Strong Convergence Theorems of the Ishikawa Process with Errors for Strictly Pseudocontractive Mapping of Browder-Petryshyn Type in

(1)

Volume 2011, Article ID 706206,13pages doi:10.1155/2011/706206

Research Article

Strong Convergence Theorems of the Ishikawa Process with Errors for Strictly Pseudocontractive Mapping of Browder-Petryshyn Type in

Banach Spaces

Yu-Chao Tang,

^{1, 2}

Yong Cai,

¹

and Li-Wei Liu

¹

1Department of Mathematics, NanChang University, Nanchang 330031, China

2Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, China

Correspondence should be addressed to Yu-Chao Tang,[email protected] Received 18 December 2010; Accepted 23 February 2011

Academic Editor: M. de la Sen

Copyrightq2011 Yu-Chao Tang et al. 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 prove several strong convergence theorems for the Ishikawa iterative sequence with errors to a fixed point of strictly pseudocontractive mapping of Browder-Petryshyn type in Banach spaces and give suﬃcient and necessary conditions for the convergence of the scheme to a fixed point of the mapping. The results presented in this work give an aﬃrmative answer to the open question raised by Zeng et al. 2006, and generalize the corresponding result of Zeng et al. 2006, Osilike and Udomene 2001, and others.

1. Introduction and Preliminaries

LetE be a real Banach space andE^∗ its dual. ·,·denotes the generalized duality pairing between E and E^∗. Let J : E → 2^E^∗ be the normalized duality mapping defined by the following:

Jx

f∈E^∗: x, f

x²f²

, ∀x∈E. 1.1

It is well known that ifEis smooth, thenJis single-valued. In this paper, we denote a single- valued selection of the normalized duality mapping by j.I denotes the identity operator.

FTis the fixed point set ofT, that is,FT {x:Txx}.

(2)

Definition 1.1see1. A mappingT :DT⊂E → Eis said to be strictly pseudocontractive if there existsλ >0 andjx−y∈Jx−y, such that

Tx−Ty, j x−y

≤x−y²−λx−y−

Tx−Ty², ∀x, y∈DT. 1.2 Remark 1.2. iWithout loss of generality, we may assumeλ ∈0,1. Inequality1.2can be written in the form

I−Tx−I−Ty, j x−y

≥λI−Tx−I−Ty². 1.3 iiIfEis a Hilbert space, then inequality1.2is equivalent to the following inequality:

Tx−Ty²≤x−y²kI−Tx−I−Ty², k1−2λ <1. 1.4 iiiT is a Lipschitz continuous mapping, that is,∃L >0, s.t,Tx−Ty ≤Lx−y. In fact, by1.3, we have

x−y ≥λx−y−

Tx−Ty

≥λTx−Ty −λx−y, 1.5

so that,

Tx−T ≤Lx−y, ∀x, y∈DT, 1.6

whereL λ1/λ.

Definition 1.3. A mappingT :DT⊂E → Eis said to be

icompact, if for any bounded sequence {xn} in DT, there exists a strongly convergent subsequence of{Txn}, or

iidemicompact, if for any bounded sequence{xn}inDT, whenever{xn−Txn}is strongly convergent, there exists a strongly convergent subsequence of{xn}.

Let us recall some important iterative processes.

Definition 1.4 Ishikawa iterative process with errors in the sense of Liu 2. Let K be a nonempty convex subset ofEwithKK⊆K. For anyx1∈K, the sequence{xn}is defined as follows:

xn1 1−αnxnαnTynun, yn

1−βn

xnβnTxnvn, n≥1, 1.7

where {αn} and {βn} are appropriate sequences in 0,1, and {un}, {vn}, are appropriate sequences inK.

(3)

Ifβn vn 0 for all n, then 1.7reduces to Mann iterative process with errors as follows:

xn1 1−αnxnαnTxnun. 1.8

Definition 1.5 Ishikawa iterative process with errors in the sense of Xu 3. Let K be a nonempty convex subset ofE. For anyx1 ∈K, the sequence{xn}is defined as follows:

xn1

1−αn−γn

xnαnTynγnun,

yn

1−βn−δn

xnβnTxnδnvn, n≥1, 1.9

where {un} and {vn} are bounded sequences in K, and {αn}, {γn}, {βn}, {δn} are real sequences in0,1satisfyingαnγn≤1,βnδn≤1, for alln≥1.

Ifβn δn 0 for alln, then 1.9reduces to Mann iterative process with errors as follows:

xn1

1−αn−γn

xnαnTxnγnun. 1.10

Remark 1.6. iIfun vn 0 in1.7orγn δn 0 in1.9, then1.7and1.9reduce to Ishikawa iterative process4,

xn1 1−αnxnαnTyn, yn

1−βn

xnβnTxn, n≥1. 1.11

iiIfun0 in1.8orγn 0 in1.10, then1.8and1.10reduce to Mann iterative process5,

xn1 1−αnxnαnTxn, n≥1. 1.12

In 1974, Rhoades 6 proved strong convergence theorem by the Mann iterative process to a fixed point of strictly pseudocontractive mapping defined on a nonempty compact convex subset of a Hilbert space. In 2001, Osilike and Udomene7proved weak and strong convergence theorems for strictly pseudocontractive mapping in a realq-uniformly smooth Banach spaceEwhich is also uniform convex.

In 2006, Zeng et al.8established the suﬃcient and necessary conditions on the strong convergence to a fixed point of strictly pseudocontractive mapping in a real q-uniformly smooth Banach space. They got the following main results.

Theorem 1.7. Letq >1 andEbe a realq-uniformly smooth Banach space, letKbe a nonempty closed convex subset ofEwithKK ⊆ K, and letT :K → Kbe a strictly pseudocontractive mapping withFT/∅. Let{un}be a bounded sequence inK. Let{αn}and{βn}be real sequences in0,1

(4)

satisfying the following conditions:

i ^∞_n1un<∞;

iiαn ≤ λq/cq^1/q−1, and ^∞_n1β^τ_n < ∞, whereτ min{1,q−1}andcqis a constant depending onq.

From an arbitraryx1∈K, let{xn}be defined by the following:

x_n1 1−αnxnαnTynun, yn

1−βn

Then {xn} converges strongly to a fixed point of T if and only if {xn} is bounded and lim inf_{n→ ∞}dxn, FT 0, wheredx, FT inf_p∈FTx−p.

In the end of Zeng et al.8, they raised an open question.

Open Question 1. Can the Ishikawa iterative process with errors1.7be extended toTheorem 1.7?

At the same year, Zeng et al.9proved the following strong convergence theorem for strictly pseudocontractive mappings.

Theorem 1.8. Letq >1 andEbe a realq-uniformly smooth Banach space. LetKbe a nonempty closed convex subset ofE, and letT :K → Kbe compact or demicompact, and strictly pseudocontractive withFT/∅. Let{un}be a bounded sequence inK. Let{αn},{βn}, and{γn}be real sequences in 0,1satisfying the following conditions:

iαnγn≤1, for alln≥1;

iilimn→ ∞αn< λq/cq^1/q−1, lim_{n→ ∞}βn<1/Land ^∞_n1αn∞;

iii ^∞_n1γn<∞and ^∞_n1αnβ^τ_n<∞, whereτ min{1,q−1}.

From an arbitraryx1∈K, let{xn}be defined by the following:

x_n1

1−αn−γn

xnαnTynγnun,

yn 1−βn

If{xn}is the bounded sequence, then{xn}converges strongly to a fixed point ofT. They raised another open question.

Open Question 2. Can the Ishikawa iterative process with errors1.9be extended toTheorem 1.8?

(5)

We have answered the Open Question 1 in 10. The purpose of this paper is to answer the OpenQuestion 2, and we prove some strong convergence theorems for strictly pseudocontractive mapping in Banach spaces, which improveTheorem 1.8in the following:

iq-uniformly smooth Banach spaces can be replaced by general Banach spaces.

iiRemove the boundedness assumption of{xn}.

iiiIterative process1.14 can be replaced by Ishikawa iterative process with errors 1.9.

Respectively, our results improve and generalize the corresponding results of Zeng el al.8, Osilike and Udomene7, and others.

In the sequel, we will need the following lemmas.

Lemma 1.9see11. Let{an},{bn},{cn}be sequences of nonnegative real numbers satisfying the inequality

an1≤1cnanbn, n≥1. 1.15 If ^∞_n1cn<∞, ^∞_n1bn<∞, we have (i) limn→ ∞anexists. (ii) In particular, if lim infn→ ∞an 0, then lim_n_{→ ∞}an0.

Lemma 1.10see 12. LetE be a Banach space and J : E → 2^E^∗ be the normalized duality mapping, then for anyx, y∈E, the following conclusions hold

ixy²≤ x²2y, jxy, for alljxy∈Jxy;

iixy²≥ x²2y, jx, for alljx∈Jx.

2. Main Results

In the rest of paper, we denote byLthe Lipschitz constant.

Lemma 2.1. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetT :K → Kbe a strictly pseudocontractive mapping withFT/∅. Letx1 ∈K;{xn}is defined by1.9and satisfying the following conditions:

iβn≤αn, δn≤γn, ^∞_n1γn<∞;

ii ^∞_n1α²_n<∞, ^∞_n1αn ∞.

Then

1there exist two sequences{rn},{sn}in0,∞, such that ^∞_n1rn < ∞, ^∞_n1sn <

∞, and

x_n1−q ≤1rnxn−qsn, ∀q∈FT, n≥1. 2.1

Furthermore, lim_n_{→ ∞}xn−qexists.

(6)

2For any integern, m≥1, there exists a constantM1>0, such that

x_nm−q ≤M1xn−qM1 nm−1

kn

sk, ∀q∈FT. 2.2

Proof. 1Letq∈FT. Since{un}and{vn}are bounded sequences inK, we have

0< M:max

sup

n≥1un−q,sup

n≥1vn−q

<∞. 2.3

SinceT is a strictly pseudocontractive mapping, byRemark 1.2i, I−Tx−I−Ty, j

x−y

≥λI−Tx−I−Ty²≥0. 2.4

By Kato13, the above inequality is equivalent to x−y ≤ x−yγ

I−Tx−I−Ty

, ∀x, y∈K, γ >0. 2.5

Letanαnγnand from1.9, we have

xn1 1−anxnanTynγn

un−Tyn

. 2.6

It follows that

xn 1anx_n1anI−Tx_n1−anxn2a²_n

xn−Tyn

an

Txn1−Tyn

γn12an

Tyn−un

. 2.7

Observe that

q 1anqanI−Tq−anq. 2.8

From2.7and2.8, we have xn−q 1an

xn1−q an

I−Tx_n1−I−Tq

−an

xn−q 2a²_n

xn−Tyn

an

Txn1−Tyn

γn12an

Tyn−un

. 2.9

(7)

By inequality2.5, we get

xn−q ≥1anx_n1−q an

1an

I−Tx_n1−I−Tq

−anxn−q −2a²_nxn−Tyn −anTxn1−Tyn

−γn12anTyn−un

≥1anxn1−q −anxn−q −2a²_nxn−Tyn

−anTx_n1−Tyn −γn12anTyn−un.

2.10

So

x_n1−q ≤ xn−q2a²_nxn−TynanTx_n1−Tyn

γn12anTyn−un. 2.11

Furthermore, setbnβnδn≤1, then

yn 1−bnxnbnTxnδnvn−Txn. 2.12

We make the following estimations.

yn−q1−bn xn−q

bn

Txn−q

δnvn−Txn

≤1 L−1bnxn−qδnvn−Txn

≤Lxn−qδnLxn−qδnM L1δnxn−qδnM,

2.13

xn−Tyn ≤ xn−qq−Tyn

≤ xn−qLyn−q

≤ xn−qL²1δnxn−qLδnM

1L²1δn

xn−qLδnM,

2.14

Tyn−un ≤Lyn−qun−q

≤L²1δnxn−q 1LδnM, 2.15

(8)

Txn1−Tyn≤Lxn1−yn Lxn−ynan

Tyn−xn

γn

un−Tyn

≤Lxn−ynLanTyn−xnLγnTyn−un

Lbnxn−TxnδnTxn−vnLanTyn−xnLγnTyn−un

≤Lbnxn−TxnLδnTxn−vnLanTyn−xnLγnTyn−un

≤L1Lbnxn−qL²δnxn−qLan

1L²1δn

xn−q L³γn1δnxn−qLδnML²anδnMLγn1LδnM.

2.16

Substituting2.14,2.15, and2.16in2.11, we obtain xn1−q ≤ xn−q2a²_n

1L²1δn

xn−qL1Lanbnxn−q anL²δnxn−qLa²_n

1L²1δn

xn−q

anL³γn1δnxn−q2a²_nLδnManLδnML²a²_nδnM Lanγn1LδnM

1rnxn−qsn,

2.17

where

rn2a²_n

1L²1δn

L1LanbnanL²δn

La²_n

1L²1δn

anL³γn1δn,

sn2a²_nLδnManLδnML²a²_nδnMLanγn1LδnM.

2.18

By conditionsiandii, we have ^∞_n1rn <∞, ^∞_n1sn <∞. It follows fromLemma 1.9 that lim_{n→ ∞}xn−qexists. This completes the proof of part1.

2Ifx≥0, then 1x≤e^x. For any integern, m≥1 and from part1, we have x_nm−q ≤1r_nm−1x_nm−1−qs_nm−1

≤e^r^nm−1e^r^nm−2xnm−2−qe^r^nm−1snm−2snm−1

· · ·

≤e ^nm−1^kn ^r^kxn−qe ^nm−1^kn ^r^k

nm−1

kn

sk

≤M1xn−qM1 nm−1

kn

sk,

2.19

whereM1e ^∞^k1^r^k. This completes the proof of part2.

(9)

Lemma 2.2. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetT :K → Kbe a strictly pseudocontractive mapping withFT/∅. Let{xn}be defined as inLemma 2.1. Then there exists a subsequencexnjof{xn}, such that

jlim→ ∞xnj−Txnj0. 2.20

Proof. Letq∈FT. It follows from1.3,1.9, andLemma 1.10ithat xn1−q²

xn−qαn

Tyn−xn

γnun−xn²

≤xn−q²2 αn

Tyn−xn

γnun−xn, j

xn1−q xn−q²−2αn

xn1−Txn1, j

xn1−q 2αn

xn1−xn, j

xn1−q 2αn

Tyn−Txn1, j

xn1−q 2γn

un−xn, j

xn1−q

≤xn−q²−2αnλxn1−Txn1²2α²_n

Tyn−xn, j

xn1−q 2αn

Tyn−Txn1, j

xn1−q

2αnγn2γn

un−xn, j

xn1−q

≤xn−q²−2αnλxn1−Txn1²2α²_nTyn−xnxn1−q 2αnLyn−xn1xn1−q

2αnγn2γn

un−xnxn1−q.

2.21

Let

kn2α²_nTyn−xnx_n1−q2αnLyn−x_n1x_n1−q

2αnγn2γn

un−xnxn1−q. 2.22

Then2.21becomes

xn1−q²≤xn−q²−2αnλxn1−Txn1²kn. 2.23 FromLemma 2.11, limn→ ∞xn−qexists. So{xn−q}is bounded. By inequalities 2.14,2.15, and2.16, the sequences{Tyn−xn},{yn−x_n1},{un−xn}are all bounded.

Notice the conditions of ^∞_n1α²_n<∞and ^∞_n1γn<∞, then ^∞_n1kn <∞. It follows from 2.23that

2αnλx_n1−Tx_n1²≤xn−q²−x_n1−q²kn, 2.24 so

2λ n

i1

αix_i1−Tx_i1²≤x1−q²ⁿ

i1

ki. 2.25

Hence, ^∞_n1αnx_n1−Tx_n1²<∞. Since ^∞_n1αn∞, so we have lim_n_{→ ∞}x_n1−Tx_n10.

(10)

By virtue ofLemma 1.10ii, we obtain x_n1−Tx_n1²

1−αn−γn

xnαnTynγnun−Txn1² xn−Txn Txn−Tx_n1 αn

Tyn−xn

γnun−xn²

≥ xn−Txn²2

Txn−Tx_n1αn

Tyn−xn

γnun−xn, jxn−Txn ,

2.26

therefore,

xn−Txn²≤ xn1−Tx_n1²2

Tx_n1−Txn, jxn−Txn 2αn

xn−Tyn, jxn−Txn 2γn

xn−un, jxn−Txn

≤ xn1−Txn1²2Txn1−Txnxn−Txn

2αnxn−Tynxn−Txn2γnxn−unxn−Txn.

2.27

Observe the right side of the above inequality, since Txn1−Txn ≤Lxn1−xn

≤αnTyn−xnγnun−xn −→0 n−→ ∞, 2.28

and{xn−Txn},{xn−Tyn},{xn−un}are all bounded. Together with lim_n_{→ ∞}xn1− Txn1 0, then lim_n_{→ ∞}xn−Txn 0, that is, there exists a subsequencexnj of{xn}, such that

Theorem 2.3. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetT :K → Kbe a strictly pseudocontractive mapping withFT/∅. Let{xn}be defined as inLemma 2.1, then{xn} converges strongly to a fixed point ofTif and only if lim inf_n_{→ ∞}dxn, FT 0, wheredx, FT inf_p∈FTx−p.

Proof. The necessity is obvious. So, we will prove the suﬃciency. From Lemma 2.11, we have

xn1−q ≤1rnxn−qsn, ∀q∈FT, n≥1. 2.30

Therefore,

dx_n1, FT≤1rndxn, FT sn. 2.31

(11)

Note that ^∞_n1rn<∞, ^∞_n1sn<∞. ByLemma 1.9and lim infn→ ∞dxn, FT 0, we get lim_n_{→ ∞}dxn, FT 0.

Next, we prove{xn}is a cauchy sequence. For eachε >0, there exists a natural number n1, such that

dxn, FT≤ ε

12M1, ∀n≥n1, 2.32

where M1 is the constant inLemma 2.1 2. Hence, there exists p1 ∈ FTand a natural numbern2> n1, such that

xn2−p1 ≤ ε 4M1,

∞ kn2

sk< ε

4M1. 2.33

FromLemma 2.12and2.33, for alln≥n2, we have

x_nm−xn ≤ x_nm−p1p1−xn

≤2M1xn2−p1M1 nm−1

kn2

skM1 n−1

kn2

sk

≤2M1 ε

4M1 2M1 ε 4M1 ε.

2.34

Hence,{xn}is a cauchy sequence. SinceKis a closed subset ofE, so{xn}converges strongly to ap∈K.

Finally, we provep∈FT. In fact, sincedp, FT 0. So, for anyε1 >0, there exists p∈FT, such thatp−p< ε1. Then we have

Tp−p ≤ Tp−pp−p

≤1Lε1. 2.35

By the arbitrary ofε1, we know thatTp−p0. Therefore,p∈FT.

A mapping T : K → K is said to satisfy ConditionA 14, if there exists a nondecreasing function f : 0,∞ → 0,∞ with f0 0, fr > 0, for all r ∈ 0,∞ such thatx−Tx ≥fdx, FTfor allx∈K.

Theorem 2.4. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetT :K → K be a strictly pseudocontractive mapping withFT/∅, and satisfy Condition(A). Let{xn}be defined as inLemma 2.1. Then{xn}converges strongly to a fixed point ofT.

(12)

Proof. ByLemma 2.2, there exists a subsequencexnjof{xn}, such that

By ConditionA, limj→ ∞fdxnj, FT 0. Sincefis a nondecreasing function andf0 0, therefore lim_j_{→ ∞}dxnj, FT 0. The rest of the proof is the same toTheorem 2.3.

Theorem 2.5. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetT :K → Kbe a compact and strictly pseudocontractive mapping withFT/∅. Let{xn}be defined as inLemma 2.1.

Then{xn}converges strongly to a fixed point ofT.

Proof. From Lemma 2.1 1, it follows that limn→ ∞xn −q exists, for any q ∈ FT. By Lemma 2.2, there exists a subsequence{xnj}of{xn}such that lim_j_{→ ∞}xnj−Txnj0. Since {xnj}is bounded andT is compact,{Txnj}has a strongly convergent subsequence. Without loss of generality, we may assume that{Txnj}converges strongly top ∈K. Next, we prove p∈FT.

xnj−p ≤ xnj−TxnjTxnj−p −→0

j−→ ∞

, 2.37

that is, lim_j_{→ ∞}xnj−p0. By the Lipschitz continuity ofT, it follows that

p−Tp ≤ p−xnjxnj−Tp −→0

j −→ ∞

. 2.38

This means thatp∈FT. By Lemmas1.9iiand2.1i, the sequence{xn}converges strongly top∈FT.

Theorem 2.6. LetKbe a nonempty closed convex subset of a real Banach spaceE. LetT :K → K be a demicompact and strictly pseudocontractive mapping withFT/∅. Let{xn}be defined as in Lemma 2.1. Then{xn}converges strongly to a fixed point ofT.

Proof. From Lemma 2.11, it follows that limn→ ∞xn − q exists, for any q ∈ FT. By Lemma 2.2, there exists a subsequence{xnj}of{xn}such that limj→ ∞xnj−Txnj0. Since {xnj}is bounded together with T being demicompact, there exists a subsequence of {xnj} which converges strongly to somep ∈K. Taking into account that limj→ ∞xnj −Txnj 0 and the Lipschitz continuity ofT, we havep∈FT. ByLemma 1.9,{xn}converges strongly top∈FT.

Acknowledgments

This work was supported by National Natural Science Foundations of China60970149and the Natural Science Foundations of Jiangxi Province2009GZS0021, 2007GQS2063.

(13)

References

1 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.

2 L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995.

3 Y. Xu, “Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 91–101, 1998.

4 S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, pp. 147–150, 1974.

5 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.

4, pp. 506–510, 1953.

6 B. E. Rhoades, “Fixed point iterations using infinite matrices,” Transactions of the American Mathematical Society, vol. 196, pp. 161–176, 1974.

7 M. O. Osilike and A. Udomene, “Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 431–445, 2001.

8 L.-C. Zeng, N.-C. Wong, and J.-C. Yao, “Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 837–849, 2006.

9 L. C. Zeng, G. M. Lee, and N. C. Wong, “Ishikawa iteration with errors for approximating fixed points of strictly pseudocontractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol. 10, no. 1, pp. 87–99, 2006.

10 Y. C. Tang and J. G. Peng, “On the strong convergence of the Ishikawa process with errors for a strictly pseudocontractive mapping of Browder-Petryshyn type,” submitted.

11 M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,” Mathematical and Computer Modelling, vol. 32, no. 10, pp.

1181–1191, 2000.

12 S.-S. Chang, “Some problems and results in the study of nonlinear analysis,” Nonlinear Analysis:

Theory, Methods & Applications, vol. 30, no. 7, pp. 4197–4208, 1997.

13 T. Kato, “Nonlinear semigroups and evolution equations,” Journal of the Mathematical Society of Japan, vol. 19, pp. 508–520, 1964.

14 H. F. Senter and W. G. Dotson Jr., “Approximating fixed points of nonexpansive mappings,”

Proceedings of the American Mathematical Society, vol. 44, no. 2, pp. 375–380, 1974.