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Iterates of q-Bernstein operators on triangular domain with all curved sides

  • Mohammad Iliyas , Asif Khan , Mohd Arif , Mohammad Mursaleen EMAIL logo and Mudassir Rashid Lone
Published/Copyright: November 30, 2022
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

In this article, Phillips-type Bernstein operators ( ℬ m , q t F ) ( t , s ) and ( ℬ n , q s F ) ( t , s ) , their products, and Boolean sum based on q-integer have been studied on a triangle with all curved sides. Furthermore, convergence of iterates of these operators have been analyzed using the weakly Picard operators technique and the contraction principle.

Keywords: Quantum calculus; Phillips q-Bernstein operators; product operators; Boolean sum operators; weakly Picard operators; contraction principle
MSC 2010: 41A05; 41A35; 41A36

1 Introduction and essential preliminaries

In 1912, S.N. Bernstein constructed sequence of polynomials (which are now famous as Bernstein polynomials) to approximate continuous function on compact interval to give a constructive proof of Weierstrass approximation theorem [1,2] by using probabilistic interpolation approach.

Further, with evolution of q-calculus (quantum calculus), Lupaş constructed a q-analogue of Bernstein operators (rational) [3]. In 1997, Phillips [4] produced another generalization of Bernstein polynomials using q-integers called as q-Bernstein polynomials. One can see a survey of the obtained results and references on the subject in [5]. The approximating operators on triangle and their basis are important from applications point of view in finite element analysis [6] and computer-aided geometric design [7,8,9].

Recently, Khan et al. constructed q-analogue of Phillips-type Bernstein operators on triangles in [10] and Lupaş-type Bernstein operators on triangles in [11]. Motivated by the work of Rus [12], on the basis of the contraction principle and the weakly Picard operators technique, we study the convergence of iterates of q-analogue of Phillips-type Bernstein operators. For relevant papers on the classical operators and their approximation properties on triangles, one can see [7,13,14, 15,16]. For other papers on convergence of the iterates, one can refer [12,17, 18,19]. For details about operators of q-Bernstein-type and its approximation properties, one can see [20,21, 22,23,24, 25,26].

In this article, we would like to attract to the Phillips q-analogue of Bernstein operators and obtain new results on triangles with all curved sides. We construct new operators, Phillips-type q-Bernstein operators, which interpolate the given function on the edges of a triangle with all curved sides. Also, we study and discuss product and Boolean sum operators for a function defined on a triangle with all curved sides. By using the contraction principle and the Weakly Picard operators technique, we study the convergence of iterates of these operators and their fixed points. For other relevant works, one can see [27,28, 29,30,31, 32,33].

To represent results by Phillips, we recall the following definitions. Let q > 0 . For any m = 0 , 1 , 2 , … , the q-integer [ m ] q is defined by

[ m ] q ≔ 1 + q + ⋯ + q m − 1 m = 1 , 2 , … , [ 0 ] q ≔ 0 ,

and the q-factorial [ m ] q ! is defined by

[ m ] q ! ≔ [ 1 ] q [ 2 ] q ⋯ [ m ] q m = 1 , 2 , … , [ 0 ] q ! = 1 .

For more details about q-calculus, like q-binomial or the Gaussian coefficient, Pascal’s identity etc., one can refer [10,34].

In the next section, we construct and study quantum analogue of operators studied in [10] on triangle with all curved sides.

2 Construction of new univariate operators using quantum calculus on triangle with all curved sides

Let F be a real-valued function defined on the triangular domain with all curved sides T h (see Figure 1 in [35]). Let ( g 2 ( s ) , s ) , ( g 3 ( s ) , s ) , and ( t , f 1 ( t ) ) , ( t , f 3 ( t ) ) be the points in which the parallel lines to the coordinates axes passing through the point ( t , s ) ∈ T h intersecting the edges Γ 1 , Γ 2 , and Γ 3 . We take the uniform partitions of the intervals [ g 2 ( s ) , g 3 ( s ) ] and [ f 1 ( t ) , f 3 ( t ) ] , t , s ∈ [ 0 , h ] .

Let

(2.1) △ m s = g 2 ( s ) + [ i ] q g 3 ( s ) − g 2 ( s ) [ m ] q ∣ i = 0 , m ¯

and

(2.2) △ n t = f 1 ( t ) + [ j ] q f 3 ( t ) − f 1 ( t ) [ n ] q ∣ j = 0 , n ¯ .

Now, we construct the new Phillips-type Bernstein operators ℬ m , q t and ℬ n , q s on triangle with all curved sides based on quantum integer as follows:

(2.3) ( ℬ m , q t F ) ( t , s ) = ∑ i = 0 m p ˜ m , i ( t , s ) F g 2 ( s ) + [ i ] q g 3 ( s ) − g 2 ( s ) [ m ] q , s , ( t , s ) ∈ T h ∣ ( 0 , h )

and

(2.4) ( ℬ n , q s F ) ( t , s ) = ∑ j = 0 n q ˜ n , j ( t , s ) f t , f 1 ( t ) + [ j ] q f 3 ( t ) − f 1 ( t ) [ n ] q , ( t , s ) ∈ T h ∣ ( h , 0 ) ,

where

(2.5) p ˜ m , i ( t , s ) = m i q [ t − g 2 ( s ) ] i ∏ r = 0 m − i − 1 g 3 ( s ) − g 2 ( s ) − q r ( t − g 2 ( s ) ) [ g 3 ( s ) − g 2 ( s ) ] m , ( t , s ) ∈ T h ∣ ( 0 , h )

and

(2.6) q ˜ n , j ( t , s ) = n j q [ s − f 1 ( t ) ] j ∏ r = 0 n − j − 1 f 3 ( t ) − f 1 ( t ) − q r ( s − f 1 ( t ) ) [ f 3 ( t ) − f 1 ( t ) ] n , ( t , s ) ∈ T h ∣ ( h , 0 ) .

Note that at vertices ( h , 0 ) and ( 0 , h ) , we assume that ( ℬ n , q s F ) ( h , 0 ) = F ( h , 0 ) and ( ℬ m , q t F ) ( 0 , h ) = F ( 0 , h ) .

Theorem 2.1

For a real-valued function F defined on T h , we have

( i ) ℬ m , q t F = F o n Γ 2 ∪ Γ 3 ;

( i i ) ℬ n , q s F = F o n Γ 1 ∪ Γ 3 ;

(2.7) ( i i i ) ( ℬ m , q t e i 0 ) ( t , s ) = t i , i = 0 , 1 ,

(2.8) ( ℬ m , q t e i j ) ( t , s ) = s j ( ℬ m , q t e i j ) ( t , s ) , i = 0 , 1 , 2 ; j ∈ N ,

(2.9) ℬ m , q t e 20 = t 2 + [ t − g 2 ( s ) ] [ g 3 ( s ) − t ] [ m ] q ,

(2.10) ( i v ) ( ℬ n , q s e 0 j ) ( t , s ) = s j , j = 0 , 1 ,

(2.11) ℬ m , q s e 02 = s 2 + [ s − f 1 ( t ) ] [ f 3 ( t ) − s ] [ n ] q ,

(2.12) ( ℬ m , q s e i j ) ( t , s ) = t i ( ℬ n , q s e 0 j ) ( t , s ) , j = 0 , 1 , 2 ; i ∈ N .

Let P m n , q = ℬ m , q t ℬ n , q s , and let Q n m , q = ℬ n , q s ℬ m , q t be the products of the operators ℬ m , q t and ℬ n , q s . We have

(2.13) ( P m n , q F ) ( s , t ) = ∑ i = 0 m ∑ j = 0 n p ˜ m , i ( t , s ) q ˜ n , j ( t i , s ) F t i , f 1 ( t i ) + [ j ] q f 3 ( t i ) − f 1 ( t i ) [ n ] q

with t i = g 2 ( s ) + [ i ] q g 3 ( s ) − g 2 ( s ) [ m ] q , respectively,

(2.14) ( Q n m , q F ) ( t , s ) = ∑ i = 0 m ∑ j = 0 n p ˜ m , i ( t , s j ) q ˜ n , j ( t , s ) F g 2 ( s j ) + [ i ] q g 3 ( s j ) − g 2 ( s j ) [ m ] q , s j

with s j = f 1 ( t ) + [ j ] q f 3 ( t ) − f 1 ( t ) [ n ] q .

Theorem 2.2

If F is real-valued function defined on T h , then

  1. ( P m n , q F ) ( V 3 ) = F ( V 3 ) , ( P m n , q F ) = F on Γ 3 and

  2. ( Q n m , q F ) ( V 3 ) = F ( V 3 ) , ( Q n m , q F ) = F on Γ 3 .

We define the Boolean sums of operators ℬ m , q t and ℬ n , q s as follows:

S m n , q ≔ ℬ m , q t ⊕ ℬ n , q s = ℬ m , q t + ℬ n , q s − ℬ m , q t ℬ n , q s ,

respectively,

T n m , q ≔ ℬ n , q s ⊕ ℬ m , q t = ℬ n , q s + ℬ m , q t − ℬ n , q s ℬ m , q t .

Theorem 2.3

For a real-valued function F defined on T h , we have

S m n , q F ∣ ∂ T h = F ∣ ∂ T h ,

T n m , q F ∣ ∂ T h = F ∣ ∂ T h .

3 Weakly Picard operators

Now we recall definition of weakly Picard operators [36].

Let ( V , d ) be metric space and B : V → V be an operator. We denote by F B ≔ { t ∈ V ∣ B ( t ) = t } for the set of fixed points of B , and I ( B ) ≔ { Z ⊂ V ∣ B ( Z ) ⊂ Z , Z ≠ Φ } for the family of nonempty invariant subsets of B . Let B 0 ≔ I V , B 1 ≔ B , … , B n + 1 ≔ B o B n , n ∈ N .

Definition 3.1

The operator B : V → V is a Picard operator if there exists v ∗ ∈ V such that

  1. F B = ( v ∗ ) ;

  2. the sequence ( B n ( v 0 ) ) n ∈ N converges to v ∗ for all v 0 ∈ V .

Definition 3.2

If B is a Weakly Picard operator, then we consider the operator B ∞ , B ∞ : V → V , defined by

B ∞ ( v ) ≔ lim n → ∞ B n ( v ) .

Definition 3.3

An operator B is a Weakly Picard operator if and only if there exists a partition of V , V = ∪ μ ∈ ∧ V μ , such that

  1. V μ ∈ I ( B ) , for all μ ∈ ∧ .

  2. B ∣ V μ : V μ → V μ is a Picard operator for all μ ∈ ∧ .

4 Iterates of Bernstein-type operators

Let us consider a real-valued function F defined on T h ; h ∈ R + . By using the contraction principle and weakly Picard operators technique, we examine the convergence of iterates of the Bernstein-type operators of their product and Boolean sum operators. The limits behavior for the iterates of some classes of positive linear operators are examined in [12,17,18, 19,35,37, 38,39]. In [17,38,39], new methods were (e.g., Korovkin of the iterates) studied for the asymptotic behavior of the iterates of positive linear operators retaining the affine functions and defined on the space of bounded real-valued functions on [ 0 , 1 ] . This technique extends the class of operators for which the limit of the iterates may be computed.

Now we investigate the convergence of the iterates of Bernstein-type operators and the convergence behavior of the iterates of Phillips-type q-Bernstein operators (2.3) and (2.4).

Theorem 4.1

The operators ℬ m , q t and ℬ n , q s are weakly Picard operators and

( ℬ m , q t , ∞ ) ( t , s ) = F ( g 3 ( s ) , s ) − F ( g 2 ( s ) , s ) g 3 ( s ) − g 2 ( s ) t + g 3 ( s ) F ( g 2 ( s ) , s ) − g 2 ( s ) F ( g 3 ( s ) , s ) g 3 ( s ) − g 2 ( s ) ( ℬ n , q s , ∞ ) ( t , s ) = F ( t , f 3 ( t ) ) − F ( t , f 1 ( t ) ) f 3 ( t ) − f 1 ( t ) s + f 3 ( t ) F ( t , f 1 ( t ) ) − f 1 ( t ) F ( t , f 3 ( t ) ) f 3 ( t ) − f 1 ( t ) .

Proof

We use interpolation properties of ℬ m , q t and ℬ n , q s (from Theorem 2.1). Let us consider

X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) = { F ∈ C ( T h ) ∣ F ( g 2 ( s ) , s ) = τ ∣ Γ 2 , F ( g 3 ( s ) , s ) = τ ∣ Γ 3 } , for s ∈ [ 0 , h ] , X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) = { F ∈ C ( T h ) ∣ F ( t , f 1 ( t ) ) = χ ∣ Γ 1 , F ( t , f 3 ( t ) ) = χ ∣ Γ 3 } , for t ∈ [ 0 , h ]

and denote

F τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) ( t , s ) ≔ τ ∣ Γ 3 − τ ∣ Γ 2 g 3 ( s ) − g 2 ( s ) t + g 3 ( s ) τ ∣ Γ 2 − g 2 ( s ) τ ∣ Γ 3 g 3 ( s ) − g 2 ( s ) F χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) ( t , s ) ≔ χ ∣ Γ 3 − χ ∣ Γ 1 f 3 ( t ) − f 1 ( t ) t + f 3 ( t ) χ ∣ Γ 1 − f 1 ( t ) χ ∣ Γ 3 f 3 ( t ) − f 1 ( t )

with τ , χ ∈ C ( T h ) .

One can easily notice the following:

  1. X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) and X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) are the closed subsets of C ( T h ) ;

  2. X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) and X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) are invariant subsets of ℬ m , q t and ℬ n , q s , respectively, for τ , χ ∈ C ( T h ) and n , m ∈ N ∗ ;

  3. C ( T h ) = ∪ τ ∈ C ( T h ) X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) and C ( T h ) = ∪ χ ∈ C ( T h ) X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) are partitions of C ( T h ) ;

  4. F τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) ∈ X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) ∩ F ℬ m , q t and F χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) ∈ X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) ∩ F ℬ n , q s , where F ℬ m , q t and F ℬ n , q s are the fixed points sets of ℬ m , q t and ℬ n , q s .

One can easily prove the statements (i) and (iii).

(ii) Using linearity of operator 2.3 and Theorem 2.1, it follows that ∀ F τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) ∈ X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) and ∀ F χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) ∈ X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) ;

we have

ℬ m , q t F τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) ≔ F τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) , ℬ n , q s F χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) ( t , s ) ≔ F χ ∣ Γ 1 , χ ∣ Γ 3 ( 1 ) ( t , s ) .

So, X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) and X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) are invariant subsets of ℬ m , q t and ℬ n , q s , respectively, for τ , χ ∈ C ( T h ) and n , m ∈ N ∗ .

Now we prove that ℬ m , q t ∣ τ ∣ Γ 2 , τ ∣ Γ 3 : X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) → X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) ( t , s ) , ℬ n , q S ∣ χ ∣ Γ 1 , χ ∣ Γ 3 : X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) → X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) ( t , s ) , are contraction maps for τ , χ ∈ C ( T h ) and n , m ∈ N ∗ . Let F , G ∈ X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) . From (3), we have, for instant fixing s ,

u m = min g 2 ( s ) ≤ t ≤ g 3 ( s ) { p ˜ m , 0 ( t , s ) + p ˜ m , m ( t , s ) } , u m = min g 2 ( s ) ≤ t ≤ g 3 ( s ) g 3 ( s ) − t g 3 ( s ) − g 2 ( s ) m + t − g 2 ( s ) g 3 ( s ) − g 2 ( s ) m .

Observe that 0 < u m < 1 .

∣ ℬ m , q t ( F ) ( t , s ) − ℬ m , q t ( G ) ( t , s ) ∣ = ∣ ℬ m , q t ( F − G ) ( t , s ) ∣ ≤ ∣ 1 − u m ∣ ‖ F − G ‖ ∞ . ∣ ∣ ℬ m , q t ( F − G ) ( t , s ) ∣ ∣ ≤ ∣ 1 − u m ∣ ‖ F − G ‖ ∞ for all F , G ∈ X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) .

Hence, ℬ m , q t ∣ X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) is a contraction for τ ∈ C ( T h ) .

Similarly, we have 0 < v n < 1 . For instance, for a fix t ,

v n = min f 1 ( t ) ≤ y ≤ f 3 ( t ) { ˜ q n , 0 ( t , s ) + q ˜ n , n ( t , s ) } , v n = min f 1 ( t ) ≤ s ≤ f 3 ( t ) f 3 ( t ) − s f 3 ( t ) − f 1 ( t ) n + s − f 1 ( t ) f 3 ( t ) − f 1 ( t ) n ,

∣ ℬ n , q s ( F ) ( t , s ) − ℬ m , q s ( G ) ( t , s ) ∣ = ∣ ℬ n , q s ( F − G ) ( t , s ) ∣ ≤ ∣ 1 − v n ∣ ‖ F − G ‖ ∞ ,

∣ ∣ ℬ n , q s ( F ) ( t , s ) − ℬ n , q s ( G ) ( t , s ) ∣ ∣ ∞ ≤ ∣ 1 − v n ∣ ‖ F − G ‖ ∞ , for all F , G ∈ X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) .

Since 0 < v n < 1 , ℬ n , q s ∣ X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) is a contraction for χ ∈ C ( T h ) .

On the other hand, ( τ ∣ Γ 3 − τ ∣ Γ 2 ) ( g 3 ( s ) − g 2 ( s ) ) ( ⋅ ) + ( g 3 ( s ) τ ∣ Γ 2 − g 2 ( s ) τ ∣ Γ 3 ) ( g 3 ( s ) − g 2 ( s ) ) ∈ X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) , and ( χ ∣ Γ 3 − χ ∣ Γ 1 ) ( f 3 ( t ) − f 1 ( t ) ) ( ⋅ ) + ( f 3 ( t ) χ ∣ Γ 1 − f 1 ( t ) χ ∣ Γ 3 ) ( f 3 ( t ) − f 1 ( t ) ) ∈ X χ ∣ Γ 1 , χ ∣ Γ 3 ( 2 ) are the fixed points of maps ℬ m , q t and ℬ n , q s , that is,

ℬ m , q t τ ∣ Γ 3 − τ ∣ Γ 2 g 3 ( s ) − g 2 ( s ) ( ⋅ ) + g 3 ( s ) τ ∣ Γ 2 − g 2 ( s ) τ ∣ Γ 3 g 3 ( s ) − g 2 ( s ) = τ ∣ Γ 3 − τ ∣ Γ 2 g 3 ( s ) − g 2 ( s ) ( ⋅ ) + g 3 ( s ) τ ∣ Γ 2 − g 2 ( s ) τ ∣ Γ 3 g 3 ( s ) − g 2 ( s ) ℬ n , q s χ ∣ Γ 3 − χ ∣ Γ 1 f 3 ( t ) − f 1 ( t ) ( ⋅ ) + f 3 ( t ) χ ∣ Γ 1 − f 1 ( t ) χ ∣ Γ 3 f 3 ( t ) − f 1 ( t ) = χ ∣ Γ 3 − χ ∣ Γ 1 f 3 ( t ) − f 1 ( t ) ( ⋅ ) + f 3 ( t ) χ ∣ Γ 1 − f 1 ( t ) χ ∣ Γ 3 f 3 ( t ) − f 1 ( t ) .

Using contraction principle, F τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) ( t , s ) ≔ ( τ ∣ Γ 3 − τ ∣ Γ 2 ) ( g 3 ( s ) − g 2 ( s ) ) ( t ) + ( g 3 ( s ) τ ∣ Γ 2 − g 2 ( s ) τ ∣ Γ 3 ) ( g 3 ( s ) − g 2 ( s ) ) is the only fixed point of ℬ m , q t in X τ ∣ Γ 2 , τ ∣ Γ 3 ( 1 ) and ℬ m , q t ∣ X τ ∣ Γ 2 ( 1 ) , τ ∣ Γ 3 is a Picard operator, with

( ℬ m , q t , ∞ ) ( t , s ) = F ( g 3 ( s ) , s ) − F ( g 2 ( s ) , s ) g 3 ( s ) − g 2 ( s ) t + g 3 ( s ) F ( g 2 ( s ) , s ) − g 2 ( s ) F ( g 3 ( s ) , s ) g 3 ( s ) − g 2 ( s ) ,

and similarly, F χ ∣ Γ 2 , χ ∣ Γ 3 ( 2 ) ( s , t ) ≔ ( χ ∣ Γ 3 − χ ∣ Γ 2 ) ( f 3 ( t ) − f 1 ( t ) ) ( s ) + ( f 3 ( t ) χ ∣ Γ 1 − f 1 ( t ) χ ∣ Γ 3 ) ( f 3 ( t ) − f 1 ( t ) ) is the unique fixed point of operator X χ ∣ Γ 2 , χ ∣ Γ 3 ( 2 ) .

( ℬ n , q s , ∞ ) ( t , s ) = F ( t , f 1 ( t ) ) − F ( t , f 2 ( t ) ) f 3 ( t ) − f 1 ( t ) s + f 3 ( t ) F ( t , f 1 ( t ) ) − f 1 ( t ) F ( t , f 3 ( t ) ) f 3 ( t ) − f 1 ( t ) .

Consequently, it is easy to notice that operators ℬ m , q t and ℬ n , q s are weakly Picard operators.□

Now we inspect the convergence of sequence of the product and Boolean sum of Phillips-type q-Bernstein-type operators.

Theorem 4.2

The operator P m n , q is a weakly Picard operator and

( P m n , q ∞ F ) ( t , s ) = 1 [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ g 3 ( s ) f 3 ( t 0 ) F ( t 0 , f 1 ( t 0 ) ) + g 2 ( s ) f 1 ( t 1 ) × F ( t 1 , f 3 ( t 1 ) ) − g 3 ( s ) f 1 ( t 1 ) × F ( t 0 , f 3 ( t 0 ) ) − g 2 ( s ) f 3 ( t 0 ) F ( t 1 , f 1 ( t 1 ) ) ] + t [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ f 1 ( t 1 ) F ( t 0 , f 3 ( t 0 ) ) + f 3 ( t 0 ) F ( t 1 , f 1 ( t 1 ) ) − f 3 ( t 0 ) F ( t 0 , f 1 ( t 0 ) ) − f 1 ( t 0 ) F ( t 1 , f 3 ( t 1 ) ) ] + s [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ g 3 ( s ) F ( t 0 , f 3 ( t 0 ) ) + g 2 ( s ) F ( t 1 , f 1 ( t 1 ) ) − g 3 ( s ) × F ( t 0 , f 1 ( t 0 ) ) − g 2 ( s ) F ( t 1 , f 3 ( t 1 ) ) ] + t s [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ F ( t 0 , f 1 ( t 0 ) ) + F ( t 1 , f 3 ( t 1 ) ) − F ( t 0 , f 3 ( t 0 ) ) − F ( t 1 , f 1 ( t 1 ) ) ] ,

with t 0 = g 2 ( s ) , t 1 = g 3 ( s ) .

