Abstract
We consider some Bernstein-type operators as well as their product and Boolean sum for a function defined on a triangle with all curved sides. Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of these operators.
1. Bernstein Type Operators
In this paper, using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of some operators introduced in [1]. Similar operators with the ones from [1] were studied in [2, 3] and [4], where the authors construct interpolation and Bernstein-type operators on triangles and squares with one and all curved sides. They studied the operators, their product and Boolean sum, as well as their interpolation properties, the order of accuracy, and the remainder of the corresponding approximation formulas.
We recall some results regarding Bernstein-type operators on a triangle with all curved sides from [1].
We denote by the triangle with all curved sides, which has the vertices , , , and the three curved sides , (along the coordinate axis), and (opposite to the vertex ); . We have that is defined by , with , , for ; is defined by , with , , for and is defined by the one-to-one functions and , where is the inverse of the function ; that is, and , with and (see Figure 1). In the sequel we denote by , for , .
Let be a real-valued function defined on and , respectively, and let be the points in which the parallel lines to the coordinate axes, passing through the point , intersecting the sides , , and . We consider the uniform partitions of the intervals and , : respectively, and the Bernstein-type operators and defined by with respectively, with
Remark 1. In Figures 2(a) and 2(b) we plot the points , and, respectively, , , for .
(a)
(b)
Theorem 2. If is a real-valued function defined on , then(i)on ,(ii)on ,(iii), , , , ; ,(iv), , .
Let , respectively, and let be the products of the operators and .
We have with , respectively, with .
Theorem 3. If is a real-valued function defined on , then(i), , on and(ii), , on .
We consider the Boolean sums of the operators and ; that is, respectively,
Theorem 4. If is a real-valued function defined on , then
2. Weakly Picard Operators
We recall some results regarding weakly Picard operators that will be used in the sequel (see, e.g., [5]).
Let be a metric space and an operator. We denote by , the fixed points set of ; , the family of the nonempty invariant subsets of ; .
Definition 5. The operator is a Picard operator if there exists such that (i); (ii)the sequence converges to for all .
Definition 6. The operator is a weakly Picard operator if the sequence converges, for all , and the limit (which may depend on ) are a fixed point of .
Definition 7. If is a weakly Picard operator, then we consider the operator , , defined by
Theorem 8. An operator is a weakly Picard operator if and only if there exists a partition of , such that (a);(b) is a Picard operator, .
3. Iterates of Bernstein Type Operators
Let be a real-valued function defined on ; .
Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of the Bernstein-type operators (3) and (5) and of their product and Boolean sum operators (7), (8), (9) and (10). The same approach for some other linear and positive operators leads to similar results in [6–12].
The limit behavior for the iterates of some classes of positive linear operators was also studied, for example, in [13–23]. In the papers [19–21] new methods were introduced (e.g., Korovkin type technique) for the study of the asymptotic behavior of the iterates of positive linear operators, positive linear operators preserving the affine functions and defined on the space of bounded real-valued functions on . This techniques enlarge the class of operators for which the limit of the iterates can be computed. In [13, 14] some methods were proposed to determine the degree of convergence for the iterates of certain positive linear operators towards the first Bernstein operator. Using the spectrum of the operators involved in [15], convergence results were proved for overiterates of certain (generalized) Bernstein-Stancu operators (see, e.g., [24–26]). In [16, 17] new techniques were introduced (infinite products, rates of convergence), based on the results from [18], in order to prove that infinite products of certain positive linear operators weakly converge to the first Bernstein operator.
Now we study the convergence of the iterates of the Bernstein-type operators (3) and (5).
Theorem 9. The operators and are weakly Picard operators and
Proof. Taking into account the interpolation properties of and (from Theorem 2), let us consider
and denote
with , .
We have the following properties:(i) and are closed subsets of ;(ii) is an invariant subset of and is an invariant subset of , for , and , ;(iii) and are partitions of ;(iv) and , where and denote the fixed points sets of and .
The statements (i) and (iii) are obvious.
(ii), by linearity of Bernstein operators and Theorem 2, it follows that and ; we have
So, and are invariant subsets of and, respectively, of , for and .
(iv), we prove that
are contractions for and .
Let . From (3) we have
where denotes the Chebyshev norm. So,
that is, is a contraction for .
Analogously we have
whence
that is, is a contraction for .
On the other hand, /, / are fixed points of and ; that is,
From the contraction principle, + is the unique fixed point of in and is a Picard operator, with
and, similarly, − is the unique fixed point of in and is a Picard operator, with
Consequently, taking into account (ii), by Theorem 8 it follows that the operators and are weakly Picard operators.
Now we study the convergence of the product and Boolean sum operators (7) and (9).
Theorem 10. The operator is a weakly Picard operator and
Proof. Let , and denote
with .
We remark that(i) is a closed subset of ;(ii) is an invariant subset of , for and ;(iii) is a partition of ;(iv), where denote the fixed points sets of .
The statements (i) and (iii) are obvious.
(ii), similarly with the proof of Theorem 9, by linearity of Bernstein operators and Theorem 3, it follows that is an invariant subset of , for and .
(iv), we prove that
is a contraction for and . Let . From [7, Lemma 8] it follows that
So,
that is, is a contraction for .
From the contraction principle we have that is the unique fixed point of in and is a Picard operator, so (26) holds. Consequently, taking into account , by Theorem 8 it follows that the operator is a weakly Picard operator.
Remark 11. We have a similar result for the operator .
Theorem 12. The operator is a weakly Picard operator and
Proof. The proof follows the same steps as in the previous theorems but using the following inequality: in order to prove that is a contraction.
Remark 13. We have a similar result for the operator .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.