Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

Volume 2009, Article ID 314581,19pages doi:10.1155/2009/314581

Research Article

Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

M. De la Sen

IIDP. Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, 48080 Bilbao, Spain

Correspondence should be addressed to M. De la Sen,[email protected] Received 18 February 2009; Accepted 27 April 2009

Recommended by Tomas Dom´ınguez Benavides

A generalization of Halpern’s iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern’s iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern’s iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.

Copyrightq2009 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.

1. Introduction

Fixed point theory is a powerful tool for investigating the convergence of the solutions of iterative discrete processes or that of the solutions of differential equations to fixed points in appropriate convex compact subsets of complete metric spaces or Banach spaces, in general, 1–12. A key point is that the equations under study are driven by contractive maps or at least by asymptotically nonexpansive maps. By that reason, the fixed point formalism is useful in stability theory to investigate the asymptotic convergence of the solution to stable attractors which are stable equilibrium points. The uniqueness of the fixed point is not required in the most general context although it can be sometimes suitable provided that only one such a point exists in some given problem. Therefore, the theory is useful for stability problems subject to multiple stable equilibrium points. Compared to Lyapunov’s stability theory, it may be a more powerful tool in cases when searching

a Lyapunov functional is a difficult task or when there exist multiple equilibrium points, 1, 12. Furthermore, it is not easy to obtain the value of the equilibrium points from that of the Lyapunov functional in the case that the last one is very involved. A generalization of the contraction principle in metric spaces by using continuous nondecreasing functions subject to an inequality-type constraint has been performed in 2. The concept ofn-times reasonable expansive mapping in a complete metric space is defined in3 and proven to possess a fixed point. In5, theT-stability of Picard’s iteration is investigated withT being a self-mapping of X whereX, dis a complete metric space. The concept of T-stability is set as follows: if a solution sequence converges to an existing fixed point ofT, then the error in terms of distance of any two consecutive values of any solution generated by Picard’s iteration converges asymptotically to zero. On the other hand, an important effort has been devoted to the investigation of Halpern’s iteration scheme and many associate extensions during the last decades see, e.g., 4, 6, 9, 10. Basic Halpern’s iteration is driven by an external sequence plus a contractive mapping whose two associate coefficient sequences sum unity for all samples, 9. Recent extensions of Halpern’s iteration to viscosity iterations have been proposed in 4, 6. In the first reference, a viscosity-type term is added as extraforcing term to the basic external sequence of Halpern’s scheme. In the second one, the external driving term is replaced with two ones, namely, a viscosity-type term plus an asymptotically nonexpansive mapping taking values on a left reversible semigroup of asymptotically nonexpansive Lipschitzian mappings on a compact convex subsetC of the Banach spaceX. The final iteration process investigated in6consists of three forcing terms, namely, a contraction onC, an asymptotically nonexpansive Lipschitzian mapping taking values in a left reversible semigroup of mappings from a subset of that of bounded functions on its dual. It is proven that the solution converges to a unique common fixed point of all the set asymptotic nonexpansive mappings for any initial conditions onC. The objective of this paper is to investigate further generalizations for Halpern’s iteration process via fixed point theory by using two more driving terms, namely, an external one taking values onC plus a nonlinear term given by a continuous nondecreasing function, subject to an inequality- type constraint as proposed in2, whose argument is the distance between pairs of points of sequences in certain complete metric space which are not necessarily directly related to the sequence solution taking values in the subsetCof the Banach spaceX. Another generalization point is that the sample-by-sample sum of the scalar coefficient sequences of all the driving terms is not necessarily unity but it converges asymptotically to unity.

2. Stability and Boundedness Properties of a Viscosity-Type Difference Equation

In this section a real difference equation scheme is investigated from a stability point of view by also discussing the existence of stable limiting finite points. The structure of such an iterative scheme supplies the structural basis for the general viscosity iterative scheme later discussed formally inSection 4in the light of contractive and asymptotically nonexpansive mappings in compact convex subsets of Banach spaces. The following well-known iterative scheme is investigated for an iterative scheme which generates real sequences.

