Coupled fractals in complete metric spaces

The aim of this paper is to present fixed set theorems, collage type and anticollage type results for single-valued operators T : X × X → X in the framework of a complete metric space X. Based on the coupled fixed point theory, existence of fixed sets, collage type and anticollage type results for iterated function systems are also presented. The results are closely related to self-similar sets theory and the mathematics of fractals. Several examples of coupled fractals illustrate our results.


Introduction
The notion of coupled fixed point appeared for the first time in some papers of Amann [1] and Opoitsev [13], while a large development of the field started after the works of Guo and Lakshmikantham [6] and Bhaskar and Lakshmikantham [4]. If (X, d) is a metric space and T : X × X → X is an operator, then, by definition, a coupled fixed point for T is a pair (x, y) ∈ X × X satisfying There are many applications of the coupled fixed point theorems for solving different problems related to systems of integral and differential equations, see [2,4,6,9,22]. and y = (y 1 , . . . , y m ), then, by definition, we write x y if and only if x i y i for i ∈ {1, 2, . . . , m}. Throughout this paper, we will make an identification between row and column vectors in R m .
Let us recall first some important preliminary concepts and results. Let X be a nonempty set. A mapping d : X × X → R m + is called a vector-valued metric on X if all the axioms of the classical metric are satisfied with respect to the component-wise partial order. Moreover, a nonempty set X endowed with a vector-valued metric d is called a generalized metric space in the sense of Perov (in short, a generalized metric space), and it will be denoted by (X, d). We notice that (P cp (X), Hd) is a generalized metric space in the sense of Perov, i.e., Hd satisfies all the axioms of the vector-valued metric on P cp (X). We point out that (P cp (X), Hd) is complete if the vector-valued metricd is complete. We also mention that the generalized metric space in the sense of Perov is a particular case of the so-called cone metric spaces (or K-metric space), see [26].
A square matrix A of real positive numbers is said to be convergent to zero if and only if all the eigenvalues of A are in the open unit disc (see, for example, [16]).
A classical result in matrix analysis is the following theorem (see, for example, [16]).
The following assertions are equivalent: (iv) the matrix (I − A) is nonsingular, and (I − A) −1 has nonnegative elements.
Theorem 2 [Perov theorem]. Let (X, d) be a complete generalized metric space, and let f : X → X be a contraction with matrix A, i.e., A ∈ M mm (R + ) converges towards zero, and Then: (ii) the sequence of successive approximations (x n ) n∈N , x n := f n (x 0 ) is convergent in X to x * for all x 0 ∈ X; (iii) one has the following estimation: Remark 1. If, in the above theorem, we take m = 1, then we obtain the well-known Banach's contraction principle with A := a ∈ (0, 1).
The following theorem is our first main result. The proof is based on the application of Banach's contraction principle on X × X endowed with a scalar type metric.
Proof. We introduce on Z := X × X the functionald : Notice thatd is a complete metric on Z.
We consider now the operator F : Z → Z given by F (x, y) := (T (x, y), T (y, x)). It is easy to prove (see [24]) that F is a contraction in (Z,d) with constant k ∈ (0, 1), i.e., As a consequence, F is continuous from (Z,d) to (Z,d).
Let us consider on P cp (Z) the fractal operator U →F (U ) generated by F . Notice that, by the continuity of F , the fractal operator is well defined, i.e.,F : P cp (Z) → P cp (Z).

Remark 2.
Using the same approach, a similar result can be obtained working on X × X with the metricd((x, y), (u, v)) := max{d(x, u), d(y, v)}. For related coupled fixed point theorems in this framework, see [23].
In the next part of this section, we will illustrate the vector-valued metric approach in coupled fixed point theory. For the proof of our next result, we need the following lemma, which is itself a result with a good potential, see Remark 3.
Let f : X → X be a contraction with a matrix A convergent to zero of the following form: Then the fractal operatorf generated by f is a (k 1 + k 2 )-contraction on (P cp (X), Hd), Proof. Since f is a contraction with matrix A, we have, for all x, y ∈ X, that .
Since the matrix A converges to zero, we get that By the classical result of Nadler [12] we obtain that the fractal operatorf generated by f is a (k 1 + k 2 )-contraction on (P cp (X), Hd). It is an open question to establish when a contraction condition with a matrix A (convergent to zero, for example) on a single-valued operator f : X → X implies that the fractal operator f : P cp (X) → P cp (X) generated by f is a contraction with matrix A on (P cp (X), H d ).
The proof of our second result involves Lemma 1 and the vector-valued metric approach.
Then the following conclusions hold: and, for any (ii) the following estimation holds: where Hd is the Hausdorff-Pompeiu generalized metric induced byd and k := k 1 + k 2 ; (iii) (the collage theorem) (iv) (the anti-collage theorem) Proof. We introduce on Z := X × X the functionald : Notice thatd is a complete generalized metric (in the sense of Perov) on Z.
We will prove now that the operator F : Z → Z given by F (x, y) := (T (x, y), T (y, x)) is a contraction in (Z,d) with a convergent to zero matrix i.e.,d(F (z), F (w)) Ad(z, w) for all z, w ∈ Z.
https://www.mii.vu.lt/NA Indeed, by the contraction condition on T we also get that On the other hand, an easy calculation shows that the eigenvalues of the matrix A are in the unit open disc. Hence, F is a contraction on (Z,d) with matrix A.
Let us now consider the fractal operatorF : P cp (Z) → P cp (Z), U →F (U ) generated by F . By the continuity of F (with respect tod) the fractal operator is well defined. Moreover, since F is a contraction with matrix A on (Z,d), we get (see Lemma 1) that F is a (k 1 + k 2 )-contraction on (P cp (Z), Hd). Hence, by Banach's contraction principlê F has a unique fixed point in P cp (Z), i.e., there exists U * ∈ P cp (Z) such that U * =F (U * ). Additionally, by the same theorem we also have that where U 0 ∈ P cp (X) is arbitrary, and U 1 :=F (U 0 ). Notice that conclusions (iii) and (iv) follow in a similar manner as above.
Remark 4. It is worth to mention that Theorem 4 also follows directly by Theorem 3. Indeed, by the hypothesis we have, for all (x, y), (u, v) ∈ X × X, that We will discuss now the case of coupled self-similar sets. We need first another auxiliary result.
Lemma 2. Let (X, d) be a metric space, and T : X × X → X be an operator. Assume that there exists k > 0 such that Then the following conclusions hold: for all A, B, U, V ∈ P cp (X).
Proof. (i), (iii), and (iv) follow immediately. Let us show (ii). For this purpose, it is enough to prove that, for all c ∈ T (A, B), there exists w ∈ T (U, V ) such that and that, for all s ∈ T (U, V ), there exists f ∈ T (A, B) such that Let c ∈ T (A, B). Then there exists a ∈ A and b ∈ B such that c = T (a, b). For a ∈ A, there exists u ∈ U such that d(a, u) H d (A, U ). In a similar way, for proving the first relation from above. In a similar way, we can prove the second relation, and the conclusion follows.
Our third main result is the following coupled self-similar set theorem.
Theorem 5. Let (X, d) be a complete metric space, and T : X × X → X be an operator. Assume that there exists k ∈ (0, 1) such that Then the following conclusions hold: (i) there exists a unique pair of coupled self-similar sets (A * , B * ) ∈ P cp (X)×P cp (X), and, for any starting point (A 0 , B 0 ) ∈ P cp (X)×P cp (X), the sequences (A n ) n∈N , (B n ) n∈N defined, for n ∈ N, by converge (with respect to H d ) to A * and respectively to B * as n → ∞; (ii) for each n ∈ N, the following estimation holds: (iii) (the collage theorem) for all A, B ∈ P cp (X), we have (iv) (the anti-collage theorem) for all A, B ∈ P cp (X), we have T (B, A) .
