nach oben

Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2013 | Research

Best proximity points for generalized proximal C-contraction mappings in metric spaces with partial orders

verfasst von: Chirasak Mongkolkeha, Yeol Je Cho, Poom Kumam

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2013

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Patentsuche

Aus

Abstract

In this paper we extend the notion of weakly C-contraction mappings to the case of non-self mappings and establish the best proximity point theorems for this class. Our results generalize the result due to Harjani et al. (Comput. Math. Appl. 61:790-796, 2011) and some other authors.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

1 Introduction and preliminaries

In 1922, Banach proved that every contractive mapping in a complete metric space has a unique fixed point, which is called Banach’s fixed point theorem or Banach’s contraction principle. Since Banach’s fixed point theorem, many authors have extended, improved and generalized this theorem in several ways and, further, some applications of Banach’s fixed point theorem can be found in [1‐6] and many others.

In 1972, Chatterjea [7] introduced the following definition.

Definition 1.1 Let

(X, d)

be a metric space. A mapping

T : X \to X

is called a C-contraction if there exists

α \in (0, \frac{1}{2})

such that, for all

x, y \in X

d (T x, T y) \leq α (d (x, T y) + d (y, T x)) .

In 2009, Choudhury [8] introduced a generalization of C-contraction given by the following definition.

Definition 1.2 Let

(X, d)

be a metric space. A mapping

T : X \to X

is called a weakly C-contraction if, for all

x, y \in X

d (T x, T y) \leq \frac{1}{2} [d (x, T y) + d (y, T x)] - ψ (d (x, T y), d (y, T x)),

(1.1)

where

ψ : {[0, \infty)}^{2} \to [0, \infty)

is a continuous and nondecreasing function such that

ψ (x, y) = 0

if and only if

x = y = 0

In 2011, Harjani et al. [9] presented some fixed point results for weakly C-contraction mappings in a complete metric space endowed with a partial order as follows.

Theorem 1.3 Let

(X, ⪯)

be a partially ordered set and suppose that there exists a metric d in X such that

(X, d)

is a complete metric space. Let

T : X \to X

be a continuous and nondecreasing mapping such that

d (T x, T y) \leq \frac{1}{2} [d (x, T y) + d (y, T x)] - ψ (d (x, T y), d (y, T x))

for

x ⪯ y

, where

ψ : {[0, \infty)}^{2} \to [0, \infty)

is a continuous and nondecreasing function such that

ψ (x, y) = 0

if and only if

x = y = 0

. If there exists

x_{0} \in X

with

x_{0} ⪯ T x_{0}

, then T has a fixed point.

On the other hand, most of the results on Banach’s fixed point theorem dilate upon the existence of a fixed point for self-mappings. Nevertheless, if T is a non-self mapping, then it is probable that the equation

T x = x

has no solution, in which case best approximation theorems explore the existence of an approximate solution, whereas best proximity point theorems analyze the existence of an approximate solution that is optimal.

A classical best approximation theorem was introduced by Fan [10], that is, if A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space B and

T : A \to B

is a continuous mapping, then there exists an element

x \in A

such that

d (x, T x) = d (T x, A)

. Afterward, several authors including Prolla [11], Reich [12], Sehgal and Singh [13, 14] have derived the extensions of Fan’s theorem in many directions. Other works on the existence of a best proximity point for some contractions can be seen in [15‐19]. In 2005, Eldred, Kirk and Veeramani [20] obtained best proximity point theorems for relatively nonexpansive mappings, and some authors have proved best proximity point theorems for several types of contractions (see, for example, [21‐26]).

Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let A and B be nonempty subsets of a metric space

(X, d)

. Now, we recall the following notions:

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-94/MediaObjects/13660_2012_Article_537_Equc_HTML.gif

A \cap B \neq \emptyset

, then

A_{0}

and

B_{0}

are nonempty. Further, it is interesting to notice that

A_{0}

and

B_{0}

are contained in the boundaries of A and B, respectively, provided A and B are closed subsets of a normed linear space such that

d (A, B) > 0

(see [27]).

Definition 1.4 A mapping

T : A \to B

is said to be increasing if

x ⪯ y ⟹ S x ⪯ S y

for all

x, y \in A

Definition 1.5 [28]

A mapping

T : A \to B

is said to be proximally order-preserving if and only if it satisfies the condition that

\begin{array}{l} x ⪯ y \\ d (u, T x) = d (A, B) \\ d (v, T y) = d (A, B) \end{array}} ⟹ u ⪯ v

(1.2)

for all

u, v, x, y \in A

It is easy to observe that for a self-mapping, the notion of a proximally order-preserving mapping reduces to that of an increasing mapping.

Definition 1.6 A point

x \in A

is called a best proximity point of the mapping

T : A \to B

d (x, T x) = d (A, B) .

In view of the fact that

d (x, T x) \geq d (A, B)

for all x in A, it can be observed that the global minimum of the mapping

x \mapsto d (x, T x)

is attained from a best proximity point. Moreover, it is easy to see that the best proximity point reduces to a fixed point if the underlying mapping T is a self-mapping.

In this paper, we introduce a new class of proximal contractions, which extends the class of weakly C-contractive mappings to the class of non-self mappings, and also give some examples to illustrate our main results. Our results extend and generalize the corresponding results given by Harjani et al. [9] and some authors in the literature.

2 Main results

In this section, we first introduce the notion of a generalized proximal C-contraction mapping and establish the best proximity point theorems.

Definition 2.1 A mapping

T : A \to B

is said to be a generalized proximal C-contraction if, for all

u, v, x, y \in A

, it satisfies

\begin{array}{l} x ⪯ y \\ d (u, T x) = d (A, B) \\ d (v, T y) = d (A, B) \end{array}} ⟹ d (u, v) \leq \frac{1}{2} (d (x, v) + d (y, u)) - ψ (d (x, v), d (y, u)),

(2.1)

where

ψ : {[0, \infty)}^{2} \to [0, \infty)

is a continuous and nondecreasing function such that

ψ (x, y) = 0

if and only if

x = y = 0

For a self-mapping, it is easy to see that equation (2.1) reduces to (1.1).

Theorem 2.2 Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let A and B be nonempty closed subsets of X such that

A_{0}

and

B_{0}

are nonempty. Let

T : A \to B

satisfy the following conditions:

(a) T is a continuous, proximally order-preserving and generalized proximal C-contraction such that

T (A_{0}) \subseteq B_{0}

;

(b) there exist elements

x_{0}

and

x_{1}

A_{0}

such that

x_{0} ⪯ x_{1}

and

d (x_{1}, T x_{0}) = d (A, B) .

Then there exists a point

x \in A

such that

d (x, T x) = d (A, B) .

Moreover, for any fixed

x_{0} \in A_{0}

, the sequence

{x_{n}}

defined by

d (x_{n + 1}, T x_{n}) = d (A, B)

converges to the point x.

Proof By the hypothesis (b), there exist

x_{0}, x_{1} \in A_{0}

such that

x_{0} ⪯ x_{1}

and

d (x_{1}, T x_{0}) = d (A, B) .

Since

T (A_{0}) \subseteq B_{0}

, there exists a point

x_{2} \in A_{0}

such that

d (x_{2}, T x_{1}) = d (A, B) .

By the proximally order-preserving property of T, we get

x_{1} ⪯ x_{2}

. Continuing this process, we can find a sequence

{x_{n}}

A_{0}

such that

x_{n - 1} ⪯ x_{n}

and

d (x_{n}, T x_{n - 1}) = d (A, B) .

Having found the point

x_{n}

, one can choose a point

x_{n + 1} \in A_{0}

such that

x_{n} ⪯ x_{n + 1}

and

d (x_{n + 1}, T x_{n}) = d (A, B) .

(2.2)

Since T is a generalized proximal C-contraction, for each

n \in N

, we have

\begin{array}{rcl} d (x_{n}, x_{n + 1}) & \leq & \frac{1}{2} (d (x_{n - 1}, x_{n + 1}) + d (x_{n}, x_{n})) - ψ (d (x_{n - 1}, x_{n + 1}), d (x_{n}, x_{n})) \\ = & \frac{1}{2} d (x_{n - 1}, x_{n + 1}) - ψ (d (x_{n - 1}, x_{n + 1}), 0) \\ \leq & \frac{1}{2} d (x_{n - 1}, x_{n + 1}) \\ \leq & \frac{1}{2} (d (x_{n - 1}, x_{n}) + d (x_{n}, x_{n + 1})) \end{array}

(2.3)

and so it follows that

d (x_{n}, x_{n + 1}) \leq d (x_{n - 1}, x_{n})

, that is, the sequence

{d (x_{n + 1}, x_{n})}

is non-increasing and bounded below. Then there exists

r \geq 0

such that

lim_{n \to \infty} d (x_{n + 1}, x_{n}) = r .

(2.4)

Taking

n \to \infty

in (2.3), we have

r \leq lim_{n \to \infty} \frac{1}{2} d (x_{n - 1}, x_{n + 1}) \leq \frac{1}{2} (r + r) = r

and so

lim_{n \to \infty} d (x_{n - 1}, x_{n + 1}) = 2 r .

(2.5)

Again, taking

n \to \infty

in (2.3) and using (2.4), (2.5) and the continuity of ψ, we get

r \leq \frac{1}{2} (2 r) = r - ψ (2 r, 0) \leq r

and hence

ψ (2 r, 0) = 0

. So, by the property of ψ, we have

r = 0

, which implies that

lim_{n \to \infty} d (x_{n + 1}, x_{n}) = 0 .

(2.6)

Next, we prove that

{x_{n}}

is a Cauchy sequence. Suppose that

{x_{n}}

is not a Cauchy sequence. Then there exist

ε > 0

and subsequences

{x_{m_{k}}}

{x_{n_{k}}}

{x_{n}}

such that

n_{k} > m_{k} \geq k

with

r_{k} : = d (x_{m_{k}}, x_{n_{k}}) \geq ε, d (x_{m_{k}}, x_{n_{k} - 1}) < ε

(2.7)

for each

k \in {1, 2, 3, \dots}

. For each

n \geq 1

, let

α_{n} : = d (x_{n + 1}, x_{n})

. So, we have

\begin{array}{rcl} ε & \leq & r_{k} \leq d (x_{m_{k}}, x_{n_{k} - 1}) + d (x_{n_{k} - 1}, x_{n_{k}}) \\ < & ε + α_{n_{k} - 1} . \end{array}

It follows from (2.6) that

lim_{k \to \infty} r_{k} = ε .

(2.8)

Notice also that

\begin{array}{rcl} r_{k} & = & d (x_{n_{k}}, x_{m_{k}}) \leq d (x_{n_{k}}, x_{m_{k} + 1}) + d (x_{m_{k} + 1}, x_{m_{k}}) \\ = & d (x_{n_{k}}, x_{m_{k} + 1}) + α_{m_{k}} \\ \leq & d (x_{n_{k}}, x_{m_{k}}) + d (x_{m_{k}}, x_{m_{k} + 1}) + α_{m_{k}} \\ = & r_{k} + α_{m_{k}} + α_{m_{k}} . \end{array}

(2.9)

Taking

k \to \infty

in (2.9), by (2.6) and (2.8), we conclude that

lim_{k \to \infty} d (x_{n_{k}}, x_{m_{k} + 1}) = ε .

(2.10)

Similarly, we can show that

lim_{k \to \infty} d (x_{m_{k}}, x_{n_{k} + 1}) = ε .

(2.11)

On the other hand, by the construction of

{x_{n}}

, we may assume that

x_{m_{k}} ⪯ x_{n_{k}}

such that

d (x_{n_{k} + 1}, T x_{n_{k}}) = d (A, B)

(2.12)

and

d (x_{m_{k} + 1}, T x_{m_{k}}) = d (A, B) .

(2.13)

By the triangle inequality, (2.12), (2.13) and the generalized proximal C-contraction of T, we have

\begin{array}{rcl} ε & \leq & r_{k} \leq d (x_{m_{k}}, x_{m_{k} + 1}) + d (x_{n_{k} + 1}, x_{n_{k}}) + d (x_{m_{k} + 1}, x_{n_{k} + 1}) \\ = & α_{m_{k}} + α_{n_{k}} + d (x_{m_{k} + 1}, x_{n_{k} + 1}) \\ \leq & α_{m_{k}} + α_{n_{k}} + \frac{1}{2} [d (x_{n_{k}}, x_{m_{k} + 1}) + d (x_{m_{k}}, x_{n_{k} + 1})] \\ - ψ (d (x_{n_{k}}, x_{m_{k} + 1}), d (x_{m_{k}}, x_{n_{k} + 1})) . \end{array}

Taking

k \to \infty

in the above inequality, by (2.6), (2.10), (2.11) and the continuity of ψ, we get

ε \leq \frac{1}{2} (ε + ε) - ψ (ε, ε) \leq ε .

Therefore,

ψ (ε, ε) = 0

. By the property of ψ, we have that

ε = 0

, which is a contradiction. Thus

{x_{n}}

is a Cauchy sequence. Since A is a closed subset of the complete metric space X, there exists

x \in A

such that

lim_{n \to \infty} x_{n} = x .

(2.14)

Letting

n \to \infty

in (2.2), by (2.14) and the continuity of T, it follows that

d (x, T x) = d (A, B) .

□

Corollary 2.3 Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let A and B be nonempty closed subsets of X such that

A_{0}

and

B_{0}

are nonempty. Let

T : A \to B

satisfy the following conditions:

(a) T is continuous, increasing such that

T (A_{0}) \subseteq B_{0}

and

\begin{array}{l} x ⪯ y \\ d (u, T x) = d (A, B) \\ d (v, T y) = d (A, B) \end{array}} ⟹ d (u, v) \leq α (d (x, v) + d (y, u)),

(2.15)

where

α \in (0, \frac{1}{2})

;

(b) there exist

x_{0}, x_{1} \in A_{0}

such that

x_{0} ⪯ x_{1}

and

d (x_{1}, T x_{0}) = d (A, B) .

Then there exists a point

x \in A

such that

d (x, T x) = d (A, B) .

Moreover, for any fixed

x_{0} \in A_{0}

, the sequence

{x_{n}}

defined by

d (x_{n + 1}, T x_{n}) = d (A, B)

converges to the point x.

Proof Let

α \in (0, \frac{1}{2})

and the function ψ in Theorem 2.2 be defined by

ψ (a, b) = (\frac{1}{2} - α) (a + b) .

Obviously, it follows that

ψ (a, b) = 0

if and only if

a = b = 0

and (2.1) become (2.15). Hence we obtain Corollary 2.3. □

For a self-mapping, the condition (b) implies that

x_{0} ⪯ T x_{0}

and so Theorem 2.2 includes the results of Harjani et al. [9] as follows.

Corollary 2.4 [9]

Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let

T : X \to X

be a continuous and nondecreasing mapping such that, for all

x, y \in X

d (T x, T y) \leq \frac{1}{2} [d (x, T y) + d (y, T x)] - ψ (d (x, T y), d (y, T x))

for

x ⪯ y

, where

ψ : {[0, \infty)}^{2} \to [0, \infty)

is a continuous and nondecreasing function such that

ψ (x, y) = 0

if and only if

x = y = 0

. If there exists

x_{0} \in X

with

x_{0} ⪯ T x_{0}

, then T has a fixed point.

Now, we give an example to illustrate Theorem 2.2.

Example 2.5 Consider the complete metric space

R^{2}

with an Euclidean metric. Let

A = {(x, 0) : x \in R}, B = {(0, y) : y \in R, y \geq 1} .

Then

d (A, B) = 1

A_{0} = {(0, 0)}

and

B_{0} = {(0, 1)}

. Define a mapping

T : A \to B

as follows:

T ((x, 0)) = (0, 1 + | x |)

for all

(x, 0) \in A

. Clearly, T is continuous and

T (A_{0}) \subseteq B_{0}

. If

x_{1} ⪯ x_{2}

and

d (u_{1}, T x_{1}) = d (A, B) = 1, d (u_{2}, T x_{2}) = d (A, B) = 1

for some

u_{1}, u_{2}, x_{1}, x_{2} \in A

, then we have

u_{1} = u_{1} = (0, 0), x_{1} = x_{2} = (0, 0) .

Therefore, T is a generalized proximal C-contraction with

ψ : {[0, \infty)}^{2} \to [0, \infty)

defined by

ψ (a, b) = \frac{1}{4} (a + b) .

Further, observe that

(0, 0) \in A

such that

d ((0, 0), T (0, 0)) = d (A, B) = 1 .

In Theorem 2.6, we do not need the condition that T is continuous. Now, we improve the condition in Theorem 2.2 to prove the new best proximity point theorem as follows.

Theorem 2.6 Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let A and B be nonempty closed subsets of X such that

A_{0}

and

B_{0}

are nonempty. Let

T : A \to B

satisfy the following conditions:

(a) T is a proximally order-preserving and generalized proximal C-contraction such that

T (A_{0}) \subseteq B_{0}

;

(b) there exist elements

x_{0}, x_{1} \in A_{0}

such that

x_{0} ⪯ x_{1}

and

d (x_{1}, T x_{0}) = d (A, B);

{x_{n}}

is an increasing sequence in A converging to x, then

x_{n} ⪯ x

for all

n \in N

Then there exists a point

x \in A

such that

d (x, T x) = d (A, B) .

Proof

As in the proof of Theorem 2.2, we have

d (x_{n + 1}, T x_{n}) = d (A, B)

(2.16)

for all

n \geq 0

. Moreover,

{x_{n}}

is a Cauchy sequence and converges to some point

x \in A

. Observe that for each

n \in N

\begin{array}{rcl} d (A, B) & = & d (x_{n + 1}, T x_{n}) \leq d (x_{n + 1}, x) + d (x, T x_{n}) \\ \leq & d (x, x_{n + 1}) + d (x, x_{n + 1}) + d (x_{n + 1}, T x_{n}) \\ \leq & d (x, x_{n + 1}) + d (x, x_{n + 1}) + d (A, B) . \end{array}

Taking

n \to \infty

in the above inequality, we obtain

{lim}_{n \to \infty} d (x, T x_{n}) = d (A, B)

and hence

x \in A_{0}

. Since

T (A_{0}) \subseteq B_{0}

, there exists

v \in A

such that

d (v, T x) = d (A, B) .

(2.17)

Next, we prove that

x = v

. By the condition (c), we have

x_{n} ⪯ x

for all

n \in N

. Using (2.16), (2.17) and the generalized proximal C-contraction of T, we have

d (x_{n + 1}, v) \leq \frac{1}{2} [d (x_{n}, v) + d (x, x_{n + 1})] - ψ (d (x_{n}, v), d (x, x_{n + 1})) .

(2.18)

Letting

n \to \infty

in (2.18), we get

d (x, v) \leq \frac{1}{2} d (x, v) - ψ (d (x, v), 0),

which implies that

d (x, v) = 0

, that is,

x = v

. If we replace v by x in (2.17), we have

d (x, T x) = d (A, B) .

□

Corollary 2.7 Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let A and B be nonempty closed subsets of X such that

A_{0}

and

B_{0}

are nonempty. Let

T : A \to B

satisfy the following conditions:

(a) T is an increasing mapping such that

T (A_{0}) \subseteq B_{0}

and

\begin{array}{l} x ⪯ y \\ d (u, T x) = d (A, B) \\ d (v, T y) = d (A, B) \end{array}} ⟹ d (u, v) \leq α (d (x, v) + d (y, u)),

(2.19)

where

α \in (0, \frac{1}{2})

;

(b) there exist

x_{0}, x_{1} \in A_{0}

such that

x_{0} ⪯ x_{1}

and

d (x_{1}, T x_{0}) = d (A, B);

{x_{n}}

is an increasing sequence in A converging to a point

x \in X

, then

x_{n} ⪯ x

for all

n \in N

Then there exists a point

x \in A

such that

d (x, T x) = d (A, B) .

Corollary 2.8 [9]

Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Assume that if

{x_{n}} \subseteq X

is a nondecreasing sequence such that

x_{n} \to x

in X, then

x_{n} ⪯ x

for all

n \in N

. Let

T : X \to X

be a nondecreasing mapping such that

d (T x, T y) \leq \frac{1}{2} [d (x, T y) + d (y, T x)] - ψ (d (x, T y), d (y, T x))

for

x ⪯ y

, where

ψ : {[0, \infty)}^{2} \to [0, \infty)

is a continuous and nondecreasing function such that

ψ (x, y) = 0

if and only if

x = y = 0

. If there exists

x_{0} \in X

with

x_{0} ⪯ T x_{0}

, then T has a fixed point.

Now, we recall the condition defined by Nieto and Rodríguez-López [3] for the uniqueness of the best proximity point in Theorems 2.2 and 2.6.

For x, y \in X, there exists z \in X which is comparable to x and y .

(2.20)

Theorem 2.9 Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let A and B be nonempty closed subsets of X and let

A_{0}

and

B_{0}

be nonempty such that

A_{0}

satisfies the condition (2.20). Let

T : A \to B

satisfy the following conditions:

(a) T is a continuous, proximally order-preserving and generalized proximal C-contraction such that

T (A_{0}) \subseteq B_{0}

;

(b) there exist elements

x_{0}

and

x_{1}

A_{0}

such that

x_{0} ⪯ x_{1}

and

d (x_{1}, T x_{0}) = d (A, B) .

Then there exists a unique point

x \in A

such that

d (x, T x) = d (A, B) .

Proof We will only prove the uniqueness of the point

x \in A

such that

d (x, T x) = d (A, B)

. Suppose that there exist x and

x^{*}

in A which are best proximity points, that is,

d (x, T x) = d (A, B) and d (x^{*}, T x^{*}) = d (A, B) .

Case I: x is comparable to

x^{*}

, that is,

x ⪯ x^{*}

(or

x^{*} ⪯ x

). By the generalized proximal C-contraction of T, we have

d (x, x^{*}) \leq \frac{1}{2} [d (x, x^{*}) + d (x^{*}, x)] - ψ (d (x, x^{*}), d (x^{*}, x)) \leq d (x^{*}, x),

which implies that

ψ (d (x, x^{*}), d (x^{*}, x)) = 0

. Using the property of ψ, we get

d (x^{*}, x) = 0

and hence

x = x^{*}

Case II: x is not comparable to

x^{*}

. Since

A_{0}

satisfies the condition (2.20), then there exists

z \in A_{0}

such that z is comparable to x and

x^{*}

, that is,

x ⪯ z

(or

z ⪯ x

) and

x^{*} ⪯ z

(or

z ⪯ x^{*}

). Suppose that

x ⪯ z

and

x^{*} ⪯ z

. Since

T (A_{0}) \subseteq B_{0}

, there exists a point

v_{0} \in A_{0}

such that

d (v_{0}, T z) = d (A, B) .

By the proximally order-preserving property of T, we get

x ⪯ v_{0}

and

x^{*} ⪯ v_{0}

. Since

T (A_{0}) \subseteq B_{0}

, there exists a point

v_{1} \in A_{0}

such that

d (v_{1}, T v_{0}) = d (A, B) .

Again, by the proximally order-preserving property of T, we get

x ⪯ v_{1}

and

x^{*} ⪯ v_{1}

. One can proceed further in a similar fashion to find

v_{n}

A_{0}

with

v_{n + 1} \in A_{0}

such that

d (v_{n + 1}, T v_{n}) = d (A, B) .

Hence

x ⪯ v_{n}

and

x^{*} ⪯ v_{n}

for all

n \in N

. By the generalized proximal C-contraction of T, we have

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-94/MediaObjects/13660_2012_Article_537_Equ23_HTML.gif

(2.21)

https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-94/MediaObjects/13660_2012_Article_537_Equ24_HTML.gif

(2.22)

It follows from (2.21), (2.22) and the property of ψ that

v_{n} \to x and v_{n} \to x^{*} as n \to \infty .

By the uniqueness of limit, we conclude that

x = x^{*}

. Other cases can we proved similarly and this completes the proof. □

Theorem 2.10 Let X be a nonempty set such that

(X, ⪯)

is a partially ordered set and let

(X, d)

be a complete metric space. Let A and B be nonempty closed subsets of X and let

A_{0}

and

B_{0}

be nonempty such that

A_{0}

satisfies the condition (2.20). Let

T : A \to B

satisfy the following conditions:

(a) T is a proximally order-preserving and generalized proximal C-contraction such that

T (A_{0}) \subseteq B_{0}

;

(b) there exist elements

x_{0}, x_{1} \in A_{0}

such that

x_{0} ⪯ x_{1}

and

d (x_{1}, T x_{0}) = d (A, B);

{x_{n}}

is an increasing sequence in A converging to x, then

x_{n} ⪯ x

for all

n \in N

Then there exists a unique point

x \in A

such that

d (x, T x) = d (A, B) .

Proof For the proof, combine the proofs of Theorems 2.6 and 2.9. □

Acknowledgements

This research was partially finished at the Department of Mathematics Education, Gyeongsang National University, Republic of Korea. Mr. Chirasak Mongkolkeha was supported by the Thailand Research Fund through the Royal Golden Jubilee Program under Grant PHD/0029/2553 for the Ph.D. program at KMUTT, Thailand. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF-2012-0008170). The third author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkut’s University of Technology Thonburi (KMUTT) (Grant No. MRG5580213).

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Vorheriger Artikel Majorization properties for certain new classes of analytic functions using the Salagean operator

Nächster Artikel Weak convergence theorems for common solutions of a system of equilibrium problems and operator equations involving nonexpansive mappings

Agarwal RP, EL-Gebeily MA, O’regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151MathSciNetCrossRef

Cho YJ:Fixed points for compatible mappings of type

(A)

. Math. Jpn. 1993, 18: 497–508.

Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5MathSciNetCrossRef

Pathak HK, Cho YJ, Kang SM: Remarks on R -weakly commuting mappings and common fixed point theorems. Bull. Korean Math. Soc. 1997, 34: 247–257.MathSciNet

Sessa S, Cho YJ: Compatible mappings and a common fixed point theorem of Chang type. Publ. Math. (Debr.) 1993, 43: 289–296.MathSciNet

Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040MathSciNetCrossRef

Chatterjea SK: Fixed point theorems. C. R. Acad. Bulgare Sci. 1972, 25: 727–730.MathSciNet

Choudhury BS: Unique fixed point theorem for weak-contractive mappings. Kathmandu Univ. J. Sci. Eng. Technol. 2009, 5: 6–13.

Harjani J, López B, Sadarangani K: Fixed point theorems for weakly C -contractive mappings in ordered metric spaces. Comput. Math. Appl. 2011, 61: 790–796. 10.1016/j.camwa.2010.12.027MathSciNetCrossRef

10.

Fan K: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 1969, 112: 234–240. 10.1007/BF01110225MathSciNetCrossRef

11.

Prolla JB: Fixed point theorems for set valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 1982–1983, 5: 449–455.MathSciNetCrossRef

12.

Reich S: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 1978, 62: 104–113. 10.1016/0022-247X(78)90222-6MathSciNetCrossRef

13.

Sehgal VM, Singh SP: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. Math. Soc. 1988, 102: 534–537.MathSciNet

14.

Sehgal VM, Singh SP: A theorem on best approximations. Numer. Funct. Anal. Optim. 1989, 10: 181–184. 10.1080/01630568908816298MathSciNetCrossRef

15.

Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70: 3665–3671. 10.1016/j.na.2008.07.022MathSciNetCrossRef

16.

Eldred AA, Veeramani P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081MathSciNetCrossRef

17.

Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 2008, 69: 3790–3794. 10.1016/j.na.2007.10.014MathSciNetCrossRef

18.

Karpagam S, Agrawal S: Best proximity point theorems for p -cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009., 2009: Article ID 197308

19.

Mongkolkeha C, Kumam P: Some common best proximity points for proximity commuting mappings. Optim. Lett. 2012. doi:10.1007/s11590–012–0525–1 in press

20.

Eldred AA, Kirk WA, Veeramani P: Proximal normal structure and relatively nonexpanisve mappings. Stud. Math. 2005, 171: 283–293. 10.4064/sm171-3-5MathSciNetCrossRef

21.

Al-Thagafi MA, Shahzad N: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 2009, 70: 1209–1216. 10.1016/j.na.2008.02.004MathSciNetCrossRef

22.

Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 2012, 155: 215–226. 10.1007/s10957-012-9991-yMathSciNetCrossRef

23.

Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380MathSciNetCrossRef

24.

Wlodarczyk K, Plebaniak R, Banach A: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 2009, 70: 3332–3341. 10.1016/j.na.2008.04.037MathSciNetCrossRef

25.

Wlodarczyk K, Plebaniak R, Banach A: Erratum to: ‘Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces’. Nonlinear Anal. 2009, 71: 3583–3586.MathSciNet

26.

Sintunavarat W, Kumam P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93. doi:10.1186/1687–1812–2012–93

27.

Sadiq Basha S, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415MathSciNetCrossRef

28.

Sadiq Basha S: Best proximity point theorems on partially ordered sets. Optim. Lett. 2012. doi:10.1007/s11590–012–0489–1

Titel: Best proximity points for generalized proximal C-contraction mappings in metric spaces with partial orders
verfasst von: Chirasak Mongkolkeha
Yeol Je Cho
Poom Kumam
Publikationsdatum: 01.12.2013
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2013
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1186/1029-242X-2013-94

Springer Professional

Abstract

Competing interests

Authors’ contributions

1 Introduction and preliminaries

2 Main results

Acknowledgements

Competing interests

Authors’ contributions

Weitere Artikel der Ausgabe 1/2013

New complexity analysis of interior-point methods for the Cartesian -SCLCP

A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces

Cesáro partial sums of certain analytic functions

Second-order duality for nondifferentiable minimax fractional programming problems with generalized convexity

Some new iterated Hardy-type inequalities: the case θ = 1

Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means