# Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Authors: M. De la Sen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach’s spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.


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## Body

## 1. Introduction

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces(X,d), which are Banach spaces if X is a vector space on a certain field F (usually R or C) and the metric is homogeneous and translation-invariant, are discussed through the paper. In this case, the metric d is also a norm and, since the space X is a vector space the complete metric space (X,d) is also a Banach space (X,∥∥). The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. Section 2 is concerned of composite operators which include sequences of pairs of contractive operators including, in general, oblique projection operators in the operator composite strip. The results are generalized in Section 3 to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limits,  such that the relevant composite sequences are also compact operators. It is proven in this paper that Banach contraction principle [1–4] guarantees the existence of unique fixed points under contractive conditions fulfilled by some relevant strips of composite operators within in the whole composite sequence of operators.


## 2. Some Results on Contractive Mappings and Fixed Points under Projection Operators

Let{Tk} be a sequence Tk:X→X of self-mappings on X, where (X,d) is a metric space and consider a sequence {Pk} of (non-necessarily orthogonal) projection operators on X of respective ranges Mk which are then closed subspaces of X, [3]. We can then consider a sequence of projection operators {PMk} with PMk:X→Mk such that Pk=PMk so that X=ImPk⊕KerPkandz=Pkx∈ImPk is in Mkfor  anyx∈Xandz-=x-z=(I-Pk)x∈KerPk for k∈N0=N∪{0}. Now, consider sequences {xk} in Xand{zk} in Mk with  zk=Pkxk such that the identities
(1)xk+1=Tkxk=Pk+1xk+1+(I-Pk+1)xk+1=zk+1+(I-Pk+1)xk+1=TkPkxk+Tk(I-Pk)xk
hold by construction for k∈N0. The subsequent result holds.Theorem 1.
Assume that(X,d) is a complete metric space with the metric d:X×X→R0+ being homogeneous and translation-invariant and 0∈X;k∈N0. The following properties hold.(i)
If all the self-mappings onX in the sequence {Tk} are nonexpansive and the sequence of projection operators {Pk} from X to the sequence of subspaces {Mk} is uniformly bounded then d(zk+2,zk+1)<∞;k∈N0.(ii)
Assume that the self-mappings onX in the subsequence {Tk}k≥n0 are contractive for some n0∈N0, that the sequence of operators {Tk} converges to T:X→X, and that the projection operator P:X→M⊂X is constant and bounded (i.e., if it is not orthogonal, i.e., it is oblique, then its norm exceeds one and it is finite) then Property (i) holds. Furthermore, ∃limk→∞d(zk+2,zk+1)=0and{zk+1=PTkxk} is a Cauchy sequence which converges to some unique limit point z(=PTx)∈M for any initial iterate x0∈X, where x(=Tx)∈X is the unique fixed point of T:X→X.(iii)
Assume that there is a strictly sequence of nonnegative integers{jk} such that the difference sequence {jk+1-jk} is uniformly bounded and has a limit J∈Nask→∞. Assume also that the associate sequence of composite self-mappings T^(jk+1+1,jk)={Tjk+1⋯Tjk+1Tjk} is contractive and that the sequence projection operators {Pk} from X to the sequence of subspaces {Mk} is uniformly bounded and has a set of subsequences each converging to a set {Pji}→P^i of projectors from X to {Mji}. Then, ∃limk→∞d(zjk+1+1,zjk+1)=0, and there is at most a finite number J of distinct Cauchy subsequences {zjk} with distinct limit points {z^1,…,z^J} in X.Proof.
Since the metric is homogeneous and translation-invariant then the complete metric space(X,d) can also be considered as a Banach space (X,∥∥) under the metric-induced norm defined as ∥x-y∥=d(x,y);forallx,y∈X. The norm of any projection operator in the considered sequence is defined as ∥Pk∥=sup∥x(∈X)≠0∥≤1(∥Pkx∥/∥x∥)=sup∥x(∈X)∥=1(∥Pkx∥/∥x∥)=sup∥x(∈X)∥=1∥Pkx∥;k∈N0. Then, if T^(k+1,j):X→X is the left-composite self-mapping T^(k+1,j)=Tk⋯Tj+1Tj for any k(≥j),j∈N0, one gets from direct calculations, by using the property that the metric is homogeneous and translation-invariant, the following relations for any iterated sequences{xk}inXand{zk} in Mk constructed as xk+1=Tkxk, zk=Pkxk with arbitrary x0∈X;k∈N:
(2)d(zk+2,zk+1)=d(Pk+2xk+2,Pk+1xk+1)=d(Pk+1xk+2,Pk+1xk+1+Pk+1xk+2-Pk+2xk+2)≤d(Pk+1xk+2,Pk+1xk+1)+d(Pk+1xk+1,Pk+1xk+1+Pk+1xk+2-Pk+2xk+2)≤d(Pk+1xk+2,Pk+1xk+1)+d(Pk+2xk+2,Pk+1xk+2)=∥Pk+1xk+2-Pk+1xk+1∥+∥Pk+2xk+2-Pk+1xk+2∥≤∥Pk+1∥∥xk+2-xk+1∥+∥Pk+2-Pk+1∥∥xk+2∥=∥Pk+1∥d(xk+1,xk+2)+∥Pk+2-Pk+1∥d(xk+2,0)=∥Pk+1∥d(Tkxk,Tk+1Tkxk)+∥Pk+2-Pk+1∥d(T^(k+2,0)x0,0)≤∥Pk+1∥d(x1,x0)+∥Pk+2-Pk+1∥d(T^(k+2,0)x0,0)≤∥Pk+1∥d(x1,x0)+(∥Pk+1∥+∥Pk+2∥)d(x0,0)≤∥Pk+1∥(d(T0x0,0)+d(x0,0))+(∥Pk+1∥+∥Pk+2∥)∥x0∥;k∈N0
since the metric is homogeneous and translation-invariant, the norm is an induced-metric norm, then∥xk∥=d(xk,0)=d(T^(k,0)x0,0), where zk∈Mk, and the self-mappings in the sequence {Tk} on X are all non-expansive; k∈N0, and the sequence of projection operators {Pk} from X to the sequence of subspaces {Mk} is uniformly bounded with supk∈N0∥Pk∥≤μ<∞. Then, one has from (2)
(3)d(zk+2,zk+1)≤4μ∥x0∥<∞;k∈N0,
where μ=1 if all the projections are orthogonal and μ>1, otherwise. Hence, Property (i). If Pk=Pk+1=Pfork∈N0 is a constant bounded projection from X to M with Mk=M⊂X being constant for k∈N0 and all the self-mappings on X of the sequence {Tk} are contractive then one gets from (2) for the real constant K=supk(≥n0)∈N0Kk such that K∈[0,1) that Property (i) holds according to the relation
(4)d(zk+2,zk+1)≤∥P∥d(Tkxk,Tk+1xk+1)≤∥P∥d(Tkxk,Tkxk+1+Tk+1xk+1-Tkxk+1)≤∥P∥d(Tkxk,Tkxk+1)+∥P∥d(Tkxk+1,Tkxk+1+Tk+1xk+1-Tkxk+1)=∥P∥(d(Tkxk,Tkxk+1)+∥P∥d(0,Tk+1xk+1-Tkxk+1))=∥P∥(d(Tkxk,Tkxk+1)+d(Tkxk+1,Tk+1xk+1))=∥P∥(∥Tkxk-Tkxk+1∥+∥Tkxk+1-Tk+1xk+1∥)≤∥P∥∥Tk∥∥xk-xk+1∥+∥P∥∥Tk-Tk+1∥∥xk+1∥≤∥P∥({xn0,T^}∥Tk∥∥xk-xk+1∥+∥Tk-Tk+1∥∥P∥a×d(T^(k+1,0)xn0,T^(k+1,0)0))≤∥P∥(K(1-Kk-n0)1-KKk-n0+1d(xn0+1,xn0)∥P∥a+K(1-Kk-n0)1-K∥Tk-Tk+1∥∥xn0∥)<∞;k(≥n0)∈N0
so that ∃limk→∞d(zk+2,zk+1)=0 from (4) for any initial value x0∈X of the iteration since Kk-n0-1→0and∥Tk-Tk+1∥→0ask→∞since{Tk}→T. Then, {zk} is Cauchy sequence which has a limit z in M, since M is closed, [4]. It is now proven that z(∈M)=Px is the unique limit point in M of any sequence of iterates, where x(=Tx)∈X is a fixed point of the self-mapping T:X→X which is unique from Banach contraction principle. It is now proven that T:X→X is contractive. Assume not so that one has if T:X→X is not contractive:
(5)d(xk+1,xk)≤d(Txk+1,Txk)=d(Tkxk+1,Tkxk+Txk-Tkxk+Tkxk+1-Txk+1)≤d(Tkxk+1,Tkxk)+d(Tkxk,Tkxk+Txk-Tkxk+Tkxk+1-Txk+1)=d(Tkxk+1,Tkxk)+d(0,Txk-Tkxk+Tkxk+1-Txk+1)=d(Tkxk+1,Tkxk)+d(Tkxk+Txk+1,Txk+Tkxk+1)≤d(Tkxk+1,Tkxk)+∥Tk-T∥(∥xk∥+∥xk+1∥)≤Kkd(xk+1,xk)+∥Tk-T∥(∥xk∥+∥xk+1∥);ak≥n0
for nonzero xkandxk+1sinceTk:X→X is contractive for k≥n0. Then, one gets, since ∥Tk-T∥→0ask→∞, that
(6)limsupk→∞[(1-Kk-∥Tk-T∥)(∥xk∥+∥xk+1∥)]=limsupk→∞[(1-Kk)(∥xk∥+∥xk+1∥)]≤0
which is a contradiction since Kk<1fork≥n0 unless {xk} converges to zero. If {xk} converges to zero then there are n1(≥n0)∈N0 and 0<λ=λ(n1)<1-supk≥n1Kk such that ∥Tk-T∥≤λ  for  all  k≥n1 since ∥Tk-T∥→0ask→∞ and some k1≥n1 such that ∥xk1∥+∥xk1+1∥>0 that yields the contradiction
(7)0<(1-Kk1-λ)(∥xk1∥+∥xk1+1∥)≤0.
Thus, if the subsequence {Tk}k≥n0 is contractive in X then its limit T:X→X is also contractive. Now, since T:X→X is contractive then its fixed point is unique since (X,d) is complete. It is clear that z=PTx is a limit point in M of any iterated sequence. Assume that it is not unique so that there are two limit points z=PTx,z^=PTx^(≠z)∈M for some x^(≠x)∈X which is not trivially a fixed point of T:X→X (since the fixed point x∈X of the contractive self-mapping T:X→X is unique if (X,d) is complete). Thus, from Banach contraction principle and since (X,d) is complete, one has
(8)0⟵d(PTkx^,PTkx)=∥PTkx^-PTkx∥≤∥P∥∥Tkx^-Tkx∥≤∥P∥Kkd(x^,x)<∞;k∈N0
as k→∞ since {Tk} converges, there is a limit self-mapping T on X:
(9)z^⟵PTkx^⟶PTkx⟶PTx=Px=z.
Thus, PTkx→z^=zask→∞. Hence a contradiction to z^≠z and then z in M is the unique limit point of PT:X→M even in the event that there is x^(≠x=Tx)∈X such that PTx^=PTx=Px=z. Property (ii) has been proven.
On the other hand, if the sequence of operators is uniformly bounded thensupk∈N0∥Pk∥≤μ<∞ and, if furthermore, the sequence of compositie mappings {T^(jk+1,jk)} is contractive with some constant K^∈[0,1) given by K^=supk∈N0(∏j=jkjk+1-1[Kj]),k∈N0, where {jk} is a strictly increasing sequence of natural numbers such that the sequence {jk+1-jk} is uniformly bounded, one has directly from (1)-(2):
(10)d(zjk+1,zjk)≤μd(xjk,T^(jk+1,jk)xjk)≤μ(K^jk[d(x1,x0)+2∥x0∥])<∞;k∈N0,d(zjk+ik,zjk)≤μd(xjk,T^(jk+1,jk)xjk)≤μ(∏j=jkjk+ik-1Kj)(K^jk[d(x1,x0)+2∥x0∥])<∞
for k∈N0, where {{ik}} is a sequence of finite sets of natural numbers satisfying j∈{ik}⇔1≤j<(jk+1-jk)+1fork∈N0. Thus, one gets from (10):
(11)∃limjk→∞d(zjk+1,zjk)=limjk→∞d(zjk+1+j,zjk)=0
for any j∈{ik} since jk→∞ask→∞ and. Since {jk+1-jk} is uniformly bounded with existing limit J∈Nask→∞,T^(jk+1+1,jk)={Tjk+1⋯Tjk+1Tjk} is contractive and the projection sequence {Pk} from X to the sequence of subspaces {Mk} is uniformly bounded while having a finite set of subsequences {Pji} converging to a set Pji→P^i of projectors from X to {Mk} for i∈J- then
(12)∃limk→∞d(zjk+1+1,zjk+1)=limk→∞d(Pjk+1+1T^(jk+1+1,0)x0,PjkT^(jk+1,0)x0=0)
so that {zjk+1+1=Pjk+1T^(jk+1+1,0)x0} is a Cauchy sequence satisfying:
(13)zjk+i⟶z^i=P^i(limk→∞(Tjk+i⋯Tjk+i-1Tjk)z^i-1)asji⟶∞,i∈M-;z^i∈(X∩⋂k=0∞Mjk+i-1)
for at most J distinct points {z^1,…,z^J} since, by hypothesis, there is a natural number J satisfying ∞>J=limk∈N0{jk+1-jk}≥1 and since {Pji}→P^i; i∈J-. Hence, Property (iii) holds.Remark 2.
The existence of somex^(≠x=Tx)∈X in the proof of Theorem 1(ii) often happens. For instance if PT:X→M is linear then x^=x+xa fulfils the relations PTx=Px=PTx^=z for any xa∈KerPT.The following auxiliary result to be then used holds.Lemma 3.
Assume that the sequences of linear self-mappings{Tn} and {Pn} converge to respective limits P and T being mappings from X to M⊂X and from X to X, respectively, in the sense that PnTnx→PTxasn→∞ from any x∈X. Let (X,∥∥) be a Banach space with the norm of any Q:X→X being defined by ∥Q∥=supx∈X,∥x∥=1∥Qx∥. For any given δ∈R+,∃n0=n0(δ) such that ∥PnTn∥≤pt+δ, where p=∥P∥ and t=∥T∥. If PT:X→X is contractive then the sequence {PnTn} of mappings from X to X is then asymptotically contractive.Proof.
Direct calculations yield(14)∥PnTn∥=sup∥x∥=1∥PTx+P(Tn-T)x+(Pn-P)Tnx∥≤∥PT∥+∥P∥∥Tn-T∥+∥Pn-P∥∥Tn∥≤∥PT∥+∥P∥∥Tn-T∥+∥Pn-P∥∥T∥+∥Pn-P∥∥Tn-T∥
and for any given ε∈R+, there are ni=ni(ε)∈N0fori=1,2 such that ∥Tn-T∥<ε for all integer n>n1and∥Pn-P∥<ε for all integer n>n2. Thus, if n3=max(n1,n2) then one has for any n>n3:
(15)∥PnTn∥≤∥PT∥+(∥P∥+∥T∥+ε)ε=pt+(p+t+ε)ε≤pt+δ.
The last inequality holds if and only ifε2+(p+t)ε-δ≤0 for any given positive real constant δ and any positive real constant ε=ε(δ) being sufficiently small to satisfyε≤(1/2)((p+t)2+4δ-p-t)≤δ/2. Since ∥Pn∥→p and ∥Tn∥→t, there are finite natural numbers n1=n1(δ,ε)=n1(ε),n2=n2(δ,ε)=n2(ε) such that ε satisfies that constraint ∥PnTn∥≤pt+δ for any integer n>n3=n3(δ,ε)=n3(ε)>max(n1,n2). If PT:X→X is contractive then pt≤K<1 for some real constant K. Thus, there is n3=n3(ε) such that pt≤K≤∥PnTn∥<K1<1 for any given real δ<K1-K and ε≤(1/2)((p+t)2+4δ-p-t)  ≤  (1/2)((p+t)2+4(K1-K)-p-t)  ≤  K1-K/2 and then the sequence {PnTn} is asymptotically contractive.Note that Lemma3 holds irrespective of the fact that one of the operators be a projection.


## 3. Results on Contractive Mappings of Sequences of Composite Bounded Operators

The results of Theorem1 are now extended to the study of contractive compositions of linear operators belonging to two sequences of bounded operators {Tki} with Tik:Dom(Tik)⊂X→Im(Tik)⊂XtoX;i=1,2 so that none of them is necessarily a projection on some subspace of X. Some preparatory results are first established. In the following, a Banach space(X,∥∥), being equivalent to a complete metric space (X,d) with a homogeneous and translation-invariant metric induced-norm d:X×X→R0+ is considered such that ∥x∥=d(x,0)=d(x+αy,αy) for any real α and any x,y∈X. The subsequent result refers to the asymptotic distances in sequences involving a convergent composite sequence of bounded linear operators.Lemma 4.
Consider a Banach space(X,∥∥), with 0∈X, being equivalent to a complete metric space (X,d) with a homogeneous and translation-invariant metric induced-norm d:X×X→R0+. Consider also a composite sequence of two sequences of bounded linear operators {Tk=T2kT1k} defined by Tk:Dom(Tk)⊂X→Im(Tk)⊂X defined by Tkx=T2k(T1kx)=T2kT1kx  for  any  x∈Dom(Tk), where Tik:Dom(Tik)⊂X→Im(Tik)⊂X;i=1,2, provided that Im(T1k)(≠∅)⊂Dom(T2k) and Im(T2k)∩Dom(T1k+1)≠∅;  for  all  k∈N0. The following properties hold.(i)
Assume that{Tik}→Ti(i=1,2). Then, limk→∞d(Tkx,T2T1x)=0;  for  all  x∈Dom(Tk);  for  all  k∈N0.(ii)
Assume that{T1k}→T1(i=1,2). Then, limk→∞d(Tkx,T2kT1x)=0;  for  all  x∈Dom(Tk);  for  all  k∈N0.(iii)
Assume that{T2k}→T2(i=1,2). Then, limk→∞d(Tkx,T2T1kx)=0;  for  all  x∈Dom(Tk);  for  all  k∈N0.(iv)
Define the operator composite sequence{T^(k+i+1,k)} of operators as T^(k+i+1,k)=Tk+i⋯Tk+1Tk with Tk=TjkkTjk-1,k⋯T2kT1k;  for  all  i,k∈N0 subject to Im(Tjk)(≠∅)⊂Dom(Tj+1,k)  for  j∈jk¯,ImTjkk(≠∅)⊂DomT1,k+1  and  1≤jk≤J(∈N0)<∞;  for  all  k∈N0. Define also the operator composite sequence of operators {T^0(k+i+1,k)} as
(16)T^0(k+i+1,k)=Tk+i0⋯Tk+10Tk0=(Tjk+ik+i0⋯Tjk+i-1,k+i0T1,k+i0)⋯(Tjkk0⋯T2k0T1k0)
for i,k∈N0, where Tjk0=Tjk if  Tjk has not a limit as k→∞ and Tjk0=Tj∞ if Tj∞=limk→∞Tjk. Then
(17)limk→∞d(T^(k+i+1,i)x,T^0(k+i+1,i)x)=0;∀x∈Dom(T10),∀i∈N0.Properties (i)–(iii) hold for any x∈Dom(T10).Proof.
Assume that{Tik}→Ti with Ti:X→X(i=1,2) since (X,d), equivalently (X,∥∥), is complete so that Tikx→Tix(∈X) strongly for i=1,2ask→∞for  anyx∈Dom(T10) since Im(T1k)(≠∅)⊂Dom(T2k)andIm(T2k)∩Dom(T1k+1)≠∅;forallk∈N0. Then, limk→∞d(Tkx,  T2T1kx)  =  0;  for  all  x∈Dom(T10), and
(18)Tkx=T2kT1kx=T2kT1x+T2k(T1k-T1)x=T2T1x+(T2k-T2)T1x+T2k(T1k-T1)x
that leads to
(19)∥(Tk-T2T1)x∥≤∥(T2k-T2)T1x∥+∥T2k(T1k-T1)x∥⟹∃limk→∞(Tk-T2T1)x=limk→∞d((Tk-T2T1)x,0)=limk→∞d(Tkx,T2T1x)=0;∀x∈Dom(T10).
Hence, Property (i). Now, assume that only {T1k} has a limit. Then,
(20)Tkx=T2kT1kx=T2kT1x+T2k(T1k-T1)x⟹∥(Tk-T2kT1)x∥≤∥T2k(T1k-T1)x∥;∀x∈Dom(Tk),∃limk→∞(Tk-T2kT1)x=limk→∞d((Tk-T2kT1)x,0)=limk→∞d(Tkx,T2kT1x)=0;∀x∈Dom(Tk).
Hence, Property (ii). Finally, assume that only {T2k} has a limit. Then,
(21)Tkx=T2kT1kx=T2T1kx+(T2k-T2)T1kx⟹∥(Tk-T2T1k)x∥≤∥(T2k-T2)T1kx∥;∀x∈Dom(Tk),∃limk→∞(Tk-T2T1k)x=limk→∞d((Tk-T2T1k)x,0)=limk→∞d(Tkx,T2T1kx)=0;∀x∈Dom(Tk).
Hence, Property (iii).
Property (iv) is direct from Properties (i)–(iii) and the associative property of composition of operators since for anyk∈N0+,  Tkx exists in Im(Tk) if Im(T1k)⊂Dom(T2k)  and  Dom(T1k+1)⊃Im(T2k)  for  x∈Dom(T10), and then, Dom(Tk)⊂Dom(T1k)  and  Im(Tk)⊃Im(T2k);  for  all  k∈N0 since
(22)T^(k+i+1,k)=Tk+i⋯Tk+1Tk=(Tjk+ik+i⋯Tjk+i-1,k+iT1,k+i)⋯(Tjkk⋯T2kT1k);∀k∈N0.
Then, for any finite i∈N, one gets
(23)∥T^(k+i+1,k)-T^0(k+i+1,k)∥≤∑j=kk+icj[o(∥Tj0-Tj∥)]≤∑j=kk+i∑ℓj=1jkcjℓj[o(∥Tℓjj0-Tℓjj∥)]⟶0ask⟶∞
for some positive finite constants cj and cjℓj since any linear operator Tℓjj0 with a limit Tℓ∞j admits a unique decomposition Tℓjj0=Tℓ∞j+T~ljj,withT~ljj→0ask→∞,for  allℓj∈j-k,j=1,2,…,k+i.The next result is concerned with the closeness of the limit operator if the sequence of operators is closed.Lemma 5.
Consider a sequence of closed linear operators{Tn} defined by Tn:Dom(Tn)⊂X→Im(Tn)⊂X in a Banach space (X,∥∥), such that Im(Tn)⊂Dom(Tn+1) with Im(Tn)∩Dom(Tn+1)≠∅, which converge to a limit operator T:Dom(T)⊂X→Im(T)⊂X. Then, such a limit is a closed operator which is bounded if all the operators of the sequence are bounded.Proof.
Note the following:(1)
{Tnxm(n)}→Txm(n)asn,m→∞ for any bounded sequence {xm(n)}⊂Dom(Tn)  since  ∃limn,m→∞∥(Tn-T)xm(n)∥  ≤  limn,m→∞∥Tn-T∥∥xm(n)∥  =  0  since  {Tn}→T. Furthermore,
(24)∥Tx∥≤∥Tnx∥+∥(Tn-T)x∥≤∥Kn+K~n∥∥x∥≤∥Kn∥∥x∥+∥K~n∥∥x∥;∀x∈X
for any n∈N0,Kn≥∥Tn∥ is a finite real sequence of constants {Kn} with Kn≤K- for any n∈N0, since {∥Tn∥} is convergent, and K~n≥∥Tn-T∥;for  alln∈N0. Again, since {∥Tn∥} is convergent, there is n0∈N such that K~≥∥K~n∥  for  any  n(≥n0)∈N0,∥Tx∥≤∥Kn∥∥x∥+∥K~n∥∥x∥≤(K+K~)∥x∥ so that T:X→X is bounded.(2)
If the bounded sequence{xm(n)}⊂Dom(Tn) converges to x(n)⊂Dom(Tn) for any n∈N0 then {Tnxm(n)}→Tnx(n)for  anyn∈N0asm→∞ since {Tn} is a closed operator for any n∈N0 so that
(25)∃limm→∞∥Tnxm(n)-Tnx(n)∥≤limm→∞∥Tn(xm(n)-x(n))∥≤∥Tn∥limm→∞∥xm(n)-x(n)∥=0.(3)
One gets combining the above points (1)-(2) that:{Tnxm(n)}→Tnx(n)→Txm(n)→Txasn,m  →  ∞if{xm(n)}→x(n)→x for any n∈N0 where {xm(n)}→{x(n)}⊂Dom(Tn)  as  m→∞, and then {xm(n)}→{x(n)}∈Dom(Tn)  as  m→∞  for  any  n∈N0  and  {xm(n)}→{x(n)}→x∈Dom(T)  as  n,m→∞ since {Tn} is a sequence of closed operators which converges. Thus, the limit of bounded converging sequences belongs to the domain of the limit operator. Furthermore, one has for any bounded sequence {xn} converging to x∈Dom(T):
(26)∥Tnxn-Tx∥=∥Txn+(Tn-T)xn-Tx∥≤∥T∥∥xn-x∥+∥Tn-T∥∥xn∥⟶0as k→∞ then Tnxn→Tx strongly so that T:Dom(T)⊂X→Im(T)⊂X is a closed operator as a result.The above result can be extended to sequences of operators not all of them being bounded provided that each of such sequences of operators can be decomposed as a composition of subsequences of composite operators such that each of such a composite subsequence is bounded. The above result can be applied to sequences of operators not all of them being bounded. It is well-known that a sequence of linear operators on a Hilbert space [5, 6] is bounded if and only if they are closed and their domain is the whole vector space X, [1, 4]. Thus, we have the following result using Lemma 5.Lemma 6.
Consider a sequence of linear bounded operators{Tn} defined by Tn:X→X in a Banach space (X,∥∥) which converge to a limit operator T:X→X. Then, such a limit is a bounded linear closed operator.Proof.
Since the operators are all bounded then their domain isX, their range is in X and are all closed. The conditions of Lemma 5 hold with
(27)X≡Im(Tn)⊂X≡Dom(Tn+1)withIm(Tn)∩Dom(Tn+1)=X≠∅.
Then, the limit T:X→X of {Tn} is also bounded and closed from Lemma 5.The subsequent result is concerned with the limit operator of a sequence of linear operators being compact if all the operators in the sequence are bounded and at least one of them is compact.Lemma 7.
The following properties hold.(i)
Consider a sequence of bounded compact linear operators{Tn} defined by Tn:Dom(Tk)⊂X→Im(Tn)⊂X in a Banach space (X,∥∥), such that Im(Tn)⊂Dom(Tn+1) with Im(Tn)∩Dom(Tn+1)≠∅, which converge to a limit operator T:Dom(T)⊂X→Im(T)⊂X. Then, such a limit is a compact operator.(ii)
Assume that the sequence{Tn} of bounded operators satisfies that there is at least one compact operator within all subsequences {Tjn,Tjn+1,…,Tjn+1-1} being subject to maxn∈N0(jn+1-jn)≤cj<∞  for some subsequence {jn}⊂N0  for any  n∈N0. Then, the composed operator T^(n,m) is compact as it is its limit provided that it exists.Proof.
We have to prove that if{xn} is bounded then {Txn} is convergent. Note that for given bounded sequences {xn(i)} and {xn(j)};  i,j,n∈N0 that
(28)∥Txn(i)-Txn(j)∥=∥(T-Tn)xn(i)+Tnxn(i)-Tnxn(j)-(T-Tn)xn(j)∥≤∥(T-Tn)xn(i)∥+∥(T-Tn)xn(j)∥+∥Tn∥∥xn(i)-xn(j)∥
and, one gets by taking subsequences {zi}⊂{xn(i)}, {zj}⊂{xn(j)}(29)∥Tzi-Tzj∥≤∥T-Tn∥(∥zi∥+∥zj∥)+∥Tnzi-Tnzj∥.
Since {Tn}→T, we can find n0,i0∈N0 such that for n(≥n0)∈N0,min(i,j)>i0, we have
(30)∥T-Tn∥<ε4c,max(∥zi∥,∥zj∥)≤Kz≤2c<∞,∥Tnzi-Tnzj∥<ε2
for any given c and ε=ε(c)∈R+, since {zi} and {zj} are bounded subsequences, and {Tnzi} converges, so that it is a Cauchy sequence, since {Tn} contains at least one compact operator. As a result, ∥Tzi-Tzj∥≤ε/2+ε/2=ε is arbitrarily small for ε being sufficiently small. Thus, {Tzi} is convergent.  Property (i) has been proven. Property (ii) follows from Property (i) and the fact that any operator composite sequence of bounded operators is a compact operator if there is at least one which is compact.Now, define the composite operatorT^(k+i+1,k):X→X;for  alli,k(≥i)∈N0 by
(31)T^(k+i+1,k)=Tk+i⋯Tk+1Tk=(Tjk+ik+i⋯Tjk+i-1,k+iT1,k+i)⋯(Tjkk⋯T2kT1k);∀x∈Dom(T10);∀k∈N0.Define also the sequence{T^0(k+i+1,k)} of composite operators as T^0(k+i+1,k)=Tk+i0⋯Tk+10Tk0;  forallk∈N0 where Tk+i0 replaces each operator in the composite operator Tk+i by its limit when such a limit exists. A result is now given based on the existence of the following limit:
(32)limk→∞d(T^(k+i+1,i)x,T^0(k+i+1,i)x)=0;∀x∈Dom(T10).The following result is obtained from Lemmas4–7.Theorem 8.
Consider the operator composite sequence{T^(k+i+1,k)};  for  all  k,i∈N0 of composed linear bounded operators in (31) on a Banach space (X,∥∥), subject to Im(Tjk)(≠∅)⊂Dom(Tj+1,k)  for  j∈jk¯,ImTjkk(≠∅)⊂DomT1,k+1  and  1≤jk≤J(∈N0)<∞;  for  all  k∈N0, and the sequence of composed linear operators {T^0(k+i+1,k)} of defined in the same way as {T^(k+i+1,k)}  as  k→∞ by replacing each operator possessing a limit by such a limit. The following properties hold.(i)
Either the sequences{T^(k+i+1,k)} and {T^0(k+i+1,k)} have limits and both limits coincide or none of them has a limit and, furthermore, and T^(k+i+1,k)  →  T^0(k+i+1,k)  as  k→∞.(ii)
If the limits of Property (i) exist and are finite then the limits of the sequences of operators{T^(k+i+1,k)}  and  {T^0(k+i+1,k)}  as  k→∞;  for  all  i∈N0 have the same set of fixed points.(iii)
Assume, in addition, that for somek∈N0, there is at least one compact operator in the composition operator T^(k+i+1,k), and that Im(Tjk)(≠∅)⊂Im-(Tjk)⊂Dom(Tj+1,k)  for  j∈jk¯,ImTjk+ℓk(≠∅)⊂Im-Tjk+ℓk(≠∅)⊂DomT1,k+1  for  0≤ℓ≤i and some i∈N0 and that all the operators are closed. If Property (i) holds with T^(k+i+1,k)→T^0(k+i+1,k)→T^*  as  k→∞; for some i∈N0  and ∥T^*∥≤K<1, then T^* is contractive and
(33)limk,n→∞∥T^(k+n(i+1),k)x-T^((k+n(i+1),k),k)y∥=limk→∞∥T^0(k+n(i+1),k)x-T^0(k+n(i+1),k)y∥=limk,n→∞∥T^*(i+1)nx-T^*(i+1)ny∥=0;∀x,y∈Dom(T10),
the sequences of composite operators {T^(k+j,j)} and {T^0(k+j,j)} converge to zero as k→∞,  and  T^*:Dom(T^*)⊂X→Im(T^*)⊂X is bounded, closed and compact and has a unique fixed point in Dom(T^*)∩Im-(T^*) to which all sequences with initial conditions in Im-(T^*) converge.(iv)
Assume that there is a (in general, nonunique) strictly increasing sequence of nonnegative integers{jk} with j0=0 and 0<jk+1-jk≤m<∞ such that
(34)∥T^(jk+2,jk)∥≤K^(jk+2,jk+1)∥T^(jk+1,jk)∥≤K∥T^(jk+1,jk)∥;∀k∈N0
for some nonnegative real sequence {Kk(jk+1,jk)};k∈N0, and some real constant K∈[0,1). Assume, in addition, that for some k∈N0, there is at least one compact operator in any composite operator T^(jk+1,jk) and that all the operators are closed. Then, the sequences of composite operators {T^(k+j,j)} and {T^0(k+j,j)} converge to zero as k→∞ for any finite j∈N0. Finally, assume that T^(jk+1,jk)→T^g*  as  k→∞. Then, T^g* is contractive, continuous, bounded, closed, and compact and has a unique fixed point in Dom(T^*)∩Im-(T^*) to which all sequences with initial conditions in Im-(T^*) converge.Proof.
Note from the definition of the sequences{T^(k+i+1,k)}and{T^0(k+i+1,k)} that for any, since
(35)∥T^0(k+i+1,k)∥≤∥T^(k+i+1,k)∥+∥T^0(k+i+1,k)-T(k+i+1,k)∥,∥T^(k+i+1,k)∥≤∥T^0(k+i+1,k)∥+∥T^0(k+i+1,k)-T(k+i+1,k)∥
so that limsupk→∞|∥T^(k+i+1,k)∥-∥T^0(k+i+1,k)∥|≤0. Then, either both sequences of operators {T^(k+i+1,k)} and {T^0(k+i+1,k)} have the same (finite or infinity) limit or none of them has a limit and T^(k+i+1,k)→T^0(k+i+1,k) as k→∞. Hence, Property (i). Property (ii) follows trivially from Property (i) for the case limk→∞T^0(k+i+1,k)=limk→∞T^(k+i+1,k)=T^*;foralli∈N0.
To prove Property (iii), note that(36)∥T^(k+2i+1,k)x-T^(k+2i+1,k)y∥≤K∥x-y∥=Kd(x,y),∥T^(k+2i+1,k+2i+1)T^(k+2i+1,k)x-T^(k+2i+1,k+2i+1)T^(k+2i+1,k)y∥≤K2∥x-y∥,⋮limn→∞∥T^(k+ni+1,k+ni+1)T^(k+ni+1,k)xa-T^(k+ni+1,k+ni+1)T^(k+ni+1,k)y∥≤Kn∥x-y∥,d(T^*nx-T^*ny)=d(limk→∞T^(k+ni+1,k)x-limk→∞T^(k+ni+1,k)y)=limk→∞d(T^(k+ni+1,k)x-T^(k+ni+1,k)y)=limk→∞d(T^(k+ni+1,k)x,T^(k+ni+1,k)y)=limk→∞d(T^0(k+ni+1,k)x,T^0(k+ni+1,k)y)=limk→∞d(T^(k+ni+1,k)x,T^0(k+ni+1,k)y)=limk→∞∥T^(k+ni+1,k)x-T^(k+ni+1,k)y∥≤Knd(x,y)
so that, sinceK∈[0,1) then the limit operator T^* on X is continuous, then bounded with
(37)limn→∞d(T^*nx-T^*ny)=d(limn→∞T^*nx-limn→∞T^*ny)=0
for any x,y∈Dom(T10). Furthermore, for any k∈N0 and the associative property of composition of operators, ∃n=n(j,k)=max(i∈Z0+:T^(k+j,j)=T^(k+j,k1)T^*n(k2,j);  for  all  j∈N0,  for  all  k(>j)∈N) with n=n(j,k) being unique for each given j∈N0,k(>j)∈N and the given operator decomposition being also trivially unique. Note that ∥T^(k+j,j)∥→0 as k→∞ for any j∈N0 since n=n(j,k)→∞ if (k-j)→∞ since ∥T^*n∥≤Kn→0asn→∞ since K∈[0,1). Similarly, it is proven that ∥T^0(k+j,j)∥→0ask→∞.
On the other hand, note the following.(1)
Any convergent sequence{xk}fork∈N0 constructed from the composed operators
(38)T^(k+i+1,k)=Tk+i⋯Tk+1Tk=(Tjk+ik+i⋯Tjk+i-1,k+iT1,k+i)⋯(Tjkk⋯T2kT1k)
for any k∈N0 as follows x1k∈Dom(T1k),x2k=T2kx1k,…,xjkk=Tjkkxjk-1,k,x1,k+1=T1,k+1xjkk,…  converges to a point x- in Dom(T^*), since all the operators in the above composite sequence of operators are closed and then the limit operator T^*:Dom(T^*)⊂X→Im(T^*)⊂Im-(T^*)⊂Dom(T^*)⊂X is also bounded and closed (from Lemma 6 and the associative property of operator compositions), and T^*x- in Im(T^*)⊂Im-(T^*) with Im-(T^*) being closed (i.e., Im(T^*) is relatively compact) since all composite sequences of operators T^(k+j,j) are compact for any given j∈N0,k(>j)∈N since at least one of the operators within any of such sequences is compact and all of them are bounded, [1, 3, 4].(2)
Any convergent sequence{T^*nx} of elements in Dom(T^*) with x∈Dom(T10) converges to some point x* in Dom(T^*), which maps to T^*x* in Im(T^*)⊂Im-(T^*)⊂Dom(T^*) which is also the limit of the same convergent sequence. Such a limit {T^*nx} has a limit in Dom(T^*)∩Im-(T^*) which is also the unique fixed point of T^*. Otherwise, if there were two distinct fixed points x* and y* then it would are x,y∈Dom(T10) such that limn→∞d(T^*nx-T^*ny)=d(x*,y*)>0, then a contradiction and hence Property (iii).
To prove Property (iv), note that strictly increasing sequence of nonnegative integers{jk} with j0=0 and 0<jk+1-jk≤m<∞ such that (39)∥T^(jk+1,0)∥≤Kjk∥T^(j1,0)∥⟶0ask⟶∞;∥T^(jk+1+mk,0)∥≤MKjk∥T^(j1,0)∥⟶0
for any sequence of nonnegative integers {mk} subject to 0≤mk<jk+2-jk+1≤m<∞;forallk∈N0 for some nonnegative real sequence {Kk(jk+1,jk)};k∈N0 and some real constant K∈[0,1). As a result the sequences of composite operators {T^(k,0)}, {T^0(k,0)}, {T^(k+j,j)}, and {T^0(k+j,j)} converge to zero as k→∞ for any finite j∈N0. On the other hand, if T^(jk+1,jk)→T^g* as,  k→∞ then T^g* is contractive. Otherwise, it would hold trivially that  liminfk→∞∥T^(k,0)∥>0, a contradiction. Thus, the limit operator T^g* is contractive and bounded and then it is also continuous as a result. T^g*:Dom(T^g*)⊂X→Im(T^g*)⊂Im-(Tg*)⊂Dom(T^g*)⊂X is closed (from Lemma 6) and compact, since all the operators in the composite sequence of operators T(jk+1,jj) are bounded and at least one of them is compact. Thus, T^g* has a unique fixed point in Dom(T^*)∩Im-(T^*) to which all sequences to which all the sequences {xk};  k∈N0 of the form x1k∈Dom(T1k),  x2k=T2kx1k,…,  xjkk=Tjkkxjk-1,k,  x1,k+1=T1,k+1xjkk,… with initial point in DomT10 converge. Hence, Theorem 8 is fully proven.Remark 9.
Note that the existence of the operator limits in Theorem8[(iii)-(iv)] is not required for each operator within the composite sequence of operators but only for certain composite strips of such operators.The subsequent result, whose proof is omitted, extends in a natural way Theorem8 through the associative property of composite operators to the case that there are subsets of composite operators having limits although each individual operator is not requested to have a limit.Theorem 10.
Consider the composite operator below:(40)T^(k+ik+1,k)=Tk+ik⋯Tk+1Tk=(Tjk+ikk+i⋯Tjk+ik-1,k+iT1,k+ik)⋯(Tjkk⋯T2kT1k),
on a Banach space (X,∥∥), subject to the following conditions.(1)
The elements of the sequences of sets{ik} are finite and each of those sets has a finite cardinal for all k∈N0, and ik→i*(<∞)  as  k→∞.(2)
Im(Tjk)(≠∅)⊂Im-(Tjk)⊂Dom(Tj+1,k)  for  j∈jk¯,ImTjk+ℓk(≠∅)⊂Im-Tjk+ℓk(≠∅)⊂DomT1,k+1  for  0≤ℓ≤ik.(3)
The elements of the sequences of nonnegative integers{jk+ik} are finite for all k∈N0,  and  jk+ℓk→jk+ℓ  for  0≤ℓk≤ik,0≤ℓ≤i*  as  k→∞.(4)
All the operators in each of the sets{Tikk:1≤ik≤jk};  for  all  k∈N0 are linear, bounded, and closed (so that all the operators are linear, bounded, and closed) and at least one of them in each set is compact.(5)
The sequences of composite operators{Tk+ℓk}  for  0≤ℓk≤ik;  for  all  k∈N0 tend to respective limit operators Tj*  for  0≤j≤i*ask→∞.
Then,T^(k+ik+1,k)→T^(k+i*+1,k)→T*=Ti*⋯T2*T1* which is linear, continuous, bounded, closed, and compact. Furthermore, if ∥Tj*∥≤Kj and K=∏j=0i*Kj<1, then T* has a unique fixed point in DomT*∩Im-T* to which all sequences {xk};  k∈N0 of the form x1k∈Dom(T1k),x2k=T2kx1k,…,xjkk=Tjkkxjk-1,k,x1,k+1=T1,k+1xjkk,…with initial point in DomT10 converge.Example 11.
This example discusses a way to use oblique projections to build composite operators with sequences of operators to take into account the approximation of the images in finite-dimensional spaces and also to take account of computing or measurement errors as well as connections with fixed point issues. Consider the complex pre-Hilbert spaceLp2(a) of square-integrable p-vector functions on [0,a] endowed with an inner product defined by the complex number 〈x,y〉;forallx,y∈Lp2(a) with associate inner product induced norm ∥x∥=〈x,x〉1/2;forallx∈Lp2(a). Consider a bounded linear operator T:Lp2(a)→Lp2(a), of norm ∥T∥=sup∥x∥=1∥Tx∥, represented by y(t)=(Tx)(t)=∑i=1∞〈y,θi〉φi(t);  for  all  t∈[0,a], where φi,θi:[0,a]→Lp2(a); i∈N are sets of linearly independent functions which define mutually reciprocal basis {φi:i∈n-} and {θi:i∈n-}, that is, 〈θi,φi〉=δij. If such basis are orthogonal then they are identical leading to y(t)=(Tx)(t)=∑i=1∞〈y,φi〉φi(t);  for  all  t∈[0,a]. We can decompose Lp2(a) uniquely as a direct sum of orthogonal subspaces as follows as Lp2(a)=Mn⊕(Lp2(a)∩Mn)  for  each  n∈N where the orthogonal projection of T:Lp2(a)→Lp2(a) on Mn⊂Lp2(a) is given by the composite operator PnTn:L2(a)→Mn of the orthogonal projection Pn:L2(a)→Mn represented by y^n(t)=(Pny-n)(t), where y-n(t)=(Tnx)(t)=∑i=1n〈y,θi〉φi(t),  for  all  t∈[0,a] is defined through the truncated operator Tn:Lp2(a)→Lp2(a) so that
(41)y^n(t)=(PnTnx)(t)=(∑i=1n〈PnTx,θi〉φi)(t)=(∑i=1n∑j=1n〈PnTφj,θi〉αjφi)(t)=∑i=1nβiφi(t),
where
(42)αi=〈x,θj〉,βi=∑j=1n〈PnTφj,θi〉αj=∑j=1n〈PnTφj,θi〉〈x,θj〉;i∈n-={1,2,…,n}.
Note that{Tn}→T and d(y,y^n)(t)→0asn→∞;forallt∈[0,a]. Now assume that y(t) is subject to some structured uncertainty y~(t)=(T~Tx)(t)=∑i=1∞〈y~,φi〉φi(t)=(∑i=1∞〈TT~x,φi〉)φi(t);  for  all  t∈[0,a] (due, for instance, to computational or measurement errors) and defined by some relative uncertainty operator T~ in T~ on Lp2(a) belonging to a family T~={T~∈Lp2(a):∥T~∥≤t~,  some  t~∈R0+} so that ymes(t)=y(t)+y~(t) and its projected value on Mn, through the orthogonal projection Mn, is
(43)y^mesn(t)=(Pn(Tn+T~nTn)x)(t)=(PnTnx)(t)+(PnT~nTn)x(t)=y^n(t)+y~n(t)=(∑i=1n〈Pn(T+T~T)x,θi〉φi)(t)=(∑i=1n∑i=1n〈PnTφj+PnT~Tφj,θi〉αjφi)(t)=(∑i=1n∑i=1n〈Pn(I+T~)Tφj,θi〉αjφi)(t)=∑i=1n(βi+β~i)φi(t);∀t∈[0,a],(44)y^mesn(t)-y^n(t)=(∑i=1n∑j=1n〈Pon(T~)Tφj,θi〉αjφi)(t)=∑i=1n(βi+β~i)φi(t);∀t∈[0,a],
where {T~n}→T~,
(45)β~i(t)=∑j=1n〈PnT~Tφj,θi〉αj(t)=∑j=1n〈PnT~Tφj,θi〉〈x,θj〉(t);i∈n-={1,2,…,n}
with Pon(T~n)=Pn(I+T~n) is an oblique operator which depends on the particular uncertainty operator T~n in the class T~ which has necessarily a norm exceeding unity while the orthogonal operator Pn has unit norm. The (non-necessarily unique) worst case in a norm deviation sense of the measured projection of y on the subspace Mn is given by
(46)y^mesnw(t)=supT~∈T~∑i=1n∑j=1n〈Ponw(T~)Tφj,θi〉αjφi(t)
so that the maximum deviation amount of the projected vector is
(47)∥y^mesnw-y^n∥=∥∑i=1n∑j=1n〈Ponw(T~)Tφj,θi〉αjφi∥=supT~∈T~∥∑i=1n∑j=1n〈Pon(T~)Tφj,θi〉αjφi∥.
If the basis{φi:i∈n-} is orthonormal then it is autoreciprocal, then all its vector functions have unit norm and
(48)y^mesnw(t)=supT~∈T~∑i=1n∑j=1n〈Ponw(T~)Tφj,φi〉αjφi(t),(49)∥y^mesnw-y^n∥=∥∑i=1n∑j=1n〈Ponw(T~)Tφj,φi〉αjφi∥=supT~∈T~∥∑i=1n∑j=1n〈Pon(T~)Tφj,φi〉αjφi∥.
The problem can be reformulated for the casea=∞forT:Lp2→Lp2 being a bounded linear operator on the Hilbert (then complete) space Lp2. Thus, T:Lp2→Lp2 is closed, since bounded, and its domain is Lp2 and it is also guaranteed to be compact from of its representation. It is clear that the operators in the sequence {Tn} are bounded, closed, compact, of closed range so that their ranges have n-finite dimension and their domain is Lp2. The orthogonal and oblique operators involved in the above discussion are all bounded and of closed ranges. Then, all the composite operators of the forms  {PnT}, {PonT},{PonT~T} and the operators in the converging sequences {PnTn}, {PonTn}, {PonT~nTn} are all bounded, closed, and compact of domain Lp2. If ∥T∥≤K<1 then for any given real ε∈(0,1-K) there is n0∈N0 such that ∥Tn∥≤K+ε<1 since {Tn}→T. Assume that the class of uncertainty operators T~n in the class T~ on Lp2 has the property ∥Pon(T~n)∥=∥I+T~n∥≤∥Ponw∥≤1/(K+ε)<1;  for  all  n>n0. Thus, the composite operators Pon(T~n)Tn are contractive if ∥I+T~n∥≤1/K+ε<1;foralln>n0 and each of such composite operators has a unique fixed point, which depends on n;foralln>n0 and which converges to the unique fixed point of the contractive operator (I+T~)Tasn→∞ from Theorem 8 since (I+T~n)Tn→(I+T~)Tasn→∞ so that d((I+T~n)nTnnz,(I+T~)nTnz)→0  and  d((I+T~)nTny,(I+T~)nTnz)→0asn→∞  for  any  y,z∈Lp2.Remark 12.
Some ideas in Example11 combining uncertainties with projections both being described through “ad hoc” operators are useful in problems of Signal Theory and Control Systems Theory, [4]. Some related problems can be combined with stability and stabilization issues of dynamic systems subject to unmodeled dynamics and/or parametrical-type uncertainties by using Lyapunov stability theory and fixed point analysis. See, for instance, [7–10]. Fixed point analysis can also be a useful technical tool when using iterative methods in numerical approaches. See, for instance, [11, 12] and references therein. It can be direct the extension of the results to a formalism concerning the replacement of fixed points by best proximity points of cyclic p-self-mappings [13–17] on unions of sets which do not intersect since best proximity points are also fixed points of certain strips of fixed length p of companion composite self-mappings Tp:⋃i∈p-Ai→⋃i∈p-Ai with themselves, the sizes p of such composite self-mappings being the number of disjoint sets Ai⊂X,  i∈p-:={1,2,…,p} in the cyclic disposal. The location of fixed points has also been approximated in some background bibliography on the field. See [18, 19] and references there in. In particular, approximated fixed points have been characterized for nonself mappings which do not possess fixed points. See, for instance, [19] and references therein.



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*Source: 325273-2013-02-17.xml*