A NEW

. In this work it was proved Matkowski’s ﬁxed point theorem. The consequences of this theorem are also presented.


Introduction
The presented work concerns Matkowski's fixed point theorem and the conclusions from this theorem.These results were used to study the limit behaviors of quotients F n+1 F n of the Fibonacci type numbers.This work thematically refers to works [2] and [3].For these studies Edelstein's fixed point theorem was used in [2], while in [3], the fixed point theorem was proved and used for the "d (f (x), f (y)) ≤ φ (d(x, y))" type mappings of the interval a, b , where the function φ is right continuous and fulfills additional conditions.
In the presented work there is a new and easy proof of Matkowski's fixed point theorem.In this theorem the function φ is not assumed to be continuous.There are also proven conclusions from this theorem.The obtained results concern the mentioned type of mappings of complete spaces.Their application is illustrated by the approximation of the golden number ϕ = 1+ √ 5 2 .
In this work, based on Matkowski's theorem, we present and demonstrate a certain extension of the Hutchinson theorem about the fixed point of the mapping determined by the so-called hyperbolic iterated functional system marked with symbol IFS.In proof of this theorem from 1981, Hutchinson applied Banach Contraction Principle.Banach's principle is a conclusion from Matkowski's theorem.It is worth adding that the basic tool enabling the construction of the so-called self-similar sets, important in fractal theory, is the Hutchinson theorem.

Fixed point theorems of the Matkowski type generalized contractions and their applications
Definition 1.A map f : (X, d) → (Y, g) of metric spaces that satisfies the inequality d (f (x), f (x )) ≤ Ld(x, x ) for some fixed constant L and all x, x ∈ X is called Lipschitzian; the smallest such L is called the Lipschitz constant λ of f .If λ < 1, the map f is called the contraction (with contraction constant λ).Definition 2. Let (X, d) be a metric space.A map f : (X, d) → (X, d) is called a Banach contraction, if there exists constant λ < 1 satisfying the inequality d (f (x), f (x )) ≤ λd(x, x ) for all x, x ∈ X. Definition 3. Let (X, d) be a metric space.For a given map φ : 0, ∞) → 0, ∞) satisfying the condition φ(t) < t for all t > 0, Definition 4. Let (X, d) be a metric space.A map f : X → X is called a Browder contraction, if f is φ-contraction for some function φ which is nondecreasing and right continuous.Definition 5. Let (X, d) be a metric space.We say, that f : X → X is a contraction of Matkowski, if f is φ-contraction for some function φ which is nondecreasing and lim n→∞ φ n (t) = 0 for any t > 0. Definition 6. Fibonacci sequence is a sequence defined recursively as follows: (sometimes formally accepted f 0 = 0 and then the recursive formula is valid for n 1).Definition 7. Fibonacci numbers are called consecutive terms of the sequence (f n ).
2, where F 1 and F 2 are given positive integers we call a Fibonacci type sequence.
For example, this sequence is the so-called Lucas sequence (l n ): 1, 3, 4, 7, 11, 18, 29, . . .These numbers can be described by a formula Definition 9. A generalized Fibonacci sequence is a sequence (G n ) defined recursively as follows: Below we present proof of Matkowski's fixed point theorem, which is one of the more general extensions of Banach Contraction Principle.In this proof we will use Cantor's intersection theorem.Before the theorem and its proof, let us note that the last two conditions of Matkowski's contraction imply the condition φ(t) < t for all t > 0 (see [1]).
Theorem 1 ([4, Theorem 3.2, 12 p.], [7]).Let (X, d) be a complete metric space.If f : X → X is the contraction of Matkowski, then f has a unique fixed point u, and We shall show that diam(D n ) → 0. For this purpose, observe first that and, by induction Consequently, using Cantor's Theorem we deduce that n 1 D n consists of a unique point u = f (u).Because from this equality we have u = f n (u) for every n ∈ N, so for any y ∈ X and for any n ∈ N, and hence f n (y) → u, when n → ∞.
Note that the above theorem can be proved in another way by considering, instead of Matkowski's contraction f , its second iteration f 2 = f • f , which is the Browder contraction (see [6]).Based on Browder's fixed point theorem (see [4,Theorem 6.10, p. 18]) f 2 has a unique fixed point u, so u is the only fixed point for f .Indeed, since u = f 2 (u), then from the equality f (u) = f 2 (f (u)) we get the fixed point f (u), so f (u) = u.It is easy to show that u is the only fixed point of f (by Theorem 1 has a useful local version: Corollary 1.Let (X, d) be a complete metric space and D = D(x 0 , r) be the set {x ∈ X : then f has a unique fixed point u, and f n (x) → u for each x ∈ D.
Proof.For any x ∈ D we have Therefore f : D → D. Since D is complete, the conclusion follows from Theorem 1.
Remark 1.Let φ(t) = λt, t ∈ 0, ∞) , λ < 1.On this assumption f : X → X in Theorem 1 is the Banach contraction, and Theorem 1 is the Banach Contraction Principle.The assumption ( * * ) in Corollary 1 takes the form Example 1 (Application of Theorem 1 to study the convergence of the quotient of neighboring terms of the Lucas sequence).Let us recall that Lucas numbers are: 1, 3, 4, 7, 11, 18, 29, 47, . . .They are terms of the sequence (l n ) starting with l 1 = 1, l 2 = 3, whose successive terms satisfy the relationship l n = l n−1 + l n−2 for n > 2. The mapping f : 4  3 , 3 → 4 3 , 3 , f (x) = 1 + 1 x is a contraction with the constant λ = 9 16 .Indeed for x, x ∈ 4 3 , 3 we have Therefore, based on Banach Contraction Principle the sequence 2 , which is the solution of the equation x = 1 + 1 x in 4 3 , 3 .Also for we have x n = f n (x 0 ) → ϕ.Therefore we finally have Example 2 (Application of Corollary 1 to study the convergence of the quotient of neighboring terms of the Fibonacci sequence).Let D = D(ϕ, 1  2 ), thus D = ϕ − 1 2 , ϕ + 1 2 .Note that the function f : D → R given by the formula f (x) = 1 + 1 x is a contraction with the constant λ = 4 5 , because from equality ϕ − 1 2 = √ 5 2 we have Hence we have Theorem 2 (compare [4], [5]).Let (X, d) be a complete metric space and let f : X → X. Suppose that there is a natural number N > 1 such that f N is the contraction of Matkowski.Then f has a unique fixed point u and the sequence of iterates f N (x) → u for each x ∈ X.
Proof.Based on Theorem 1 f N has a unique fixed point u = f N (u).
Proof of the second part of the thesis is analogous to the last part of the proof of Theorem 1 (comp.( * )).

Example 3 (Application of Theorem 2 to study the convergence of the quotients F n+1
F n of the Fibonacci type sequence (F n )).We will justify that successive quotients F n+1 F n of terms of the Fibonacci type sequence (F n ) approach the value of ϕ.We will assume that the initial terms F 1 and F 2 of this sequence, which are natural numbers, satisfy the inequality Therefore f 2 is the contraction with constant λ = 1 4 .We can now apply Theorem 2 assuming φ(t) = 1 4 t, t 0. By Theorem 2 f has in 1, 2 a unique fixed point u, and the sequence of iterates f n (y 0 ) → u for each Remark 2. It is worth adding that, using Banach Contraction Principle as a conclusion from the fixed point theorem of Matkowski, we can study the limit behavior of the quotients G n+1 G n of the corresponding terms of the generalized Fibonacci sequence (G n ) (see [3]).
Let K(X) be a family of non-empty and compact subset of the metric space (X, d).In the set K(X) we define the metric using the definition: an epsilon extension of the set A we call the set A ε = {x ∈ X; d(a, x) ≤ ε for some a ∈ A}.
A ε is also called the ε-envelope of the set A.
It can be shown that the function d H : K(X) × K(X) → 0, ∞) given by the formula is a metric.We call it the Hausdorff metric on the set K(X). (K(X), d H ) is a complete metric space, if (X, d) is a complete metric space.Let the mapping F : K(X) → K(X) be given by the formula , where f i : X → X, i = 1, . . ., k are functions.Theorem 3. If all functions f i : X → X, i ∈ {1, . . ., k} are Matkowski contractions for the same non-decreasing function φ : 0, ∞) → 0, ∞), then the mapping F : K(X) → K(X) is the Matkowski contraction with the function φ (also the same).
Proof.Since every function f i is the Matkowski contraction with φ, so for any p, q ∈ X and i = 1, . . ., k we have d (f i (p), f i (q)) ≤ φ (d(p, q)).Let A, B ∈ K(X) and let δ = d H (A, B).Then for every p ∈ A there exists such q ∈ B that d(p, q) ≤ δ.Therefore for each i we have d (f i (p), f i (q)) ≤ φ(δ).It follows that f i (A) is a set contained in the epsilon extension f i (B) for ε = φ(δ).
So we have We will now present one of the extensions of Huchinson's theorem on the fixed point of mapping F which concerned the Banach contraction system {f 1 , . . ., f k }.Theorem 4. If the space (X, d) is complete and the mapping F : K(X) → K(X) is defined by the formula where each function f i (i = 1, . . ., k) is the Matkowski contraction with the same non-decreasing function φ : 0, ∞) → 0, ∞), then there exists exactly one set A * ∈ K(X) such that ( * * * ) Moreover, for any K 0 ∈ K(X) the iteration sequence (F n (K 0 )) converges to A * relative to the Hausdorff metric.Sets A * ∈ K(X) satisfying the condition ( * * * ) are called self-similar (relative to f 1 , . . ., f k ) or fractals.
Proof.It is enough to recall that: (i) (K(X), d H ) is a complete space, (ii) F : K(X) → K(X) is the Matkowski contraction and refer to Theorem 1. Remark 3. If X is the Euclidean space (R n , d) and F is the Matkowski contraction with the function φ of the form φ(t) = λt, λ < 1, t 0, then we can obtain, among others, the Cantor set.Namely let S be the family of all closed nonempty subsets of the unit interval 0, 1 .Let f : S → S be a transformation that assigns to each set A ∈ S the set F (A) = 1  3 A ∪ 2 3 + 1 3 A .Let's put D 0 = 0, 1 .Finding successive iterations of the transformation F of set D 0 we get: