Alternative proofs of some classical metric fixed point theorems by using approximate fixed point sequences

The notion of approximate fixed point sequence, emphasized in Chidume (Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer-Verlag London, Ltd., London, 2009), is a very useful tool in proving convergence theorems for fixed point iterative schemes in the class of nonexpansive-type 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 nonexpansive-type mappings.
To exemplify this, let K be a nonempty closed convex subset of a real Banach space X and T : K → K a nonexpansive map, i.e., a map satisfying For arbitrary x 0 , u ∈ K , let {x n } be the Halpern-type iterative sequence defined by where λ n ∈ [0, 1] and S = (1 − δ)I + δT , for δ ∈ (0, 1) (I denotes the identity map). If X has uniformly Gâteaux differentiable norm and {λ 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].
In the following examples, (X, d) is supposed to be a metric space.
Example 2.7 (Zamfirescu [66]) Any Zamfirescu mapping, i.e., any mapping T : X → X for which there exist a, b, c ≥ 0 satisfying a < 1, b < 1/2, c < 1/2 such that for each x, y ∈ X at least one of the following conditions is true: is a graphic contraction with Example 2.8 (Ćirić [24]) Any strongĆirić quasi contraction, i.e., any mapping T : X → X satisfying, for all x, y ∈ X , for some h ∈ [0, 1), is a graphic contraction with α = h. Example 2.9 (Hardy and Rogers [32]) Any Hardy and Rogers contraction, i.e., any mapping T : X → X satisfying, for all x, y ∈ X , for a 1 , a 2 , a 3 , a 4 , a 5 ≥ 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 → X satisfying where a ∈ [0, 1) and L ≥ 0, is a graphic contraction with α = 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].
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 → X admits an approximate fixed point sequence.
Proof Denote Obviously, δ ≥ 0. Asume δ > 0. Then, since D 1 ⊆ D, by using (2) we get which, by the definition of infimum, shows that there exists a sequence {x n } ⊂ X such that 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 ≥ 0, for some x 0 ∈ 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 → X be given by T . 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 α ≥ 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 → 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 converges to some p ∈ Fi x (T ), for any x 0 ∈ X . If T is a weakly Picard operator and Fi x (T ) = {p}, then T is called a Picard operator.
Our first main result in this section is an alternative proof of the well-knownĆirić-Reich-Rus 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 → X be aĆirić-Reich-Rus contraction. Then T is a Picard operator.
Hence, by Lemma 2.12, there exists an approximate fixed point sequence {x n } with respect to T , that is, a sequence {x n } ⊂ 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 theĆirić-Reich-Rus condition (5), we obtain which, by (14) and (15), proves that T p = p, i.e., Fi x (T ) = ∅. Assume that q = p is another fixed point of T . Then, by (5) a contradiction. This proves that Fi x (T ) = {p}. Now, let {y n } ⊂ X be the Picard iteration defined by y 0 ∈ X and Then, by (5) one obtains which, by induction, yields This proves that {y n } converges to p as n → ∞. So, T is a Picard operator.
Corollary 2.15 (Banach [4], [16]) Let (X, d) be a complete metric space and T : X → X a Banach contraction. Then T is a Picard operator.
Proof Any Banach contraction is aĆirić-Reich-Rus contraction with the constant b = 0. We apply Theorem 2.14 and get the conclusion.
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 which, by (19) and (20), proves that T p = p.
which, by (19) and (20), also proves that T p = p, i.e., and this proves that {y n } converges to p, for any starting point y 0 ∈ X .
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 T p = p, i.e., Fi x (T ) = ∅. Assume that q = p is another fixed point of T . Then, by (7) 0 < d( p, q) = d(T p, T q) ≤ 2c · d( p, q) < d( p, q), a contradiction. This proves that Fi x (T ) = {p}. Now, let {y n } ⊂ X be defined by y 0 ∈ X and Then, by Example 2.6 one obtains d(y n+1 , p) ≤ α · d(y n , p), n ≥ 0 which, by induction, yields d(y n , p) ≤ α n · d(y 0 , p), n ≥ 0, (26) and this proves that {y n } converges to p, for any starting point y 0 ∈ X . (Zamfirescu [66]) Let (X, d) be a complete metric space and T : X → X a Zamfirescu mapping. Then T is a Picard operator.

Theorem 2.19
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 } ⊂ X such that Now, if for x n , x m ∈ X and T we have condition (i) in Example 2.7 satisfied, then If for x n , x m ∈ X and T we have condition (ii) in Example 2.7 satisfied, then while, if for x n , x m ∈ 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 , if, for x n and p, (i) holds if, for x n and p, (ii) holds , if, for x n and p, (iii) holds, which, by (27) and (31), proves that T p = p, i.e., Fi x (T ) = ∅. Assume that q = 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 Fi x (T ) = {p}. Now, let {y n } ⊂ X be defined by y 0 ∈ X and y n+1 = T y n , n ≥ 0.
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 so-called 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 ρ two metrics on X and T : X → X a mapping. Suppose that , y), for each x, y ∈ X; (ii) (X, d) is a complete metric space; (iii) T : X → X is continuous with respect to the metric d; (iv) T is a contraction mapping with respect to the metric ρ, with contraction coefficient a ∈ [0, 1).

Then T is a Picard operator.
Proof By assumption (iv), T is a graphic contraction with respect to the metric ρ, with α = 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 } ⊂ 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, ρ).  A more general Maia-type result, which generalizes Theorem 2.14, is given by the following: Theorem 3.3 Let X be a nonempty set, d and ρ two metrics on X and T : X → X a mapping. Suppose that

is a complete metric space;
(iii) T : X → X is continuous with respect to the metric d; (iv) T is aĆirić-Reich-Rus contraction with respect to the metric ρ, with contraction coefficients a, b ∈ [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 ρ, with α = a + b Then, by Lemma 2.12, there exists an approximate fixed point sequence {x n } with respect to T , i.e., a sequence {x n } ⊂ X such that ρ(x n , T x n ) → 0, as n → ∞.
For this sequence, by theĆirić-Reich-Rus contraction condition valid for all x, y ∈ X , we obtain which, by virtue of (35), shows that {x n } is a Cauchy sequence in the metric space (X, ρ 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 single-valued 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 → 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) Fi x (T ) = ∅ and there exists a metric on X with respect to which T is asymptotically regular.
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.
Funding The first author acknowledges the support provided by the Technical University of Cluj-Napoca, North University Center at Baia Mare.

Declarations
Data availability Not applicable.

Conflict of interest
There are no competing interests.