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Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type Fixed Point Theory and Applications 2011, 2011:102

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Fixed point-type results for a class of extended cyclic self-mappings under three

general weak contractive conditions of rational type

Fixed Point Theory and Applications 2011, 2011:102 doi:10.1186/1687-1812-2011-102

Manuel De la Sen (wepdepam@lg.ehu.es)Ravi P Agarwal (Agarwal@tamuk.edu)

ISSN 1687-1812

Article type Research

Submission date 12 September 2011

Acceptance date 21 December 2011

Publication date 21 December 2011

Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/102

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Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type

Manuel De la Sen*1 and Ravi P Agarwal2

1

Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus

of Leioa (Bizkaia) – Aptdo 644-Bilbao, 48080-Bilbao, Spain

2-best proximity points in each of the subsets

A general contractive condition has been proposed in [1, 2] for mappings on a partially ordered metric space Some results about the existence of a fixed point and then its

contractive condition proposed in [3] includes as particular cases several of the previously proposed ones [1, 4–12], including Banach principle [5] and Kannan fixed point theorems [4, 8, 9, 11] The rational contractive conditions of [1, 2] are applicable only on distinct points of the considered metric spaces In particular, the fixed point theory for Kannan mappings is extended in [4] by the use of a non-increasing function affecting the contractive condition and the best constant to ensure a fixed point is also obtained Three fixed point theorems which extended the fixed point theory for Kannan mappings were stated and proved in [11] More attention has been paid to the investigation of standard contractive and Meir-Keeler-type contractive 2-cyclic self-mappings T:A∪B→A∪B defined on subsets A,B⊆X and, in general, p-cyclic self-

mappings T :Ui∈p A i→Ui∈p A i defined on any number of subsets A i ⊂X , { , , , p}

p

i∈ := 1 2 , where (X , d) is a metric space (see, for instance [13–22]). More recent investigation about cyclic self-mappings is being devoted to its characterization in partially ordered spaces and also to the formal extension of the contractive condition through the use of more general strictly increasing functions of the distance between adjacent subsets In particular, the uniqueness of the best proximity points to which all the sequences of iterates of composed self-mappings T2:A∪B→A∪B converge is proven in [14] for the extension of the contractive principle for cyclic self-mappings in uniformly convex Banach spaces (then being strictly convex and reflexive [23]) if the subsets A,B⊂X in the metric space (X , d), or in the Banach space (X , ), where the 2-

cyclic self-mappings are defined, are both non-empty, convex and closed The research in [14] is centred on the case of the cyclic self-mapping being defined on the union of two subsets of the metric space Those results are extended in [15] for Meir-Keeler cyclic contraction maps and, in general, for the self-mapping T :Ui∈p A i→Ui∈p A i be a p( )≥ 2 -cyclic self-mapping being defined on any number of subsets of the metric space with

{ , , , p}

p:= 1 2 Also, the concept of best proximity points of (in general) non-mappings S , T:A→B relating non-empty subsets of metric spaces in the case that such maps do not have common fixed points has recently been investigated in [24, 25] Such an approach is extended in [26] to a mapping structure being referred to as K-cyclic mapping with contractive constant k<1 /2 In [27], the basic properties of cyclic self-mappings under a rational-type of contractive condition weighted by point-to-point-

self-dependent continuous functions are investigated On the other hand, some extensions of Krasnoselskii-type theorems and general rational contractive conditions to cyclic self-mappings have recently been given in [28, 29] while the study of stability through fixed point theory of Caputo linear fractional systems has been provided in [30] Finally, promising results are being obtained concerning fixed point theory for multivalued maps

(see, for instance [31–33])

This manuscript is devoted to the investigation of several modifications of rational type

of the φ-contractive condition of [21, 22] for a class of 2-cyclic self-mappings on empty convex and closed subsets A , B⊂X The contractive modification is of rational type and includes the nondecreasing function associated with the ϕ-contractions The existence and uniqueness of two best proximity points, one in each of the subsets

non-points coincide with the unique fixed point in the intersection of both the sets

2 Basic properties of some modified constraints of 2-cyclic ϕ

-contractions

Let (X , d ) be a metric space and consider two non-empty subsets A and B of X Let

