Fibre Contraction Principle with respect to an Iterative Algorithm

Abstract

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=1_𝑋, 𝑓¹=𝑓,…,𝑓^𝑛+1=𝑓∘𝑓^𝑛, 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 𝐹_𝑓∶={𝑥∈𝑋∣𝑓(𝑥)=𝑥} we denote the fixed point set of the operator 𝑓.

Let 𝑋 be a nonempty set, 𝑠(𝑋)∶={(𝑥_𝑛)_𝑛∈ℕ∣𝑥_𝑛∈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 𝑥_𝑛=𝑥, for all 𝑛∈ℕ, then (𝑥_𝑛)_𝑛∈ℕ∈𝑐(𝑋) and Lim(𝑥_𝑛)_𝑛∈ℕ=𝑥,(ii)if (𝑥_𝑛)_𝑛∈ℕ∈𝑐(𝑋) and Lim(𝑥_𝑛)_𝑛∈ℕ=𝑥, then for all subsequences, (𝑥_𝑛𝑖)_𝑖∈ℕ, of (𝑥_𝑛)_𝑛∈ℕ we have that (𝑥_𝑛𝑖)_𝑖∈ℕ∈𝑐(𝑋) and Lim(𝑥_𝑛𝑖)_𝑖∈ℕ=𝑥.

By definition an element of 𝑐(𝑋) is convergent sequence, 𝑥∶=Lim(𝑥_𝑛)_𝑛∈ℕ is the limit of this sequence and we write 𝑥_𝑛→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)∈ℝ^𝑚₊), 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 Fréchet [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}, 𝑃_𝑏(𝑋)∶={𝑌∈𝑃(𝑋)∣𝑌 is bounded},  𝑃_𝑐𝑙(𝑋)∶={𝑌∈𝑃(𝑋)∣𝑌 is closed}, 𝑃_{𝑏,𝑐𝑙}(𝑋)∶=𝑃_𝑏(𝑋)∩𝑃_𝑐𝑙(𝑋).

If 𝑋 is a Banach space, then 𝑃_𝑐𝑣(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)𝐹_𝑓={𝑥^∗};(ii)𝑓^𝑛(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 (𝑓^𝑛(𝑥))_𝑛∈𝑁 converges for all 𝑥∈𝑋 and the limit (which may depend on 𝑥) is a fixed point of 𝑓.

If 𝑓∶𝑋→𝑋 is a WPO, then we may define the operator 𝑓^∞∶𝑋→𝑋 by 𝑓^∞(𝑥)∶=lim_𝑛→∞𝑓^𝑛(𝑥). (1) If 𝑓∶𝑋→𝑋 is a PO, then 𝑓^∞(𝑥)=𝑥^∗, for all 𝑥∈𝑋.

The following problem has been considered in [1].

Problem 3 (fibre Picard operator problem). Let (𝑋,→−¹) and (𝑌,→−²) 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: 𝑓𝑔(𝑥).𝑓∶𝑋×𝑌→𝑋×𝑌,𝑥,𝑦∶=,ℎ𝑥,𝑦 (2) 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 𝐹_𝑓=𝑔^∞(𝑥),ℎ𝑔^∞(𝑥),⋅^∞𝑦∶𝑥∈𝑋,𝑦∈𝑌. (3) Moreover, if 𝑔 is a PO, then 𝑓 is a PO and 𝐹_𝑓={(𝑥^∗,𝑦^∗)}, where 𝐹_𝑔={𝑥^∗} and 𝐹_{ℎ(𝑥∗,⋅)}={𝑦^∗}.

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], Tămăşan [9], Rus [6, 10], Şerban [11–13], Andrász [14], Bacoţiu [15], Petruşel et al. [16, 17], Chiş-Novac et al. [18], Ilea and Otrocol [19], Dobriţoiu [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 𝑓_𝐺∶𝑋→𝑋 defined by 𝑓_𝐺(𝑥)=𝐺𝑥,𝑓(𝑥). (4)

Definition 5 (Rus [25]). We call 𝑓_𝐺 an admissible perturbation of 𝑓 corresponding to 𝐺 if 𝐺 satisfies(𝐴₁)  𝐺(𝑥,𝑥)=𝑥, ∀𝑥∈𝑋;(𝐴₂)  𝑥,𝑦∈𝑋, 𝐺(𝑥,𝑦)=𝑥 implies 𝑥=𝑦.

We remark that 𝐹_𝑓𝐺=𝐹_𝑓, (5) but, in general, 𝐹_𝑓^𝑛_𝐺≠𝐹_𝑓^𝑛,𝑛≥2. (6)

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 𝐺𝑥,𝑦∶=(1−𝜆)𝑥+𝜆𝑦. (7)
Then 𝑓_𝐺 is an admissible perturbation of 𝑓. We will denote 𝑓_𝐺 by 𝑓_𝜆 and call it the Krasnoselskii perturbation of 𝑓. The corresponding iterative algorithm generated by the Krasnoselskii perturbation of 𝑓 is 𝑥₀∈𝑋,𝑥_𝑛+1=(1−𝜆)𝑥_𝑛+𝜆𝑓𝑥_𝑛. (8)

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.

Example 7 (GK-algorithm (Rus [25])). We consider the iterative algorithm 𝑥₀∈𝑋,𝑥_𝑛+1=𝐺𝑥_𝑛,𝑓𝑥_𝑛,𝑛∈ℕ. (9)
By definition, this iterative process is convergent if and only if 𝑥_𝑛⟶𝑥^∗𝑥₀∈𝐹_𝑓,as𝑛⟶∞,∀𝑥₀∈𝑋. (10) We remark that 𝑥_𝑛=𝑓^𝑛_𝐺(𝑥₀). So, this algorithm is convergent if and only if 𝑓_𝐺 is WPO. If 𝑓_𝐺 is WPO and an admissible perturbation of 𝑓, then 𝑓^∞_𝐺∶𝑋⟶𝐹_𝑓 (11) 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 𝑓𝑥,𝑦∶=𝑔(𝑥),ℎ𝑥,𝑦. (12) Let 𝐺₁∶𝑋×𝑋→𝑋 and 𝐺₂∶𝑌×𝑌→𝑌 satisfying (𝐴₁) and (𝐴₂). Then 𝑔_𝐺1∶𝑋→𝑋, 𝑔_𝐺1(𝑥)∶=𝐺₁𝑥,𝑔(𝑥), (13) is an admissible perturbation of 𝑔 and, for 𝑥∈𝑋, ℎ(𝑥,⋅)_𝐺2∶𝑌→𝑌, ℎ(𝑥,⋅)_𝐺2𝑦∶=𝐺₂𝑦,ℎ𝑥,𝑦 (14) is an admissible perturbation of ℎ(𝑥,⋅). In these settings, we have the corresponding iterative algorithm 𝑥_𝑛+1=𝑔_𝐺1𝑥_𝑛=𝐺₁𝑥_𝑛,𝑔𝑥_𝑛,𝑦_𝑛+1=ℎ𝑥_𝑛,⋅_𝐺2𝑦_𝑛=𝐺₂𝑦_𝑛,ℎ𝑥_𝑛,𝑦_𝑛 (15) for starting points 𝑥₀∈𝑋, 𝑦₀∈𝑌, and the operator 𝑓_𝐺1𝐺2∶𝑋×𝑌→𝑋×𝑌𝑓_𝐺1𝐺2𝑥,𝑦∶=𝑔_𝐺1(𝑥),ℎ(𝑥,⋅)_𝐺2𝑦 (16) is an admissible perturbation of 𝑓 and 𝐹_𝑓=𝐹_{𝑓𝐺1𝐺2}. (17) 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 (𝐴₁) and (𝐴₂). One supposes that(i)𝑔_𝐺1 is a WPO;(ii)ℎ(𝑥,⋅)_𝐺2∶𝑌→𝑌 is an 𝛼-contraction, for all 𝑥∈𝑋;(iii)𝑓_𝐺1𝐺2 is continuous.
Then(a)𝐹_𝑓=𝐹_{𝑓𝐺1𝐺2}={(𝑔^∞_𝐺1(𝑥),(ℎ(𝑔^∞_𝐺1(𝑥),⋅)_𝐺2)^∞(𝑦))∶𝑥∈𝑋,𝑦∈𝑌}; (b)𝑓_𝐺1𝐺2 is a WPO; that is, the iterative algorithm, defined by (15), is convergent to a fixed point of 𝑓;(c)if 𝑔_𝐺1 is a PO, then 𝑓_𝐺1𝐺2 is a PO.

Proof. (a) From (i) and the fact that 𝐺₁ satisfies (𝐴₁) and (𝐴₂) we have that 𝑔^∞_𝐺1(𝑥)∈𝐹_𝑔. From (ii) we have that ℎ(𝑥,⋅)_𝐺2 is PO for a fixed 𝑥∈𝑋, so 𝐹_{ℎ(𝑥,⋅)𝐺2}={𝑦^∗(𝑥)} and (ℎ(𝑥,⋅)_𝐺2)^∞(𝑦)=𝑦^∗(𝑥) for all 𝑦∈𝑌. But 𝐺₂, also, satisfies (𝐴₁) and (𝐴₂); therefore 𝐹_{ℎ(𝑥,⋅)}=𝐹_{ℎ(𝑥,⋅)𝐺2}={𝑦^∗(𝑥)}. Thus, 𝑔^∞_𝐺1(𝑥),𝑦^∗𝑔^∞_𝐺1(𝑥=𝑔_∞𝐺1)(𝑥),ℎ𝑔^∞_𝐺1(𝑥),⋅_𝐺2^∞𝑦∈𝐹_{𝑓𝐺1𝐺2}=𝐹_𝑓. (18) It is easy to see that the triangular operator 𝑓_𝐺1𝐺2=(𝑔_𝐺1,ℎ(𝑥,⋅)_𝐺2) satisfies the conditions from the fibre contraction principle and, thus, we get the conclusion.

Let us consider, in particular, the following iterative algorithm 𝑥_𝑛+1=1−𝜆₁𝑥_𝑛+𝜆₁𝑔𝑥_𝑛,𝑦_𝑛+1=ℎ𝑥_𝑛,⋅_𝜆2𝑦_𝑛=1−𝜆₂𝑦_𝑛+𝜆₂ℎ𝑥_𝑛,𝑦_𝑛 (19) for starting points 𝑥₀∈𝑋, 𝑦₀∈𝑌, 𝜆₁,𝜆₂∈]0;1[, 𝑔∶𝑋→𝑋, ℎ∶𝑋×𝑌→𝑌, 𝑋∈𝑃_𝑐𝑣(𝑉₁), 𝑌∈𝑃_𝑐𝑣(𝑉₂), 𝑉₁, 𝑉₂ are vectorial spaces. The operator 𝑓_𝜆1𝜆2∶𝑋×𝑌→𝑋×𝑌𝑓_𝜆1𝜆2𝑥,𝑦∶=𝑔_𝜆1(𝑥),ℎ(𝑥,⋅)_𝜆2𝑦 (20) is an admissible perturbation of 𝑓 and 𝐹_𝑓=𝐹_{𝑓𝜆1𝜆2}. (21)

Corollary 9. Let 𝑉₁ be a Hilbert space, (𝑉₂,‖⋅‖) a Banach space, and 𝑋∈𝑃_{𝑐𝑙,𝑐𝑣}(𝑉₁), 𝑌∈𝑃_{𝑐𝑙,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 𝜆₁∈]0;1[ such that 𝑔_𝜆1 is WPO;(ii)there exists 𝜆₂∈]0;1[ such that ℎ(𝑥,⋅)_𝜆2∶𝑌→𝑌 is an 𝛼-contraction, for all 𝑥∈𝑋;(iii)𝑔 and ℎ are continuous.
Then(a)𝐹_𝑓=𝐹_{𝑓𝜆1𝜆2}={(𝑔^∞_𝜆1(𝑥),(ℎ(𝑔^∞_𝜆1(𝑥),⋅)_𝜆2)^∞(𝑦))∶𝑥∈𝑋,𝑦∈𝑌};(b)𝑓_𝜆1𝜆2 is a WPO; that is, the iterative algorithm defined by (19) is convergent to a fixed point of 𝑓;(c)if 𝑔_𝜆1 is a PO, then 𝑓_𝜆1𝜆2 is a PO.

Proof. In this case we take 𝐺₁∶𝑋×𝑋→𝑋, 𝐺₁(𝑥₁,𝑥₂)=(1−𝜆₁)𝑥₁+𝜆₁𝑥₂, 𝐺₂∶𝑌×𝑌→𝑌, 𝐺₂(𝑦₁,𝑦₂)=(1−𝜆₂)𝑦₁+𝜆₂𝑦₂. Since 𝑔 and ℎ are continuous, then 𝑓_𝜆1𝜆2 is continuous and from Theorem 8 we get the conclusion.

Corollary 10. Let 𝑉₁ be a real Hilbert space, (𝑉₂,‖⋅‖) a Banach space, and 𝑋∈𝑃_{𝑏,𝑐𝑙,𝑐𝑣}(𝑉₁), 𝑌∈𝑃_{𝑐𝑙,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 𝜆₂∈]0;1[ such that ℎ(𝑥,⋅)_𝜆2∶𝑌→𝑌 is an 𝛼-contraction, for all 𝑥∈𝑋;(iii)ℎ is continuous.
Then(a)𝐹_𝑓=𝐹_{𝑓𝜆1𝜆2}={(𝑔^∞_𝜆1(𝑥),(ℎ(𝑔^∞_𝜆1(𝑥),⋅)_𝜆2)^∞(𝑦))∶𝑥∈𝑋,𝑦∈𝑌}, for any 𝜆₁∈]0;1[;(b)𝑓_𝜆1𝜆2 is a WPO; that is, the iterative algorithm, defined by (19), is convergent to a fixed point of 𝑓;(c)if 𝐹_𝑔𝜆1={𝑥^∗} then 𝑓_𝜆1𝜆2 is a PO.

Proof. In this case we take 𝐺₁∶𝑋×𝑋→𝑋, 𝐺₁(𝑥₁,𝑥₂)=(1−𝜆₁)𝑥₁+𝜆₁𝑥₂, 𝐺₂∶𝑌×𝑌→𝑌, 𝐺₂(𝑦₁,𝑦₂)=(1−𝜆₂)𝑦₁+𝜆₂𝑦₂. Condition (i) implies that 𝑥_𝑛=𝑔^𝑛_𝜆1(𝑥₀) converges (strongly) to a fixed point of 𝑔 for any 𝜆₁∈]0;1[, (see Theorem 3.2 in [26]), so 𝑔_𝜆1 is WPO. Since 𝑔 is nonexpansive and ℎ is continuous, then, also, 𝑓_𝜆1𝜆2 is continuous and from Corollary 9 we obtain the conclusion.

Corollary 11. Let 𝑉₁ be a real Hilbert space with the inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖, (𝑉₂,|⋅|) a Banach space and 𝑋∈𝑃_{𝑐𝑙,𝑐𝑣}(𝑉₁), 𝑌∈𝑃_{𝑐𝑙,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 generalized pseudocontraction; that is, there exists, 𝑟>0𝑔𝑥₁−𝑔𝑥₂,𝑥₁−𝑥₂≤𝑟²𝑥₁−𝑥₂², (22) for all 𝑥₁,𝑥₂∈𝑋, with 0<𝑟<1;(ii)  𝑔 is 𝑙_𝑔-Lipschitz, with 𝑙_𝑔>1;(iii)  there exists 𝜆₂∈]0;1[ such that ℎ(𝑥,⋅)_𝜆2∶𝑌→𝑌 is an 𝛼-contraction, for all 𝑥∈𝑋;(iv)   ℎ is continuous.
Then 𝑓_𝜆1𝜆2 is a PO, for all 𝜆₁∈]0;1[, satisfying 0<𝜆₁<2(1−𝑟)1−2𝑟+𝑙²_𝑔; (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 𝑔_𝜆1 is 𝑙_𝑔𝜆1-contraction for any 𝜆₁∈]0;1[ satisfying (23), with 𝑙_𝑔𝜆1=((1−𝜆₁)²+2𝜆₁(1−𝜆₁)𝑟+𝜆²₁𝑙_𝑔)^1/2 (see Theorem 3.6, in [26]), so 𝑔_𝜆1 is PO and 𝐹_𝑔=𝐹_𝑔𝜆1={𝑥^∗}. From the continuity of 𝑔 and ℎ we have that, also, 𝑓_𝜆1𝜆2 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 1𝑥=𝑥,𝑦=𝑥²+2𝑥𝑦 (24) on 𝑋×𝑌=[1/2;2]×[1/5;24]. We have the operators 𝑔∶𝑋→𝑋𝑔(𝑥)=1𝑥 (25) and ℎ∶𝑋×𝑌→𝑌ℎ𝑥,𝑦=𝑥²+2𝑥𝑦. (26) It is clear that a solution of the system (24) is a fixed point of the triangular operator 𝑓∶𝑋×𝑌→𝑋×𝑌𝑓𝑥,𝑦=𝑔(𝑥),ℎ𝑥,𝑦. (27) 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 𝜆₁∈]0;2/5[ and 𝜆₂∈]0;2/51[.

