On the Convergence for an Iterative Method for Quasivariational Inclusions

Volume 2010, Article ID 278973,11pages doi:10.1155/2010/278973

Research Article

On the Convergence for an Iterative Method for Quasivariational Inclusions

Yu Li

and Changqun Wu

1Research Institute of Management Science and Engineering, Henan University, Kaifeng 475004, China

2School of Business and Administration, Henan University, Kaifeng 475004, China

Correspondence should be addressed to Changqun Wu,[email protected] Received 27 September 2009; Accepted 13 December 2009

Academic Editor: Tomonari Suzuki

Copyrightq2010 Y. Li and C. Wu. 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 introduce an iterative algorithm for finding a common element of the set of solutions of quasivariational inclusion problems and of the set of fixed points of strict pseudocontractions in the framework Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.

1. Introduction and Preliminaries

Throughout this paper, we always assume that H is a real Hilbert space with the inner product ·,·and the norm · . Let S : H → H be a nonlinear mapping. In this paper, we useFSto denote the fixed point set ofS.

Recall the following definitions.

1The mappingSis said to be contractive with the coefficientα∈0,1if

Sx−Sy≤αx−y, ∀x, y∈H. 1.1 2The mappingSis said to be nonexpansive if

Sx−Sy≤x−y, ∀x, y∈H. 1.2

3The mappingSis said to be strictly pseudocontractive with the coefficientk∈0,1if Sx−Sy²≤x−y²kI−Sx−I−Sy², ∀x, y∈H. 1.3

4The mappingSis said to be pseudocontractive if

Sx−Sy²≤x−y²I−Sx−I−Sy², ∀x, y∈H. 1.4 Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. See, for example,1–6and the references therein.

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mappingSon a real Hilbert spaceH:

x∈FSmin 1

2Ax, x −hx, 1.5

whereAis a linear bounded and strongly positive operator andhis a potential function for γfi.e.,hx γfxforx∈H.

Recently, Marino and Xu2studied the following iterative scheme:

x0∈H, x_n1 I−αnASxnαnγfxn, n≥0. 1.6 They proved that the sequence{xn}generated in the above iterative scheme converges strongly to the unique solution of the variational inequality:

A−γf

x^∗, x−x^∗

≥0, x∈FS, 1.7

which is the optimality condition for the minimization problem1.5.

Next, letB:H → Hbe a nonlinear mapping. Recall the following definitions.

1The mappingBis said to be monotone if for eachx, y∈H, we have Bx−By, x−y

≥0. 1.8

2Bis said to beμ-strongly monotone if Bx−By, x−y

≥μx−y², ∀x, y∈H. 1.9 3The mapping Bis said to beμ-inverse-strongly monotone if there exists a constant

μ >0 such that

Bx−By, x−y

≥μBx−By², ∀x, y∈H. 1.10

4The mappingBis said to be relaxedδ-cocoercive if there exists a constantδ >0 such that

Bx−By, x−y

≥−δBx−By², ∀x, y∈H. 1.11 5The mappingBis said to be relaxedδ, r-cocoercive if there exist two constantsδ, r >

0 such that

Bx−By, x−y

≥−δBx−By²rx−y², ∀x, y∈H. 1.12 6Recall also that a set-valued mapping M : H → 2^His called monotone if for all x, y ∈H,f ∈Mxandg ∈Myimplyx−y, f−g ≥0.The monotone mapping M:H → 2^His maximal if the graph ofGMofT is not properly contained in the graph of any other monotone mapping.

The so-called quasi-variational inclusion problem is to find au∈Hfor a given element f∈Hsuch that

f∈BuMu, 1.13

whereB:H → HandM:H → 2^Hare two nonlinear mappings. See, for example,7–12.

A special case of the problem1.13is to find an elementu∈Hsuch that

0∈BuMu. 1.14

In this paper, we use V IH, B, M to denote the solution of the problem 1.14. A number of problems arising in structural analysis, mechanics, and economic can be studied in the framework of this class of variational inclusions.

Next, we consider two special cases of the problem1.14.

AIf M ∂φ : H → 2^H, where φ : H → R∪ {∞} is a proper convex lower semicontinuous function and ∂φ is the subdifferential of φ, then the variational inclusion problem1.14is equivalent to findingu∈Hsuch that

Bu, v−uφv−φu≥0, ∀v∈H, 1.15

which is said to be the mixed quasi-variational inequality. See, for example,7,8for more details.

BIfφis the indicator function ofC,then the variational inclusion problem1.14is equivalent to the classical variational inequality problem, denoted byV IC, B, to findu∈Csuch that

Bu, v−u ≥0, ∀v∈C. 1.16

For finding a common element of the set of fixed points of a nonexpansive mapping and of the set of solutions to the variational inequality 1.16, Iiduka and Takahashi 13 proved the following theorem.

Theorem IT. LetC be a closed convex subset of a real Hilbert space H. Let B be anα-inverse- strongly monotone mapping ofCintoHand letSbe a nonexpansive mapping ofCinto itself such thatFS∩V IC, B/∅. Suppose thatx₁x∈Cand{xn}is given by

x_n1αnx 1−αnSPCxn−λnBxn 1.17 for everyn1,2, . . . ,where{αn}is a sequence in0,1and{λn}is a sequence ina, b. If{αn}and {λn}are chosen so that{λn} ∈a, bfor somea, bwith 0< a < b <2α,

nlim→ ∞αn0, ^∞

n1

αn∞, ^∞

n1

|αn1−αn|<∞, ^∞

n1

|λn1−λn|<∞, 1.18

then{xn}converges strongly toP_{FS∩V IC,B}x.

Recently, Zhang et al.11 considered the problem1.14. To be more precise, they proved the following theorem.

Theorem ZLC. Let H be a real Hilbert space, B : H → H an α-inverse-strongly monotone mapping,M:H → 2^Ha maximal monotone mapping, andS:H → Ha nonexpansive mapping.

Suppose that the setFS∩V IH, B, M/∅, whereV IH, B, Mis the set of solutions of variational inclusion1.14. Suppose thatx₀x∈Hand{xn}is the sequence defined by

x_n1αnx0 1−αnSyn,

y_nJ_M,λxn−λBx_n, n≥0, 1.19 whereλ∈0,2αand{αn}is a sequence in0,1satisfying the following conditions:

alimn→ ∞α_n0,_∞

n1α_n∞;

b_∞

n0|αn1−αn|<∞.

Then{xn}converges strongly toPFS∩V IH,B,Mx₀.

In this paper, motivated by the research work going on in this direction, see, for instance, 2, 3, 7–21, we introduce an iterative method for finding a common element of the set of fixed points of a strict pseudocontraction and of the set of solutions to the problem1.14with multivalued maximal monotone mapping and relaxedδ, r-cocoercive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.

In order to prove our main results, we need the following conceptions and lemmas.

Definition 1.1see11. LetM : H → 2^H be a multivalued maximal monotone mapping.

Then the single-valued mappingJM,λ :H → Hdefined byJM,λu IλM⁻¹u,for all u ∈H, is called the resolvent operator associated withM, whereλis any positive number andIis the identity mapping.

Lemma 1.2see4. Assume that{αn}is a sequence of nonnegative real numbers such thatα_n1≤ 1−γ_nαnδ_n,where{γn}is a sequence in0,1and{δn}is a sequence such that

a_∞

n1γn∞;

blim sup_{n→ ∞}δ_n/γ_n≤0 or_∞

n1|δn|<∞.

Then lim_{n→ ∞}α_n0.

Lemma 1.3see22. Let{xn}and{yn}be bounded sequences in a Banach spaceXand let{βn}be a sequence in0,1with 0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1. Suppose thatx_n1 1−βnyn β_nx_nfor alln≥0 and lim sup_{n→ ∞}y_n1−y_n − x_n1−x_n≤0.Then lim_{n→ ∞}yn−x_n0.

Lemma 1.4see11. u∈His a solution of variational inclusion1.14if and only ifuJM,λu− λBu,for allλ >0, that is,

V IH, B, M FJ_M,λI−λB, ∀λ >0. 1.20 Lemma 1.5 see 11. The resolvent operator JM,λ associated with M is single-valued and nonexpansive for allλ >0.

Lemma 1.6see23. LetCbe a closed convex subset of a strictly convex Banach spaceE. LetSand T be two nonexpansive mappings onC. Suppose thatFT∩FSis nonempty. Then a mappingR onCdefined byRxaSx 1−aTx, wherea∈0,1, forx∈Cis well defined and nonexpansive andFR FT∩FSholds.

Lemma 1.7see24. LetHbe a real Hilbert space, letCbe a nonempty closed convex subset ofH, and letS:C → Cbe a nonexpansive mapping. ThenI−Sis demiclosed at zero.

Lemma 1.8see25. LetC be a nonempty closed convex subset of a real Hilbert spaceH and T :C → Cak-strict pseudocontraction. DefineS:C → HbySxαx 1−αTxfor eachx∈C.

Then, asα∈k,1,Sis nonexpansive such thatFS FT.

2. Main Results

Theorem 2.1. LetHbe a real Hilbert space andM:H → 2^Ha maximal monotone mapping. Let B : H → H be a relaxedδ, r-cocoercive andν-Lipschitz continuous mapping, andSak-strict pseudocontraction with a fixed point. Define a mappingSk : H → HbySkx kx 1−kSx.

Letfbe a contraction ofHinto itself with the contractive coefficientα0< α <1,andAa strongly positive linear bounded self-joint operator with the coefficientγ >0. Assume that 0 < γ < γ/αand Ω FS∩V IH, B, M/∅. Letx₁∈Hand{xn}be a sequence generated by

ynJM,λxn−λBxn, x_n1α_nγfx_n β_nx_n 1−β_n

I−α_nA μS_kx_n 1−μ

y_n

, ∀n≥1, 2.1

where{αn}and{βn}are sequences in0,1. Assume thatλ ∈ 0,2r−δν²/ν²,r > δν². If the control consequences{αn}and{βn}satisfy the following restrictions:

C10< a≤βn≤b <1, for alln≥1, C2limn→ ∞αn0 and_∞

n1αn∞,

then{xn}converges strongly toz∈Ω,which solves uniquely the following variational inequality:

A−γf

z, z−x^∗

≤0, ∀x^∗∈Ω. 2.2

Equivalently, one hasP_ΩI−Aγfzz.

Proof. The uniqueness of the solution of the variational inequality2.2is a consequence of the strong monotonicity ofA−γf. Suppose thatz1 ∈Ωandz2∈Ωboth are solutions to2.2;

thenA−γfz1, z₁−z₂ ≤0 andA−γfz2, z₂−z₁ ≤0.Adding up the two inequalities, we see that

A−γf z1−

A−γf

z2, z1−z2

≤0. 2.3

The strong monotonicity of A −γf see 2, Lemma 2.3 implies that z₁ z₂ and the uniqueness is proved. Below we usezto denote the unique solution of2.2.

Next, we show that the mappingI−λBis nonexpansive. Indeed, for allx, y∈H, one see from the conditionλ∈0,2r−γμ²/μ²that

I−λBx−I−λBy² x−y−λBx−By²

x−y²−2λBx−By, x−yλ²Bx−By²

≤x−y²−2λ

−δBx−By²rx−y²

λ²ν²x−y²

1λ²ν²−2λr2λδν²x−y²

≤x−y²,

2.4

which implies that the mappingI−λBis nonexpansive. Takingx^∗∈Ω,we havex^∗J_M,λx^∗− λBx^∗.It follows fromLemma 1.5that

yn−x^∗JM,λxn−λBxn−JM,λx^∗−λBx^∗ ≤ xn−x^∗. 2.5 Note that from the conditionsC1andC2, we may assume, without loss of generality, that αn ≤ 1−βnA⁻¹ for all n ≥ 1. SinceAis a strongly positive linear bounded self-adjoint operator, we haveA sup{|Ax, x|:x∈ H,x 1}. Now forx∈ Hwithx 1, we see that

1−βn

I−αnA x, x

1−βn−αnAx, x ≥1−βn−αnA ≥0; 2.6

that is,1−βnI−αnAis positive. It follows that 1−β_n

I−α_nAsup

1−β_n

I−α_nA

x, x:x∈C,x1 sup

1−β_n−α_nAx, x:x∈C,x1

≤1−βn−αnγ.

2.7

Sett_nμS_kx_n 1−μyn. FromLemma 1.8, we see thatS_kis nonexpansive. It follows from 2.5that

tn−x^∗ ≤μSkxn−x^∗

1−μyn−x^∗≤ xn−x^∗. 2.8 From2.7and2.8, we arrive at

xn1−x^∗

α_nγfxn β_nx_n 1−β_n

I−α_nA

t_n−x^∗

≤αnγfxn−Ax^∗βnxn−x^∗1−βn

I−αnAtn−x^∗

≤αnγfxn−Ax^∗βnxn−x^∗

1−βn−αnγ

tn−x^∗

≤α_nγfxn−γfx^∗α_nγfx^∗−Ax^∗β_nxn−x^∗

1−βn−αnγ

xn−x^∗

≤αα_nγx_n−x^∗α_nγfx^∗−Ax^∗β_nx_n−x^∗

1−β_n−α_nγ

xn−x^∗ 1−αn

γ−αγ

xn−x^∗αnγfx^∗−Ax^∗.

2.9

By simple inductions, one obtains thatxn−x^∗ ≤ max{x1−x^∗,γfx^∗−Ax^∗/γ−αγ}, which gives that the sequence{xn}is bounded, so are{yn}and{tn}.

On the other hand, we see from the nonexpansivity of the mappingsJM,λthat

y_n1−ynJM,λxn1−λBx_n1−JM,λxn−λBxn ≤ xn1−xn. 2.10

It follows that

tn1−tnμSkx_n1 1−μ

y_n1− μSkxn 1−μ

yn

≤μSkx_n1−S_kx_n

1−μy_n1−y_n

≤ x_n1−xn.

2.11

Setting

x_n1 1−βn

enβnxn, ∀n≥1, 2.12

we see that e_n1−en

α_n1γfx_n1 1−β_n1

I−α_n1A t_n

1−β_n1 −α_nγfxn 1−β_n

I−α_nA t_n 1−βn

α_n1

1−β_n1 γfxn1−Atn

t_n1− α_n

1−βn γfxn−Atn

−tn.

2.13

It follows that

e_n1−e_n ≤ α_n1

1−β_n1γfx_n1−At_n α_n

1−β_nγfxn−At_nt_n1−t_n, 2.14 which combines with2.11yields that

e_n1−e_n − x_n1−x_n ≤ α_n1

1−β_n1γfx_n1−At_n αn

1−β_nγfxn−At_n. 2.15 It follows from the conditionsC1andC2that lim sup_{n→ ∞}en1−en − xn1−xn≤0.

Hence, fromLemma 1.3, one obtains limn→ ∞en−x_n0.From2.12, one hasx_n1−x_n 1−β_nen−x_n.Thanks to the conditionC1, we see that

nlim→ ∞x_n1−x_n0. 2.16

On the other hand, we have

x_n1−x_nα_nγfxn β_nx_n 1−β_n

I−α_nA t_n−x_n αn

γfxn−Atn

1−βn

tn−xn. 2.17

It follows that

1−βn

t_n−x_n ≤ x_n1−x_nα_nγfx_n−At_n. 2.18

From the conditionsC1andC2and2.16, we see that

nlim→ ∞tn−xn0. 2.19

Next, we prove that lim sup_{n→ ∞}γf−Az, xn−z ≤0,wherezP_ΩI−A−γfz.

To see this, we choose a subsequence{xni}of{xn}such that lim sup

n→ ∞

γf−A

z, x_n−z lim

i→ ∞

γf−A

z, x_ni−z

. 2.20

Since{xni}is bounded, there exists a subsequence{xn_ij}of {xni} which converges weakly to w. Without loss of generality, we can assume that x_ni w.Next, we show that w ∈ FS∩V IH, M, B. Define a mappingDby

DxμS_kx 1−μ

J_M,λI−λB, ∀x∈H. 2.21

In view ofLemma 1.6, we see thatDis nonexpansive such that

FD FSk∩FJM,λI−λB FS∩V IH, B, M. 2.22

From2.19, we obtain limn→ ∞Dxni−xni 0.It follows fromLemma 1.7thatw ∈FD.

That is,w∈FS∩V IH, M, B.Thanks to2.20, we arrive at lim sup

n→ ∞

γf−A

z, xn−z lim

i→ ∞

γf−A

z, xni−z

γf−A

z, w−z

≤0. 2.23

Finally, we show thatxn → z,asn → ∞.Indeed, we have xn1−z²

αnγfxn β_nx_n 1−β_n

I−α_nA

t_n−z, x_n1−z α_nγfxn−Az, x_n1−zβ_nxn−z, x_n1−z

1−βn

I−αnA

tn−z, xn1−z

≤α_nγfxn−fz, x_n1−zα_nγfz−Az, x_n1−z βnxn−zxn1−z

1−βn−αnγ

xn−zxn1−z

≤ γα 2 αn

xn−z²xn1−z²

αnγfz−Az, x_n1−z

1−αnγ

xn−zxn1−z

≤ γα 2 αn

xn−z²xn1−z²

αnγfz−Az, x_n1−z

1−αnγ 2

xn−z²x_n1−z²

1−α_n γ−αγ

2 xn−z²1

2xn1−z²αnγfz−Az, x_n1−z,

2.24

which implies that

x_n1−z²≤ 1−α_n

γ−αγ

xn−z²2α_n

γfz−Az, x_n1−z

. 2.25

From the conditionC2,2.23, and usingLemma 1.2, we see that limn→ ∞xn−z0.This completes the proof.

Lettingγ 1 andA I, the identity mapping, we can obtain fromTheorem 2.1the following result immediately.

Corollary 2.2. LetHbe a real Hilbert space andM:H → 2^H a maximal monotone mapping. Let B : H → H be a relaxedδ, r-cocoercive andν-Lipschitz continuous mapping, andSak-strict pseudocontraction with a fixed point. Define a mappingS_k : H → HbyS_kx kx 1−kSx.

Letf be a contraction ofH into itself with the contractive coefficientα0 < α < 1.Assume that Ω FS∩V IH, B, M/∅. Letx1∈Hand{xn}be a sequence generated by

ynJM,λxn−λBxn, x_n1α_nfx_n β_nx_n

1−β_n−α_n μS_kx_n 1−μ

y_n

, ∀n≥1, 2.26 where{αn}and{βn}are sequences in0,1. Assume thatλ ∈ 0,2r−δν²/ν²,r > δν². If the control consequences{αn}and{βn}satisfy the following restrictions:

C10< a≤β_n≤b <1, for alln≥1, C2limn→ ∞αn0 and_∞

n1αn∞, then{xn}converges strongly toz∈Ω.

Remark 2.3. Corollary 2.2improvesTheorem 2.1of Zhang et al.11in the following sense:

1from nonexpansive mappings to strict pseudocontractions;

2the analysis technique used in this paper is different from11’s: the proof is also more concise than11’s;

3the restriction imposed on the parameter{αn}is relaxed.

Acknowledgments

The authors are extremely grateful to the referee for useful suggestions that improved the contents of the paper. This work was supported by Important Science and Technology Research Project of Henan province, China092102210134.

References

1 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no.

1-2, pp. 33–56, 1998.

2 G. Marino and H. K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”

Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.

3 X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1033–1046, 2008.

4 H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol.

66, no. 1, pp. 240–256, 2002.

5 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.

6 I. Yamada, N. Ogura, Y. Yamashita, and K. Sakaniwa, “Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 19, no.

1-2, pp. 165–190, 1998.

7 M. A. Noor and K. I. Noor, “Sensitivity analysis for quasi-variational inclusions,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 290–299, 1999.

8 M. A. Noor, “Generalized set-valued variational inclusions and resolvent equations,” Journal of Mathematical Analysis and Applications, vol. 228, no. 1, pp. 206–220, 1998.

9 S. Plubtieng and W. Sriprad, “A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 567147, 20 pages, 2009.

10 J. W. Peng, Y. Wang, D. S. Shyu, and J.-C. Yao, “Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems,” Journal of Inequalities and Applications, vol. 2008, Article ID 720371, 15 pages, 2008.

11 S. S. Zhang, J. H. W. Lee, and C. K. Chan, “Algorithms of common solutions to quasi variational inclusion and fixed point problems,” Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 571–581, 2008.

12 L. C. Zhao, S. S. Chang, and M. Liu, “Viscosity approximation algorithms of common solutions for fixed points of infinite nonexpansive mappings and quasi-variational inclusion problems,”

Communications on Applied Nonlinear Analysis, vol. 15, no. 3, pp. 83–98, 2008.

13 H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse- strongly monotone mappings,” Nonlinear Analysis: Theory, Methods& Applications, vol. 61, no. 3, pp.

341–350, 2005.

14 S. S. Chang, “Set-valued variational inclusions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 248, no. 2, pp. 438–454, 2000.

15 X. Qin and Y. Su, “Approximation of a zero point of accretive operator in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 415–424, 2007.

16 X. Qin, S. Y. Cho, and S. M. Kang, “Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 231–240, 2009.

17 X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.

18 X. Qin, Y. J. Cho, and S. M. Kang, “Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications,” Nonlinear Analysis: Theory, Methods and Applications, vol. 72, no. 1, pp. 99–112, 2010.

19 X. Qin, S. S. Chang, and Y. J. Cho, “Iterative methods for generalized equilibrium problems and fixed point problems with applications,” Nonlinear Analysis: Real World Applications. In press.

20 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.

21 H. Zhou, “Convergence theorems of fixed points fork-strict pseudo-contractions in Hilbert spaces,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 2, pp. 456–462, 2008.

22 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non- expansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005.

23 R. E. Bruck Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,”

Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973.

24 F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.

25 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.