The equivalence of Mann iteration and Ishikawa iteration for non Lipschitzian operators

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THE EQUIVALENCE OF MANN ITERATION AND ISHIKAWA

ITERATION FOR NON-LIPSCHITZIAN OPERATORS

B. E. RHOADES and ¸STEFAN M. ¸SOLTUZ

Received 25 November 2002

We show that the convergence of Mann iteration is equivalent to the convergence of Ishikawa iteration for various classes of non-Lipschitzian operators.

2000 Mathematics Subject Classiﬁcation: 47H10.

1. Introduction. LetXbe a real Banach space,Ba nonempty, convex subset ofX, andT:B→Ban operator. Letu1,x1∈B. The following iteration is known as Mann iteration (see [6]):

un+1=

1−αnun+αnT un. (1.1)

The sequence(αn)n⊂(0,1)is convergent such that

lim n→∞αn=0,

∞

n=1

αn= ∞. (1.2)

Ishikawa iteration is given by (see [5])

xn+1=1−αnxn+αnT yn,

yn=1−βnxn+βnT xn, n=1,2,....

(1.3)

The sequences(αn)n,(βn)n⊂(0,1)are convergent such that

lim

n→∞αn=0, nlim→∞βn=0, ∞

n=1

αn= ∞. (1.4)

Moreover, the sequence(αn)nfrom (1.3) is the same as in (1.1).

The mapJ:X→2X∗given byJx:= {f∈X∗_:_x,f₌_x2_,_f₌_x_}_{, for}

allx∈X, is called thenormalized duality mapping. The Hahn-Banach theorem assures thatJx≠∅, for all x∈X. It is easy to see that we havej(x),y ≤ xy, for allx,y∈Xand for allj(x)∈J(x).

Definition 1.1. Let X be a real Banach space and let B be a nonempty subset. A map T :B →B is called strongly pseudocontractive if there exist

k∈(0,1)and aj(x−y)∈J(x−y)such that

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A mapS:D(S)→Xis called strongly accretive if there existk∈(0,1)and a

j(x−y)∈J(x−y)such that

Sx−Sy,j(x−y)≥kx−y2_, _∀_x,y_∈_D(S). _(1.6)

In (1.5), whenk=1,T is calledpseudocontractive. In (1.6), whenk=0,S is calledaccretive. We denote byIthe identity map.

Remark1.2. (i) The operatorT is (strongly) pseudocontractive map if and only if(I−T )is (strongly) accretive.

(ii) IfS is accretive map, thenT=f−S is strongly pseudocontractive map.

Remark 1.2(i) is obvious from (1.5) and (1.6). ForRemark 1.2(ii), supposing thatx,y∈Bandj(x−y)∈J(x−y), one obtains

Sx−Sy,j(x−y)≥0⇐⇒(f−T )x−(f−T )y,j(x−y)≥0

⇐⇒T x−T y,j(x−y)≤0≤kx−y2_, (1.7)

for allk∈(0,1).

In [10], it was shown that the Mann and Ishikawa iterations are equivalent for various classes of Lipschitzian operators. We prove here the equivalence for non-Lipschitzian operators. For this purpose, we need several lemmas.

Lemma1.3 [11]. Let(an)n be a nonnegative sequence which satisﬁes the following inequality:

an+1≤

1−λnan+σn, (1.8)

whereλn∈(0,1), for alln∈N,∞n=1λn=∞, andσn=o(λn). Thenlimn→∞an=0.

It is known thatJ(x)=∂φ(x), whereφ(x)=(1/2)x2_and_∂φ(x)_denotes

the subdiﬀerential ofφ(x)atx, so the following inequality is satisﬁed, see also [1, Lemma 2.1] or [9, Lemma 1].

Lemma1.4[1,9]. IfXis a real Banach space, then the following relation is true:

x+y2_≤_x2₊₂_y,j(x₊_y)_, _∀_x,y_∈_X, _∀_j(x₊_y)_∈_J(x₊_y). _(1.9)

2. Main result. We are now able to prove the following result.

Theorem2.1. LetXbe a real Banach space with a uniformly convex dual andBa nonempty, closed, convex, and bounded subset ofX. LetT:B→Bbe a continuous and strongly pseudocontractive operator. Then foru1=x1∈B, the following assertions are equivalent:

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THE EQUIVALENCE OF MANN ITERATION... 2647 Proof. Deimling [3, Corollary 1] assures the existence of a ﬁxed point. The uniqueness of the ﬁxed point comes from (1.5). BecauseX∗_{is uniformly}

convex, the duality map is single valued (see, e.g., [4]). Using (1.1), (1.3), and

Lemma 1.4, we get _x_n₊₁₋_u_n₊₁2

=1−αnxn−un+αnT yn−T un2

≤1−αn2xn−un2+2αnT yn−T un,Jxn+1−un+1

=1−αn2xn−un2

+2αnT yn−T un,Jxn+1−un+1

−Jyn−un +2αnT yn−T un,Jyn−un

≤1−αn2xn−un2+2αnkyn−un2 +2αnT yn−T un,Jxn+1−un+1

−Jyn−un ≤1−αn2xn−un2+2αnkyn−un2

+2αnT yn−T unJxn+1−un+1

−Jyn−un ≤1−αn2xn−un2+2αnkyn−un2

+2αnM1Jxn+1−un+1−Jyn−un,

(2.1)

for some positive constantM1. Observe that(T yn−T un)nis bounded. We now prove that

Jxn+1−un+1

−Jyn−un →0 (n→ ∞). (2.2)

Deimling [4, Proposition 12.3, page 115] assures that whenX∗ _{is uniformly}

convex,Jis uniformly continuous on every bounded set ofX. To prove (2.2), it is suﬃcient to see that

_x_n₊₁₋_u_n₊₁

−yn−un =xn+1−yn−un+1−un

=−αnxn+αnT yn+βnxn−βnT xn+αnun−αnT un ≤αnxn+T yn+un+T un+βnxn+T xn ≤αn+βnM →0 (n → ∞),

(2.3)

where

M=sup n

_x_n₊_{T y}_n₊_u_n₊_{T u}_n_,_x_n₊_{T x}_n_<_∞_. (2.4)

The sequences(un)n,(xn)n,(T xn)n,(T un)n, and(T yn)nare bounded be-ing in the bounded setB. Hence one can see that theMabove is ﬁnite and (2.2) holds. We deﬁne

σn:=2αnM1Jxn+1−un+1

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Again, using (1.1) and (1.3), we get _y_n₋_u_n2

=1−βnxn−un+βnT xn−un2

≤1−βn2xn−un2+2βnT xn−un,Jyn−un ≤xn−un2+βnM2.

(2.6)

The last inequality is true because(T xn−un,J(yn−un))nis bounded, with a constantM2>0. Replacing (2.5) and (2.6) in (2.1), we obtain

_x_n₊₁₋_u_n₊₁2

≤1−αn2xn−un2+2αnkxn−un2 +σn+αn(2k)βnM2

=1−2(1−k)αn+α2nxn−un2+oαn.

(2.7)

The condition limn→∞αn=0 implies the existence of ann0such that, for all

n≥n0, we have

αn≤(1−k). (2.8)

Substituting (2.8) into (2.7), we get

1−2(1−k)αn+α2n≤1−2(1−k)αn+(1−k)αn=1−(1−k)αn. (2.9)

Finally,

_x_n₊₁₋_u_n₊₁2

≤1−(1−k)αnxn−un2+oαn. (2.10)

With an := xn−un2, λn :=(1−k)αn ∈(0,1), and using Lemma 1.3, we obtain lim_n_→∞an=limn→∞xn−un2=0, that is,

lim

n→∞xn−un=0. (2.11)

Letx∗_{be the ﬁxed point of}_T_{. Suppose that lim}

n→∞un=x∗. The inequality

0≤x∗₋_x

n≤un−x∗+xn−un (2.12)

and (2.11) imply that lim_n_→∞xn=x∗. Analogously, limn→∞xn=x∗ implies limn→∞un=x∗.

Remark2.2. (i) IfT has a ﬁxed point, thenTheorem 2.1holds without the continuity ofT.

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THE EQUIVALENCE OF MANN ITERATION... 2649

Theorem 2.1does not completely generalize the Lipschitzian case from [10] because the operator there is not necessarily bounded.

Theorem2.3[10]. LetKbe a closed convex (not necessary bounded) subset of an arbitrary Banach space X and letT be a Lipschitzian pseudocontrac-tive selfmap ofK. We consider Mann iteration and Ishikawa iteration with the same initial point and with the conditionslimn→∞αn=0,limn→∞βn=0, and ∞

n=1αn= ∞. Letx∗∈F(T ). Then the following conditions are equivalent: (i) Mann iteration (1.1) converges tox∗_∈_{F(T )}_;

(ii) Ishikawa iteration (1.3) converges tox∗_∈_{F(T )}_.

