De la Sen,manuel.delasen@ehu.es Received 13 May 2009; Accepted 31 August 2009 Recommended by Andrzej Szulkin In recent literature concerning fixed point theory for self-mappings T : X →
Trang 1Volume 2009, Article ID 815637, 25 pages
Institute of Research and Development of Processes, Faculty of Science and Technology,
University of the Basque Country, Campus de Leioa (Bizkaia), Apertado 644 de Bilbao, 48080 Bilbao, Spain
Correspondence should be addressed to M De la Sen,manuel.delasen@ehu.es
Received 13 May 2009; Accepted 31 August 2009
Recommended by Andrzej Szulkin
In recent literature concerning fixed point theory for self-mappings T : X → X in metric spaces
X, d, there are some new concepts which can be mutually related so that the inherent properties
of each one might be combined for such self-mappings Self-mappings T : X → X can be referred to, for instance, as Kannan-mappings, reasonable expansive mappings, and Picard T-
stable mappings Some relations between such concepts subject either to sufficient, necessary,
or necessary and sufficient conditions are obtained so that in certain self-mappings can exhibitcombined properties being inherent to each of its various characterizations
Copyrightq 2009 M De la Sen This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited
1 The properties of the so-called n-times reasonably expansive mapping are
investigated in1 in complete metric spaces X, d as those fulfilling the property that dx, T n x ≥ βdx, Tx for some real constant β > 1 The conditions for the
existence of fixed points in such mappings are investigated
2 Strong convergence of the wellknown Halpern’s iteration and variants is gated in2,8 and several the references therein
investi-3 Fixed point techniques have been recently used in 4 for the investigation of globalstability of a wide class of time-delay dynamic systems which are modeled byfunctional equations
Trang 24 Generalized contractive mappings have been investigated in 5 and referencestherein, weakly contractive and nonexpansive mappings are investigated in6 andreferences therein.
5 The existence of fixed points of Liptchitzian semigroups has been investigated, forinstance, in3
6 Picard’s T-stability is discussed in 9 related to the convergence of perturbediterations to the same fixed points as the nominal iteration under certain conditions
in a complete metric space
7 The so-called Kannan mappings in 10 are recently investigated in 11,12 andreferences therein
LetX, d be a metric space Consider a self-mapping T : X → X The basic concepts
used through the manuscript are the subsequent ones:
1 T : X → X is k-contractive, following the contraction Banach’s principle, if there exists a real constant k ∈ 0, 1 such that
this manuscript as the L, m property for some real constants L ∈ R0 and m ∈
0, 1 ∈ R0seeDefinition 1.2in what follows, then Picard’s iteration is T-stable iflimk→ ∞d y k1, Ty k 0
The following result is direct
Proposition 1.1 If a self-mapping T : X → X is k-contractive, then it is also k -contractive;
∀k ∈ k, 1.
If a self-mapping T : X → X is α-Kannan, then it is also α -Kannan; ∀α ∈ α, 1/2.
The so- called theL, m-property is defined as follows.
Definition 1.2 A self-mapping T : X → X with F T / ∅ possesses the L, m-property for some
real constants L ∈ R0 and m ∈ 0, 1 ∈ R0 if d Tx, q ≤ Ldx, Tx mdx, q; ∀q ∈ FT,
∀x ∈ X.
The above property has been introduced in [ 9 ] to discuss the T-stability of Picard’s iteration If the L, m-property is fulfilled in a complete metric space and, furthermore, lim k→ ∞d y k1, Ty k 0,
then Picard’s iteration x k1 Tx k is T-stable defined as d x k1, Ty k → 0 as k → ∞ ⇒ x k → q ∈
F T as k → ∞ The main results obtained in this paper rely on the following features.
Trang 31 In fact k-contractive mappings T : X → X are α-Kannan self-mappings and vice-versa
under certain mutual constraints between the constants k and α, [ 10 – 12 ] A necessary and sufficient condition for both properties to hold is given Some of such constraints are obtained in the manuscript The existence of fixed points and their potential uniqueness is discussed accordingly under completeness of the metric space, [ 1 4 , 8 10 , 13 ].
2 If T : X → X is n (Z n ≥ 2)-times reasonable expansive self-mapping then it cannot be
contractive as expected but it is α-Kannan under certain constraints The converse is also true under certain constraints Some of such constraints referred to are obtained explicitly
in the manuscript The existence of fixed points is also discussed for two types of n (Z
n ≥ 2)-times reasonable expansive self-mappings proposed in [ 1 ].
3 The L, m-property guaranteeing Picard’s T-stability of iterative schemes, under the added
condition lim k→ ∞d y k1, Ty k 0, is compatible with both contractive self-mappings
and α-Kannan ones under certain constraints A sufficient condition that as self-mapping possessing the L, m-property is α-Kannan is also given It may be also fulfilled by n (Z
n ≥ 2)-times reasonable expansive self-mappings.
α-Kannan Mappings, and the L − m-Property
It is of interest to establish when a k-contractive mapping is also α-Kannan and viceversa.
Theorem 2.1 The following properties hold:
i if T : X → X is k-contractive with k ∈ 0, 1/3 then it is α-Kannan with α k/1 − k,
ii T : X → X is k-contractive and α-Kannan if and only if
Trang 4iv if T : X → X is k-contractive and α-Kannan with k / 0, and 0 / α < k then the
1− k
d x, Tx dTy, y
2.3
are feasible for all x, y in X.
Proof i Since T : X → X is k-contractive, then
so that T : X → X is α-Kannan with α k/1 − k provided that k/1 − k < 1/2 ⇔ k < 1/3.
As a result, if T : X → X is k-contractive with k ∈ 0, 1/3, then it is also k/1 − k-Kannan.
ii It is direct if T : X → X is k-contractive and α-Kannan with k / 0 and α / 0 For
α k 0, the result holds trivially.
iii Proceed by contradiction Assume that the inequality holds for x, y ∈ X ∩ FT with x y where FT is the empty or nonempty set of fixed points of T Since x y, the inequality leads to 2αd x, Tx 0 This implies that dx, Tx 0 since α / 0 However,
d x, Tx > 0; ∀x /∈ FT, what is a contradiction Therefore, the inequality cannot cold in X.
iv The first inequality can potentially hold even for the set of fixed points
Furthermore, one gets from the triangle inequality for the distance dx, Tx ≤ dx, y
for all x, y ∈ X since α < k Also, by using dy, Ty ≤ dx, y dx, Tx, one gets dx, y ≤
α/ k − αdx, Tx dx, Ty As a result, the second inequality follows by combining both
partial results The third inequality follows from the second one and Propertyi Property
iv has been proven
Theorem 2.1ii leads to the subsequent result
Trang 5Corollary 2.2 If T : X → X is k-contractive and α-Kannan, then
inequalities together yield the result
The following two results follows directly fromTheorem 2.1iii for y Tx.
Corollary 2.3 If T : X → X is k-contractive and α-Kannan with k > α / 0, then the inequality
d Tx, T2x ≤ k − α/αdx, Tx cannot hold ∀x ∈ X.
Corollary 2.4 If T : X → X is k-contractive and α-Kannan with α > k / 0, then the inequality
α − kdx, Tx αdTx, T2x ≤ 0 cannot hold for x ∈ X ∩ FT.
The following three results follows directly fromTheorem 2.1iv for y Tx.
Corollary 2.5 If T : X → X is k-contractive and α-Kannan with k > α / 0, then the inequality
is feasible from the first feasible inequality inTheorem 2.1ii ∀x ∈ X and y Tx.
Corollary 2.6 If T : X → X is k-contractive and α-Kannan with k > 2α / 0, then the inequality
is feasible from the second feasible inequality inTheorem 2.1ii ∀x ∈ X and y Tx.
Corollary 2.7 If T : X → X is k-contractive and α-Kannan with k > 2α / 0, then the inequality
Trang 6Remark 2.8 It turns out from Definition 1.2 that if T : X → X has the L, m property for some real
constants L∈ R0and m ∈ 0, 1 ∈ R0, then it has also the L0, m0; ∀L0 ∈ L, ∞, ∀m0∈ m, 1.
The subsequent result is concerned with some joint L, m, α-Kannan and k-contractiveness of a
self-mapping T : X → X.
Theorem 2.9 The following properties hold:
i T : X → X is α-Kannan if it has the L, m-property for any real constants L and m which
satisfy the constraints α L m/1 − m, 0 ≤ L < 1 − 3m/2, 0 ≤ m < 1/3,
ii assume that T : X → X is k-contractive Then, it is also k/1 − k-Kannan and it
possesses the k − m/1 − k, m-property for any real constant m which satisfies 0 ≤
m ≤ k < 1/3,
iii assume that T : X → X is α-Kannan and FT / ∅ Then T : X → X has the L,
m-property with L α 2/1 − α and ∀m ∈ 0, 1 ∩ R,
iv assume that T : X → X is k-contractive with k ∈ 0, 1/3 ∩ R and FT / ∅ Then
T : X → X is k/1 − k-Kannan and it has the L, m-property with L 2 − 3k/1 −
Thus, T : X → X is α-Kannan with α : L m/1 − m < 1/2 which holds if 0 ≤ L <
1−3m/2 and 0 ≤ m < 1/3 Property i is proven Furthermore, if T : X → X is k-contractive then it is also α-Kannan if α k/1 − k with k < 1/3 fromTheorem 2.1ii Then, T : X → X
is k-contractive, α-Kannan, and it has the L, m-property if α : Lm/1−m k/1−k < 1/2 which holds for k : Lm/L1 α/1α < 1/3 if 0 ≤ L : k−m/1−k < 1−3m/2
and 0 ≤ m ≤ k < 1/3 which is already fulfilled since T : X → X is α-Kannan with the
k − m/1 − k, m-property Property ii has been proven.
iii By using the triangle inequality for distances and taking x ∈ X and q ∈ FT, one
Trang 7for any real constant m ∈ 0, 1 after using the subsequent relation:
Further results concerning α-Kannan mappings follow below.
Theorem 2.10 Assume that T : X → X is α-Kannan Then, the following properties hold:
iv if X, d is a complete metric space and T : X → X is k-contractive for some k ∈ 0, 1/3
or if it is α-Kannan and k-contractive, then z limj→ ∞T n j x ∈ X is independent of x;
∀x ∈ X, ∀n ∈ Zso that F T {z} consists of a unique fixed point.
Trang 8Proof Proceed by complete induction by assuming that d Tx, T j1x ≤ j
i1α/
1 − α i d x, Tx; ∀x ∈ X, ∀j ∈ n − 1 : { 1, 2, , n − 1} Since T : X → X is α-Kannan, take
y T n x so that one gets from the triangle inequality for distances and the above assumption
α 1 − α/1 − 2α so that dTx, T n1x ≤ 1 − α/1 − 2αdx, Tx; ∀x ∈ X and the proof
of Propertyi is complete
Propertyii.1 follows from Property i, since T is α-Kannan, by taking into account that it is k-contractive Propertyii.2 follows directly from Property i andTheorem 2.1i.Propertyii.3 follows from
0≤ d T j x, T n j1 x
≤ 1− α
1− 2α d x, Tx
lim sup
Propertyiv follows directly from Properties ii and iii from the uniqueness of the
fixed point Banach’s contraction mapping principle since T is a strict contraction.
Proposition 2.11 If T : X → X is α-Kannan, then dTx, x ≤ 1 − α/1 − 2αdTx, T2x ;
∀x ∈ X If, in addition, T : X → X is k-contractive, then 1 − 2α/1 − α ≤ k < 1.
Proof It holds that
Trang 9for all x ∈ X by using the triangle property of distances andTheorem 2.10i The first part ofthe result has been proven The second part of the result follows since
Remark 2.12 If T : X → X is k-contractive and α-Kannan, it follows from Corollary 2.2 and
Proposition 2.11 that 1 > min k, α/1 − α ≥ 1 − α/1 − α, α/1 − α; ∀x ∈ X.
Proposition 2.13 If T : X → X is α-Kannan then dTx, T n1x ≤ 1 − α/1− 2α2d Tx, T2x ;
.Proof The upper-bound for d Tx, x has been obtained inProposition 2.11 Its lower-bound
1−α/αdTx, T2x follows fromTheorem 2.10i subject to 1−α/α ≤ 1−α/1−2α which
holds∀x ∈ X if and only if α ≥ 1/3 The proof is complete.
α-Kannan-Mappings, and a Class of Expansive Mappings
Definition 3.1 see 1 Let X, d be a complete metric space Also, T : X → X is said to be an n
(Z n ≥ 2)-times reasonable expansive self-mapping if there exists a real constant β > 1 such that
d x, T n x ≥ βdx, Tx; ∀x ∈ X, Z n ≥ 2. 3.1
.
Theorem 3.2 Let X, d be a complete metric space Assume that T : X → X is a continuous
surjective self-mapping which is continuous everywhere in X and α-Kannan while it also satisfies
d T n−1x, T n x ≥ βdx, Tx for some real constant β > 1, some n ≥ 2 ∈ Z, ∀x ∈ X (i.e., T : X →
X is n (Z n ≥ 2) times reasonable expansive self-mapping) Then, the following properties hold if
Trang 10β > 1/ 1 − α:
i dx, Tx ≤ β1 − α − α/β1 − αdT n−1x, T n x ; ∀x ∈ X,
ii T : X → X has a unique fixed point in X,
iii T : X → X has a fixed point in X even if it is not α-Kannan.
Proof Since T : X → X is α-Kannan and it satisfies dT n−1x, T n x ≥ βdx, Tx; some real constant β > 1, some n≥ 2 ∈ Z,∀x ∈ X, then
where g : X → X is the identity mapping on X; that is, gx x; ∀x ∈ X, f : X → X
is defined by f x Tx Tgx; ∀x ∈ X and then it is a surjective mapping since T is
surjective and the functional ϕ : Im T ⊂ X → R0is defined as ϕx β1−α−α/β 1−
α n−2
j0d T j x, T j1x It turns out that ϕ : ImT ⊂ X → R0is continuous everywhere onits definition domainand then lower semicontinuous bounded from below as a result since
the distance mapping d : X ×X → R0is continuous on X Then, T : X → X has a fixed point
in X in1, Lemma 2.4, even if T : X → X is not α-Kannan, since f is surjective on X, g is the
identity mapping on X, and ϕ is lower semicontinuous bounded from below The fixed point
is unique sinceX, d is a complete metric space Properties ii-iii have been proven.
The subsequent result gives necessary conditions forTheorem 3.2to hold as well as asufficient condition for such a necessary condition to hold
Trang 11Theorem 3.3 Let (X, d) be a complete metric space Assume that T : X → X is a surjective
self-mapping which is continuous everywhere in X which satisfies d T n−1x, T n x ≥ βdx, Tx for some
real constant β > 1, some n≥ 2 ∈ Z, ∀x ∈ X The following holds i The following zero limit
and such limits superior and inferior coincide as existing limits and are zero.
Proof i Assume that Property i does not hold Then, T has not a fixed point in X what
contradictsTheorem 3.2iii Thus, Property i holds
ii The condition dT n−1x, T n x ≥ βdx, Tx; ∀x ∈ X together with the α-Kannan
Trang 12for all x ∈ X, ∀j ∈ Zso that dT j n x, T j n1 x − dT j n−1 x, T j n x → 0 as Z j → ∞ is a a
sufficient condition for Property i to hold The necessary condition for the above sufficient
to hold follows directly from the constraint βdx, Tx ≥ αdT n−1x, T n1x ; ∀x ∈ X.
iii It follows since the subsequent constraints follow directly from the hypotheses
and T : X → X has a fixed point
Theorem 3.2 may be generalized by generalizing the inequality dT n−1x, T n x ≥
βd x, Tx to eventually involve other powers of T, not necessarily being respectively identical
ton − 1 and n, as follows.
Theorem 3.4 Let X, d be a complete metric space Then, the following properties hold.
i assume that T : X → X is a surjective self-mapping which is continuous everywhere in X
for some real constants β n −j−1 > 1 ; ∀j ∈ J, some n ≥ 2 ∈ Z, ∀x ∈ X, then, T : X → X has at least
a fixed point in X and it may eventually possess δ = card J ≥ 1 fixed points in X.
ii if Property (i) holds for J n − 1 ∪ {0} then T : X → X has at least a fixed point in X