Theorem 2.1. Consider the difference equation:

x_k1 βkxk 1−βk

zk 2.1

such that the error sequence{ek: xk−zk}is generated by

e_k1 β_ke_k−z_k1, 2.2

for allk∈Z0: N∪ {0}, wherez_k: z_k1−z_k.

Assume thatx0andz0are bounded real constants and 0≤βk<1; for allk∈Z0. Then, the following properties hold.

iThe real sequences{xk},{zk}, and{ek}are uniformly bounded if 0≤e_k≤2x_k/1−β_kif xk>0 and 2xk/1−βk ≤ek≤0 ifxk≤0; for allk∈Z0. If, furthermore, 0< ek<2xk/1−βk ifx_k>0 and 2x_k/1−β_k< e_k≤0, ifx_k≤0, withe_k 0 if and only ifx_k 0; for allk∈Z₀, then the sequences{xk},{zk}, and{ek}converge asymptotically to the zero equilibrium point ask → ∞ and{|xk|}is monotonically decreasing.

iiLet the real sequence{k}be defined byk: z_k1/ek z_k1−zk/xk−zkifxk /zk

and_k 1 ifx_k z_k(what implies thatz_k1 x_k1 x_k z_kfrom2.1and_k 1). Then,{ek}is uniformly bounded ifk∈βk−1,1βk; for allk∈Z0. If, furthermore,k∈βk−1,1βk; for allk∈Z₀thenek → 0 ask → ∞.

iiiLet x₀ ≥ 0 and let{zk}a positive real sequence (i.e., all its elements are nonnegative real constants). Definek : z_k1/ek if xk/zk and k 1 if xk zk. Then,{xk} is a positive real sequence and{ek}is uniformly bounded ifk ∈ 0,1−βk; for allk ∈ Z₀. If, furthermore, _k∈0,1−β_k; for allk∈Z₀, thene_k → 0 ask → ∞.

ivIf |βk| ≤ 1; for allk ∈ Z0 and _∞

k 0|zk| < ∞, then |xk| < ∞; for allk ∈ Z0. If

|βk| ≤β <1 and|zk|<∞; for allk∈Z₀, then|xk|<∞; for allk ∈Z₀. If|βk| ≤β <1/12β0 and|zk| ≤β₀|xk|<∞; for allk ∈Z₀for someβ₀ ∈R : {z∈R : z >0}, withR₀ : {z∈R : z≥0} R∪ {0}, then|xk|<∞; for allk∈Z0andx_k → 0 ask → ∞.

v(Corollary to Venter’s theorem, [7]). Assume thatβk∈0,1,for allk∈Z0,1−βk → 0 ask → ∞and_k

j 01−β_j → ∞(what implyβ_k → 1 ask → ∞and the sequence{βk}has only a finite set of unity values). Assume also thatx₀ ≥ 0 and{zk}is a nonnegative real sequence with _∞

k 01−βkzk<∞. Thenxk → 0 ask → ∞.

vi(Suzuki [8]; see also Saeidi [6]). Let{βk}be a sequence in0,1with 0<lim inf_{k→ ∞}β_k≤ lim sup_{k→ ∞}β_k <1, and let{xk}and{zk}be bounded sequences. Then, lim sup_{k→ ∞}|zk1−z_k| −

|xk1−xk|≤0.

vii(Halpern [9]; see Hu [4]). Letz_kbez_k P x_k; for allk∈Z₀in2.1subject tox₀∈C, β_k ∈ 0,1; for allk ∈Z0 withP : C → Cbeing a nonexpansive self-mapping onC. Thus,{xk} converges weakly to a fixed point of P in the framework of Hilbert spaces endowed with the inner productx, P x , for allx∈X, ifβ_k k^−βfor anyβ∈0,1.

Proof. iDirect calculations with2.1lead to x²_k1−x²_k

β²_k−1 x_k²

1−β_k2

x²_ke²_k−2x_ke_k 2β_k

1−β_k

x_kxk−e_k 1−β_k2

e²_k−2 1−β_k

x_ke_k 1−βk

₂

|ek| −2 1−βk

xk sgn ek

|ek| ifek/0

2.3

so thatx²_k1 ≤ x_k² if1−β_k² e_ksgne_k ≤21−β_kxksgne_k, and equivalently, if1−β_k|ek| ≤ 2|xk|ande_kx_k xk−z_kxk≥0 withe_k/0, and

x²_k1−x²_k 0 ife_k x_k−z_k 0. 2.4

Thus,x²_k1 ≤ x²_k ≤ x₀² < ∞,|ek| ≤ 2|xk|/1−βk ≤ 2|x0|/1−βk < ∞ and|zk| |xk1 − β_kx_k/1−β_k| ≤1β_k/1−β_k|x0|<∞; for allk∈Z₀. If, in addition,1−β_k|ek|<2|xk| ande_kx_k xk−z_kxk ≥0 withe_k/0 thenx_k → 0 and{|xk|}is a monotonically decreasing sequence,zk → 0 andek → 0 ask → ∞. Propertyihas been proven.

iiDirect calculations with2.2yield forek/0, e_k1² −e²_k

β²_k−1_k²−2βkk

e²_k≤0 ifgk: ²_k−2βkkβ²_k−1≤0. 2.5 Sincegkis a convex parabolagk≤0 for all ∈k1, k2if real constantskiexist such thatgki 0;i 1,2. The parabola zeros arek1,2 βk±1 so thate²_k1 ≤ e²_k ≤ e²₀ < ∞if _k ∈βk−1, β_k1. Ife_k 0, thene_k1 −z_k1 z_k−z_k1 x_k1−z_k1 e_k 0 with_k 1.

Thus,e²_k1 ≤e_k² ≤e₀² < ∞ifk ∈ βk−1, βk1, for allk ∈Z0. Ifk ∈βk−1, βk1, then ek → 0 ask → ∞. Propertyiihas been proven.

iiiIf{zk}is positive then{xk}is positive from direct calculations through2.1. The second part follows directly from Propertyii by restricting k ∈ 0, βk1 for uniform boundedness of{ek}andk∈0, βk1for its asymptotic convergence to zero in the case of nonzeroe_k.

ivIf|βk| ≤ 1; for allk ∈ Z0 and _∞

k 0|zk| < ∞, then from recursive evaluation of 2.1:

|xk|

k j 0

βj

x0^k

j 0 k j1

β

1−βj

zj

≤ |x0| x0^k

j 0

zj

<∞; ∀k∈Z0. 2.6 If,|βk| ≤β <1 and|zk|<∞; for allk∈Z₀, then

|xk| ≤β^kx₀

k j 0

k j1

β^k−

1−β_j z_j

≤β^kx₀ 2 1−β

1−β^k−1

max0≤j≤kz_j

≤ |x0| 2 1−βmax

0≤j≤kzj

<∞; ∀k∈Z0.

2.7

If|βk| ≤β <1/12β₀and|zk| ≤β₀|xk|<∞, for allk∈Z₀for someβ₀∈R₀: {0/z∈R}, then|xk1| ≤β|xk|2ββ0|xk| ≤12β0β|xk|<|xk|, for allk∈Z0; thus,{|xk|}is monotonically strictly decreasing so that it converges asymptotically to zero.

Equation2.1under the form

x_k1 β_kx_k 1−β_k

P x_k 2.8

withx₀ ∈ Cand P : C → Cbeing a nonexpansive self-mapping onCunder the weak or

strong convergence conditions ofTheorem 2.1viiis known as Halpern’s iteration4, which is a particular case of the generalized viscosity iterative scheme studied in the subsequent sections. Theorem 2.1vi extends stability Venter’s theorem which is useful in recursive stochastic estimation theory when investigating the asymptotic expectation of the norm- squared parametrical estimation error7. Note that the stability result of this section has been derived by using discrete Lyapunov’s stability theorem with Lyapunov’s sequence {Vk : x²_k}what guarantees global asymptotic stability to the zero equilibrium point if it is strictly monotonically decreasing onR and to global stabilitystated essentially in terms of uniform boundedness of the sequence{xk}if it is monotonically decreasing onR. The links between Lyapunov’s stability and fixed point theory are clearsee, e.g.,1,2. However, fixed point theory is a more powerful tool in the case of uncertain problems since it copes more easily with the existence of multiple stable equilibrium points and with nonlinear mappings.

Note that the results ofTheorem 2.1may be further formalized in the context of fixed point theory by defining a complete metric spaceR, d, respectively, R₀, dfor the particular results being applicable to a positive system under nonnegative initial conditions, with the Euclidean metrics defined bydxk, z_k |xk−z_k|.

3. Some Definitions and Background as Preparatory Tools for Section 4

The four subsequent definitions are then used in the results established and proven in Section 4.

Definition 3.1. Sis a left reversible semigroup ifaS∩bS /∅; for alla, b∈S.

It is possible to define a partial preordering relation “≺” bya≺b ⇔aS ⊃ bS; for all a, b∈Sfor any semigroupS. Thus,∃c aa bb∈S, for some existingaandb∈S, such thataS∩bS ⊇ cS ⇒ a ≺ c∧b ≺ cifS is left reversible. The semigroupS is said to be left-amenable if it has a left-invariant mean and it is then left reversible,6,13.

Definition 3.2see6,13. S : {Ts:s∈S}is said to be a representation of a left reversible semigroup S as Lipschitzian mappings onC if Tsis a Lipschitzian mapping on Cwith Lipschitz constantksand, furthermore,Tst TsTt; for alls, t∈S.

The representation S : {Ts : s ∈ S} may be nonexpansive, asymptotically nonexpansive, contractive and asymptotically contractive according to Definitions3.3and 3.4which follow.

Definition 3.3. A representation S : {Ts : s ∈ S} of a left reversible semigroup S as Lipschitzian mappings onC, a nonempty weakly compact convex subset ofX, with Lipschitz constants{ks:s∈S}is said to be a nonexpansiveresp., asymptotically nonexpansive,6 semigroup onCif it holds the uniform Lipschitzian conditionks≤1resp., limSks≤1 on the Lipschitz constants.

Definition 3.4. A representation S : {Ts : s ∈ S} of a left reversible semigroup S as Lipschitzian mappings onCwith Lipschitz constants{ks:s∈S}is said to be a contractive resp., asymptotically contractive semigroup on C if it holds the uniform Lipschitzian conditionks≤δ <1resp., limSks≤δ <1on the Lipschitz constants.

The iteration process3.1is subject to a forcing term generated by a set of Lipschitzian mappingsSTμk:Z^∗×C → Cwhere{μk}is a sequence of means onZ⊂^∞S, with the subsetZdefined inDefinition 3.5belowcontaining unity, where^∞Sis the Banach space of all bounded functions onSendowed with the supremum norm, such thatμk :Z → Z^∗ whereZ^∗is the dual ofZ.

Definition 3.5. The real sequence{μk}is a sequence of means onZifμk μ_k1 1.

Some particular characterizations of sequences of means to be invoked later on in the results ofSection 4are now given in the definitions which follow.

Definition 3.6. The sequence of means{μk}onZ⊂^∞Sis

1left invariant ifμsf μf; for alls ∈S, for allf ∈Z, for allμ∈ {μk}inZ^∗for s∈^∞S;

2strongly left regular if lim_αs^∗ μ_α−μ_α 0, for alls ∈ S, where_s^∗is the adjoint operator ofs∈^∞Sdefined bysft fst; for allt∈S, for allf ∈^∞S.

Parallel definitions follow for right-invariant and strongly right-amenable sequences of means. Z is said to be leftresp., right-amenable if it has a leftresp., right-invariant mean. A general viscosity iteration process considered in6is the following:

x_k1 α_kfxk β_kx_kγ_kT μ_k

x_k; ∀k∈Z₀, 3.1

where

ithe real sequences{αk},{βk}, and{γk}have elements in0,1of sum being identity, for allk∈Z0;

iiS : {Ts :s ∈S}is a representation of a left reversible semigroup with identity Sbeing asymptotically nonexpansive, on a compact convex subsetCof a smooth Banach space, with respect to a left-regular sequence of means defined on an appropriate invariant subspace of^∞S;

iiifis a contraction onC.

It has been proven that the solution of the sequence converges strongly to a unique common fixed point of the representationS which is the solution of a variational inequality6. The viscosity iteration process3.1generalizes that proposed in13forαk 0 andγk 1−βk

and also that proposed in14,15withβ_k 0,γ_k 1−β_k andTμk T; for allk ∈ Z₀. Halpern’s iteration is obtained by replacingγ_kTμk → 1−α_kuandβ_k 0 in3.1by using the formalism of Hilbert spaces, for allk ∈ Z0see, e.g.,4,9,10. There has been proven the weak convergence of the sequence{xk}to a fixed point of T for any given u, x₀ ∈ C if α_k k^−αforα ∈ 0,1 9, also proven to converge strongly to one such a point ifα_k → 0 andαk1−αk/α²_k1 → 0 ask → ∞, and_∞

k 0αk ∞10. On the other hand, note that if α_k 0,γ_k 1−β_k, andz_k Tμkxkwithx_k∈R, for allk∈Z₀, then the resulting particular iteration process3.1becomes the difference equation2.1discussed inTheorem 2.1from a stability point of view provided that the boundedness of the solution is ensured on some convex compact setC⊂R; for allk∈Z₀.

4. Boundedness and Convergence Properties of a More General Difference Equation

The viscosity iteration process 3.1 is generalized in this section by including two more forcing terms not being directly related to the solution sequence. One of them being dependent on a nondecreasing distance-valued function related to a complete metric space while the other forcing term is governed by an external sequence{δkr}. Furthermore the sum of the four terms of the scalar sequences{αk},{βk}, and{γk}and{δk}at each sample is not necessarily unity but it is asymptotically convergent to unity.

The following generalized viscosity iterative scheme, which is a more general difference equation than3.1, is considered in the sequel

x_k1 αkfxk βkxkγkT μk

xk _s

k

i 1

νikϕi

d

ωk, ω_k−p δkr

; ∀k∈Z₀, 4.1

for allx0 ∈ Cfor a sequence of given finite numbers {sk}with sk ∈ Z0 if sk 0, then the corresponding sum is dropped offwhich can be rewritten as2.1if 0 < β_k < 1; for all k ∈ Z0 except possibly for a finite number of values of the sequence{βk} what implies 0<lim infk→ ∞βk≤lim sup_{k→ ∞}βk<1by defining the sequence

zk

1 1−βk

αkfxk γkT μk

xk _s

k

i 1

νikϕi

d

ωk, ω_k−p δkr

4.2

withx0∈C, where

i{μk}is a strongly left-regular sequence of means on Z⊂^∞S, that is,μ_k∈Z^∗. See Definition 3.5;

iiSis a left reversible semigroup represented as Lipschitzian mappings onCbyS : {Ts:s∈S}.

The iterative scheme is subject to the following assumptions.

Assumption 1. 1{αk},{γk}, and{δk}are real sequences in0,1,{βk}is a real sequence in 0,1, and{νik}are sequences inR₀, for alli∈k: {1,2, . . . , k}for some givenk∈Z≡N : Z0\ {0}andr∈R.

2limk→ ∞αk limn→ ∞δk 0, lim inf_{k→ ∞}γk>0.

3lim_{k→ ∞k}

j 1α_j ∞,lim_{k→ ∞k}

j 1δ_j<∞.

40<lim inf_{k→ ∞}β_k≤lim sup_{k→ ∞}β_k<1.

5αkβkγkδk 1 1−βkεk; for allk ∈Z0 with{εk}being a bounded real sequence satisfyingε_k≥1/βk−1and lim_{k→ ∞}ε_k 0.

6 f is a contraction on a nonempty compact convex subset C, of diameter d_C diamC : sup{x−y : x, y ∈ C},of a Banach space X, of topological dualX^∗, which is smooth, that is, its normalized duality mappingJ :X → 2^X∗ ⊂X^∗fromXinto the family of

nonemptyby the Hahn-Banach theorem6,11, weak-star compact convex subsets ofX^∗, defined by

Jx:

x^∗∈X^∗:x^∗x x, x^∗ x^∗2 x²

⊂X^∗, ∀x∈X 4.3

is single valued.

7 The representationS : {Ts : s ∈ S} of the left reversible semigroupS with identity is asymptotically nonexpansive onCseeDefinition 3.3with respect to{μk}, with μ_k∈Z^∗which is strongly left regular so that it fulfils lim_{k→ ∞}μ_k1−μ_k 0.

8lim sup_{k→ ∞}sup_x,y∈CTμkx−Tμky − x−y/minαk, δ_k≤0.

9 W, dis a complete metric space andQ:W → Wis a self-mapping satisfying the inequality

ϕ_i d

Qy, Qz

≤ϕ_i d

y, z

−φ_i d

y, z

; ∀y, z∈W, 4.4

whereϕ_i, φ_i ∈ R₀ → R₀, for alli ∈ k are continuous monotone nondecreasing functions satisfyingϕit φit 0 if and only ift 0; for alli∈k.

10{ωk} is a sequence inW generated asω_k1 Qωk,k ∈ Z₀ withω0 ∈ W and p∈Zis a finite given number.

Note thatAssumption 14is stronger than the conditions imposed on the sequence {βk}inTheorem 2.1for2.1. However, the whole viscosity iteration is much more general than the iterative equation 2.1. Three generalizations compared to existing schemes of this class are that an extracoefficient sequence{δk} is added to the set of usual coefficient sequences and that the exact constraint for the sum of coefficients αkβkγk δk being unity for allkis replaced by a limit-type constraintα_kβ_kγ_kδ_k → 1 ask → ∞while during the transient such a constraint can exceed unity or be below unity at each sample seeAssumption 15. Another generalization is the inclusion of a nonnegative term with generalized contractive mappingQ :W → W involving another iterative scheme evolving on another, and in general distinct, complete metric space W, d see Assumptions 19 and110. Some boundedness and convergence properties of the iterative process4.1are formulated and proven in the subsequent result.

Theorem 4.1. The difference iterative scheme 4.1 and equivalently the difference equation2.1 subject to4.2possess the following properties underAssumption 1.

imaxsup_k∈Z0|xk|,sup_k∈Z0|Tμkxk| < ∞; for allx0 ∈ C. Also, xk < ∞ and Tμkxk < ∞ for any norm defined on the smooth Banach spaceX and there exists a nonempty bounded compact convex setC0 ⊆ C ⊂ X such that the solution of 4.2 is permanent in C0, for all k ≥ k0 and some sufficiently large finite k0 ∈ Z₀ with max_k≥k0xk,Tμkxk≤d_C0: diamC₀.

iilimk→ ∞Tμkxk−xk 0 andxk → zk → γkTμkxk/1−βk → Tμkxk → x^∗∈ C₀ask → ∞.

iii

∞>|x^∗−x0| lim

k→ ∞

k j 0

x_j1−x_j

∞ j 0

αjf

xj

βj−1

xjγjT μj

xj _s

j

i 1

νijϕi

d

ωj, ω_j−p δjr

.

4.5

ivAssume that{xk} ∈Csuch that each sequence elementx_k∈R^m₀(the first closed orthant of R^m); for allk∈Z0, for somem∈Zso that4.1is a positive viscosity iteration scheme.

Then,

iv.1{xk}is a nonnegative sequence (i.e., all its components are nonnegative for allk≥0, for allx0∈C), denoted asxk≥0; for allk≥0.

iv.2Property (i) holds forC₀⊆Cand Property (ii) also holds for a limiting pointx^∗∈C₀. iv.3Property (iii) becomes

∞> |x^∗−x0|

∞ j 0

αjf

xj

γjT μj

xj _sj

i 1

νijϕi

d

ωj, ω_j−p δjr

−^∞

j 0

1−βj

xj

4.6

what implies that either

∞ j 0

α_jf

x_j γ_jT

μ_j x_j

_sj

i 1

ν_ijϕ_i d

ω_j, ω_j−p δ_jr

<∞, ∞

j 0

1−βj

xj

<∞

4.7

or

lim sup

k→ ∞

k j 0

αjf

xj

γjT μj

xj _s

j

i 1

νijϕi

d

ωj, ω_j−p δjr

∞,

lim sup

k→ ∞

∞ j 0

1−β_j x_j

∞.

4.8

Proof. From 4.2and substituting the real sequence {γk} from the constraint Assumption 15, we have the following:

z_k1−zk 1 1−βk1

α_k1fxk1γk1T μ_k1

x_{k1 s}

k1

i 1

ν_i,k1ϕi

d

ω_k1, ω_k1−p δk1r

− 1 1−β_k

αkfxk γkT μk

xk _s

k

i 1

νi,kϕi

d

ωk, ω_k−p δkr

1 1−β_k1

α_k1fxk1 1

1−β_k1

ε_k1−α_k1−β_k1−δ_k1 T

μ_k1 x_k1

_s

k1

i 1

ν_i,k1ϕ_i d

ω_k1, ω_k1−p δ_k1r

− 1 1−β_k

α_kfxk

1 1−β_k

ε_k−α_k−β_k−δ_k T

μ_k x_k

_s

k

i 1

ν_i,kϕ_i d

ω_k, ω_k−p δ_kr

1−α_k1δ_k1 1−β_k1 ε_k1

T

μ_k1 x_k1−

1−αkδk

1− β_k εk

T

μk

xk

α_k1

1−β_k1fx_k1− α_k 1−βkfxk

δ_k1

1−β_k1 − δ_k 1−βk

r

1 1−β_k1

_s k1

i 1

ν_i,k1ϕi

d

ω_k1, ω_k1−p

− 1 1−βk

_s k

i 1

νi,kϕi

d

ωk, ω_k−p . 4.9

Thus,

z_k1−z_k ≤T μ_k1

x_k1−T μ_k

x_k

α_k1δ_k1 1−β_k1 ε_k1

T

μ_k1 x_k1−

αkδk

1−βk εk

T

μk

xk

K₁αkα_k1 δ_kδ_k1|r|K₂s ν; ∀k≥k₀

≤T μ_k1

x_k1−T μ_k

x_k1T μ_k

x_k1−T μ_k

x_k α_k1δ_k1K1ε_k1T

μ_k1

x_k1−αkδ_kK1ε_kT μ_k

x_k Kαkα_k1K1 δ_kδ_k1|r|K₂s ν; ∀k≥k₀

≤T μ_k1

x_k1−T μk

x_k1T μk

x_k1−T μk

xk αkδkK1εk1ρk

T μ_k1

x_k1−T μk

xk Kαkα_k1K1 δ_kδ_k1|r|K₂s ν; ∀k≥k₀

≤1 α_kδ_kK1ε_kT μ_k1

x_k1−T μ_k

x_k1 1 α_kδ_kK1ε_kT

μ_k

x_k1−T μ_k

x_k α_kδ_kK1ε_kρ_kT

μ_k1

x_k1−T μ_k

x_k Kαkα_k1K1 δkδ_k1|r|K2s ν; ∀k≥k0,

4.10

wherek0∈Z0is an arbitrary finite sufficiently large integer, and s sk0: max

k≥k0

sk, ν νk0: max

k≥k0

maxi∈sk

νik, ρk: α_k1δ_k1−αk−δkK1ε_k1−εk; ∀k∈Z0,

K: 1

1−lim sup_{k→ ∞}βk−εβ

<∞, K₁ K₁x0, k₀: sup

k≥k0

fxk≤sup

x∈C

fx<∞,

∞> K₂ K₂ω0, k₀: 2sk0νk0sup

k≥k0

max

i∈sk

ϕ_i d

ω_k, ω_k−p

−→0 ask₀ −→ ∞ 4.11 since the functionsϕiare continuous onR0withϕi0 0 anddωk, ω_k−p → 0 ask → ∞, 2withεβ>0 being prefixed and arbitrarily small. The constantsK, K1,andK2are finite for sufficiently largek∈Z₀since lim sup_{k→ ∞}β_k<1Assumption 14,fis a contraction onC Assumption 16, andQis a self-mapping onWsatisfyingAssumption 19. Sinceαk → 0, δk → 0 andεk → 0 ask → ∞from Assumptions 11and15andK1is finite,ρk → 0 as k → ∞and|ρk| ≤ρk0; for allk≥k₀being arbitrarily small sincek₀is arbitrarily large. Since fromAssumption 17,S is an asymptotically nonexpansive semigroup onC, andα_k → 0, δk → 0, andεk → 0 ask → ∞:

1 αkδkK1εkT μk

x_k1−T μk

xk α_kδ_kK1ε_kρ_kT

μ_k1

x_k1−T μ_k

x_k

≤1ςkxk1−xkξk, ∀k≥k0

4.12

withR₀ς_k, ξ_k → 0 ask → ∞. One gets from4.12into4.10, zk1−z_k ≤1 α_kδ_kK1ε_kT

μ_k1

x_k1−T μ_k

x_k1 1ς_kxk1−x_k ξkKαkα_k1K1 δkδ_k1|r|s νK2ω0, k; ∀k≥k0

4.13

what implies that lim sup

k→ ∞ z_k1−z_k − x_k1−x_k

≤lim sup

k→ ∞ zk1−zk −ςkxk1−xk

≤lim sup

k→ ∞

1 α_kδ_kK1ε_kT μ_k1

x_k1−T μ_k

x_k1 ξkKαkα_k1K1 δkδ_k1|r|s νK2ω0, k 0 ⇒ lim

k→ ∞xk−z_k 0

4.14

see8since Tμk1xk1−Tμkxk1 → 0 ask → ∞since {xk} is inCand{μk}is a strongly left-regular sequence of means onXsuch that limk→ ∞μ_k1−μk 0; furthermore, α_k → 0,δ_k → 0,ε_k → 0,ς_k → 0,ξ_k → 0 ask → ∞andK₂ω0, k → 0 ask → ∞. Thus, from4.14and using the above technical result in8for difference equations of the class 2.1 see also2, it follows that

klim→ ∞xk1−x_k lim

k→ ∞

1−β_k

xk−z_k 0 ⇒ lim

k→ ∞xk1−x_k lim

k→ ∞xk−z_k 0

⇒x_k1−→x_k−→z_k−→ γkT μk

xk

1−β_k ask−→ ∞

4.15

since 0< lim inf_{k→ ∞}β_k ≤lim sup_{k→ ∞}β_k <1 fromAssumption 14sinceα_k → 0,δ_k → 0, andεk → 0 ask → ∞. From4.1,

x_k1−x_k α_kfxk

1−β_k T

μ_k

x_k−x_k

1−β_k

ε_k−α_k−δ_k T

μ_k x_k

_s k

i 1

ν_ikϕ_i d

ω_k, ω_k−p δ_kr

; ∀k∈Z₀ 4.16

so that T

μ_k

x_k−x_k 1 1− β_k

xk1−xkαkfxk−T μk

xk 1−βk

εk−αk−δkT μk

xk

_s k

i 1

νikϕi

d

ωk, ω_k−p δk|r|

; ∀k∈Z0.

4.17

UsingAssumption 1and using4.15into4.17yield

klim→ ∞T μk

xk−xk 0 ⇒xk−→T μk

xk ask → ∞ 4.18