3. Further equivalences. LetSbe a strongly accretive operator. We consider when the equationSx=fhas a solution for a givenf∈X. It easy to see that

T x=x+f−Sx, ∀x∈X, (3.1)

is a strongly pseudocontractive operator. A ﬁxed point forT is the solution ofSx=f, and conversely.Theorem 2.1assures that the convergence of Mann and Ishikawa iterations to the ﬁxed point ofT are equivalent for bounded strongly pseudocontractive maps. A similar result holds for the convergence of Mann and Ishikawa iterations to the solution ofSx=f. Suppose that the op-eratorSis strongly accretive. It is well known that ifSis bounded,(I−S)could be unbounded. Take, for example,S:R→B=[−1,1]withS(x)=(1/2)cosx. According to [2], the map(I−S)(x)=x−(1/2)cosx is strongly accretive and (I−S)(R)=R. Thus, if B is bounded andx ∈ B does not mean that

T x=x−Sx+f∈B. For the same(αn)n,(βn)n⊂(0,1)as in (1.4), iterations (1.1) and (1.3) become

xn+1=

1−αnxn+αnf+(I−S)yn,

yn=1−βnxn+βnf+(I−S)xn, n=1,2,...,

(3.2)

un+1=1−αnun+αnf+(I−S)un, n=1,2,.... (3.3)

The existence of the solution forSx=f whenS is a continuous and strongly accretive operator results from [8]. This argument andRemark 2.2(ii) lead us to the following corollary.

Corollary3.1. LetXbe a real Banach space with a uniformly convex dual andBa nonempty, convex, and closed subset ofX. LetS:B→Bbe a continuous and strongly accretive operator and let(xn)n, given by (3.2), be bounded. Then, foru1=x1∈B, the following assertions are equivalent:

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LetSbe an accretive operator. FromRemark 1.2(ii), the operatorT x=f−Sx

is strongly pseudocontractive for a givenf∈X. A solution forT x=xbecomes a solution forx+Sx=f. For(αn)n,(βn)n⊂(0,1), as in (1.4), iterations (1.1) and (1.3) become

xn+1=

1−αnxn+αnf−Syn,

yn=1−βnxn+βnf−Sxn, n=1,2,..., (3.4)

un+1=

1−αnun+αnf−Sun, n=1,2,.... (3.5)

The existence of a solution for this equation follows from [7]. We are now able to give the following result.

Corollary3.2. LetXbe a real Banach space with a uniformly convex dual andBa nonempty, convex, and closed subset ofX. LetS:B→Bbe a continuous and accretive operator and let (xn)n, given by (3.4), be bounded. Then, for

u1=x1∈B, the following assertions are equivalent:

(a) Mann iteration (3.5) converges to the solution ofx+Sx=f; (b) Ishikawa iteration (3.4) converges to the solution ofx+Sx=f.

Observe that ifSis not continuous, and the equationsSx=f, respectively,

x+Sx=f, have solutions, thenCorollary 3.1, respectively,Corollary 3.2hold. We remark that all the results from this paper hold in the multivalued case, provided that these multivalued maps admit appropriate single-valued selections.

References

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[5] S. Ishikawa,Fixed points by a new iteration method, Proc. Amer. Math. Soc.44

(1974), 147–150.

[6] W. R. Mann,Mean value methods in iteration, Proc. Amer. Math. Soc.4(1953), 506–510.

[7] R. H. Martin Jr.,A global existence theorem for autonomous diﬀerential equations in a Banach space, Proc. Amer. Math. Soc.26(1970), 307–314.

[8] C. H. Morales,Surjectivity theorems for multivalued mappings of accretive type, Comment. Math. Univ. Carolin.26(1985), no. 2, 397–413.

[9] C. H. Morales and J. S. Jung,Convergence of paths for pseudocontractive map-pings in Banach spaces, Proc. Amer. Math. Soc.128(2000), no. 11, 3411– 3419.

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THE EQUIVALENCE OF MANN ITERATION... 2651 [11] X. Weng,Fixed point iteration for local strictly pseudocontractive mapping, Proc.

Amer. Math. Soc.113(1991), no. 3, 727–731.

B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

E-mail address:rhoades@indiana.edu

¸

Stefan M. ¸Soltuz: “T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania