Stable Iteration Procedures in Metric Spaces which Generalize a Picard-Type Iteration

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Volume 2010, Article ID 953091,15pages doi:10.1155/2010/953091

Research Article

Stable Iteration Procedures in Metric Spaces which

Generalize a Picard-Type Iteration

M. De la Sen

IIDP, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), Aptdo. 644, Bilbao, Spain

Correspondence should be addressed to M. De la Sen,manuel.delasen@ehu.es

Received 25 March 2010; Accepted 11 July 2010

Academic Editor: Dominguez Benavides

Copyrightq2010 M. De la Sen. 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.

This paper investigates the stability of iteration procedures defined by continuous functions acting on self-maps in continuous metric spaces. Some of the obtained results extend the contraction principle to the use of altering-distance functions and extended altering-distance functions, the last ones being piecewise continuous. The conditions for the maps to be contractive for the achievement of stability of the iteration process can be relaxed to the fulfilment of being large contractions or to be subject to altering-distance functions or extended altering functions.

1. Introduction

Banach contraction principle is a very basic and useful result of Mathematical Analysis

1–7. Basic applications of this principle are related to stability of both continuous-time and discrete-time dynamic systems4,8, including the case of high-complexity models for dynamic systems consisting of functional diﬀerential equations by the presence of delays

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exhaustivelysee, e.g.,5,19–22. The Picard and approximate Picard methods have been also used in classical papers for proving the existence and uniqueness of solutions in many diﬀerential equations including those of Sobolev typesee, e.g.,23.

This paper presents some generalizations of results concerning the stability of iterations in the sense that the iteration scheme subject to error sequences converges asymptotically to its nominal fixed point provided that the iteration error converges asymptotically to zero. Several generalizations are discussed in the framework of stability of iteration schemes in complete metric spaces including:

athe use of altering-distance functions Definition 1.1 1, 2, and the so-called then defined extended altering functions Definition 2.1in Section 2 where the continuous altering functions are allowed to be piecewise continuous;

bthe use of iteration schemes which are based on continuous functions which modify the Picard iteration scheme5,6;

cthe removal of the common hypothesis in the context ofT-stability that the set of fixed points of the iteration scheme is nonempty by guaranteeing that this is in fact true under contractive mappings, large contractions, or altering- and extended altering-distance functions,1–4,6.

Definition 1.1see1 altering-distance function. A monotone nondecreasing functionϕ ∈ C0R0,R0, withϕx 0, if and only ifx0, is said to be an altering-distance function.

If X, d is a complete metric space, T : X → X is a self-mapping on X, and

ϕdTx, Ty ≤ cϕdx, y, for allx, y ∈ X and some real constantc ∈ 0,1, thenT has a unique fixed point1,2. This result is extendable to the use of monotone nondecreasing functionsϕ∈C0R0,R0satisfying

ϕdTx, Ty≤ϕdx, y−φdx, y, ∀x, y∈X, 1.1

for some monotone nondecreasing functionϕ ∈ C0R0,R0satisfyingφt ϕt 0 ⇔ t 0. Those results are directly extended to monotone nondecreasing piecewise continuous functions being continuous at “0” after a preliminary “ad hoc” definition in the subsequent section.

2. Fixed Point Properties Related to Altering- and

Extended Altering-Distance Functions

Since ϕ ∈ P C0R0,R0 but continuous at t 0, it can possess bounded isolated discontinuities onR and it is necessary to reflect this fact in the notation as follows. The leftresp., rightlimit ofϕ att dx, yis simply denoted byϕdx, y, instead of using the more cumbersome classical notationϕdx, y− resp., byϕdx, yinstead of using the more cumbersomeϕdx, y. Sinceϕis an extended altering-distance function, then continuous att 0, ϕ0 ϕ0 0. Ifϕ is continuous at a givent dx, y > 0, then

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Definition 2.1extended altering-distance function. A monotone nondecreasing functionϕ∈ P C0R0,R0being continuous at “0”, withϕx 0, if and only ifx 0, is said to be an

extended altering-distance function.

Theorem 2.2. LetX, dbe a complete metric space andT :X → Xbe a self-mapping onX. Then, the following properties hold.

iAssume thatϕ∈C0R0,R0is an altering-distance function such thatϕdTx, Ty≤ cϕdx, y, for allx, y ∈Xfor some real constantc∈0,1. ThenT has a unique fixed point [1].

iiAssume that ϕ ∈ P C0R0,R0 is an extended altering-distance function such that ϕdTx, Ty≤cdx, yϕdx, yandϕdTx, Ty≤cdx, yϕdx, y, for allx, y∈Xfor some real function

c∈P C0R0,R0∩0,1 2.1

defined by

cdx, y

⎧ ⎪ ⎨ ⎪ ⎩

1−φ

dx, y

ϕdx, y, if x /y,

0, if xy,

cdx, y

⎧ ⎪ ⎨ ⎪ ⎩

1− φ

dx, y

ϕdx, y, if x /y,

0, if xy,

2.2

for allx, y∈Xfor some monotone nondecreasing functionφ∈C0R0,R0satisfyingφt< ϕt, for allt∈R andφt ϕt 0, if and only ift 0.Thenc :R0 → R0∩0,1is monotone

nondecreasing andT has a unique fixed point. In particular, ifϕdx, y dx, yso that

cdx, y

⎧ ⎪ ⎨ ⎪ ⎩

1−φ

dx, y

dx, y , ifx /y,

0, ifxy,

2.3

for all x, y ∈ X for some monotone nondecreasing function φ ∈ C0R0,R0 satisfying φdx, y < dx, y,for all_{x, y /}x ∈ X andφdx, y 0, if and only ify x ∈ X, thenT

has a unique fixed point.

Proof of Property (ii). Note that cdx, x 1 − limy→xφdx, y/ϕdx, y 1 −

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“0” withφ0 ϕ0 0. Note that after taking left and right limits at each nonnegative real argument

1> cdx, y 1−φ

dx, y

ϕdx, y ≥c

_d_{x, y}₁₋ φ

dx, y ϕdx, y

≥cdx, y1−φ

dx, y ϕdx, y,

cdz, z 1− φdz, z

ϕdz, z cdz, z 1−

φdz, z ϕdz, z 0,

2.4

for allx, x/x, y, y/y, z∈X, such thatdx, y≥dx, y, since

0< φdx, y< ϕdx, y≤ϕdx, y≤ϕdx, y > φdx, y

ϕdz, z ϕdz, z φdz, z 0, ∀x, x/x, y, y/y∈X.

2.5

Then,c:R0 → R0∩0,1is monotone nondecreasing from simple inspection of the above

properties. Thus,

ϕdTx, Ty≤cϕdx, y≤cdx, yϕdx, y≤cdx, yϕdx, y 2.6

so that

ϕdTx, Ty≤ϕdTx, Ty≤cdx, yϕdx, y< ϕdx, y, ∀x/y∈X. 2.7

Now, it is proven by contradiction that there is no ε ∈ R such that dx, y ≥ ε, for any

givenx_/y∈X.Take two arbitraryx0_/y0∈X.Assume thatϕdTj_{x0, T}j_y0_≥_ε_{, for all}

j ∈k∪ {0}, and some givenk ∈Z, so that ifϕdTjx0, Tjy0≥εalso for allj ∈Z0, then

for someε0∈R0,

ε≤ϕdTkNx0, TKNy0 ≤

N

j1

cdTkjx0, Tkjy0 ϕdx, y

εε0N

j1

cdTkjx0, Tkjy0 ,

2.8

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what leads to a contradiction. Thus, there is noε∈Rsuch thatϕdTkx0, Tky0≥ε, for all k∈Z0for any givenx0/y0∈X. As a result, the subsequent relations are true:

lim

k→ ∞

k

j1

cdTjx0, Tjy0 0

⇒ lim

k→ ∞ϕ

dTkx0, Tky0 lim

k→ ∞ϕ

_d_Tk_{x0, T}k_y0 ₀_⇐⇒ _lim k→ ∞d

Tkx0, Tky0 0, 2.9

for allx0_/y0 ∈ X, with the above limits sinceϕtis continuous at t 0 andϕt 0, if and only ift 0. Furthermore, any sequence{xk}withxk Tkx, for allx∈X, is a Cauchy

sequence since for any arbitrarily small prefixed constantε∈R, there exist sequences{Nk},

{nk}, and{mk}of nonnegative integers satisfyingZ0 Nk → ∞;nk > Nk, mk > nk, such

that

ϕdTmknk1_{x, T}nk1_x

≤nk

j1

cdTmkj_{x, T}j_x

≤ε. 2.10

Thus, there is a unique z ∈ clX which is in FT, the set of fixed points of T, that is,z Tz limk→ ∞Tkx, for all x ∈ X. SinceX, dis a complete metric space and the sequence

{xk}with xk Tkxis a Cauchy sequence, for all x ∈ X,then X ⊃ FT {z}. It holds

trivially that all the above proof also holds for special case ϕdx, y dx, yand some monotone nondecreasing function φ ∈ C0R0,R0 satisfying φdx, y < dx, y i.e., T :X → Xis a weak contractionas may be proven3 see also24. Propertyiihas been fully proven.

Theorem2.2might be linked to the concept of large contraction which is less restrictive than that of contraction. The related discussion follows.

Definition 2.3 see4 large contraction. LetX, dbe a complete metric space. Then, the self-mapping T : X → X on X is said to be a large contraction, if dTx, Ty < dx, y, for allx/y ∈ X, and if for any given ε ∈ R, such thatdTx, Ty ≥ ε, then there exist δδε∈0,1, such thatdTx, Ty≤δdx, y.

It turns out that a contraction is also a large contraction with δ ∈ 0,1 being independent ofε in Definition2.3. The following result proves that the self-mappingT on

Xsatisfying Theorem2.2iiis a large contraction.

Proposition 2.4. Let X, d be a complete metric space and T : X → X be a self-mapping on

X. Ifϕ ∈ P C0R0,R0is a modified altering-distance function which satisfies the conditions of

Theorem2.2(ii), thenT is a large contraction.

Proof. Given ϕdTx, Ty ≤ cdx, yϕdx, y < ϕdx, y, for all x/y ∈ X, since

cdx, y < 1, if dx, y > 0. Since ϕ ∈ P C0R0,R0 is an extended altering-distance

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∃δ δε < 1, such thatdTx, Ty ≤δdx, y. Takex/y∈ X, such thatdx, y > 0, and assume thatdTx, Ty≥dx, y.Sinceϕ∈P C0R0,R0is monotone nondecreasing, then ϕdTx, Ty≥ϕdx, y>0, for allx_/y∈X, and one also gets that

ϕdTx, Ty≥maxϕdTx, Ty, ϕdx, y≥ϕdx, y, ∀x/y∈X. 2.11

Then,

ϕdx, y> ϕdx, y−φdx, y≥ϕdTx, Ty≥ϕdx, y>0,

ϕdx, y> ϕdx, y−φdx, y≥ϕdTx, Ty≥ϕdx, y≥ϕdx, y>0, 2.12

which are two contradictions. Thus,ρdx, y>0⇒dTx, Ty≤δρdx, y< ρ, for some

δρ<1 andTis a large contraction.

It is now proven that the sequence{dx, Tk_x_}_{is uniformly bounded if Theorem}_2.2_ii

holds.

Proposition 2.5. Let X, d be a complete metric space and T : X → X be a self-mapping on

X. Ifϕ ∈ P C0R0,R0is a modified altering-distance function which satisfies the conditions of Theorem2.2(ii), thendx, Tk_x_≤_{L <}_∞_{, for all}_x_∈_X_and_k_∈_Z

0.

Proof. Proceed by contradiction by assuming that dx, Tk_x _≤ _{L <} _{∞, for all} _x _∈ _X _and

k ∈ Z0, is false so that{dx, Tkx},k ∈ Z0 is unbounded. Thus, there is a subsequence

{Tjk}

jk∈Zα⊂Z0 of self-mappings onX, withZαjk → ∞, asZ0 k → ∞, such that the real

subsequence{dx, Tjkx}

jk∈Zαis strictly monotone increasing so that it diverges to∞, so that

Lk1dx, Tk1x> LkandLk → ∞, asZαk → ∞. Sinceϕ∈P C0R0,R0is monotone

nondecreasing, one gets

ϕLk1≥max

ϕLk1, ϕLk

≥ϕLk, 2.13

sinceϕ∈P C0R0,R0is monotone nondecreasing. Since the inequalities are nonstrict, the

above subsequences might either converge to nonnegative real limitsϕ∞x, zandϕ∞x, z,

or diverge to∞. The event thatϕ∞x, zandϕ∞x, zare one finite and the other infinity is

not possible sinceϕ∈P C0R0,R0so that any existing discontinuity is a finite-jump type

discontinuity. Thus, both limits are either finite, although eventually distinct or both are∞ so thatϕLk → ϕ∞x, z ≤ ∞andϕLk → ϕ_∞x, z ≤ ∞and simultaneously finite or

infinityasZαk → ∞, wherezzx Tx∈Xfor the givenx∈X. Such azalways exists

inXfor each givenx∈XsinceT is a self-mapping onX.Then,

∞ ≥ϕ∞x, z←−ϕLk1≤ϕLk−φLk−→ϕ∞x, z−φLk, asZαk−→ ∞,

2.14

which is a contradiction unlessφLk → 0, asZαk → ∞ ⇒Lk → 0, asZαk → ∞, since

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false and then{dx, Tk_x_},_k _∈_Z

0being unbounded fails so that the contradiction follows.

The right-limit convergence ϕLk → ϕ∞x, z ≤ ∞leads to the same conclusion. As a

result, there is nox∈ xsuch that{dx, Tk_x_},_k _∈_Z

0 is unbounded and the result is fully

proven.

An alternative proof to that of Theorem2.2iirelated to the existence of a unique fixed point inX, follows directly by using Theorem 1.2.4 in4sinceT is a large contraction and the sequence{dx, Tk_x_}_{is uniformly bounded}_Propositions_2.4_and_2.5_.

Proposition 2.6. Let x, d be a complete metric space andT : X → X be a large contraction. If ϕ ∈ P C0R0,R0 is a modified altering-distance function which satisfies the conditions of

Theorem2.2(ii), thenT has a unique fixed point inX.

Proof. T is a large contraction from Proposition 2.4, since it fulfils Theorem 2.2ii. Also,

dx, Tk_x_≤_{L <} _{∞, for all}_x_∈ _X_and_k _∈_Z

0 from Proposition2.5. Thus, from4, Theorem

1.2.4,T has a unique fixed point inX.

The following result is a direct consequence of Theorem2.2, Propositions2.4and2.5.

Proposition 2.7. LetX, dbe a complete metric space andT :X → Xbe a weak contraction onX. Then,dx, Tk_x_≤_{L <}_∞_{, for all}_x_∈_X_and_k_∈_Z

0andT has a unique fixed point onX.

3. f, T

-Stability Related to a Class of Nonlinear Iterations Related to

Distance and Altering-Distance Functions

Assume that X, d is a complete metric space, T : X → X is a self-mapping onX. The iteration process xk1 fTxk has a fixed point ifFf, T : {z ∈ x : z fTz}/∅. A

necessary condition forf : X → X to have a fixed point is that it to be injective. The T -stability of the Picard iteration has been investigated in a set of paperssee, e.g.,5,19,20. The Picard iteration is said to beT-stable if limk→ ∞dxk1, Txk 0 ⇒ limk→ ∞xk z∈ X,

for allx0∈X. The subsequent result is an extension of a previous one in5for the so-called

f, T-stability of the iterationxk1 fTxk, for all k ∈ Z0 if the pairf, T satisfies the

so-calledL, hproperty defined by

dfTx, q≤LdfTx, xhdx, q 3.1

for allx∈X;q∈Ff, T:{z∈X :zfTz}, with 0≤h <1 andL≥0, provided that the set of fixed pointsFf, Tis nonempty. Iff :X → Xis identity, then the above property is stated asT satisfying theL, hproperty.

Theorem 3.1. Assume that

1 X, dis a complete metric space,f :X → Xis a continuous mapping, andT :X → X

is a self-mapping onXsuch that the set of fixed pointsFf, Tof the iteration procedure

xk1 fTxk, for allk∈Z0, is nonempty;

2the pair f, T satisfies the L, h property; that is, dfTx, q ≤ LdfT, x, x

hdx, q; for allq∈Ff, T:{z∈X:zfTz}with0≤h <1andL≥0;

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Then, the iteration procedurexk1fTxk, for allk∈Z0, isf, T-stable and it possesses a unique

fixed point.

Proof. For any givenq ∈Ff, T, which exists sinceFf, T_/∅, and for allxk ∈ X such that

xxk1fTxk εkwith{εk}being the computation error sequence, one has

dfTxk, q

≤LdfTxk, xk

hdxk, q

≤LhdfTxk, xk

hdq, fTxk

⇒1−hdfTxk, q

≤LhdfTxk, xk

.

3.2

Since 0≤h <1,L≥0, anddfTxk, xk → 0, ask → ∞, then,

dfTxk, q

−→dfTxk, xk

−→0, ask−→ ∞ 3.3

from3.2,5,6. Also,dq, xk≤dxk, fTxk dq, fTxk → 0, ask → ∞ ⇒dq, xk →

0, ask → ∞. Also,

dfTxk, xk1

≤dfTxk, q

dq, xk1

≤LhdfTxk, xk

hdq, fTxk

dq, xk1

≤LhdfTxk, xk

hdq, fTxk

dq, xk

dxk, xk1

−→dxk, xk1,

3.4

ask → ∞, sincedq, xk → dfTxk, q → dfTxk, xk → 0, ask → ∞. From the above

inequalities eitherdfTxk, xk1 → dxk, xk1 → 0, ask → ∞, or lim infk→ ∞dxk, xk1>

0, withdfTxk, xk → 0,ask → ∞, but in this second case,dq, xk → 0, ask → ∞,

is false so that q /∈Ff, T. Then, dfTxk, xk1 → dxk, xk1 → 0, as k → ∞. Thus,

dfTxk, xk1 → dfTxk, xk → 0, ask → ∞from3.4. Then,xk1 → fTxk → xk

andfT, xk → q, ask → ∞. Sincef :X → X is injective,xk → q, ask → ∞so thatf is

T-stable. It is proven by contradiction that the fixed point of the iteration procedurexk1 fTxk, for allk∈Z0is unique. Assume that there existsp /q∈Ff, T. Then,

0_/dp, qdfTp, q≤Ldp, fTphdp, qhdp, q< dp, q0, 3.5

what is impossible if_{p /}q. Then,Ff, T {q}.

Theorem3.1is now extended by extending theL, hproperty of the pairf, Tto that of the tripleϕ, f, T, whereϕ:R0 → R0is an appropriate continuous function.

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2ϕ : R0 → R0 is continuous, satisfiesϕx 0 ⇔ x 0, possesses the subadditive

property, and, furthermore, the triple ϕ, f, T satisfies the L, h property defined by

ϕdfTx, q ≤ LϕdfT, x, x +hϕdx, q;for all q ∈ Ff, T : {z ∈ X :

zfTz}with0≤h <1andL≥0;

3limk→ ∞dfTxk, xk 0, for allx0∈X.

Then, the iteration procedurexk1fTxk, for allk∈Z0isf, T-stable and it possesses a unique

fixed point.

Proof. For any givenq ∈Ff, T, which exists sinceFf, T_/∅, and for allxk ∈ X such that

xxk1 fTxk εk, with{εk}being the computation error sequence and, sinceϕ:R0 →

R0possesses the sub-additive property, one has

ϕdfTxk, q

≤LϕdfT, xk, xk

hϕdxk, q

≤LϕdfT, xk, xk

hϕdxk, fT, xk

dfT, xk, q

≤LϕdfT, xk, xk

hϕdxk, fT, xk

ϕdfT, xk, q

⇒1−hϕdfTxk, q

≤LhϕdfTxk, xk

⇒ 0←ϕdfTxk, q

≤ Lh 1−hϕ

dfTxk, xk

−→0 ask−→ ∞ 3.6

according to Hypothesis3since 0 ≤ h < 1 andLh ≥ 0 from Hypothesis2. Sinceϕ :

R0 → R0 is everywhere continuous and satisfiesϕx 0 ⇔ x 0, thendfTxk, q →

dfTxk, xk → 0 ask → ∞. Also,dq, xk ≤ dxk, fTxk dq, fTxk → 0 ask →

∞ ⇒dq, xk → 0 ask → ∞. The remaining of the proof follows with the same arguments

as in that of Theorem3.1.

xk1 fTxk, for allk∈Z0, is nonempty;

2ϕ :R0 → R0 andφ : R0 → R0 are both continuous and monotone nondecreasing

while satisfyingϕx φx 0 ⇔x 0, and, furthermore, the quadrupleϕ, φ, f, T satisfies the L, h property: ϕdfTxk, q ≤ ϕLdfTxk, xk hdxk, q −

φLdfTxk, xk hdxk, q, for allx ∈ X; and for allq ∈ Ff, T : {z ∈ X :

zfTz}with0≤h <1andL≥0;

3limk→ ∞ϕdfTxk, q 0, for allx0∈X.

Then, the iteration procedurexk1fTxk, for allk∈Z0isf, T-stable and it possesses a unique

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Proof. Since ϕ : R0 → R0 and ψ : R0 → R0 are both continuous and monotone

nondecreasing then Hypothesis2implies thatdfTx, q ≤ LdfT, x, x hdx, q, for all k ∈ Z0 and x ∈ X; for all q ∈ Ff, T : {z ∈ X : z fTz} with 0 ≤ h < 1

andL ≥ 0 which is Hypothesis2of Theorem3.1. Furthermore, limk→ ∞ϕdfTxk, q

ϕlimk→ ∞dfTxk, q 0⇒limk→ ∞dfTxk, q 0 from the continuity ofϕ:R0 → R0

everywhere within its definition domainR₀ and its propertyϕx ψx 0 ⇔ x 0.

Thus, the proof follows as in Theorem 3.1 since Hypothesis 1 to 3 of this theorem hold.

The following direct particular result of Theorems3.1to3.3follows.

Corollary 3.4. Theorems3.1,3.2, and3.3hold “mutatis-mutandis” stated for the functionf :X → Xbeing the identity mapping onXandFI, T FT/∅.

Corollary3.4referred to Theorem3.1was first proven in5. It is now of interest the removal of the condition of the set of fixed points to be nonempty by guaranteeing that is in fact nonempty consisting of a unique element under extra contractive properties of the pair

f, T. The following result holds.

Theorem 3.5. The following two properties hold.

iConsider the Picard T-iteration process xk1 Txk, for allx ∈ X and k ∈ Z0. IfT

satisfies the L, hproperty while it is ak-contraction (i.e., a contractive mapping with constant0≤k <1, thenFT {q}for someq∈X, and, furthermore,

dTx, q≤ Lh

1−hdx, Tx, d

T2x, q ≤ kLh

1−h dTx, x, 3.7

dT2x, Tx ≤min

kLdx, Tx kh1dx, q, Lkh1dx, Tx

hh1dx, q,

LkLh

1−h

dTx, x hdx, q.

3.8

iiConsider the iteration processxk1 fTxk, for allx ∈ X andk ∈ Z0. IfT satisfies

theL, hproperty while the pairf, Tis ak-contraction (i.e., a contractive mapping with constant0≤k <1, thenFf, T {q}for someq∈x, and, furthermore,

dfTx, q≤ Lh

1−hd

x, fTx, dfT2x , q ≤ kLh

1−h d

fTx, x, 3.9

dfT2x , fTx ≤min

kLdx, fTxkh1dx, q, Lkh1dx, fTx

hh1dx, q,

LkLh

1−h

dTx, x hdx, q.

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Proof. iEquation3.7follows from theL, hproperty leading to

dTx, q≤Ldx, Tx hdx, q≤Ldx, Tx hdx, Tx hdTx, q

⇒ dTx, q≤ Lh

1−hdx, Tx;d

T2x, q

≤ Lh 1−hd

T2x, Tx ≤kLh

1−hdTx, x,

3.11

sinceT isk-contractive, so that it possesses a unique fixed pointq ∈ X, and it satisfies the

L, h property with 0 ≤ h < 1. Equation 3.8 follows directly from the following three inequalities:

dT2x, Tx ≤kdTx, x≤kdTx, qdx, q

≤kLdx, Tx hdx, qkdx, q≤kLdx, Tx kh1dx, q

3.12

ForxzTx, for allx∈X, one gets

dT2x, Tx ≤dT2x, q dq, Tx

dTz, qdq, z≤Ldz, Tz h1dz, q

≤Lkdx, Tx h1Ldx, Tx hdx, q

≤Lkh1dx, Tx hh1dx, q

3.13

after using theL, hproperty, and

dT2x, Tx ≤dT2x, q dTx, q≤ kLh

1−h dTx, x d

Tx, q

≤ kLh

1−h dTx, x LdTx, x hd

x, q

≤

LkLh

1−h

dTx, x hdx, q

3.14

again with the use of theL, hproperty.

iiEquation3.9follows from theL, hproperty leading to

dfTx, q≤Ldx, fTxhdx, q≤Ldx, fTxhdx, fTxhdfTx, q

⇒dfTx, q≤ Lh

1−hd

x, fTx;

dfT2x , q ≤ Lh

1−hd

fT2x , fTx ≤kLh

1−hd

fTx, x,

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since the pairf, Tisk-contractive and it satisfies theL, hproperty with 0≤h <1. Equation

3.10follows directly from the following three inequalities:

dfT2x , fTx ≤kdfTx, x≤kdq, fTxdx, q

≤kLdfTx, xhdx, qkhdx, q

≤kLdx, fTxkh1dx, q,

3.16

since the pairf,Tisk-contractive. Then,

lim

k→ ∞d

fTk1x , fTkx d

lim

k→ ∞Tf

Tkx , lim

k→ ∞f

Tkx 0⇒qTq lim

k→ ∞f

Tkx , 3.17

for allx∈Xand someq∈Xindependent ofx0∈X.Ff, T {q}. Therefore, for anyx∈X, one gets

dfT2x , fTx ≤dfT2x , q dq, fTx

≤LdfT2x , fTx h1dfTx, q

≤LkdfTx, x h1LdfTx, xhdx, q

≤Lkh1dfTx, xhh1dx, q

3.18

after using theL, hproperty, and

dfT2x , fTx ≤dfT2x , q dfTx, q

≤ kLh 1−h d

fTx, xdfTx, q

≤ kLh 1−h d

fTx, xLdfTx, xhdx, q

≤

LkLh

1−h

dfTx, xhdx, q

3.19

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Theorem 3.5ii allows directly extending Theorem 3.1as follows by removing the requirement of the set of fixed points to be nonempty with a unique element since this is guaranteed by the Banach contractive mapping principle.

Theorem 3.6. Assume that:x, dis a complete metric space,f :X → Xis a continuous mapping onX, andT :X → Xis a self-mapping onXsuch that the pairf, Tisk-contractive and satisfies theL, hproperty (provided that the set of fixed points is nonempty), that is,

dfT2x , fTy ≤kdfTx, y, dfTx, q≤LdfTx, xhdx, q, ∀x, y∈X, 3.20

withqbeing any fixed point; that is,q∈Ff, T⊂Xand some real constants0≤k <1,0≤h <1, andL≥0, and, furthermore,

lim

k→ ∞d

fTxk, xk

0, ∀x0∈X. 3.21

Then, the iteration procedurexk1 fTxk, for allk ∈Z0isf, T-stable withqbeing its unique

fixed point, that is,x⊃Ff, T {q}/∅.

A direct particular case of Theorem 3.5 applies directly to the case of f being the identity map onXvia Theorem3.5i.

Corollary 3.7. Assume thatX, dis a complete metric space, andT : X → X is ak-contractive self-mapping onXsatisfying theL, hproperty, that is,

dT2x, Tx ≤kdTx, q; dTx, q≤LdTx, x hdx, q, ∀x∈X, 3.22

withqbeing any fixed point; that is,q∈Ff, T⊂xand some real constants0≤k <1,0≤h <1, and L ≥ 0, and, furthermore,limk→ ∞dTxk, xk 0, for allx0 ∈ X. Then, the Picard iteration

procedurexk1 Txk, for allk ∈ Z0, is T-stable withqbeing its unique fixed point, that is,X ⊃ FT {q}/∅.

Theorems3.5and3.6and Corollary3.7are directly extendable to the case that the pair

f, Tis a large contraction. Also, Theorem3.6and Corollary3.7can be extended directly for the use of distance functions or extended altering-distance functions as follows.

1 X, dis a complete metric space,f:X → Xis a continuous mapping onX, andT :X → Xis a self-mapping onX;

2ϕ ∈P C0R0,R0is an extended altering-distance function (Definition2.1) satisfying

the conditions of Theorem 2.2(ii) with some monotone nondecreasing function ϕ ∈ C0R0,R0satisfyingφt < ϕt, for allt ∈ R, andφt ϕt 0, if and only

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3the quadrupleϕ, φ, f, Tsatisfies the followingL, hproperty:

ϕdfTxk, q

≤ϕLdfTxk, xk

hdxk, q

−φLdfTxk, xk

hdxk, q

, 3.23

for allx∈X, andq∈Ff, T:{z∈X :zfTz}with real constants0≤h <1and

L≥0;

Then, the iteration procedurexk1 fTxk, for allk∈Z0, isf, T-stable withqbeing its unique

fixed point, that is,X⊃Ff, T {q}/∅.

Corollary 3.9. Assume that Assumptions (1) and (4) of Theorem3.8hold, and, furthermore,

1ϕ ∈ C0R0,R0 is an altering-distance function satisfying the conditions of

Theorem2.2(i), for allt∈R, satisfyingφ0 0, if and only ift0; 2the tripleϕ, f, Tsatisfies the followingL, hproperty:

ϕdfTxk, q

≤ϕLdfTxk, xk

hdxk, q

, 3.24

for allx∈X, andq∈Ff, T:{z∈X :zfTz}with real constants0≤h <1and

L≥0;

Then, the iteration procedurexk1 fTxk, for allk ∈Z0isf, T-stable withqbeing its unique

fixed point, that is,X⊃Ff, T {q}/∅.

Acknowledgments

The author is very grateful to the Spanish Ministry of Education for its partial support of this work through Grant DPI2009-07197. He is also grateful to the Basque Government for its support through Grants IT378-10, SAIOTEK S-PE08UN15, and SAIOTEK SPE07UN04.

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