# Fibre Contraction Principle with respect to an Iterative Algorithm

Authors: Marcel-Adrian Şerban
Journal: Journal of Operators
 (2013)

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## 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.


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## Body

## 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.LetX 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)
ifxn=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 ofc(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 anL-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}, Pb(X)≔{Y∈P(X)∣Y is bounded},Pcl(X)≔{Y∈P(X)∣Y is closed}, Pb,cl(X)≔Pb(X)∩Pcl(X).IfX is a Banach space, then Pcv(X)≔{Y∈P(X)∣Y is convex}Let(X,→) be an L-space.Definition 1.
An operatorf: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 operatorf: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.Iff:X→X is a WPO, then we may define the operator f∞:X→X by
(1)f∞(x)≔limn→∞fn(x).
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.
Letg: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 operatorf defined as follows:
(2)f:X×Y→X×Y,f(x,y)≔(g(x),h(x,y)).
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.
Thenf is a WPO and
(3)Ff={(g∞(x),(h(g∞(x),·))∞(y)):x∈X,y∈Y}.
Moreover, if g is a PO, then f is a PO and Ff={(x*,y*)}, where Fg={x*} and Fh(x*,·)={y*}.Theorem4 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 theadmissible 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

LetX be a nonempty set, and f:X→X, G:X×X→X be two operators. We consider the operator fG:X→X defined by
(4)fG(x)=G(x,f(x)).Definition 5 (Rus [25]).
We callfG 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(5)FfG=Ff,
but, in general,
(6)FfGn≠Ffn,n≥2.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
(7)G(x,y)≔(1-λ)x+λy.
ThenfG 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
(8)x0∈X,xn+1=(1-λ)xn+λf(xn).It is known that the Krasnoselskii iteration is convergent to a fixed point of the operatorsf: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(9)x0∈X,xn+1=G(xn,f(xn)),n∈ℕ.
By definition, this iterative process is convergent if and only if(10)xn⟶x*(x0)∈Ff,asn⟶∞,∀x0∈X.
We remark that xn=fGn(x0). So, this algorithm is convergent if and only if fG is WPO. If fG is WPO and an admissible perturbation of f, then
(11)fG∞:X⟶Ff
is a set retraction.We call this algorithm, Krasnoselskii algorithm corresponding toG or GK-algorithm.For other examples of iterative algorithms see Rus [25].


## 3. Fibre Contraction Principle for Admissible Perturbations

LetX and Y be two nonempty sets, g:X→X, h:X×Y→Y, and f:X×Y→X×Y the triangular operator
(12)f(x,y)≔(g(x),h(x,y)).
Let G1:X×X→X and G2:Y×Y→Y satisfying (A1) and (A2). Then gG1:X→X,
(13)gG1(x)≔G1(x,g(x)),
is an admissible perturbation of g and, for x∈X, h(x,·)G2:Y→Y,
(14)h(x,·)G2(y)≔G2(y,h(x,y))
is an admissible perturbation of h(x,·). In these settings, we have the corresponding iterative algorithm
(15)xn+1=gG1(xn)=G1(xn,g(xn)),yn+1=h(xn,·)G2(yn)=G2(yn,h(xn,yn))
for starting points x0∈X, y0∈Y, and the operator fG1G2:X×Y→X×Y(16)fG1G2(x,y)≔(gG1(x),h(x,·)G2(y))
is an admissible perturbation of f and
(17)Ff=FfG1G2.
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)
gG1 is a WPO;(ii)
h(x,·)G2:Y→Y is an α-contraction, for all x∈X;(iii)
fG1G2 is continuous.
Then(a)
Ff=FfG1G2={(gG1∞(x),(h(gG1∞(x),·)G2)∞(y)):x∈X,y∈Y};(b)
fG1G2 is a WPO; that is, the iterative algorithm, defined by (15), is convergent to a fixed point of f;(c)
ifgG1 is a PO, then fG1G2 is a PO.Proof.
(a) From (i) and the fact thatG1 satisfies (A1) and (A2) we have that gG1∞(x)∈Fg. From (ii) we have that h(x,·)G2 is PO for a fixed x∈X, so Fh(x,·)G2={y*(x)} and (h(x,·)G2)∞(y)=y*(x) for all y∈Y. But G2, also, satisfies (A1) and (A2); therefore Fh(x,·)=Fh(x,·)G2={y*(x)}. Thus,
(18)(gG1∞(x),y*(gG1∞(x)))=(gG1∞(x),(h(gG1∞(x),·)G2)∞(y))∈FfG1G2=Ff.
It is easy to see that the triangular operator fG1G2=(gG1,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(19)xn+1=(1-λ1)xn+λ1g(xn),yn+1=h(xn,·)λ2(yn)=(1-λ2)yn+λ2h(xn,yn)
for starting points x0∈X, y0∈Y, λ1,λ2∈]0;1[, g:X→X, h:X×Y→Y, X∈Pcv(V1), Y∈Pcv(V2), V1, V2 are vectorial spaces. The operator fλ1λ2:X×Y→X×Y(20)fλ1λ2(x,y)≔(gλ1(x),h(x,·)λ2(y))
is an admissible perturbation of f and
(21)Ff=Ffλ1λ2.Corollary 9.
LetV1 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 existsλ1∈]0;1[ such that gλ1 is WPO;(ii)
there existsλ2∈]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)
Ff=Ffλ1λ2={(gλ1∞(x),(h(gλ1∞(x),·)λ2)∞(y)):x∈X,y∈Y};(b)
fλ1λ2 is a WPO; that is, the iterative algorithm defined by (19) is convergent to a fixed point of f;(c)
ifgλ1 is a PO, then fλ1λ2 is a PO.Proof.
In this case we takeG1: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 fλ1λ2 is continuous and from Theorem 8 we get the conclusion.Corollary 10.
LetV1 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 existsλ2∈]0;1[ such that h(x,·)λ2:Y→Y is an α-contraction, for all x∈X;(iii)
h is continuous.
Then(a)
Ff=Ffλ1λ2={(gλ1∞(x),(h(gλ1∞(x),·)λ2)∞(y)):x∈X,y∈Y}, for any λ1∈]0;1[;(b)
fλ1λ2 is a WPO; that is, the iterative algorithm, defined by (19), is convergent to a fixed point of f;(c)
ifFgλ1={x*} then fλ1λ2 is a PO.Proof.
In this case we takeG1: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 xn=gλ1n(x0) converges (strongly) to a fixed point of g for any λ1∈]0;1[, (see Theorem 3.2 in [26]), so gλ1 is WPO. Since g is nonexpansive and h is continuous, then, also, fλ1λ2 is continuous and from Corollary 9 we obtain the conclusion.Corollary 11.
LetV1 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:(i)
g is a generalized pseudocontraction; that is, there exists, r>0(22)〈g(x1)-g(x2),x1-x2〉≤r2∥x1-x2∥2,for allx1,x2∈X, with 0<r<1;(ii)
g is lg-Lipschitz, with lg>1;(iii)
there existsλ2∈]0;1[ such that h(x,·)λ2:Y→Y is an α-contraction, for all x∈X;(iv)
h is continuous.
Thenfλ1λ2 is a PO, for all λ1∈]0;1[, satisfying
(23)0<λ1<2(1-r)1-2r+lg2;
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 thatgλ1 is lgλ1-contraction for any λ1∈]0;1[ satisfying (23), with lgλ1=((1-λ1)2+2λ1(1-λ1)r+λ12lg)1/2 (see Theorem 3.6, in [26]), so gλ1 is PO and Fg=Fgλ1={x*}. From the continuity of g and h we have that, also, fλ1λ2 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(24)x=1x,y=x2+2xy
on X×Y=[1/2;2]×[1/5;24]. We have the operators g:X→X(25)g(x)=1x
and h:X×Y→Y(26)h(x,y)=x2+2xy.
It is clear that a solution of the system (24) is a fixed point of the triangular operator f:X×Y→X×Y(27)f(x,y)=(g(x),h(x,y)).
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 thatg 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
(28){x0,1x0,x0,1x0,…},
so it is not convergent. For λ1∈]0;1[ we have
(29)gλ1(x)=(1-λ1)x+λ1g(x)=(1-λ1)x+λ1x,gλ1′(x)=1-λ1-λ1x2;
therefore
(30)|gλ1′(x)|≤max{|1-5λ1|,|1-54λ1|}.
We have the following cases:(i)
if0<λ1<1/5, then lgλ1=1-(5/4)λ1 and lgλ1∈(3/4;1);(ii)
if1/5≤λ1<8/25, then lgλ1=1-(5/4)λ1 and lgλ1∈(3/5;3/4];(iii)
if8/25≤λ1<2/5, then lgλ1=5λ1-1 and lgλ1∈[3/5;1);(iv)
if2/5≤λ1<1, then lgλ1=5λ1-1>1, so gλ1 is not a contraction.
For anyλ1∈]0;2/5[ we get that gλ1 is a contraction, so gλ1 is PO. The smallest contraction constant lgλ1=3/5 is obtained for λ1=8/25, so for this choice we get the best rate convergence for (xn).
Forh:X×Y→Y we have
(31)|∂h∂y(x,y)|=2xy2≤50,
for all (x,y)∈X×Y; thus lh(x,·)=50 which implies that h(x,·) is not a contraction. For λ2∈]0;1[ we obtain
(32)h(x,·)λ2(y)=(1-λ2)y+λ2h(x,y)=(1-λ2)y+λ2x2+λ2xy,∂h(x,·)λ2(y)∂y=1-λ2-λ2xy2,
so
(33)|∂h(x,·)λ2(y)∂y|≤max{|1-51λ2|,|1-11531152λ2|}.
We have the following cases:(i)
if0<λ2<1/51, then lh(x,·)λ2=1-(1153/1152)λ2 and lh(x,·)λ2∈(57599/58752;1)≅(0.98038;1);(ii)
if1/51≤λ2<2304/59905, then lh(x,·)λ2=1-(1153/1152)λ2, and lh(x,·)λ2∈(57599/59905;57599/58752]≅(0.96151;0.98038];(iii)
if2304/59905≤λ2<2/51, then lh(x,·)λ2=51λ2-1, and lh(x,·)λ2∈[57599/59905;1)≅[0.96151;1);(iv)
if2/51≤λ2<1, then lh(x,·)λ2=51λ2-1>1, so h(x,·)λ2 is not a contraction.
For anyλ2∈]0;2/51[ we get that h(x,·)λ2 is a contraction. The smallest contraction constant lh(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(34)x(t,μ)=F(t,x(t,μ),μ),t∈[a;b],μ∈I⊂ℝ,
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 Theorem8 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 existsλ∈]0;1[ such that
(35)|∂∂u[(1-λ)u+λF(t,u,μ)]|≤q<1,for allt∈[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.
LetX≔C([a;b]×I) and g:X→X be defined by
(36)g(x)(t,μ)=F(t,x(t,μ),μ),t∈[a;b],μ∈I.
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.
Let us suppose thatx*(t,·)∈C1(I). We have from
(37)x*(t,μ)=F(t,x*(t,μ),μ)
that
(38)∂x*∂μ(t,μ)=∂F∂u(t,x*(t,μ),μ)·∂x*∂μ(t,μ)+∂F∂μ(t,x*(t,μ),μ).
This leads us to consider the following operator h:X×X→X defined by
(39)h(x,y)(t,μ)=∂F∂u(t,x(t,μ),μ)·y(t,μ)+∂F∂μ(t,x(t,μ),μ).
So, we have the operator f:X×X→X×X(40)(x,y)⟼(g(x),h(x,y))
and the operator fλ:X×X→X×X(41)fλ(x,y)=(gλ(x),hλ(x,y))=((1-λ)x+λg(x),(1-λ)y+λh(x,y)).
In our conditions, from Theorem 8 we have that fλ is PO and Ffλ={(x*,y*)}.
Let us takex0∈C([a;b]×I) such that ∂x0/∂μ∈C([a;b]×I) and consider the sequence
(42)xn+1=gλ(xn),yn+1=hλ(xn,yn)
with y0=∂x0/∂μ. Since y0=∂x0/∂μ, we can prove by induction that yn=∂xn/∂μ. So,
(43)xn→∥·∥∞x*asn⟶+∞,yn=∂xn∂μ→∥·∥∞y*asn⟶+∞.
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].



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