SELF-SIMILAR SETS IN CONVEX METRIC SPACES

. The purpose of this paper is to present some existence and uniqueness results for self-similar sets in convex complete metric spaces. MSC 2000. 47H10, 58F13.


INTRODUCTION
Let (X, d) be a complete metric space and f i , i ∈ {1, . . ., m} be singlevalued mappings of X into itself.Let P cp (X) be the space of all nonempty compact subsets of X and denote by H the Hausdorff-Pompeiu metric on P cp (X).If we define the operator T : (P cp (X), H) → (P cp (X), H) by the formula T (Y ) = m i=1 f i (Y ), then T is called the fractal operator generated by the so-called iterated functions system f = {f 1 , f 2 , ..., f m }.Any fixed point of T is, by definition, a self-similar set for the iterated functions system f = {f 1 , f 2 , ..., f m }.It is known that, in many cases, the Hausdorff dimension of a self-similar is not an integer.For this reason, in all these cases selfsimilar sets are fractals and P cp (X) is the space of fractals.Moreover, selfsimilar sets among the fractals form an important class, since many of them have computable Hausdorff dimensions.Regarding the existence of self-similar sets, if f i are α i -contractions for i ∈ {1, . . ., m} then the operator T is a max(α i | i ∈ {1, . . ., m})-contraction having a unique fixed point.Hence the iterated functions system f = {f 1 , f 2 , ..., f m } has a unique self-similar set (this is a result of Hutchinson and Barnsley, see for example Petruşel [9], Yamaguti, Hata, Kigami [13]).Same conclusion holds, one hand when f i , i ∈ {1, . . ., m} are ϕ-contractions (see I.A.Rus [11]) and on the other hand when f i , i ∈ {1, . . ., m} are Meir-Keeler type operators (see Petruşel [7], [10]).
The purpose of this paper is to study the existence and uniqueness of selfsimilar sets for iterated functions systems on convex complete metric spaces.The multi-valued case is also considered.

MAIN RESULTS
Let (X, d) be a metric space and f : X → X be a single-valued operator.We will consider first some contraction-type conditions for the operator f .Definition 1.The operator f : X → X is said to be: Let us observe that, the condition (i) implies (ii), (i) implies (iii), (iii) implies each of the conditions (ii) (iv), (v) and (vi).Jachymski (see [2]) proved that the reverse implications, i.e. (iv) implies (iii) and (iv) implies (ii), are, in general, not true.Definition 2. A metric space is said to be metrically convex if for every distinct points x, y ∈ X there exists z ∈ X such that d(x, y) = d(x, z) + d(z, y) and x = z = y.
For example, every normalized space and any of its convex subsets is metrically convex.An important property of such spaces is: d) is a complete and metrically convex metric space, then for every distinct points x, y ∈ X and for each λ ∈]0, In what follows we use the terminology "convex metric space" for a metrically convex metric space.
The following result is proved by Petruşel [7].
Theorem 4. Let (X, d) be a complete metric space and f i : X → X, i ∈ {1, 2, . . ., m} be a finite family of Meir-Keeler type operators.Then: a) the fractal operator T : (P cp (X), H) → (P cp (X), H) generated by the iterated functions system f = {f 1 , f 2 , ..., f m } is a Meir-Keller type operator.
b) the iterated function system f = {f 1 , f 2 , ..., f m } has a unique selfsimilar set A * , having the property that for each compact subset A 0 of X the sequence of successive approximations (T n (A 0 )) n∈N converges to A * .
The following theorem gives us the equivalence of some generalized contractions conditions (see Matkowski and Wegrzyk [4]): Theorem 5. Let (X, d) be a convex complete metric space and (Y, ρ) be a metric space.Let f : X → Y be an arbitrary function.Then  Using the above result, we obtain the first main result of this paper: Theorem 6.Let (X, d) be a convex complete metric space and f i : X → X, i ∈ {1, 2, . . ., m} be a finite family of Matkowski-Wegrzyk type operators.Then the iterated functions system f = {f 1 , f 2 , ..., f m } has a unique self-similar set A * .Moreover, for each compact subset A 0 of X the sequence of successive approximations (T n (A 0 )) n∈N converges to A * .Proof.From Theorem 5 we have that f = {f 1 , f 2 , ..., f m } is an iterated function system having the property that each function f i satisfies to a contraction type condition.From the classical result of Hutchinson and Barnsley we obtain that the fractal operator T is a contraction too from P cp (X) to itself.Hence, by Banach contraction principle we get the desired conclusion.The proof is complete.
Let us remark that the above theorem can be proved, via Theorem 5, using Theorem 4 instead of Banach contraction principle.
Remark.The similarity dimension d of a self-similar set A * corresponding to an iterated functions system f = {f 1 , f 2 , ..., f m }, where f i is an α icontraction, for each i ∈ {1, 2, ..., m}, is defined as the unique positive root of the equation m i=1 α d i = 1.It is easy to see now that the similarity dimension of a self-similar set generated by a finite family of Matkowski-Wegrzyk type operators on a convex complete metric space can be calculated in the same way.
Let us consider now the multi-valued case.Let F 1 , . . ., F m : X → P cp (X) be a finite family of upper semi-continuous (briefly u.s.c.) multi-valued operators.We define the multi-fractal operator T F generated by the following iterated multi-functions system F = (F 1 , F 2 , . . ., F m ), by the following relation: In this framework, a nonempty compact subset A * of X is said to be a self-similar set for the iterated multifunctions system F = (F 1 , F 2 , . . ., F m ) if and only if it is a fixed point for the associated multi-fractal operator.The following notions are needed in the sequel.
Definition 7. The multi-valued operator F : X → P cp (X) is said to be: A similar discussion with the single-valued case can be done also for the multi-valued setting.
Regarding the existence and uniqueness of self-similar sets for iterated multifunctions systems, it is well-known that a finite family of multi-valued contractions has an unique self-similar set.Moreover, for the case of multi-valued Meir-Keeler operators the following result holds (see Petruşel [7]): Theorem 8. Let (X, d) be a complete metric space and F i : X → P cp (X), i ∈ {1, . . ., m} be a finite family of multi-valued Meir-Keeler type operators.Then: a) the multi-fractal operator T F : P cp (X) → P cp (X) is a Meir-Keeler type operator.b) the iterated multi-functions system F = (F 1 , F 2 , . . ., F m ) has a unique self-similar set A * .Moreover, for each compact subset A 0 of X the sequence of successive approximations (T n F (A 0 )) n∈N converges to A * .The equivalence between the following generalized contractions conditions is proved in Mot ¸[5]: Theorem 9. Let (X, d) be a convex complete metric space and (Y, ρ) be a metric space.Let F : X → P cp (Y ) be multi-function.Then the following assertions are equivalent: (i) F is a multi-valued Meir-Keeler type operator; (ii) F is a multi-valued Rakotch type operator; (iii) F is a multi-valued Boyd-Wong type operator; (iv) F is a multi-valued Matkowski-Wegrzyk type operator.
From Theorem 8 and Theorem 9 we obtain: Theorem 10.Let (X, d) be a convex complete metric space and F i : X → P cp (X), i ∈ {1, 2, . . ., m} be a finite family of multi-valued Matkowski-Wegrzyk type operators.Then the iterated multi-functions system F = {F 1 , F 2 , . . ., F m } has a unique self-similar set A * .Moreover, for each compact subset A 0 of X the sequence of successive approximations (T n (A 0 )) n∈N converges to A * .

Matkowski-Wegrzyk type operator if
Moreover, if f fulfils one of the above conditions, then the function k in (iii) is strictly increasing, concave and continuously differentiable in [0, ∞[ and the function k in (ii) is continuous.
(iii) f is a Boyd-Wong type operator; (iv) f is a Matkowski-Wegrzyk type operator.