Proof. We have that 𝑔 is 𝑙_𝑔-Lipschitz, with 𝑙_𝑔=4, so 𝑔 is not a contraction. Notice that, for any 𝑥₀∈𝑋 and 𝑥₀≠1, the Picard iteration is an oscillatory sequence 𝑥₀,1𝑥₀,𝑥₀,1𝑥₀,…, (28) so it is not convergent. For 𝜆₁∈]0;1[ we have 𝑔_𝜆1(𝑥)=1−𝜆₁𝑥+𝜆₁𝑔(𝑥)=1−𝜆₁𝜆𝑥+¹𝑥,𝑔^′_𝜆1(𝑥)=1−𝜆₁−𝜆₁𝑥²; (29) therefore 𝑔^′_𝜆1(𝑥)≤max1−5𝜆₁,51−4𝜆₁. (30)
We have the following cases:(i)if 0<𝜆₁<1/5, then 𝑙_𝑔𝜆1=1−(5/4)𝜆₁ and 𝑙_𝑔𝜆1∈(3/4;1);(ii)if 1/5≤𝜆₁<8/25, then 𝑙_𝑔𝜆1=1−(5/4)𝜆₁ and 𝑙_𝑔𝜆1∈(3/5;3/4];(iii)if 8/25≤𝜆₁<2/5, then 𝑙_𝑔𝜆1=5𝜆₁−1 and 𝑙_𝑔𝜆1∈[3/5;1);(iv)if 2/5≤𝜆₁<1, then 𝑙_𝑔𝜆1=5𝜆₁−1>1, so 𝑔_𝜆1 is not a contraction.
For any 𝜆₁∈]0;2/5[ we get that 𝑔_𝜆1 is a contraction, so 𝑔_𝜆1 is PO. The smallest contraction constant 𝑙_𝑔𝜆1=3/5 is obtained for 𝜆₁=8/25, so for this choice we get the best rate convergence for (𝑥_𝑛).
For ℎ∶𝑋×𝑌→𝑌 we have 𝜕ℎ=𝜕𝑦𝑥,𝑦2𝑥𝑦²≤50, (31) for all (𝑥,𝑦)∈𝑋×𝑌; thus 𝑙_{ℎ(𝑥,⋅)}=50 which implies that ℎ(𝑥,⋅) is not a contraction. For 𝜆₂∈]0;1[ we obtain ℎ(𝑥,⋅)_𝜆2𝑦=1−𝜆₂𝑦+𝜆₂ℎ=𝑥,𝑦1−𝜆₂𝑦+𝜆₂𝑥²+𝜆₂𝑥𝑦,𝜕ℎ(𝑥,⋅)_𝜆2𝑦𝜕𝑦=1−𝜆₂−𝜆₂𝑥𝑦², (32) so 𝜕ℎ(𝑥,⋅)_𝜆2𝑦𝜕𝑦≤max1−51𝜆₂,1−1153𝜆1152². (33) We have the following cases:(i)if 0<𝜆₂<1/51, then 𝑙_{ℎ(𝑥,⋅)𝜆2}=1−(1153/1152)𝜆₂ and 𝑙_{ℎ(𝑥,⋅)𝜆2}∈(57599/58752;1)≅(0.98038;1);(ii)if 1/51≤𝜆₂<2304/59905, then 𝑙_{ℎ(𝑥,⋅)𝜆2}=1−(1153/1152)𝜆₂, and 𝑙_{ℎ(𝑥,⋅)𝜆2}∈(57599/59905;57599/58752]≅(0.96151;0.98038];(iii)if 2304/59905≤𝜆₂<2/51, then 𝑙_{ℎ(𝑥,⋅)𝜆2}=51𝜆₂−1, and 𝑙_{ℎ(𝑥,⋅)𝜆2}∈[57599/59905;1)≅[0.96151;1);(iv)if 2/51≤𝜆₂<1, then 𝑙_{ℎ(𝑥,⋅)𝜆2}=51𝜆₂−1>1, so ℎ(𝑥,⋅)_𝜆2 is not a contraction.
For any 𝜆₂∈]0;2/51[ we get that ℎ(𝑥,⋅)_𝜆2 is a contraction. The smallest contraction constant 𝑙_{ℎ(𝑥,⋅)𝜆2}=57599/59905≅0.96151 is obtain for 𝜆₂=2304/59905, so the fastest iterative algorithm (19) is obtained for 𝜆₁=8/25 and 𝜆₂=2304/59905.

4. Application

Let us consider the following functional equation with parameter 𝑥𝑡,𝜇=𝐹𝑡,𝑥𝑡,𝜇,𝜇,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(i)  𝐹∈𝐶([𝑎;𝑏]×ℝ×𝐼);(ii)  𝐹(𝑡,⋅,⋅)∈𝐶¹(ℝ×𝐼) for all 𝑡∈[𝑎;𝑏];(iii)  there exists 𝜆∈]0;1[ such that 𝜕)𝜕𝑢(1−𝜆𝑢+𝜆𝐹𝑡,𝑢,𝜇≤𝑞<1, (35) for all 𝑡∈[𝑎;𝑏], 𝑢∈ℝ, 𝜇∈𝐼.
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 𝑔(𝑥)𝑡,𝜇=𝐹𝑡,𝑥𝑡,𝜇,𝜇,𝑡∈[𝑎;𝑏],𝜇∈𝐼. (36) We consider on 𝐶([𝑎;𝑏]×𝐼) the supremum norm ‖⋅‖_∞. From (i), (ii), and (iii) we have that 𝑔_𝜆 is 𝑞-contraction, so 𝑔_𝜆 is PO; that is, (34) has in 𝑋 a unique solution 𝑥^∗ and 𝑔^𝑛_𝜆(𝑥₀)→𝑥^∗ as 𝑛→+∞ for all 𝑥₀∈𝑋.
Let us suppose that 𝑥^∗(𝑡,⋅)∈𝐶¹(𝐼). We have from 𝑥^∗𝑡,𝜇=𝐹𝑡,𝑥^∗𝑡,𝜇,𝜇 (37) that 𝜕𝑥^∗=𝜕𝜇𝑡,𝜇𝜕𝐹𝜕𝑢𝑡,𝑥^∗𝑡,𝜇,𝜇⋅𝜕𝑥^∗+𝜕𝜇𝑡,𝜇𝜕𝐹𝜕𝜇𝑡,𝑥^∗𝑡,𝜇,𝜇. (38) This leads us to consider the following operator ℎ∶𝑋×𝑋→𝑋 defined by ℎ𝑥,𝑦𝑡,𝜇=𝜕𝐹+𝜕𝑢𝑡,𝑥𝑡,𝜇,𝜇⋅𝑦𝑡,𝜇𝜕𝐹.𝜕𝜇𝑡,𝑥𝑡,𝜇,𝜇 (39) So, we have the operator 𝑓∶𝑋×𝑋→𝑋×𝑋𝑥,𝑦⟼𝑔(𝑥),ℎ𝑥,𝑦 (40) and the operator 𝑓_𝜆∶𝑋×𝑋→𝑋×𝑋𝑓_𝜆𝑥,𝑦=𝑔_𝜆(𝑥),ℎ_𝜆=)(𝑥),().𝑥,𝑦(1−𝜆𝑥+𝜆𝑔1−𝜆𝑦+𝜆ℎ𝑥,𝑦 (41) In our conditions, from Theorem 8 we have that 𝑓_𝜆 is PO and 𝐹_𝑓𝜆={(𝑥^∗,𝑦^∗)}.
Let us take 𝑥₀∈𝐶([𝑎;𝑏]×𝐼) such that 𝜕𝑥₀/𝜕𝜇∈𝐶([𝑎;𝑏]×𝐼) and consider the sequence 𝑥_𝑛+1=𝑔_𝜆𝑥_𝑛,𝑦_𝑛+1=ℎ_𝜆𝑥_𝑛,𝑦_𝑛 (42) with 𝑦₀=𝜕𝑥₀/𝜕𝜇. Since 𝑦₀=𝜕𝑥₀/𝜕𝜇, we can prove by induction that 𝑦_𝑛=𝜕𝑥_𝑛/𝜕𝜇. So, 𝑥−−−→_𝑛^‖⋅‖∞𝑥^∗𝑦as𝑛⟶+∞,_𝑛=𝜕𝑥_𝑛𝜕𝜇^‖⋅‖∞−−−→𝑦^∗as𝑛⟶+∞. (43) From this it follows that 𝑥^∗(𝑡,⋅)∈𝐶¹(𝐼) and 𝑦^∗=𝜕𝑥^∗/𝜕𝜇.

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.