Volume 2010, Article ID 470141,6pages doi:10.1155/2010/470141
Research Article
A Counterexample on a Theorem by Khojasteh, Goodarzi, and Razani
Ivan D. Arandelovi ´c
1and Dragoljub J. Ke ˇcki ´c
21Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Beograd, Serbia
2Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia
Correspondence should be addressed to Ivan D. Arandelovi´c,[email protected] Received 27 June 2010; Revised 1 September 2010; Accepted 1 September 2010
Academic Editor: L. G ´orniewicz
Copyrightq2010 I. D. Arandelovi´c and D. J. Keˇcki´c. 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.
In the paper by Khojasteh et al. 2010, the authors tried to generalize Branciari’s theorem, introducing the new integral type contraction mappings. In this note we give a counterexample on their main statementTheorem 2.9. Also we give a comment explaining what the mistake in the proof is, and suggesting what conditions might be appropriate in generalizing fixed point results to cone spaces, where the cone is taken from the infinite dimensional space.
1. Introduction
In the paper 1, Branciari proved the following fixed point theorem with integral-type contraction condition.
Theorem 1.1. LetX, dbe a complete metric space,α∈0,1, andf :X → Xis a mapping such that for allx, y∈X,
dfx,fy
0
φtdt≤α dx,y
0
φtdt, 1.1
whereφ : 0,∞ → 0,∞is nonnegative measurable mapping, having finite integral on each compact subset of0,∞such that for eachε > 0,ε
0φtdt > 0. Thenf has a unique fixed point a∈X, such that for eachx∈X, limn→ ∞fnxa.
There are many generalizations of fixed point results to the so-called cone metric spaces, introduced by several Russian authors in mid-20th. These spaces are re-introduced
by Huang and Zhang2. In the same paper, the notion of convergent and Cauchy sequences are given.
Definition 1.2. LetEbe a Banach space. ByΘwe denote the zero element ofE. A subsetP of Eis called a cone if
1Pis closed, nonempty, andP /{Θ};
2a, b∈R,a, b >0, andx, y∈P implyaxby∈P;
3P∩−P {Θ}.
Given a coneP ⊆ E, we define partial ordering≤onEwith respect to P byx ≤yif and only ify−x∈P. We will writex < yto indicate thatx≤yandx /y, whereasxywill stand fory−x∈intP interior ofP.
We say thatPis a solid cone if and only if intP /∅.
LetP be a solid cone inEand let≤be the corresponding partial ordering onEwith respect toP. We say thatP is a normal cone if and only if there exists a real numberK >0 such thatΘ≤x≤yimplies
x ≤Ky 1.2
for eachx, y∈P. The least positiveKsatisfying1.2is called the normal constant ofP. Definition 1.3. LetXbe a nonempty set. Suppose that a mappingd:X×X → Esatisfies:
1 Θ≤dx, yfor allx, y∈Xanddx, y Θif and only ifxy;
2dx, y dy, xfor allx, y∈X;
3dx, y≤dx, z dz, yfor allx, y, z∈X.
Then,dis called a cone metric onX, andX, dis called a cone metric space.
Definition 1.4. LetX, dbe a solid cone metric space, letx∈X, and letxnbe a sequence in X. Then
1 xnconverges toxif for everyc∈intP there exists a positive integerNsuch that for alln≥Ndxn, xc. We denote this by limxnxorxn → x;
2 xnis a cone Cauchy sequences if for everyc∈intPthere exists a positive integer Nsuch that for allm, n≥Ndxm, xnc;
3 X, dis a complete cone metric space if every Cauchy sequence is convergent.
In the paper3Khojasteh et al. tried to generalize Branciari fixed point result to the cone metric spaces. They introduce the concept of integration along the intervala, b {ta 1−tb|0≤t≤1} ⊆Pas a limit of Cauchy sums.
Definition 1.5see3. LetP be a normal solid cone, and letφ : P → P. We say that φis integrable ona, bif and only if the following sums:
Ln−1
i0
φxixi−xi1,
Un−1
i0
φxi1xi−xi1
1.3
converge to the same element ofP, wherexk, xk1form a partition ofa, b. Clearly,a, b stands fora, b\ {b}. This element is denoted by
b
a
φdp. 1.4
We say thatφis subadditive if and only if for anya, b∈Pthere holds ab
Θ φdp≤ a
Θφdp b
Θφdp. 1.5 Using this concept, they stated the following statementTheorem 2.9 in3.
Theorem 1.6 see3. LetX, dbe a complete cone metric space and let P be a normal cone.
Suppose thatφ:P → Pis a nonvanishing map which is subadditive cone integrable on eacha, b⊆ Pand such that for eachε0,ε
0φdp0. Iff:X → Xis a map such that for allx, y∈X, dfx,fy
Θ φdp≤α dx,y
Θ φdp, 1.6 for someα∈0,1, thenfhas a unique fixed point inX.
However, the last statement is not true. This will be proved in the next section.
2. Constructing the Counterexample
Consider the Banach space
ECR0,1 {x:0,1−→R|xis continuous}, 2.1 with the normxsup0≤t≤1|xt|, and the cone
P {x∈E| ∀t∈0,1xt≥0}. 2.2
It is obvious thatPis a normal solid cone with normal constant equals to 1.
Consider the set
X{x∈E|x0 1, x1 0}, 2.3
and the mappingd:X×X → P given by
d x, y
|td x, y
t xt−yt, ∀t∈0,1or more simplyd x, y
x−y. 2.4 Proposition 2.1. aintP {x∈P | ∃δ >0 for allt∈0,1xt≥δ}.
b X, dis a cone metric space.
cA sequencexn∈Xis convergent (inX, d) toxif and only ifxn−x → 0. Alsoxnis Cauchy sequence (inX, d) if and only ifxnis a Cauchy sequence with respect to norm inX⊆E.
d X, dis a complete cone metric space.
Proof. aandbobvious.
cLetxn → xinX, d, and letε >0. Then, the functionεt ≡ε ∈intP, and we have that for alln≥n0and for allt∈0,1there holds
|xnt−xt|dxn, x≤εt≡ε. 2.5 Letxn−x → 0, and letct∈intP. Then there existsδ >0 such that for allt∈0,1, ct≥δ. Also for alln≥n0there holds
dxn, x |xnt−xt| ≤ δ
2. 2.6
Hence,ct−dxnt, xt≥δ/2 implyingdxn, xc.
The similar argument proves the second part of the statement concerning Cauchy sequences.
dThe setX can be represented asX Λ−10 {1}∩Λ−11 {0}, whereΛt : E → R is the bounded linear functional given byΛtx xt. Therefore,Xis a closed subset ofEand hence complete in the norm. By partcof this proposition, it implies thatXis a complete cone metric space.
Lete : 0,1 → R denote the function identically equal to 1. Consider the mapping φ:P → Pgiven by
φx 1
x 1
0
xtdt·e, 2.7
forx / ΘandφΘ Θ.
Proposition 2.2. aφis integrable on every segmenta, b⊆Pandx
Θφdp1
0xtdt·e.
bφis a nonvanishing subadditive function such that for allε0 there holdsε
0φdp0.
Proof. aThe integrability ofφona, b/Θfollows immediately from its continuity. Further, letxkbe a partition ofΘ, x. Thenxj tjxfor some partitiontjof0,1, and we have
limLlim
n−1
j0
φ tjx
x tj1−tj
lim
n−1
j0 1 0
xtdt·e
tj1−tj
1
0
xtdt·e,
2.8
and similarly limU1
0xtdt·e.
bFollows from the parta.
Proposition 2.3. Let the spaceX, dbe defined by2.1and 2.3. LetF : E → Ebe given by Fxt x2tfor 0≤t≤1/2, andFxt x1, otherwise, and letf F|X.
The spaceX, dtogether with the mappingsfandφsatisfies all assumptions ofTheorem 1.6.
On the other hand,fhas no fixed point.
Proof. We only have to check the inequality1.6. Note that for allz ∈ X and allt ≥ 1/2 we haveFzt 0. Also, note thatF is a linear mapping, andF|x| |Fx|. Therefore dFx, Fy |Fx−Fy||Fx−y|F|x−y| Fdx, y. Thus1.6becomes
Fdx,y
Θ φdp≤α dx,y
Θ φdp. 2.9 Taking into accountProposition 2.2, parta, we have for allz∈P
Fz
Θ φdp 1
0
Fztdt·e 1/2
0
z2tdt·e
1 2
1
0
zsds·e 1 2
z
Θφdp.
2.10
Puttingzdx, y, we obtain
Fdx,y
Θ φdp 1 2
dx,y
Θ φdp, 2.11 which completes the proof of the first statement.
On the other hand,fhas no fixed point. Namely, if we suppose thatxis a fixed point forf, it means thatxt≡0 for allt >1/2, and moreoverxt≡0 for allt >1/4, and also for allt >1/2n, by induction. By continuity ofx, it follows thatx0 0 implyingx /∈X!
3. A Comment
The mistake in the proof ofTheorem 1.6given in3is in the following. The authors from dxn1,xn
0 φdp → 0 conclude thatdxn1, xn → 0 also, which is unjustifiable. The original Branciari’s proof1deals with one-dimensional integral, and such conclusion is valid due to the implication
ε >0⇒ ε
0
φtdt >0 3.1
and the existence of the total ordering on R. However, in infinite dimensional case, such conclusion invokes continuity of the function inverse to x → x
0 φdp. Even for the linear mappings this is not always true, but only under additional assumption that initial mapping is bijective. This asserts the well known Banach open mapping theorem. In the absence of some generalization of the open mapping theorem to nonlinear case, it is necessary to include continuity of the inverse function in the assumptions, as it was done in4.
References
1 A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531–536, 2002.
2 L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,”
Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.
3 F. Khojasteh, Z. Goodarzi, and A. Razani, “Some fixed point theorems of integral type contraction in cone metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 189684, 13 pages, 2010.
4 F. Sabetghadam and H. P. Masiha, “Common fixed points for generalizedϕ-pair mappings on cone metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 718340, 8 pages, 2010.