Proof

Let X ζ , η , θ , ϑ = { F ∈ C ( T h ) ∣ F ( t 0 , f 1 ( t 0 ) ) = ζ , F ( t 1 , f 1 ( t 1 ) ) = η , F ( t 1 , f 1 ( t 1 ) ) = θ , F ( t 0 , f 3 ( t 0 ) ) = ϑ } and denote

F ζ , η , θ , ϑ ( s , t ) ≔ ( g 3 ( s ) f 3 ( t 0 ) ζ + g 2 ( s ) f 1 ( t 1 ) θ − g 3 ( s ) f 1 ( t 1 ) ϑ − g 2 ( s ) f 3 ( t 0 ) η ) × ( [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] ) − 1 + f 1 ( t 1 ) ϑ + f 3 ( t 0 ) η − f 3 ( t 0 ) ζ − f 1 ( ( t 0 ) ) θ [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] t + g 3 ( s ) ϑ + g 2 ( s ) θ − g 3 ( s ) ζ − g 2 ( ( s ) ) θ [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] s + ζ + θ − η − ϑ [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] t s

with ζ , η , θ , ϑ ∈ R .

We observe that

  1. X ζ , η , θ , ϑ is closed subset of C ( T h ) ;

  2. X ζ , η , θ , ϑ is an invariant subset of P m n , q for ζ , η , θ , ϑ ∈ R and n , m ∈ N ∗ ;

  3. C ( T h ) = ∪ ζ , η , θ , ϑ X ζ , η , θ , ϑ is a partition of C ( T h ) ;

  4. F ζ , η , θ , ϑ ∈ X ζ , η , θ , ϑ ∩ F P m n , q , where F P m n , q indicate the fixed points sets of P m n , q .

The statements (i) and (iii) are easy to follow.

(ii) Similarly the same idea of proof of Theorem 4.1, by linearity of Phillips-type-q Bernstein operators and Theorem 2.2, it is easy to observe that X ζ , η , θ , ϑ is invariant subset of P n m , q for ζ , η , θ , ϑ ∈ R and n , m ∈ N ∗

(iv) Now we show that P m n , q ∣ X ζ , η , θ , ϑ : X ζ , η , θ , ϑ → X ζ , η , θ , ϑ is a contraction for ζ , η , θ , ϑ ∈ R n , m ∈ N ∗ . Let F , G ∈ X ζ , η , θ , ϑ . With the same idea as used in [19], it follows that, since

j m n = u m v n , j m n = min g 2 ( s ) ≤ t ≤ g 3 ( s ) t − g 2 ( s ) g 3 ( s ) − g 2 ( s ) m + g 3 ( s ) − t g 3 ( s ) − g 2 ( s ) m min f 1 ( t ) ≤ s ≤ f 3 ( t ) s − f 1 ( t ) f 3 ( t ) − f 2 ( t ) n + f 3 ( t ) − s f 3 ( t ) − f 2 ( t ) n ,

we have

∣ P m n , q ( F ) ( t , s ) − P m n , q ( G ) ( t , s ) ∣ = ∣ P m n , q ( F − G ) ( t , s ) ∣ ≤ ∣ 1 − u m v n ∣ ‖ F − G ‖ ∞ ≤ ∣ 1 − j m n ∣ ‖ F − G ‖ ∞ .

Since 0 < j m n < 1 , P m n , q ∣ X ζ , η , θ , ϑ is a contraction for ζ , η , θ , ϑ ∈ R .

So, ‖ P m n , q ( F ) ( t , s ) − P m n , q ( G ) ( t , s ) ‖ ∞ ≤ ( 1 − j m n ) ‖ F − G ‖ ∞ that P n m , q ∣ X ζ , η , θ , ϑ is the contraction for ζ , η , θ , ϑ ∈ R .

With the help of the contraction principle, F ζ , η , θ , ϑ is the only fixed point of P m n , q in X ζ , η , θ , ϑ and P m n , q ∣ X ζ , η , θ , ϑ is a Picard operator. It is easy to note that the operator P m n , q is the weakly Picard operator.□

Remark 4.3

We have a analogously similar result for operator Q n m , q .

Theorem 4.4

The operator S m n , q is weakly Picard operator

( S m n , q ∞ F ) ( t , s ) = F ( g 3 ( s ) , s ) − F ( g 2 ( s ) , s ) g 3 ( s ) − g 2 ( s ) t + g 3 ( s ) F ( g 2 ( s ) , s ) − g 2 ( s ) F ( g 3 ( s ) , s ) g 3 ( s ) − g 2 ( s ) + F ( t , f 3 ( t ) ) − F ( t , f 1 ( t ) ) f 3 ( t ) − f 1 ( t ) s + f 3 ( t ) F ( t , f 1 ( t ) ) − f 1 ( t ) F ( t , f 3 ( t ) ) f 3 ( t ) − f 1 ( t ) − 1 [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ g 3 ( s ) f 3 ( t 0 ) F ( t 0 , f 1 ( t 0 ) ) + g 2 ( s ) f 1 ( t 1 ) × F ( t 1 , f 3 ( t 1 ) ) − g 3 ( s ) f 1 ( t 1 ) × F ( t 0 , f 3 ( t 0 ) ) − g 2 ( s ) f 3 ( t 0 ) F ( t 1 , f 1 ( t 1 ) ) ] − t [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ f 1 ( t 1 ) F ( t 0 , f 3 ( t 0 ) ) + f 3 ( t 0 ) F ( t 1 , f 1 ( t 1 ) ) − f 3 ( t 0 ) F ( t 0 , f 1 ( t 0 ) ) − f 1 ( t 0 ) F ( t 1 , f 3 ( t 1 ) ) ] − s [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ g 3 ( s ) F ( t 0 , f 3 ( t 0 ) ) + g 2 ( s ) F ( t 1 , f 1 ( t 1 ) − g 3 ( s ) ) × F ( t 0 , f 1 ( t 0 ) ) − g 2 ( s ) F ( t 1 , f 3 ( t 1 ) ) ] − t s [ g 3 ( s ) − g 2 ( s ) ] [ f 3 ( t ) − f 1 ( t ) ] × [ F ( t 0 , f 1 ( t 0 ) ) + F ( t 1 , f 3 ( t 1 ) ) − F ( t 0 , f 3 ( t 0 ) ) − F ( t 1 , f 1 ( t 1 ) ) ] ,

with t 0 = g 2 ( s ) , t 1 = g 3 ( s ) .

Proof

It is easy to prove in a similar manner as done in the previous theorem and following inequality.

□ ‖ S m n , q ( F ) ( t , s ) − S m n , q ( G ) ( t , s ) ‖ ∞ ≤ [ 1 − ( u m + v n − j m n ) ] ‖ F − G ‖ ∞ .

Remark 4.5

Similar result can be achieved for the operator T n m , q .

Acknowledgment

Not applicable.

  1. Funding information: Not applicable.

  2. Author contributions: A.K.: supervision and formal analysis. M.I. and M.A.: writing of the original draft. M.R.L.: verification. M.M.: editing and final writing. All the authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they have no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-04-27
Revised: 2022-09-22
Accepted: 2022-10-06
Published Online: 2022-11-30

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2022-0173
Keywords for this article
Quantum calculus; Phillips q-Bernstein operators; product operators; Boolean sum operators; weakly Picard operators; contraction principle
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