On Some New Jungck–Fisher–Wardowski Type Fixed Point Results

: Many authors used the concept of F− contraction introduced by Wardowski in 2012 in order to deﬁne and prove new results on ﬁxed points in complete metric spaces. In some later papers (for example Proinov P.D., J. Fixed Point Theory Appl. (2020)22:21, it is shown that conditions (F2) and (F3) are not necessary to prove Wardowski’s results. In this article we use a new approach in proving that the Picard–Jungck sequence is a Cauchy one. It helps us obtain new Jungck–Fisher–Wardowski type results using Wardowski’s condition (F1) only, but in a way that differs from the previous approaches. Along with that, we came to several new contractive conditions not known in the ﬁxed point theory so far. With the new results presented in the article, we generalize, extend, unify and enrich methods presented in the literature that we cite.


Introduction and Preliminaries
In 1976, Jungck [1] proved the following result. Theorem 1. Let T and I be commuting mappings of a complete metric space (Υ, d Υ ) into itself that satisfy the inequality d Υ (T x, T y) ≤ λd Υ (I x, I y) for all x, y ∈ Υ, where 0 < λ < 1. If the range of I contains the range of T and if I is continuous, then T and I have a unique common fixed point.
In 1981, Fisher [2] proved the common fixed point theorem for four mappings and thus obtained a genuine generalization of Jungck's result from 1976. Theorem 2. Let S, I and T , J be pairs of commuting mappings of a complete metric space (Υ, d Υ ) into itself that satisfies d Υ (S x, T y) ≤ λd Υ (I x, J y) for all x, y ∈ Υ, where 0 < λ < 1. If S x ∈ J (Υ) and T x ∈ I (Υ) for each x ∈ Υ and if I and J are continuous, then all mappings S, T , I and J have a unique common fixed point.

Remark 1.
It is obvious that both previous Theorems holds true for λ = 0 alike. In addition, it is evident that Theorems 1 and 2 genuinely generalize the famous Banach contraction principle [3].
Additionally, in [17] Wardowski proved and generalized the Banach contraction principle in the following form: Theorem 3. Let (Υ, d Υ ) be a complete metric space and A : Υ → Υ an F − contraction. Then A has a unique fixed point, say x * in Υ and for every x ∈ Υ the sequence {A p x} , p ∈ N converges to x * .
. Then A has a unique fixed point.
is not a Cauchy sequence, then there exist ε > 0 and two sequences {p (s)} and {q (s)} of positive integers such that p (s) > q (s) > s and the sequences: tend to ε + , as s → +∞.
Proof. Suppose the opposite, let and that is a contradiction.
At the end of this section, let us recall the following terms and results (for more information, see [25,26]). Let A and B be self mappings of a nonempty set Υ. If y = A x = B x for some x ∈ Υ, then x is called a coincidence point of A and B, and y is called a point of coincidence of A and B. A pair of

Proposition 1. [26]
If weakly compatible self mappings A and B of a set Υ have a unique point of coincidence y = A x = B x, then y is a unique common fixed point of A and B.

Results
In the following theorem, we bring forward first of our results for four self-mappings in a complete metric space.
Theorem 5. Let (S, I) and (T , J ) be a pair of compatible self-mappings of a complete metric space (Υ, d Υ ) into itself and F : (0, +∞) → (−∞, +∞) is a strictly increasing mapping such that and τ is a given positive constant. If I, J , S and T are continuous and if S (Υ) ⊆ J (Υ) , T (Υ) ⊆ I (Υ) then mappings I, J , S and T have a unique common fixed point.
Proof. First of all we show the uniqueness of a possible common fixed point. Suppose that I, J , S and T have two distinct common fixed points x and y in Υ. Since d Υ ( x, y) = d Υ (S x, T y) > 0 we get according to (6): where Hence, Since τ > 0 and x = y we get a contradiction. So, if there exists a common fixed point, it is unique. We further prove the existence of this common fixed point. Let In general, there are x 2p+1 and x 2p+2 in Υ such that J x 2p+1 = S x 2p and I . Replacing x and y respectively with x 2p and x 2p+1 in (6) we obtain where Hence, (9) transforms into It is clear that max Finally, since F is a strictly increasing mapping and d Υ z 2p , z 2p+1 < d Υ z 2p−1 , z 2p for all p ∈ N we have Similarly, replacing x with x 2p+2 and y with x 2p+1 in (6), it follows d Υ z 2p+2 , z 2p+1 < d Υ z 2p+1 , z 2p , for all p ∈ N. So, d Υ z p+1 , z p < d Υ z p , z p−1 (12) for all p ∈ N, which, further, implies that lim p→+∞ d Υ z p+1 , z p = d * Υ ≥ 0. If d * Υ > 0 from (11) follows and that is a contradiction. Hence, lim p→+∞ d Υ z p+1 , z p = 0. To prove that { z p } is a Cauchy sequence, it suffices proving that for the sequence { z 2p }. Indeed, according to Lemma 1, puting x = x 2p(s) , y = x 2q(s)+1 in (6), we get where when s → +∞. Taking the limit in (13) as s → +∞, we get a contradiction Thus, { z p } is a Cauchy sequence in a complete metric space (Υ, d Υ ). Having in mind that Υ is a complete, we conclude that there exists z ∈ Υ such that lim p→+∞ z p = z or Further, due to the continuity and compatibility of mappings J and T , we obtain and J x 2p+1 converge to the same z, so due to their compatibility, we obtain where Now, (15) can be written in the form which is a contradiction. Therefore, S z = T z. This further entails equality and where and now (18) can be written as which is a contradiction. Therefore, it must be S z = T w. Hence, T w = w and from (17) it follows that w is a common fixed point for T and J . Similarly as in previous case, assumption S w = T z implies a contradiction, since from (6) we get Therefore, S w = T z. Suppose, further, that S w = w. Then from (16) it follows that w is a common fixed point for S and J . We proved that w is unique common fixed point for S, T , I and J .
It is worth to notice that Theorem 5 generalizes Theorems 1 and 2 in several directions. Namely, putting J = I and S = T in (6) we get the following Jungck-Wardowski type result: Theorem 6. Let (T , I) be a pair of compatible self-mappings of a complete metric space (Υ, d Υ ) into itself and F : (0, +∞) → (−∞, +∞) is strictly increasing mapping such that for all x, y ∈ Υ with d Υ (T x, T y) > 0, where τ is a given positive constant. If T and I are continuous and T (Υ) ⊆ I (Υ) then T , I have a unique common fixed point.
in (6) we also find Theorem 6 to be true.
As a result of Theorems 5 and 6 in the following we introduce new contractive conditions that complement the ones given in [11,[27][28][29][30].

Corollary 1.
Suppose that (S, I) and (T , J ) are the pairs of compatible self-mappings of a complete metric space (Υ, d Υ ) into itself such that for all x, y ∈ Υ with d Υ (S x, T y) > 0 there exist τ i > 0, i = 1, 7 and the following inequalities hold true: The next example supports Theorem 5. In fact, it is a modification of an example given in [31].
then due to the continuity of S and I it follows only for x = 0. Similarly we can prove that the second pair (T , J ) is compatible. Furthermore, it is easy to show that both pairs are not commuting. For x, y ∈ Υ, we obtain Putting τ = ln 6561 82 , F (ω) = ln ω we get that the condition (6) holds true. Based on Theorem 5, this means that the mappings I, J , S and T have a unique common fixed point x = 0.
Finaly, we believe that the following problem may be interesting for some future research : Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.