Since T is continuous,T is well defined, i.e.,T (A, B) ∈ P cp (X) for all A, B ∈ P cp (X). By (iv) in Lemma 2 we get thatT satisfies all the assumptions of Theorem 3.7 in [24]. Hence, there exists a unique pair (A * , B * ) ∈ P cp (X)×P cp (X) such that A * =T (A * , B * ) and B * =T (B * , A * ). These relations show that (A * , B * ) defines a pair of coupled selfsimilar sets for T . The second conclusion also follows by [24,Thm. 3.7]. Conclusion (iii) follows by (ii), while (iv) is a consequence of the following estimations: A similar approach can be considered by working with the following contraction condition on the operator T : where k ∈ (0, 1). For example, we have the following results.
Lemma 3. Let (X, d) be a metric space, and let T : X ×X → X be an operator. Assume that there exists k > 0 such that Then the following conclusions hold: (iv) for all A, B, U, V ∈ P cp (X), we have Proof. We will prove conclusion (ii). For this purpose, it is enough to prove that, for all c ∈ T (A, B), there exists w ∈ T (U, V ) such that and that, for all s ∈ T (U, V ), there exists f ∈ T (A, B) such that Let c ∈ T (A, B). Then there exists a ∈ A and b ∈ B such that c = T (a, b). For a ∈ A, there exists u ∈ U such that d(a, u) H d (A, U ). In a similar way, proving the first relation from above. In a similar way, we can prove the above second relation and the conclusion follows.
For our next theorems, we need the following result, which was essentially proved in [23], see Theorem 5.
Theorem 6. Let (X, d) be a complete metric space. Let T : X × X → X be an operator. Assume that there exists k ∈ (0, 1) such that, for all (x, y), (u, v) ∈ X × X, we have Then the following conclusions hold: (i) there exists a unique solution (x * , y * ) ∈ X × X of the coupled fixed point problem (1), and, for any initial point (x 0 , y 0 ) ∈ X × X, the sequences (x n ) n∈N , (y n ) n∈N defined, for n ∈ N, by converge to x * and respectively to y * as n → ∞; (ii) for all n ∈ N * , the following estimation holds: Based on Lemma 3 and Theorem 6, we can prove the following coupled self-similar set theorem.
Theorem 7. Let (X, d) be a complete metric space and T : X × X → X be an operator. Assume that there exists k ∈ (0, 1) such that Then the following conclusions hold: https://www.mii.vu.lt/NA (i) there exists a unique coupled self-similar pair (A * , B * ) ∈ P cp (X) × P cp (X), and, for any starting point (A 0 , B 0 ) ∈ P cp (X)×P cp (X), the sequences (A n ) n∈N , (B n ) n∈N defined, for n ∈ N, by converge (with respect to H d ) to A * and respectively to B * as n → ∞; (ii) the following estimation holds: Proof. As in the proof of Theorem 5, the operatorT : P cp (X) × P cp (X) → P cp (X)

Iterated function systems and coupled self-similar pairs
The purpose of this section is to discuss coupled fixed point properties for iterated function system of operators f i : X × X → X, where i ∈ {1, 2, . . . , m}.
Let (X, d) be a metric space, and f i : X × X → X (where i ∈ {1, 2, . . . , m}) be continuous operators. We denote by F : P cp (X) × P cp (X) → P cp (X) given by . . , m}. We will call F the fractal operator generated by the iterated function system (f 1 , . . . , f m ).
By the continuity of f i the operator F is well defined. Then we have the following result.
Theorem 8. Let (X, d) be a complete metric space, and f i : X × X → X be operators such that, for each i ∈ {1, 2, . . . , m}, there exists k i ∈ (0, 1) satisfying Then the following conclusions hold: (i) the fractal operator F generated by the iterated function system (f 1 , . . . , f m ) satisfies the following condition: (ii) there exists a unique pair (A * , B * ) ∈ P cp (X) × P cp (X) such that (iii) for any pair (A 0 , B 0 ) ∈ P cp (X) × P cp (X), the sequence ((A n , B n )) n∈N ⊂ P cp (X) × P cp (X) given by converges to (A * , B * ) ∈ P cp (X) × P cp (X) as n → ∞.
Proof. Notice first that, by the contraction condition, each operator f i is continuous for each i ∈ {1, 2, . . . , m}.