# Characterization for the Convergence of Krasnoselskij Iteration for Non-Lipschitzian Operators

Authors: Ştefan M. Şoltuz; B. E. Rhoades
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We establish the convergence of Krasnoselskij iteration for various
classes of non-Lipschitzian operators.


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## Body

## 1. Introduction

LetX be a real
Banach space; B a nonempty,
convex  subset of X; and T:B→B an operator.
Let x0∈B. The following
iteration is known as Krasnoselskij iteration (see [1]):(1.1)xn+1=(1−λ)xn+λTxn.The mapJ:X→2X∗ given by Jx:={f∈X∗:〈x,f〉=∥x∥2,∥f∥=∥x∥},  for all x∈X, is called the
normalized duality mapping. It is easy to see that we have(1.2)〈y,j(x)〉≤∥x∥∥y∥,∀x,y∈X,∀j(x)∈J(x).Denote(1.3)Ψ:={ψ∣ψ:[0,+∞)→[0,+∞)is astrictly increasing map withψ(0)=0}.Definition 1.1.
LetX
be a real
Banach space, and let  B
be a nonempty
subset of  X. A map  T:B→B  is called uniformly pseudocontractive if there exists a map  ψ∈Ψ and j(x−y)∈J(x−y) such
that(1.4)〈Tx−Ty,j(x−y)〉≤∥x−y∥2−ψ(∥x−y∥),∀x,y∈B. A map S:X→X is called
uniformly accretive if there exists a map  ψ∈Ψ and j(x−y)∈J(x−y) such
that(1.5)〈Sx−Sy,j(x−y)〉≥ψ(∥x−y∥),∀x,y∈X.Takingψ(a):=ψ(a)⋅a, for all a∈[0,+∞),   (ψ∈Ψ), reduces to the
usual definitions of ψ-strongly
pseudocontractive and ψ-strongly
accretive. Taking ψ(a):=γ⋅a2, γ∈(0,1), for all   a∈[0,+∞),   (ψ∈Ψ), we get the
usual definitions of strongly pseudocontractive and strongly accretive.
Therefore, the class of strongly pseudocontractive maps is included stricly in
the class of ψ-strongly
pseudocontractive maps. The example from [2] shows that this inclusion is
proper. Remark, further, that the class of ψ-strongly
pseudocontractive maps is also included strictly in the class of uniformly
pseudocontractive maps (see also [3]).We will give a characterization for the convergence
of (1.1) when applied to
uniformly pseudocontractive operators. For this purpose, we need the following
lemma similar to  [4, Lemma 1]. Next, ℕ denotes the set
of all natural numbers.Lemma 1.2.
Let{an}  be a positive
bounded sequence and assume that there exists  n0∈ℕ such that (1.6)an+1≤(1−λ)an+λan+1−λψ(an+1)an+1+λεn,∀n≥n0,  where λ∈(0,1), εn≥0, for all n∈ℕ  and  limn→∞εn=0.  Then  limn→∞an=0.Proof.
There exists anM>0 such that an≤M,  for all n∈ℕ. Denote a:=liminfan. We will prove
that a=0. Suppose on the
contrary that a>0. Then there
exists an N1∈ℕ such
that(1.7)an≥a2,∀n≥N1.From limn→∞εn=0, we know that
there exists an N2∈ℕ such
that(1.8)εn≤ψ(a/2)2M,∀n≥N2.Set N0:=max{N1,N2}. Using the fact
that −(1/M)≥−(1/an+1), we get the
following:(1.9)an+1≤(1−λ)an+λan+1−λψ(an+1)an+1+λεn≤(1−λ)an+λan+1−λψ(a/2)M+λψ(a/2)2M≤(1−λ)an+λan+1−λψ(a/2)2M,which implies that (1−λ)an+1≤(1−λ)an−λ((ψ(a/2))/2M), or(1.10)an+1≤an−λ1−λψ(a/2)2M≤an−λψ(a/2)2M,since −(λ/(1−λ))≤−λ. Thus λ(ψ(a/2))/2M≤an−an+1, which implies
that ∑λ<∞, in
contradiction to ∑λ=∞. Therefore, liminfan=0. Hence there
exists a subsequence {anj}⊂{an} such that limj→∞anj=0. Fix ε>0. Then there exists an n3∈ℕ such
that(1.11)anj<ε4,∀j≥n3.Also there exists an n4∈ℕ such
that(1.12)εn<ψ(ε/4)2M,∀n≥n4.Define n0:=max{n3,n4,N0}. We claim that anj+k<ε/4 for each j>n0 and each k>0. Suppose not.
Then there exists an n0 and a k>0 such that(1.13)anj+k≥ε4.For this nj, let k denote the
smallest positive integer for which (1.13) is true. Then anj+k−1≤ε/4.
From (1.6), (1.14)anj+k≤(1−λ)anj+k−1+λanj+k−λψ(anj+k)anj+k+λεnj+k−1≤(1−λ)anj+k−1+λanj+k−λψ(ε/4)anj+k+λψ(ε/4)2M≤(1−λ)anj+k−1+λanj+k−λψ(ε/4)2M,which implies that anj+k≤(ε/4)−(λ/(1−λ))(ψ(ε/4)/2M). This leads to
the contradiction:(1.15)ε4≤anj+k≤ε4−λ1−λψ(ε/4)2M<ε4.Therefore, anj+k<ε/4, for all k∈ℕ, and each j>n0, hence limn→∞an=0.


## 2. Main Result

Theorem 2.1.
LetX be a real
Banach space, B a nonempty,
closed, convex, bounded subset of X. Let T:B→B be a uniformly
pseudocontractive and uniformly continuous operator with F(T)≠∅. Then for x0∈B, the
Krasnoselskij iteration (1.1) converges to
the fixed point of T if and only if limn→∞∥xn+1−xn∥=0.Proof.
SinceT is a self-map
of B, which is
bounded and convex, then, from (1.1), each xn∈B, so {xn} is bounded for
each n∈ℕ. Uniqueness of
the fixed point follows from (1.4). If {xn} converges to
the fixed point of T, that is, limn→∞xn=x∗, then,
obviously, limn→∞∥xn+1−xn∥=0. Conversely, we
will prove that if limn→∞∥xn+1−xn∥=0, then limn→∞xn=x∗. Suppose that xn=x∗ for some n∈ℕ. Then from (1.1), it follows that xm=x∗ for each m>n, and the theorem
is proved. Now suppose that xn≠x∗ for each n∈ℕ. Using (1.1) and (1.2),(2.1)∥xn+1−x∗∥2=〈xn+1−x∗,j(xn+1−x∗)〉=〈(1−λ)(xn−x∗)+λ(Txn−Tx∗),j(xn+1−x∗)〉=(1−λ)〈(xn−x∗),j(xn+1−x∗)〉+λ〈Txn−Tx∗,j(xn+1−x∗)〉≤(1−λ)∥xn−x∗∥∥xn+1−x∗∥+λ〈Txn+1−Tx∗,j(xn+1−x∗)〉+λ〈Txn−Txn+1,j(xn+1−x∗)〉≤(1−λ)∥xn−x∗∥∥xn+1−x∗∥+λ∥xn+1−x∗∥2−λψ(∥xn+1−x∗∥)+λ∥Txn−Txn+1∥∥xn+1−x∗∥≤∥xn+1−x∗∥((1−λ)∥xn−x∗∥+λ∥xn+1−x∗∥−λψ(∥xn+1−x∗∥)∥xn+1−x∗∥+λ∥Txn−Txn+1∥).
Hence(2.2)∥xn+1−x∗∥≤(1−λ)∥xn−x∗∥+λ∥xn+1−x∗∥−λψ(∥xn+1−x∗∥)∥xn+1−x∗∥+λ∥Txn−Txn+1∥.Since limn→∞∥xn+1−xn∥=0 and T is uniformly
continuous, it follows that (2.3)limn→∞∥Txn−Txn+1∥=0.Set an=∥xn−x∗∥,  εn=∥Txn−Txn+1∥ and use Lemma 1.2
to obtain the conlcusion.Remark 2.2.
(1) IfB is not bounded,
then Theorem 2.1 holds under the assumption that {xn} is bounded.
(2) IfT(B) is bounded,
then {xn} is bounded.
(3) IfT is strongly
pseudocontractive, then automatically F(T)≠∅.


## 3. Further Results

LetI denote the
identity map. A map T:B→B is called
pseudocontractive if there exists j(x−y)∈J(x−y) such that 〈Tx−Ty,j(x−y)〉≤∥x−y∥2.Remark 3.1.
The operatorT is a (uniformly, strongly) pseudocontractive map if and only if (I−T) is a (uniformly, strongly) accretive map.Remark 3.2.
(1) LetT,S:X→X, and let f∈X be given. A
fixed point for the map Tx=f+(I−S)x, for all x∈X, is a solution
for Sx=f.
(2)   Letf∈X be a given
point. If S is an accretive
map, then T=f−S is a strongly
pseudocontractive map.Consider Krasnoselskij iteration withTx=f+(I−S)x,(3.1)xn+1=(1−λ)xn+λ(f+(I−S)xn).Remarks 3.1 and 3.2 and Theorem 2.1
lead to the following result.Corollary 3.3.
LetX be a real
Banach space and let S:X→X be a uniformly
accretive and uniformly continuous operator, with (I−S)(X) bounded.
Suppose that Sx=f has a solution.
Then for any x0∈X, the
Krasnoselskij iteration (3.1) converges to
the solution of Sx=f if and only if limn→∞∥xn+1−xn∥=0.LetS be an accretive
operator. The operator Tx=f−Sx is strongly
pseudocontractive for a given f∈X. A solution for Tx=x becomes a
solution for x+Sx=f. Consider
Krasnoselskij iteration with Tx:=f−Sx,(3.2)xn+1=(1−λ)xn+λ(f−Sxn).Again, using Remarks 3.1 and 3.2
and Theorem 2.1, we obtain the following result.Corollary 3.4.
LetX be a real
Banach space and let S:X→X be an accretive
and uniformly continuous operator, with (I−S)(X) bounded.
Suppose that x+Sx=f has a solution.
Then for x0∈X, the
Krasnoselskij iteration (3.2) converges to
the solution of x+Sx=f if and only if limn→∞∥xn+1−xn∥=0.Remark 3.5.
If (1.4) holds for all x∈B and y:=x∗∈F(T), then such a map
is called uniformly hemicontractive. It is trivial to see that our
results hold for the uniformly hemicontractive maps.



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*Source: 630589-2008-03-26.xml*