B A

B

A

T: ∪ → ∪ be a 2-cyclic self-mapping, i.e., T( )A ⊆B and T( )B ⊆A Suppose, in addition, that T:A∪B→A∪B is a 2-cyclic modified weak ϕ -contraction (see [21, 22]) for some non-decreasing function ϕ :R0+ →R0+ subject to the rational modified ϕ -

contractive constraint:

( )

( ) ( ) (x , y) (d(x , y) (d(x , y) ) ) ( )D

d

Ty , y d Tx , x d y

, x d

Ty , y d Tx , x d

, x d k lim x T , x T

n n n n

Note that (2.1) is, in particular, a so-called 2-cyclic ϕ -contraction if α = 0 and( ) (t α)t

ϕ = 1 − for some real constant α ∈[0 ,1) since ϕ :R0+ →R0+ is strictly increasing [1]

We refer to “modified weak ϕ -contraction” for (2.1) in the particular case α ≥ 0,β ≥ 0,

condition obtained from (2.1) with α = 0, β = 1 , and ϕ :R0+ →R0+ being strictly increasing, that is,

min and α + β < 1 with ϕ :R0+→R0+ being non-decreasing Then, the following properties hold:

(i) Assume that ϕ( )D ≥D

k k

x m

If d(x,Tx) is finite and, in particular, if x and Tx in A∪B are finite then the sequences

n n x

T are bounded sequences where T n x∈A and T n+1x∈B if x∈A

and n is even, T n x∈B and T n+1x∈B if x∈B and n is even

Proof: Take y=Tx so that Ty=T2x Since ϕ :R0+ →R0+ is non-decreasing ϕ( )x ≥ ϕ( )D for

− +

=

−

−

− +

n

n

k D x , Tx kd k k D x , Tx d k x T

+ +

∞

→

1 0

1

1 ,

m n

i

i m

n n m n m n n

k k

D x , Tx d k lim x T x T

d

sup

lim

( )( ) ( )D

k

k lim

k

D

m n n

n

; ∀x∈A∪B Hence,

Property (i) follows from (2.9) and (2.10) since ϕ( )D ≥D and d(x,Tx)≥D; ∀x∈A∪B, since

B A

B

A

T: ∪ → ∪ is a 2-cyclic self-mapping and ϕ :R0+→R0+ is non-decreasing Now, it

follows from triangle inequality for distances and (2.9a) that:

1 1

1 1

1 ,

i

i n

i i n

i

i i n

k D

Tx , x d k x

T x T d x

Tx , x

1

k

k k

Tx , x d k

k

ϕ 1 1

1 1 1 1

n

n x

Concerning the case that A and B intersect, we have the following existence and

uniqueness result of fixed points:

Theorem 2.2 If ϕ( )D = D= 0 (i.e., A0∩B0 ≠ ∅) then ∃ ( + +1 + )= 0

∞

→

x T , x T d lim n m n m n

and

k

Tx , x d

non-to which all the sequences { }T n x n∈N0, which are Cauchy sequences, converge;

T is a Cauchy sequence, ∀x∈A∪B, then being

bounded and also convergent in A∩B as n→ ∞ since (X , d) is complete and Aand B are

non-empty, closed, and convex Thus, lim T n x z A B

T

lim

n n

Thus, z=y Hence, the theorem □

Now, the contractive condition (2.1) is modified as follows:

( )

( ) ( ) (x , y) (d(x , y) (d(x , y) ) ) ( )D

d

Ty , y d Tx , x d y

, x d

Ty , y d Tx , x d

for x , y(≠ x)∈X, where min(α0,β0)≥ 0, min(α0,β0)> 0 , and α0+ β0≤ 1 Note that in the former contractive condition (2.1), α + β < 1 Thus, for any non-negative real constants

d

Ty , y d Tx , x d y

, x d

Ty , y d Tx , x d

d

Ty , y d Tx , x d y

, x d

Ty , y d Tx ,

x

d

ϕ β

β ϕ

β in (2.1) whose sum can equalize unityα0+ β0= 1

Lemma 2.3 Assume that T:A∪B→A∪B is a cyclic self-map satisfying the contractive condition (2.13) with min(α0,β0)≥ 0, α0+ β0≤ 1 , and ϕ :R0+→R0+ is non-decreasing Assume also that

, Tx

α ϕ

−

−

−

≤ 1

1

0 , α ≤ α0 and β ≤ β0 with α + β < 1.Then, the following properties hold:

(i) D lim sup d(T n m x , T n m x) ( ) (D )D

n

β α β α

(ii) If ϕ( ) (D = 1 + α + β − α0− β 0)D then lim d(T n m x , T n m x) D

n n x

T are bounded sequences, where T n x∈A and T n+1x∈B if x∈A

and n is even and T n x∈B and T n+1x∈B if x∈B and n is even

Proof: Since ϕ :R0+→R0+ is non-decreasing then ϕ( )x ≥ ϕ( )D for x(∈R0+)≥D Note also

α

β α

α ϕ

1

1

M x , Tx d x , Tx d Tx

, x T d Tx

α β α β α ϕ

β β ϕ

=

≤

−

− +

−

−

−

≤ 1

1

0 One gets from (2.13) and (2.17) the following modifications of (2.9) and (2.10) by taking y=Tx, Ty=T2x, and successive iterates by composition of the self-mapping T:A∪B→A∪B:

n

n

+

− +

(Tx , x) ( )D M

≤ ϕ ; ∀x∈A∪B, ∀n∈N0: =N∪{ }0 (2.18)

m n

i

i m

n

n

k k

D D

x , Tx d

1 1

1

i i n

i i n

i

i i n

k M

D Tx

, x d k x

T x T d x

D Tx , x

Tx , x

n n x

T are bounded for any finite x∈A∪B Property (iii) has

Theorem 2.4 If ϕ( )D = D= 0 then ∃ ( + +1 + )= 0

∞

→

x T , x T d lim n m n m n

; ∀x∈A∪B Furthermore, if

(X , d ) is complete and both A and B are non-empty, closed, and convex then there is a

unique fixed point z∈A∩B of T:A∪B→A∪B to which all the sequences { }n∈N0

n x

which are Cauchy sequences, converge; ∀x∈A∪B

Proof guideline: It is identical to that of Theorem 2.2 by using ϕ( )D =D=M0=M = 0 and the fact that from (2.17) α0= α and β0 = β with 0 ≤ α + β < 1 if there is a pair (x , Tx)∈A×B∪B×A such that d(Tx , x)= ϕ(d(Tx , x) ); d(T2x , Tx)= ϕ(d(T2x , Tx) ); ∀x∈A∪B

Remark 2.5 Note that Lemma 2.2 (ii) for ϕ( )D ≤D (ϕ( )D <D if α + β ≤ α0+ β0≤ 1) leads to

an identical result as Lemma 2.1 (i) for ϕ( )D =D and α + β < 1 consisting in proving that

D≤ ϕ since d(T2x , Tx)≥D and d(T x , x)≥D; ∀x∈A∪B On the other hand, note from

Lemma 2.2, Equation (2.14)that ( ) 0

1

1

M D

D

β α

α ϕ

−

−

−

−

≥ , and one also gets from (2.18) for n

β α

M D

β α

Remark 2.7 Note that Lemmas 2.2 and 2.3 apply for non-decreasing functions

ϕ ϕ

ϕ ϕ

x sup lim

x

and

having a finite limit:

Lemma 2.8 Assume that T:A∪B→A∪B is a cyclic self-map satisfying the contractive condition (2.21) with min(α ,β)≥ 0, α + β < 1 , and ϕ :R0+ →R0+ is non-decreasing having a

finite limit ϕ( )= ϕ

∞

→

x lim

ϕ ϕ

x

sup

lim

x

Then, the following properties hold:

(i) The following relations are fulfilled:

β α ϕ β α

ϕ ϕ

2

x T , x T d D

β α ϕ

β α

ϕ ϕ

2

x T , x T d sup lim D

ϕ

− +

2

Tx , x T d Tx , x T d Tx , x T d

ϕ α

ϕ ϕ

T , x T d x T , x T

what implies the necessary condition ( )D D

β α

β α ϕ

β α

ϕ ϕ

x

sup

lim

x

; ∀x∈R+, by construction, then d(T n 1+x , T n x) is bounded; ∀n∈N

since, otherwise, a contradiction to (2.24) holds Since ϕ :R0+→R0+ is non-decreasing and has a finite limit ϕ ≥ ϕ( )x ≥ 0; ∀x∈R0+ (ϕ = 0 if and only if ϕ :R0+ →R0+ is identically zero), thus ϕ ≥ ϕ( )D ≥ 0 Then, (2.22)–(2.23) hold and Property (i) has been proven On the other hand, one gets from (2.25), since ϕ :R0+ →R0+ is sub-additive and nondecreasing and has a finite limit, that:

= +

x , x T

1

n i i n

i

i

k D

Tx , x d Tx , x d

k

α

ϕ ϕ

i

n

k D

Tx , x d Tx , x d

1

1 1

1

1

α

ϕ ϕ

k

k k

Tx , x

1

1 1

D k

k Tx

, x d Tx

ϕ ϕ

1

1 1

k

k Tx

, x d Tx , x d k

k x

Then the sequences { }T n x n∈N0 and { 1 } 0

N

∈ +

n n x

T are both bounded for any x∈A∪B Hence, the first part of Property (ii) If ϕ :R0+ →R0+ is identically zero then ( )= 0

from (2.23) Hence, the

The existence and uniqueness of a fixed point in A∩B if A and B are non-empty, closed,

and convex and (X , d) is complete follows in the subsequent result as its counterpart in Theorem 2.2 modified cyclic φ-contractive constraint (2.21):

Theorem 2.9 if (X , d ) is complete and A and Bintersect and are non-empty, closed, and convex then there is a unique fixed point z∈A∩B of T:A∪B→A∪B to which all the sequences { }n∈N0

n x

T , which are Cauchy sequences, converge; ∀x∈A∪B. □

Remark 2.7 Note that the nondecreasing function ϕ :R0+→R0+ of the contractive condition (2.21) is not monotone increasing under Lemma 2.5 since it possesses a finite

Remark 2.8 The case of T:A∪B→A∪B being a φ-contraction, namely, (Tx , Ty) d(x , y) (d(x , y) ) ( )D

d ≤ − ϕ + ϕ with strictly increasing ϕ :R0+→R0+; ∀x∈A∪B, [1, 2] implies , since ϕ( )x = 0 if and only if x= 0, implies the relation

(Tx , Ty) d(x , y) ( )D d(x , y) ( )D

d ≤ β1 + ϕ < + ϕ ; ∀x , y( )≠x ∈A∪B (2.29)

for some real constant 0 ≤ β1= β1(x , y)< 1; ∀x , y( )≠x ∈A∪B so that proceeding recursively:

i i n

Remark 2.9 Note that the constraint (2.1) implies in Lemma 2.1 and Theorem 2.2 that (1 − α − β) ( ) (ϕ D ≤ 1 − α − β)D what implies ϕ( )D ≤ D if max(α,β)> 0 since 0 ≤ α + β < 1 However, such a constraint in Lemma 2.3 and Theorem 3.4 implies that (1 − α0− β0) ( ) (ϕ D ≤ 1 − α0− β0)D □

This section considers the contractive conditions (2.1) and (2.21) for the case A ∩ B= ∅ For such a case, Lemmas 2.1, 2.3, and 2.8 still hold However, Theorems 2.2, 2.4, and 2.9

do not further hold since fixed points in A∩B cannot exist Thus, the investigation is centred in the existence of best proximity points It has been proven in [1] that if

B A

B

A

T: ∪ → ∪ isa cyclic φ-contraction with A and B being weakly closed subsets of a

n→ ∞ 2 , = : = ∈ for some y∈A and x∈B

then the sequence { 2 } 0

N

∈

n n

x

T has a convergent subsequence [14]) Theorem 2.2 extends

via Lemma 2.1 as follows for the case when A and B do not intersect, in general:

Theorem 3.1 Assume that T:A∪B→A∪B is a modified weak φ-contraction, that is, a cyclic self-map satisfying the contractive condition (2.1) subject to the constraints (α ,β)≥ 0

min and α + β < 1 with ϕ :R0+→R0+ being nondecreasing with ϕ( )D =D Assume also that A and B are non-empty closed and convex subsets of a uniformly convex Banach space (X, ) Then, there exist two unique best proximity points z∈A, y∈B of

B A

T converge to z and y for all x∈A , respectively, to y and z for

all x∈B If A ∩ B≠ ∅ then z= y∈A∩B is the unique fixed point of T:A∪B→A∪B

Proof: If D= 0, i.e., A and B intersect then this result reduces to Theorem 2.2 with the

best proximity points being coincident and equal to the unique fixed point Consider the

case that A and B do not intersect, that is, D> 0 and take x∈A∪B Assume with no loss

in generality that x∈A It follows, since A and B are non-empty and closed, A is convex

and Lemma 3.1 (i) that: