an approximate fixed point sequence \(\{x_n\}\) with respect to T, i.e., a sequence \(\{x_n\}\subset X\) such that
For this sequence, by the Ćirić–Reich–Rus contraction condition
valid for all \(x,y \in X\), we obtain
which, by virtue of (35), shows that \(\{x_n\}\) is a Cauchy sequence in the metric space \((X,\rho )\).
By (i), it follows that \(\{x_n\}\) is a Cauchy sequence in the metric space (X, d), too, and by (ii) we deduce that it converges with respect to the metric d. Let
Now, by (iii) we obtain that \(p\in Fix\,(T)\) and by (iv) that \(Fix\,(T)=\{p\}\). \(\square \)
Remark 3.4
Note that in the case of Ćirić–Reich–Rus contractions, condition (ii) in Theorem 3.3 is not always satisfied, because these mappings are in general not continuous, see the examples in [11].
If in Theorem 3.3 we have \(\rho \equiv d\), then one obtains the Ćirić–Reich–Rus fixed point theorem (Theorem 2.14).
Conclusions

1.
We presented simple and unified alternative proofs, based on the concepts of graphic contraction and approximate fixed point sequence, for some classic metric fixed point theorems emerging from Picard–Banach contraction mapping principle: Kannan fixed point theorem (Kannan [34]); ĆirićReichRus fixed point theorem (Ćirić [24], Reich [51], Rus [55]); Bianchini fixed point theorem (Bianchini [62]); Chatterjea fixed point theorem (Chatterjea [17]) and Maia’s fixed point theorem (Maia [43]).

2.
Similar proofs could be given for other important classes of contractivetype mappings that are related to the Banach contractions: strong Ćirić quasi contractions (see Example 2.8); Hardy and Rogers contractions (see Example 2.9) etc. which are left as exercises for the reader.

3.
In connection with Maiatype fixed point theorems, it is an open problem to find weaker conditions than the continuity of the mapping T involved in Theorems 3.1 and 3.3.

4.
The technique of proof used in the present paper, essentially based on the concepts of graphic contraction and approximate fixed point sequence, could also be nontrivially applied to other classes of self and nonself singlevalued mappings in the literature on metric fixed point theory, see [5, 7,8,9,10,11,12, 14, 20, 23, 28, 31, 36, 38, 39, 42, 44, 46, 50, 60, 61, 63] etc.

5.
There exists another important technique for proving metric fixed point theorems which is based on the property of asymptotic regularity of the mappings, see [14, 29,30,31], and which is naturally closely related but independent to the technique emphasized in the current paper, in view of Theorem 3.1 in [13], which shows that, for a nonempty set X and a mapping \(T:X\rightarrow X\), the following statements are equivalent:

(a)
there exists a complete metric on X with respect to which T is a continuous graphic contraction;

(b)
\(Fix\,(T)\ne \emptyset \) and there exists a metric on X with respect to which T is asymptotically regular.

(a)
So, by also having in view Remark 2.13 (3), it would be very important to compare directly the two methods, the one based on graphic contractions (and approximate fixed point sequences) and the other based on asymptotical regularity, for some concrete classes of mappings to establish, if possible, which one is more reliable.
For example, in the case of Kannan mappings, one can compare the proof of Corollary 2.16 to the proof of the corresponding result in [29]–[31] and conclude that the two methods exhibit slightly different facets of the fixed point problem under study.
References
Ait Mansour, M.; Bahraoui, M.A.; El Bekkali, A.: Metric regularity and Lyusternik–Graves theorem via approximate fixed points of setvalued maps in noncomplete metric spaces. Set Valued Var. Anal. 30(1), 233–256 (2022)
AminiHarandi, A.; Fakhar, M.; Hajisharifi, H.R.: Approximate fixed points of \(\alpha \)nonexpansive mappings. J. Math. Anal. Appl. 467(2), 1168–1173 (2018)
Bachar, M.; Khamsi, M.A.: On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces. Fixed Point Theory Appl. 160, 11 (2015)
Banach, S.: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math. 3, 133–181 (1922)
Berinde, V.: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9(1), 43–53 (2004)
Berinde, V.: Iterative Approximation of Fixed Points, Springer (2007)
Berinde, V.: General constructive fixed point theorems for Ćirićtype almost contractions in metric spaces. Carpathian J. Math. 24(2), 10–19 (2008)
Berinde, V.: Approximating fixed points of implicit almost contractions. Hacettepe J. Math. Stat. 41(1), 93–102 (2012)
Berinde, V.; Păcurar, M.: Iterative Approximation of Fixed Points of Singlevalued Almost Contractions, in Fixed Point Theory and Graph Theory, 29–97. Elsevier/Academic Press, Amsterdam (2016)
Berinde, V., Păcurar, M.: Approximating fixed points of enriched contractions in Banach spaces. J. Fixed Point Theory Appl. 22(2), 10 (2020) (Paper No. 38)
Berinde, V.; P\(\breve{{\rm a}}\)curar, M.: Fixed point theorems for enriched Ćirić–Reich–Rus contractions in Banach spaces and convex metric spaces. Carpathian J. Math. 37(2), 173–184 (2021)
Berinde, V., P\(\breve{{\rm a}}\)curar, M.: A new class of unsaturated mappings: Ciric–Reich–Rus contractions, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 30 (2022) (accepted).
Berinde, V.; Rus, I.A.: Asymptotic regularity, fixed points and successive approximations. Filomat 34(3), 965–981 (2020)
Bisht, R.K.: A remark on asymptotic regularity and fixed point property. Filomat 33(14), 4665–4671 (2019)
Browder, F.E.; Petryshyn, W.V.: The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Am. Math. Soc. 72, 571–575 (1966)
Caccioppoli, R.: Un teorema generale sull’esistenza di elementi uniti in una transformazione funzionale. Rend. Accad. Lincei. 11, 794–799 (1930)
Chatterjea, S.K.: Fixedpoint theorems. C. R. Acad. Bulgare Sci. 25, 727–730 (1972)
Chidume, C.: Geometric properties of Banach spaces and nonlinear iterations, vol. 1965. Lecture Notes in Mathematics. SpringerVerlag, London Ltd, London (2009)
Chidume, C.E.; Chidume, C.O.: Iterative approximation of fixed points of nonexpansive mappings. J. Math. Anal. Appl. 318(1), 288–295 (2006)
Chidume, C.E.; Minjibir, M.S.: Krasnoselskii algorithm for fixed points of multivalued quasinonexpansive mappings in certain Banach spaces. Fixed Point Theory 17(2), 301–311 (2016)
Chidume, C.E.; Zegeye, H.: Approximate fixed point sequences and convergence theorems for asymptotically pseudocontractive mappings. J. Math. Anal. Appl. 278(2), 354–366 (2003)
Chidume, C.E.; Zegeye, H.: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc. Am. Math. Soc. 132(3), 831–840 (2004)
Chuensupantharat, N.; Gopal, D.: On Caristi’s fixed point theorem in metric spaces with a graph. Carpathian J. Math. 36(2), 259–268 (2020)
Ćirić, L. B.: Generalized contractions and fixedpoint theorems. Publ. Inst. Math. (Beograd) (N.S.) 12(26), 19–26 (1971)
Ćirić, L.B.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45, 267–273 (1974)
Dung, N. V., Radenovic, S.: Remarks on the approximate fixed point sequence of \((\alpha ,\beta )\)maps. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115(4), 7 (2021) (Paper No. 193)
Fuan, S., Ullah, R., Rahmat, G., Numan, M., Butt, S. I., Ge, X.: Approximate fixed point sequences of an evolution family on a metric space. J. Math. Art. ID 1647193, 6 (2020)
GarciaFalset, J.; LlorensFuster, E.; Sims, B.: Fixed point theory for almost convex functions. Nonlinear Anal. 32(5), 601–608 (1998)
Górnicki, J.: Fixed point theorems for Kannan type mappings. J. Fixed Point Theory Appl. 19, 2145–2152 (2017)
Górnicki, J.: Various extensions of Kannan’s fixed point theorem. J. Fixed Point Theory Appl. 20(1), 12 (2018) (Paper No. 20)
Górnicki, J.: Remarks on asymptotic regularity and fixed points. J. Fixed Point Theory Appl. 21(1), 20 (2019) (Paper No. 29)
Hardy, G.E.; Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201–206 (1973)
Joseph, J.E.; Kwack, M.H.: Alternative approaches to proofs of contraction mapping fixed point theorems. Missouri J. Math. Sci. 11(3), 167–175 (1999)
Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1968)
Kannan, R.: Some results on fixed points. II. Am. Math. Mon. 76, 405–408 (1969)
Karapınar, E.; Petruşel, A.; Petruşel, G.: On admissible hybrid Geraghty contractions. Carpathian J. Math. 36(3), 433–442 (2020)
Khamsi, M. A.: La propriété du point fixe dans les espaces de Banach avec base inconditionnelle. (French) Math. Ann. 277(4), 727–734 (1987)
Khamsi, M.A.: Approximate fixed point sequences of nonlinear semigroups in metric spaces. Can. Math. Bull. 58(2), 297–305 (2015)
Khan, A. R., Oyetunbi, D. M.: On some mappings with a unique common fixed point. J. Fixed Point Theory Appl. 22(2), 7 (2020) (Paper No. 47)
Kirk, W.A.: Approximate fixed points of nonexpansive maps. Fixed Point Theory 10(2), 275–288 (2009)
Lin, P.K.: Unconditional bases and fixed points of nonexpansive mappings. Pacific J. Math. 116(1), 69–76 (1985)
Llorens Fuster, E., Moreno Gálvez, E.: The fixed point theory for some generalized nonexpansive mappings. Abstr. Appl. Anal. Art. ID 435686, 15 (2011)
Maia, M.: Un’osservazione sulle contrazioni metriche. Rend. Sem. Mat. Univ. Padova 40, 139–143 (1968)
MartínezMoreno, J.; Calderón, K.; Kumam, P.; Rojas, E. Approximating fixed points of Suzuki \((\alpha ,\beta )\)nonexpansive mappings in ordered hyperbolic metric spaces. Advances in metric fixed point theory and applications, 365–383, Springer, Singapore (2021)
Meszaros, J.: A comparison of various definitions of contractive type mappings. Bull. Calcutta Math. Soc. 84(2), 167–194 (1992)
Mitrović, Z.D.; Radenović, S.; Reich, S.; Zaslavski, A.J.: Iterating nonlinear contractive mappings in Banach spaces. Carpathian J. Math. 36(2), 287–294 (2020)
Păcurar, M.: Iterative methods for fixed point approximation. Editura Risoprint, ClujNapoca (2009)
Petruşel, A. Rus, I. A.: Graphic contraction principle and applications. Mathematical analysis and applications, 411–432, Springer Optim. Appl., 154, Springer, Cham (2019)
Pourrazi, S.: Approximate fixed points of \(\alpha \)\(\psi \)contractive setvalued maps and application. J. Adv. Math. Stud. 14(2), 303–310 (2021)
Puiwong, J.; Saejung, S.: On convergence theorems for singlevalued and multivalued mappings in \(p\)uniformly convex metric spaces. Carpathian J. Math. 37(3), 513–527 (2021)
Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14, 121–124 (1971)
Reich, S.; Zaslavski, A.J.: Approximate fixed points of nonexpansive mappings in unbounded sets. J. Fixed Point Theory Appl. 13(2), 627–632 (2013)
Reich, S.; Zaslavski, A.J.: Approximate fixed points of nonexpansive setvalued mappings in unbounded sets. J. Nonlinear Convex Anal. 16(9), 1707–1716 (2015)
Rhoades, B.E.: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)
Rus, I.A.: Some fixed point theorems in metric spaces. Rend. Istit. Mat. Univ. Trieste 3(1971), 169–172 (1972)
Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, ClujNapoca (2001)
Rus, I.A.: Picard operators and applications. Sci. Math. Jpn. 58(1), 191–219 (2003)
Rus, I.A.; Petruşel, A.; Petruşel, G.: Fixed Point Theory. Cluj University Press, ClujNapoca (2008)
Samet, B.; Vetro, C.; Vetro, F.: Approximate fixed points of setvalued mapping in \(b\)metric space. J. Nonlinear Sci. Appl. 9(6), 3760–3772 (2016)
Som, S.; Petruşel, A.; Garai, H.; Dey, L.K.: Some characterizations of Reich and Chatterjea type nonexpansive mappings. J. Fixed Point Theory Appl. 21(4), 21 (2019)
Suantai, S.; Chumpungam, D.; Sarnmeta, P.: Existence of fixed points of weak enriched nonexpansive mappings in Banach spaces. Carpathian J. Math. 37(2), 287–294 (2021)
Tiberio Bianchini, R. M., Su un problema di S. Reich riguardante la teoria dei punti fissi. (Italian) Boll. Un. Mat. Ital. (4). 5, 103–108 (1972)
Tiwari, R.; Khan, M.S.; Rani, S.; Rakočević, V.: On \((\psi, \varphi )^2\)contractive maps. Carpathian J. Math. 36(2), 303–312 (2020)
Van Dung, N., Radenovic, S.: Remarks on the approximate fixed point sequence of \((\alpha ,\beta \)maps. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115, 193 (2021). https://doiorg.am.enformation.ro/10.1007/s1339802101142z.
Wei, L.; Zhou, H.Y.: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive mappings in Banach spaces. Acta Anal. Funct. Appl. 9(1), 29–39 (2007)
Zamfirescu, T.: Fix point theorems in metric spaces. Arch. Math. (Basel) 23, 292–298 (1972)
Zaslavski, A.J.: Approximate fixed points of nonexpansive mappings on hyperbolic spaces. Linear Nonlinear Anal. 5(3), 517–524 (2019)
Acknowledgements
The paper is dedicated to the memory of Professor Charles E. Chidume (19472021) who relaunched the interest for the study of iterative approximation of fixed points in the early 1990s and has contributed with major results to this area of research.
Funding
The first author acknowledges the support provided by the Technical University of ClujNapoca, North University Center at Baia Mare.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Data availability
Not applicable.
Conflict of interest
There are no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Berinde, V., P̆acurar, M. Alternative proofs of some classical metric fixed point theorems by using approximate fixed point sequences. Arab. J. Math. (2022). https://doi.org/10.1007/s40065022003986
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40065022003986
Mathematics Subject Classification
 47H09
 47H10
 54H25