Fibre Contraction Principle with respect to an Iterative Algorithm

Hindawi Publishing Corporation Journal of Operators Volume 2013, Article ID 408791, 6 pages http://dx.doi.org/10.1155/2013/408791

Research Article Fibre Contraction Principle with respect to an Iterative Algorithm Marcel-Adrian Ferban Department of Mathematics, Babes¸-Bolyai University, 1 M. Kog˘alniceanu, 400048 Cluj-Napoca, Romania Correspondence should be addressed to Marcel-Adrian S¸erban; [email protected] Received 6 March 2013; Accepted 2 September 2013 Academic Editor: Lingju Kong Copyright © 2013 Marcel-Adrian S¸erban. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We apply the fibre contraction principle in the case of a general iterative algorithm to approximate the fixed point of triangular operator using the admissible perturbation. A simple example and an application to a functional equation with parameter are given in order to illustrate the abstract results and to show the role of admissible perturbations.

1. Introduction We will use the notations and notions from [1]. Let 𝑓 : 𝑋 → 𝑋 be an operator; then 𝑓0 = 1𝑋 , 𝑓1 = 𝑓, . . . , 𝑓𝑛+1 = 𝑓 ∘ 𝑓𝑛 , 𝑛 ∈ N, denote the iterate operators of 𝑓. By 𝐼(𝑓) := {𝑌 ⊂ 𝑋 | 𝑓(𝑌) ⊆ 𝑌} we denote the set of all nonempty invariant subsets of 𝑓. By 𝐹𝑓 := {𝑥 ∈ 𝑋 | 𝑓(𝑥) = 𝑥} we denote the fixed point set of the operator 𝑓. Let 𝑋 be a nonempty set, 𝑠(𝑋) := {(𝑥𝑛 )𝑛∈N | 𝑥𝑛 ∈ 𝑋, 𝑛 ∈ N}, 𝑐(𝑋) ⊂ 𝑠(𝑋) a subset of 𝑠(𝑋), and Lim : 𝑐(𝑋) → 𝑋 an operator. By definition the triple (𝑋, 𝑐(𝑋), Lim) is called an 𝐿-space if the following conditions are satisfied: (i) if 𝑥𝑛 = 𝑥, for all 𝑛 ∈ N, then (𝑥𝑛 )𝑛∈N ∈ 𝑐(𝑋) and Lim(𝑥𝑛 )𝑛∈N = 𝑥, (ii) if (𝑥𝑛 )𝑛∈N ∈ 𝑐(𝑋) and Lim(𝑥𝑛 )𝑛∈N = 𝑥, then for all subsequences, (𝑥𝑛𝑖 )𝑖∈N , of (𝑥𝑛 )𝑛∈N we have that (𝑥𝑛𝑖 )𝑖∈N ∈ 𝑐(𝑋) and Lim(𝑥𝑛𝑖 )𝑖∈N = 𝑥. By definition an element of 𝑐(𝑋) is convergent sequence, 𝑥 := Lim(𝑥𝑛 )𝑛∈N is the limit of this sequence and we write 𝑥𝑛 → 𝑥 as 𝑛 → ∞. In what follows we will denote an 𝐿-space by (𝑋, → ). Actually, an 𝐿-space is any set endowed with a structure implying a notion of convergence for sequences. For example, Hausdorff topological spaces, metric spaces, generalized metric spaces in Perov’s sense (i.e., 𝑑(𝑥, 𝑦) ∈ R𝑚 + ), generalized metric spaces in Luxemburg’s sense (i.e., 𝑑(𝑥, 𝑦) ∈ R+ ∪

{+∞}), 𝐾-metric spaces (i.e., 𝑑(𝑥, 𝑦) ∈ 𝐾, where 𝐾 is a cone in an ordered Banach space), gauge spaces, 2-metric spaces, 𝐷-𝑅-spaces ([2, 3]), probabilistic metric spaces, syntopogenous spaces are such 𝐿-spaces. For more details see Fr´echet [4], Blumenthal [5], and Rus [1]. Let (𝑋, 𝑑) be a metric space. We will use the following symbols: P(𝑋) = {𝑌 | 𝑌 ⊂ 𝑋}, 𝑃(𝑋) = {𝑌 ⊂ 𝑋 | 𝑌 is nonempty}, 𝑃𝑏 (𝑋) := {𝑌 ∈ 𝑃(𝑋) | 𝑌 is bounded}, 𝑃𝑐𝑙 (𝑋) := {𝑌 ∈ 𝑃(𝑋) | 𝑌 is closed}, 𝑃𝑏,𝑐𝑙 (𝑋) := 𝑃𝑏 (𝑋)∩ 𝑃𝑐𝑙 (𝑋). If 𝑋 is a Banach space, then 𝑃𝑐V (𝑋) := {𝑌 ∈ 𝑃(𝑋) | 𝑌 is convex}. Let (𝑋, → ) be an 𝐿-space. Definition 1. An operator 𝑓 : 𝑋 → 𝑋 is called a Picard operator (briefly PO) if (i) 𝐹𝑓 = {𝑥∗ }; (ii) 𝑓𝑛 (𝑥) → 𝑥∗ as 𝑛 → ∞, for all 𝑥 ∈ 𝑋. Definition 2. An operator 𝑓 : 𝑋 → 𝑋 is said to be a weakly Picard operator (briefly WPO) if the sequence (𝑓𝑛 (𝑥))𝑛∈𝑁 converges for all 𝑥 ∈ 𝑋 and the limit (which may depend on 𝑥) is a fixed point of 𝑓.

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If 𝑓 : 𝑋 → 𝑋 is a WPO, then we may define the operator 𝑓∞ : 𝑋 → 𝑋 by 𝑓∞ (𝑥) := lim 𝑓𝑛 (𝑥) .

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𝑛→∞

If 𝑓 : 𝑋 → 𝑋 is a PO, then 𝑓∞ (𝑥) = 𝑥∗ , for all 𝑥 ∈ 𝑋. The following problem has been considered in [1].

Definition 5 (Rus [25]). We call 𝑓𝐺 an admissible perturbation of 𝑓 corresponding to 𝐺 if 𝐺 satisfies (𝐴 1 ) 𝐺(𝑥, 𝑥) = 𝑥, ∀𝑥 ∈ 𝑋; (𝐴 2 ) 𝑥, 𝑦 ∈ 𝑋, 𝐺(𝑥, 𝑦) = 𝑥 implies 𝑥 = 𝑦. We remark that

1

󳨀 ) and Problem 3 (fibre Picard operator problem). Let (𝑋, →

𝐹𝑓𝐺 = 𝐹𝑓 ,

2

(𝑌, → 󳨀 ) be two 𝐿-spaces. Let 𝑔 : 𝑋 → 𝑋 be a WPO and let ℎ : 𝑋 × 𝑌 → 𝑌 be such that ℎ(𝑥, ⋅) : 𝑌 → 𝑌 is a WPO for every 𝑥 ∈ 𝑋. Consider the triangular operator 𝑓 defined as follows: 𝑓 : 𝑋 × 𝑌 → 𝑋 × 𝑌, 𝑓 (𝑥, 𝑦) := (𝑔 (𝑥) , ℎ (𝑥, 𝑦)) .

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but, in general, 𝐹𝑓𝐺𝑛 ≠ 𝐹𝑓𝑛 ,

𝑛 ≥ 2.

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Example 6 (Krasnoselskii). Let (𝑉, +, R) be a vectorial space, 𝑋 ⊂ 𝑉 a convex subset, 𝜆 ∈]0, 1[, 𝑓 : 𝑋 → 𝑋 and 𝐺 : 𝑋 × 𝑋 → 𝑋, defined by 𝐺 (𝑥, 𝑦) := (1 − 𝜆) 𝑥 + 𝜆𝑦.

In which conditions is 𝑓 a WPO? An answer to this problem is the following result. Theorem 4 (fibre contraction principle (Rus [1, 6])). Let (𝑋, → ) be an 𝐿-space and let (𝑌, 𝜌) be a complete metric space. Let 𝑔 : 𝑋 → 𝑋, ℎ : 𝑋 × 𝑌 → 𝑌 be two operators and 𝑓 : 𝑋 × 𝑌 → 𝑋×𝑌 the triangular operator, 𝑓(𝑥, 𝑦) := (𝑔(𝑥), ℎ(𝑥, 𝑦)). Assume that the following conditions are satisfied: (i) 𝑔 is a WPO; (ii) ℎ(𝑥, ⋅) : 𝑌 → 𝑌 is an 𝛼-contraction, for all 𝑥 ∈ 𝑋; (iii) 𝑓 is continuous. Then 𝑓 is a WPO and

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Then 𝑓𝐺 is an admissible perturbation of 𝑓. We will denote 𝑓𝐺 by 𝑓𝜆 and call it the Krasnoselskii perturbation of 𝑓. The corresponding iterative algorithm generated by the Krasnoselskii perturbation of 𝑓 is 𝑥0 ∈ 𝑋,

𝑥𝑛+1 = (1 − 𝜆) 𝑥𝑛 + 𝜆𝑓 (𝑥𝑛 ) .

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It is known that the Krasnoselskii iteration is convergent to a fixed point of the operators 𝑓 : 𝑋 → 𝑋 in the case when 𝑋 is a bounded closed convex subset of a Hilbert space and 𝑓 is a nonexpansive and demicompact operator (see [26]). Let (𝑋, → ) be an 𝐿-space, 𝑓 : 𝑋 → 𝑋, and 𝐺 : 𝑋 × 𝑋 → 𝑋.

𝐹𝑓 = {(𝑔∞ (𝑥) , (ℎ (𝑔∞ (𝑥) , ⋅)) (𝑦)) : 𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌} . (3)

Example 7 (GK-algorithm (Rus [25])). We consider the iterative algorithm

Moreover, if 𝑔 is a PO, then 𝑓 is a PO and 𝐹𝑓 = {(𝑥∗ , 𝑦∗ )}, where 𝐹𝑔 = {𝑥∗ } and 𝐹ℎ(𝑥∗ ,⋅) = {𝑦∗ }.

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∞

Theorem 4 generalizes the result of Hirsch and Pugh [7]. The fibre contraction principle is used in order to prove the differentiability of the solutions for some operatorial equations with respect to parameters. For more considerations on fiber WPOs and applications see Sotomayor [8], T˘am˘a¸san [9], Rus [6, 10], S¸erban [11–13], Andr´asz [14], Bacot¸iu [15], Petrus¸el et al. [16, 17], Chis¸-Novac et al. [18], Ilea and Otrocol [19], Dobrit¸oiu [20–22], Olaru [23], and Barreira and Valls [24]. The aim of this paper is to establish some new fixed point theorems for triangular operators using the admissible perturbations. This notion was introduced by Rus in [25] and gives the advantage to obtain new iterative approximations of the fixed point for such operators.

2. Admissible Perturbations of an Operator Let 𝑋 be a nonempty set, and 𝑓 : 𝑋 → 𝑋, 𝐺 : 𝑋 × 𝑋 → 𝑋 be two operators. We consider the operator 𝑓𝐺 : 𝑋 → 𝑋 defined by 𝑓𝐺 (𝑥) = 𝐺 (𝑥, 𝑓 (𝑥)) .

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𝑥0 ∈ 𝑋,

𝑥𝑛+1 = 𝐺 (𝑥𝑛 , 𝑓 (𝑥𝑛 )) ,

𝑛 ∈ N.

By definition, this iterative process is convergent if and only if 𝑥𝑛 󳨀→ 𝑥∗ (𝑥0 ) ∈ 𝐹𝑓 ,

as 𝑛 󳨀→ ∞, ∀𝑥0 ∈ 𝑋.

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𝑓𝐺𝑛 (𝑥0 ). So, this algorithm is convergent

We remark that 𝑥𝑛 = if and only if 𝑓𝐺 is WPO. If 𝑓𝐺 is WPO and an admissible perturbation of 𝑓, then 𝑓𝐺∞ : 𝑋 󳨀→ 𝐹𝑓

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is a set retraction. We call this algorithm, Krasnoselskii algorithm corresponding to 𝐺 or 𝐺𝐾-algorithm. For other examples of iterative algorithms see Rus [25].

3. Fibre Contraction Principle for Admissible Perturbations Let 𝑋 and 𝑌 be two nonempty sets, 𝑔 : 𝑋 → 𝑋, ℎ : 𝑋 × 𝑌 → 𝑌, and 𝑓 : 𝑋 × 𝑌 → 𝑋 × 𝑌 the triangular operator 𝑓 (𝑥, 𝑦) := (𝑔 (𝑥) , ℎ (𝑥, 𝑦)) .

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3

Let 𝐺1 : 𝑋 × 𝑋 → 𝑋 and 𝐺2 : 𝑌 × 𝑌 → 𝑌 satisfying (𝐴 1 ) and (𝐴 2 ). Then 𝑔𝐺1 : 𝑋 → 𝑋, 𝑔𝐺1 (𝑥) := 𝐺1 (𝑥, 𝑔 (𝑥)) ,

Let us consider, in particular, the following iterative algorithm 𝑥𝑛+1 = (1 − 𝜆 1 ) 𝑥𝑛 + 𝜆 1 𝑔 (𝑥𝑛 ) ,

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is an admissible perturbation of 𝑔 and, for 𝑥 ∈ 𝑋, ℎ(𝑥, ⋅)𝐺2 : 𝑌 → 𝑌, ℎ(𝑥, ⋅)𝐺2 (𝑦) := 𝐺2 (𝑦, ℎ (𝑥, 𝑦))

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is an admissible perturbation of ℎ(𝑥, ⋅). In these settings, we have the corresponding iterative algorithm

𝑦𝑛+1 = ℎ(𝑥𝑛 , ⋅)𝜆 2 (𝑦𝑛 ) = (1 − 𝜆 2 ) 𝑦𝑛 + 𝜆 2 ℎ (𝑥𝑛 , 𝑦𝑛 )

(19)

for starting points 𝑥0 ∈ 𝑋, 𝑦0 ∈ 𝑌, 𝜆 1 , 𝜆 2 ∈]0; 1[, 𝑔 : 𝑋 → 𝑋, ℎ : 𝑋 × 𝑌 → 𝑌, 𝑋 ∈ 𝑃𝑐V (𝑉1 ), 𝑌 ∈ 𝑃𝑐V (𝑉2 ), 𝑉1 , 𝑉2 are vectorial spaces. The operator 𝑓𝜆 1 𝜆 2 : 𝑋 × 𝑌 → 𝑋 × 𝑌 𝑓𝜆 1 𝜆 2 (𝑥, 𝑦) := (𝑔𝜆 1 (𝑥) , ℎ(𝑥, ⋅)𝜆 2 (𝑦))

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is an admissible perturbation of 𝑓 and

𝑥𝑛+1 = 𝑔𝐺1 (𝑥𝑛 ) = 𝐺1 (𝑥𝑛 , 𝑔 (𝑥𝑛 )) , 𝑦𝑛+1 = ℎ(𝑥𝑛 , ⋅)𝐺2 (𝑦𝑛 ) = 𝐺2 (𝑦𝑛 , ℎ (𝑥𝑛 , 𝑦𝑛 ))

(15)

for starting points 𝑥0 ∈ 𝑋, 𝑦0 ∈ 𝑌, and the operator 𝑓𝐺1 𝐺2 : 𝑋×𝑌 → 𝑋×𝑌 𝑓𝐺1 𝐺2 (𝑥, 𝑦) := (𝑔𝐺1 (𝑥) , ℎ(𝑥, ⋅)𝐺2 (𝑦))

(16)

is an admissible perturbation of 𝑓 and 𝐹𝑓 = 𝐹𝑓𝐺 𝐺 .

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1 2

From the fibre contraction principle, we have the following. Theorem 8. Let (𝑋, → ) be an 𝐿-space and let (𝑌, 𝜌) be a complete metric space. Let 𝑔 : 𝑋 → 𝑋, ℎ : 𝑋 × 𝑌 → 𝑌 be two operators, 𝑓 : 𝑋 × 𝑌 → 𝑋 × 𝑌 the triangular operator 𝑓(𝑥, 𝑦) := (𝑔(𝑥), ℎ(𝑥, 𝑦)), 𝐺1 : 𝑋 × 𝑋 → 𝑋, and 𝐺2 : 𝑌 × 𝑌 → 𝑌 satisfying (𝐴 1 ) and (𝐴 2 ). One supposes that

𝐹𝑓 = 𝐹𝑓𝜆

1 𝜆2

.

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Corollary 9. Let 𝑉1 be a Hilbert space, (𝑉2 , ‖ ⋅ ‖) a Banach space, and 𝑋 ∈ 𝑃𝑐𝑙,𝑐V (𝑉1 ), 𝑌 ∈ 𝑃𝑐𝑙,𝑐V (𝑉2 ). Let 𝑔 : 𝑋 → 𝑋, ℎ : 𝑋 × 𝑌 → 𝑌 be two operators and 𝑓 : 𝑋 × 𝑌 → 𝑋 × 𝑌 the triangular operator 𝑓(𝑥, 𝑦) := (𝑔(𝑥), ℎ(𝑥, 𝑦)). One supposes that (i) there exists 𝜆 1 ∈]0; 1[ such that 𝑔𝜆 1 is WPO;

(ii) there exists 𝜆 2 ∈]0; 1[ such that ℎ(𝑥, ⋅)𝜆 2 : 𝑌 → 𝑌 is an 𝛼-contraction, for all 𝑥 ∈ 𝑋; (iii) 𝑔 and ℎ are continuous. Then ∞

(a) 𝐹𝑓 = 𝐹𝑓𝜆 𝜆 = {(𝑔𝜆∞1 (𝑥), (ℎ(𝑔𝜆∞1 (𝑥), ⋅)𝜆 ) (𝑦)) : 𝑥 ∈ 𝑋, 1 2 2 𝑦 ∈ 𝑌};

(i) 𝑔𝐺1 is a WPO;

(b) 𝑓𝜆 1 𝜆 2 is a WPO; that is, the iterative algorithm defined by (19) is convergent to a fixed point of 𝑓;

(ii) ℎ(𝑥, ⋅)𝐺2 : 𝑌 → 𝑌 is an 𝛼-contraction, for all 𝑥 ∈ 𝑋;

(c) if 𝑔𝜆 1 is a PO, then 𝑓𝜆 1 𝜆 2 is a PO.

(iii) 𝑓𝐺1 𝐺2 is continuous. Then ∞

(a) 𝐹𝑓 = 𝐹𝑓𝐺 𝐺 = {(𝑔𝐺∞1 (𝑥), (ℎ(𝑔𝐺∞1 (𝑥), ⋅)𝐺 ) (𝑦)) : 𝑥 ∈ 𝑋, 1 2 2 𝑦 ∈ 𝑌}; (b) 𝑓𝐺1 𝐺2 is a WPO; that is, the iterative algorithm, defined by (15), is convergent to a fixed point of 𝑓; (c) if 𝑔𝐺1 is a PO, then 𝑓𝐺1 𝐺2 is a PO. Proof. (a) From (i) and the fact that 𝐺1 satisfies (𝐴 1 ) and (𝐴 2 ) we have that 𝑔𝐺∞1 (𝑥) ∈ 𝐹𝑔 . From (ii) we have that ℎ(𝑥, ⋅)𝐺2 is PO for a fixed 𝑥 ∈ 𝑋, so 𝐹ℎ(𝑥,⋅)𝐺 = {𝑦∗ (𝑥)} and 2 (ℎ(𝑥, ⋅)𝐺2 )∞ (𝑦) = 𝑦∗ (𝑥) for all 𝑦 ∈ 𝑌. But 𝐺2 , also, satisfies (𝐴 1 ) and (𝐴 2 ); therefore 𝐹ℎ(𝑥,⋅) = 𝐹ℎ(𝑥,⋅)𝐺 = {𝑦∗ (𝑥)}. Thus, 2

=

(𝑥) , (ℎ(𝑔𝐺∞1

Corollary 10. Let 𝑉1 be a real Hilbert space, (𝑉2 , ‖⋅‖) a Banach space, and 𝑋 ∈ 𝑃𝑏,𝑐𝑙,𝑐V (𝑉1 ), 𝑌 ∈ 𝑃𝑐𝑙,𝑐V (𝑉2 ). Let 𝑔 : 𝑋 → 𝑋, ℎ : 𝑋 × 𝑌 → 𝑌 be two operators and 𝑓 : 𝑋 × 𝑌 → 𝑋 × 𝑌 the triangular operator 𝑓(𝑥, 𝑦) := (𝑔(𝑥), ℎ(𝑥, 𝑦)). One supposes that: (i) 𝑔 is a nonexpansive and demicompact operator; (ii) there exists 𝜆 2 ∈]0; 1[ such that ℎ(𝑥, ⋅)𝜆 2 : 𝑌 → 𝑌 is an 𝛼-contraction, for all 𝑥 ∈ 𝑋; (iii) ℎ is continuous. Then

(𝑔𝐺∞1 (𝑥) , 𝑦∗ (𝑔𝐺∞1 (𝑥))) (𝑔𝐺∞1

Proof. In this case we take 𝐺1 : 𝑋×𝑋 → 𝑋, 𝐺1 (𝑥1 , 𝑥2 ) = (1− 𝜆 1 )𝑥1 +𝜆 1 𝑥2 , 𝐺2 : 𝑌×𝑌 → 𝑌, 𝐺2 (𝑦1 , 𝑦2 ) = (1−𝜆 2 )𝑦1 +𝜆 2 𝑦2 . Since 𝑔 and ℎ are continuous, then 𝑓𝜆 1 𝜆 2 is continuous and from Theorem 8 we get the conclusion.

∞

(𝑥) , ⋅)𝐺 ) (𝑦)) ∈ 𝐹𝑓𝐺 𝐺 = 𝐹𝑓 . 2

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1 2

It is easy to see that the triangular operator 𝑓𝐺1 𝐺2 = (𝑔𝐺1 , ℎ(𝑥, ⋅)𝐺2 ) satisfies the conditions from the fibre contraction principle and, thus, we get the conclusion.

∞

(a) 𝐹𝑓 = 𝐹𝑓𝜆 𝜆 = {(𝑔𝜆∞1 (𝑥), (ℎ(𝑔𝜆∞1 (𝑥), ⋅)𝜆 ) (𝑦)) : 𝑥 ∈ 𝑋, 1 2 2 𝑦 ∈ 𝑌}, for any 𝜆 1 ∈]0; 1[; (b) 𝑓𝜆 1 𝜆 2 is a WPO; that is, the iterative algorithm, defined by (19), is convergent to a fixed point of 𝑓; (c) if 𝐹𝑔𝜆 = {𝑥∗ } then 𝑓𝜆 1 𝜆 2 is a PO. 1

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Proof. In this case we take 𝐺1 : 𝑋 × 𝑋 → 𝑋, 𝐺1 (𝑥1 , 𝑥2 ) = (1 − 𝜆 1 )𝑥1 + 𝜆 1 𝑥2 , 𝐺2 : 𝑌 × 𝑌 → 𝑌, 𝐺2 (𝑦1 , 𝑦2 ) = (1 − 𝜆 2 )𝑦1 + 𝜆 2 𝑦2 . Condition (i) implies that 𝑥𝑛 = 𝑔𝜆𝑛 1 (𝑥0 ) converges (strongly) to a fixed point of 𝑔 for any 𝜆 1 ∈]0; 1[, (see Theorem 3.2 in [26]), so 𝑔𝜆 1 is WPO. Since 𝑔 is nonexpansive and ℎ is continuous, then, also, 𝑓𝜆 1 𝜆 2 is continuous and from Corollary 9 we obtain the conclusion.

and ℎ : 𝑋 × 𝑌 → 𝑌

Corollary 11. Let 𝑉1 be a real Hilbert space with the inner product ⟨⋅, ⋅⟩ and the norm ‖ ⋅ ‖, (𝑉2 , | ⋅ |) a Banach space and 𝑋 ∈ 𝑃𝑐𝑙,𝑐V (𝑉1 ), 𝑌 ∈ 𝑃𝑐𝑙,𝑐V (𝑉2 ). Let 𝑔 : 𝑋 → 𝑋, ℎ : 𝑋 × 𝑌 → 𝑌 be two operators and 𝑓 : 𝑋 × 𝑌 → 𝑋 × 𝑌 the triangular operator 𝑓(𝑥, 𝑦) := (𝑔(𝑥), ℎ(𝑥, 𝑦)). One supposes that:

The system (24) has a unique solution (𝑥∗ , 𝑦∗ ) ∈ 𝑋 × 𝑌, (𝑥∗ , 𝑦∗ ) = (1, 2), and the iterative algorithm (19) is convergent to (𝑥∗ , 𝑦∗ ) for any 𝜆 1 ∈]0; 2/5[ and 𝜆 2 ∈]0; 2/51[.

(i) 𝑔 is a generalized pseudocontraction; that is, there exists, 𝑟 > 0 󵄩 󵄩2 ⟨𝑔 (𝑥1 ) − 𝑔 (𝑥2 ) , 𝑥1 − 𝑥2 ⟩ ≤ 𝑟2 󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩 ,

ℎ (𝑥, 𝑦) = 𝑥2 +

2𝑥 . 𝑦

(26)

It is clear that a solution of the system (24) is a fixed point of the triangular operator 𝑓 : 𝑋 × 𝑌 → 𝑋 × 𝑌 𝑓 (𝑥, 𝑦) = (𝑔 (𝑥) , ℎ (𝑥, 𝑦)) .

(27)

Proof. We have that 𝑔 is 𝑙𝑔 -Lipschitz, with 𝑙𝑔 = 4, so 𝑔 is not a contraction. Notice that, for any 𝑥0 ∈ 𝑋 and 𝑥0 ≠ 1, the Picard iteration is an oscillatory sequence {𝑥0 ,

(22)

1 1 , 𝑥 , , . . .} , 𝑥0 0 𝑥0

(28)

so it is not convergent. For 𝜆 1 ∈]0; 1[ we have

for all 𝑥1 , 𝑥2 ∈ 𝑋, with 0 < 𝑟 < 1; (ii) 𝑔 is 𝑙𝑔 -Lipschitz, with 𝑙𝑔 > 1;

𝑔𝜆 1 (𝑥) = (1 − 𝜆 1 ) 𝑥 + 𝜆 1 𝑔 (𝑥) = (1 − 𝜆 1 ) 𝑥 +

(iii) there exists 𝜆 2 ∈]0; 1[ such that ℎ(𝑥, ⋅)𝜆 2 : 𝑌 → 𝑌 is an 𝛼-contraction, for all 𝑥 ∈ 𝑋; (iv) ℎ is continuous.

𝑔𝜆󸀠 1

𝜆1 , 𝑥

𝜆 (𝑥) = 1 − 𝜆 1 − 21 ; 𝑥

(29)

therefore 󵄨󵄨 5 󵄨󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑔𝜆 (𝑥)󵄨󵄨󵄨 ≤ max {󵄨󵄨󵄨󵄨1 − 5𝜆 1 󵄨󵄨󵄨󵄨 , 󵄨󵄨󵄨1 − 𝜆 1 󵄨󵄨󵄨} . 󵄨󵄨 󵄨 󵄨 1 4 󵄨󵄨

Then 𝑓𝜆 1 𝜆 2 is a PO, for all 𝜆 1 ∈]0; 1[, satisfying 2 (1 − 𝑟) 0 < 𝜆1 < ; 1 − 2𝑟 + 𝑙𝑔2

(23)

that is, the iterative algorithm, defined by (19), is convergent to a unique fixed point of 𝑓, (𝑥∗ , 𝑦∗ ) ∈ 𝑋 × 𝑌. Proof. Conditions (i) and (ii) imply that 𝑔𝜆 1 is 𝑙𝑔𝜆 -contra1

ction for any 𝜆 1 ∈]0; 1[ satisfying (23), with 𝑙𝑔𝜆 = ((1 − 𝜆 1 )2 + 1

2𝜆 1 (1 − 𝜆 1 )𝑟 + 𝜆21 𝑙𝑔 )1/2 (see Theorem 3.6, in [26]), so 𝑔𝜆 1 is PO and 𝐹𝑔 = 𝐹𝑔𝜆 = {𝑥∗ }. From the continuity of 𝑔 and ℎ we 1 have that, also, 𝑓𝜆 1 𝜆 2 is continuous and from Corollary 9 we get the conclusion. The following example is a simple illustration of the above abstract results and also shows the importance of the admissible perturbations for a faster convergence of Picard iterations. Example 12. Let us consider the following system 𝑥=

1 , 𝑥

𝑦 = 𝑥2 +

2𝑥 𝑦

(24)

1 𝑥

We have the following cases: (i) if 0 < 𝜆 1 < 1/5, then 𝑙𝑔𝜆 = 1 − (5/4)𝜆 1 and 𝑙𝑔𝜆 ∈ 1 1 (3/4; 1); (ii) if 1/5 ≤ 𝜆 1 < 8/25, then 𝑙𝑔𝜆 = 1 − (5/4)𝜆 1 and 𝑙𝑔𝜆 ∈ 1 1 (3/5; 3/4]; (iii) if 8/25 ≤ 𝜆 1 < 2/5, then 𝑙𝑔𝜆 = 5𝜆 1 − 1 and 𝑙𝑔𝜆 ∈ 1 1 [3/5; 1); (iv) if 2/5 ≤ 𝜆 1 < 1, then 𝑙𝑔𝜆 = 5𝜆 1 − 1 > 1, so 𝑔𝜆 1 is not 1 a contraction. For any 𝜆 1 ∈]0; 2/5[ we get that 𝑔𝜆 1 is a contraction, so 𝑔𝜆 1 is PO. The smallest contraction constant 𝑙𝑔𝜆 = 3/5 is 1 obtained for 𝜆 1 = 8/25, so for this choice we get the best rate convergence for (𝑥𝑛 ). For ℎ : 𝑋 × 𝑌 → 𝑌 we have 󵄨󵄨 󵄨󵄨 𝜕ℎ 󵄨󵄨󵄨 (𝑥, 𝑦)󵄨󵄨󵄨 = 2𝑥 ≤ 50, (31) 󵄨󵄨 𝑦2 󵄨󵄨 𝜕𝑦 󵄨 󵄨 for all (𝑥, 𝑦) ∈ 𝑋×𝑌; thus 𝑙ℎ(𝑥,⋅) = 50 which implies that ℎ(𝑥, ⋅) is not a contraction. For 𝜆 2 ∈]0; 1[ we obtain ℎ(𝑥, ⋅)𝜆 2 (𝑦) = (1 − 𝜆 2 ) 𝑦 + 𝜆 2 ℎ (𝑥, 𝑦)

on 𝑋 × 𝑌 = [1/2; 2] × [1/5; 24]. We have the operators 𝑔 : 𝑋 → 𝑋 𝑔 (𝑥) =

(30)

(25)

= (1 − 𝜆 2 ) 𝑦 + 𝜆 2 𝑥2 + 𝜕ℎ(𝑥, ⋅)𝜆 2 (𝑦) 𝜕𝑦

= 1 − 𝜆2 −

𝜆2𝑥 , 𝑦2

𝜆2𝑥 , 𝑦

(32)

Journal of Operators

5 Let us suppose that 𝑥∗ (𝑡, ⋅) ∈ 𝐶1 (𝐼). We have from

so 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 𝜕ℎ(𝑥, ⋅)𝜆 2 (𝑦) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ≤ max {󵄨󵄨󵄨1 − 51𝜆 2 󵄨󵄨󵄨 , 󵄨󵄨󵄨󵄨1 − 1153 𝜆 2 󵄨󵄨󵄨󵄨} . 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨 𝜕𝑦 1152 󵄨󵄨 󵄨 󵄨󵄨 󵄨

𝑥∗ (𝑡, 𝜇) = 𝐹 (𝑡, 𝑥∗ (𝑡, 𝜇) , 𝜇)

(33) that

We have the following cases: (i) if 0 < 𝜆 2 < 1/51, then 𝑙ℎ(𝑥,⋅)𝜆 = 1 − (1153/1152)𝜆 2 2 and 𝑙ℎ(𝑥,⋅)𝜆 ∈ (57599/58752; 1) ≅ (0.98038; 1);

𝜕𝑥∗ 𝜕𝑥∗ 𝜕𝐹 (𝑡, 𝜇) = (𝑡, 𝑥∗ (𝑡, 𝜇) , 𝜇) ⋅ (𝑡, 𝜇) 𝜕𝜇 𝜕𝑢 𝜕𝜇 𝜕𝐹 (𝑡, 𝑥∗ (𝑡, 𝜇) , 𝜇) . + 𝜕𝜇

2

(ii) if 1/51 ≤ 𝜆 2 < 2304/59905, then 𝑙ℎ(𝑥,⋅)𝜆 = 1 − 2 (1153/1152)𝜆 2 , and 𝑙ℎ(𝑥,⋅)𝜆 ∈ (57599/59905; 57599/ 2 58752] ≅ (0.961 51; 0.980 38]; (iii) if 2304/59905 ≤ 𝜆 2 < 2/51, then 𝑙ℎ(𝑥,⋅)𝜆 = 51𝜆 2 − 1, 2 and 𝑙ℎ(𝑥,⋅)𝜆 ∈ [57599/59905; 1) ≅ [0.961 51; 1); 2

For any 𝜆 2 ∈]0; 2/51[ we get that ℎ(𝑥, ⋅)𝜆 2 is a contraction. The smallest contraction constant 𝑙ℎ(𝑥,⋅)𝜆 = 57599/59905 ≅ 2 0.961 51 is obtain for 𝜆 2 = 2304/59905, so the fastest iterative algorithm (19) is obtained for 𝜆 1 = 8/25 and 𝜆 2 = 2304/59905.

(38)

This leads us to consider the following operator ℎ : 𝑋 × 𝑋 → 𝑋 defined by ℎ (𝑥, 𝑦) (𝑡, 𝜇) =

(iv) if 2/51 ≤ 𝜆 2 < 1, then 𝑙ℎ(𝑥,⋅)𝜆 = 51𝜆 2 − 1 > 1, so 2 ℎ(𝑥, ⋅)𝜆 2 is not a contraction.

𝜕𝐹 (𝑡, 𝑥 (𝑡, 𝜇) , 𝜇) ⋅ 𝑦 (𝑡, 𝜇) 𝜕𝑢 𝜕𝐹 + (𝑡, 𝑥 (𝑡, 𝜇) , 𝜇) . 𝜕𝜇

(39)

So, we have the operator 𝑓 : 𝑋 × 𝑋 → 𝑋 × 𝑋 (𝑥, 𝑦) 󳨃󳨀→ (𝑔 (𝑥) , ℎ (𝑥, 𝑦))

(40)

and the operator 𝑓𝜆 : 𝑋 × 𝑋 → 𝑋 × 𝑋 𝑓𝜆 (𝑥, 𝑦) = (𝑔𝜆 (𝑥) , ℎ𝜆 (𝑥, 𝑦))

4. Application Let us consider the following functional equation with parameter 𝑥 (𝑡, 𝜇) = 𝐹 (𝑡, 𝑥 (𝑡, 𝜇) , 𝜇) ,

(37)

𝑡 ∈ [𝑎; 𝑏] , 𝜇 ∈ 𝐼 ⊂ R,

(34)

where 𝐹 : [𝑎; 𝑏]×R×𝐼 → R and 𝐼 ⊂ R is a compact interval. Such type of equations arise from several classes of integral equations with parameter, initial value and boundary value problems for ordinary differential equations with parameter. From Theorem 8 we have the following. Theorem 13. Corresponding to (34) one supposes that

(ii) 𝐹(𝑡, ⋅, ⋅) ∈ 𝐶1 (R × 𝐼) for all 𝑡 ∈ [𝑎; 𝑏];

𝑦𝑛+1 = ℎ𝜆 (𝑥𝑛 , 𝑦𝑛 )

‖⋅‖∞

(35)

(42)

𝑦𝑛 =

as 𝑛 󳨀→ +∞,

𝜕𝑥𝑛 ‖⋅‖∞ ∗ 󳨀󳨀󳨀→ 𝑦 𝜕𝜇

(43) as 𝑛 󳨀→ +∞.

From this it follows that 𝑥∗ (𝑡, ⋅) ∈ 𝐶1 (𝐼) and 𝑦∗ = 𝜕𝑥∗ /𝜕𝜇.

Then (34) has in 𝐶([𝑎; 𝑏] × 𝐼) a unique solution 𝑥∗ and 𝑥 (𝑡, ⋅) ∈ 𝐶1 (𝐼) for all 𝑡 ∈ [𝑎; 𝑏]. ∗

Proof. Let 𝑋 := 𝐶([𝑎; 𝑏] × 𝐼) and 𝑔 : 𝑋 → 𝑋 be defined by 𝑡 ∈ [𝑎; 𝑏] , 𝜇 ∈ 𝐼.

𝑥𝑛+1 = 𝑔𝜆 (𝑥𝑛 ) ,

𝑥𝑛 󳨀󳨀󳨀→ 𝑥∗

for all 𝑡 ∈ [𝑎; 𝑏], 𝑢 ∈ R, 𝜇 ∈ 𝐼.

𝑔 (𝑥) (𝑡, 𝜇) = 𝐹 (𝑡, 𝑥 (𝑡, 𝜇) , 𝜇) ,

In our conditions, from Theorem 8 we have that 𝑓𝜆 is PO and 𝐹𝑓𝜆 = {(𝑥∗ , 𝑦∗ )}. Let us take 𝑥0 ∈ 𝐶([𝑎; 𝑏]×𝐼) such that 𝜕𝑥0 /𝜕𝜇 ∈ 𝐶([𝑎; 𝑏]× 𝐼) and consider the sequence

with 𝑦0 = 𝜕𝑥0 /𝜕𝜇. Since 𝑦0 = 𝜕𝑥0 /𝜕𝜇, we can prove by induction that 𝑦𝑛 = 𝜕𝑥𝑛 /𝜕𝜇. So,

(i) 𝐹 ∈ 𝐶([𝑎; 𝑏] × R × 𝐼); (iii) there exists 𝜆 ∈]0; 1[ such that 󵄨󵄨 𝜕 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 [(1 − 𝜆) 𝑢 + 𝜆𝐹 (𝑡, 𝑢, 𝜇)]󵄨󵄨󵄨 ≤ 𝑞 < 1, 󵄨󵄨 𝜕𝑢 󵄨󵄨

= ((1 − 𝜆) 𝑥 + 𝜆𝑔 (𝑥) , (1 − 𝜆) 𝑦 + 𝜆ℎ (𝑥, 𝑦)) . (41)

(36)

We consider on 𝐶([𝑎; 𝑏]×𝐼) the supremum norm ‖ ⋅ ‖∞ . From (i), (ii), and (iii) we have that 𝑔𝜆 is 𝑞-contraction, so 𝑔𝜆 is PO; that is, (34) has in 𝑋 a unique solution 𝑥∗ and 𝑔𝜆𝑛 (𝑥0 ) → 𝑥∗ as 𝑛 → +∞ for all 𝑥0 ∈ 𝑋.

Remark 14. For (34) in the case where the admissible perturbation is not used, see [18].

Acknowledgments The author would like to thank Professor Ioan A. Rus for giving useful suggestions and comments for the improvement of this paper. This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0094.

6

References [1] I. A. Rus, “Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 58, no. 1, pp. 191–219, 2003. [2] H. von Trotha, “Contractivity in certain D-R spaces,” Mathematische Nachrichten, vol. 101, pp. 207–213, 1981. [3] H. von Trotha, “Structure properties of 𝐷 − 𝑅 spaces,” Polska Akademia Nauk, vol. 184, 72 pages, 1981. [4] M. Fr´echet, Les Espaces Abstraits, Gauthier-Villars, Paris, France, 1928. [5] L. M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, Oxford, UK, 1953. [6] I. A. Rus, “Fiber Picard operators theorem and applications,” Universitatis Babes¸-Bolyai, vol. 44, no. 3, pp. 89–97, 1999. [7] M. W. Hirsch and C. C. Pugh, “Stable manifolds and hyperbolic sets,” in Global Analysis, pp. 133–163, American Mathematical Society, Providence, RI, USA, 1970. [8] J. Sotomayor, “Smooth dependence of solutions of differential equations on initial data: a simple proof,” Boletim da Sociedade Brasileira de Matem´atica, vol. 4, no. 1, pp. 55–59, 1973. [9] A. T˘am˘a¸san, “Differentiability with respect to lag for nonlinear pantograph equations,” Pure Mathematics and Applications, vol. 9, no. 1-2, pp. 215–220, 1998. [10] I. A. Rus, “A fiber generalized contraction theorem and applications,” Mathematica, vol. 41(64), no. 1, pp. 85–90, 1999. [11] M.-A. S¸erban, “Fiber 𝜑-contractions,” Universitatis Babes¸-Bolyai, vol. 44, no. 3, pp. 99–108, 1999. [12] M.-A. S¸erban, The Fixed Point Theory for the Operators on Cartesian Product, Cluj University Press, Cluj-Napoca, Romania, 2002. [13] M.-A. S¸erban, “Fiber contraction theorem in generalized metric spaces,” Automation Computers Applied Mathematics, vol. 16, no. 1-2, pp. 9–14, 2007. [14] S. Andr´asz, “Fibre 𝜑−contraction on generalized metric spaces and applications,” Mathematica, vol. 45(68), no. 1, pp. 3–8, 2003. [15] C. Bacot¸iu, “Fiber Picard operators,” Seminar on Fixed Point Theory Cluj-Napoca, vol. 1, pp. 5–8, 2000. [16] A. Petrus¸el, I. A. Rus, and M. A. S¸erban, “Fixed points for operators on generalized metric spaces,” Cubo, vol. 10, no. 4, pp. 45–66, 2008. [17] I. A. Rus, A. Petrus¸el, and M. A. S¸erban, “Fibre Picard operators on gauge spaces and applications,” Zeitschrift f¨ur Analysis und ihre Anwendungen, vol. 27, no. 4, pp. 407–423, 2008. [18] A. Chis¸-Novac, R. Precup, and I. A. Rus, “Data dependence of fixed points for non-self generalized contractions,” Fixed Point Theory, vol. 10, no. 1, pp. 73–87, 2009. [19] V. Ilea and D. Otrocol, “On a D. V. Ionescu problem for functional-differential equations,” Fixed Point Theory, vol. 10, no. 1, pp. 125–140, 2009. [20] M. Dobrit¸oiu, “An integral equation with modified argument,” Universitatis Babes¸-Bolyai, vol. 49, no. 3, pp. 27–33, 2004. [21] M. Dobrit¸oiu, “Properties of the solution of an integral equation with modified argument,” Carpathian Journal of Mathematics, vol. 23, no. 1-2, pp. 77–80, 2007. [22] M. Dobrit¸oiu, W. W. Kecs, and A. Toma, “An application of the fiber generalized contractions theorem,” WSEAS Transactions on Mathematics, vol. 5, no. 12, pp. 1330–1335, 2006. [23] I. M. Olaru, “An integral equation via weakly Picard operators,” Fixed Point Theory, vol. 11, no. 1, pp. 97–106, 2010.

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