Abstract
In this paper, we first introduce two new classes of -contractions of the first and second kinds and establish some related new fixed point and best proximity point theorems in preordered metric spaces. Our theorems subsume the corresponding recent results of Samet (J. Optim. Theory Appl. (2013), doi:10.1007/s10957-013-0269-9) and extend and generalize many of the well-known results in the literature. An example is also provided to support our main results.
MSC: 47H10, 41A65.
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1 Introduction and preliminaries
Given a metric space and a self-mapping T on X, the theory on the existence of a solution to the equation of the form has gained impetus because of its applicability to solve many interesting problems that can be formulated as ordinary differential equations, matrix equations etc. For some recent fixed point results, see [1–6] and references therein. Let A and B be nonempty subsets of X, and let be a non-self mapping. The equation is unlikely to have a solution, because of the fact that a solution of the preceding equation demands the nonemptiness of . Eventually, it is quite natural to seek an approximate solution x that is optimal in the sense that the distance is minimum. The well-known best approximation theorem, due to Fan [7], states that if A is a nonempty, compact, and convex subset of a normed linear space X and T is a continuous function from A to X, then there exists a point x in A such that . Such a point x is called a best approximant point of T in A. Many generalizations and extensions of this theorem appeared in the literature (see [8–11] and references therein).
Best proximity problem for the pairs is to find an element such that , where . Since is a lower bound for the function on A, then the solutions of the best proximity problem are the minimum points of the function on A. Every solution of the best proximity problem is said to be a best proximity point of T in A. Moreover, if then every best proximity point of T is a fixed point. According to this fact, many authors by motivation of well-known fixed point results obtained sufficient conditions to solving best proximity problems; for more details, see [12–27] and the references therein.
Existence of best proximity and fixed points in partially ordered metric spaces has been considered recently by many authors (see [6, 13, 20, 28]). Recently Samet [29] studied the existence of best proximity points for a class of non-self almost -contractive mappings. In this work we define two new classes of contractions called -contractions of the first and second kind and establish some related new fixed point results in the setting of preordered metric spaces, and then we derive some new best proximity point theorems for these new classes of non-self contractive mappings. The presented theorems extend and generalize many of the well-known fixed point and best proximity point results.
2 Fixed point theory
Definition 2.1 Let be a metric space, and let .
-
(a)
Denote by Ω the family of functions such that , for each and for each sequence in X with,
-
(b)
Denote by Δ the family of functions such that δ is continuous and if for some , then ;
-
(c)
Denote by Φ the family of non-decreasing functions such that for each ;
-
(d)
Denote by Σ the family of functions such that for each and satisfies
(1) -
(e)
Denote by Ψ the family of non-decreasing functions such that for each ;
-
(f)
Denote by Λ the family of non-decreasing and upper semicontinuous from the right functions such that for each ;
-
(g)
Let Θ be a collection of the following functions:
, ;
, ;
, .
Lemma 2.2 Let be a metric space. Then the following statements hold:
-
(i)
,
-
(ii)
,
-
(iii)
,
-
(iv)
,
-
(v)
.
Proof Let be a sequence in X. To prove (i), assume that for each , where . Since ϕ is non-decreasing, then by induction we get
Then, for each sufficiently large , we have
and so is a Cauchy sequence.
(ii) Let us suppose that for each , where for each and α satisfying (1). Then
for each . Since for each , then is a non-increasing sequence of non-negative numbers and so is convergent to a real number, say . We will show that . On the contrary, assume that . Then from (2) we get
a contradiction and so . To show that is a Cauchy sequence, on the contrary assume that . Thus there exist subsequences and such that . Then, by the triangle inequality, we get
Then
for each . From the above, we obtain . Then from (1) we get , a contradiction. Therefore, is a Cauchy sequence.
(iii) Notice first that for each . To see this, suppose that there exists with , then since ψ is non-decreasing, we see that for all and it is a contradiction with for each . Note also that .
Now assume that for each , where . Since ψ is non-decreasing, then by induction we get
Let be fixed. Choose such that
Now we have
Also we have
So, by induction, for each , we have
This implies that is Cauchy and the proof of (iii) is complete.
(iv) For each , we have for each (see Remark 2.2 in [30]). Then the conclusion follows from (iii).
(v) obviously holds. □
Let X be a nonempty set. A preorder ⪯ on X is a binary relation which is reflexive and transitive. Let be a preordered set, and let be a mapping. We say that T is non-decreasing if for each , .
Definition 2.3 Let be a preordered set and d be a metric on X. We say that is regular if and only if the following condition holds:
Definition 2.4 Let be a preordered metric space, and let and be arbitrary mappings.
-
(a)
A mapping is said to be -contraction of the first kind if for all with ,
-
(b)
A mapping is said to be -contraction of the second kind if for all with ,
where .
Remark 2.5 If , that is, for each , then -contractions of the first and second kind are called -contractions of the first and second kind in brief. The class of -contraction maps of the first and second kind include the mappings with condition (B) [3] and almost generalized contractions [6], respectively.
Theorem 2.6 Let be a complete preordered metric space, and let be a mapping. Suppose that the following conditions hold:
-
(i)
T is continuous or is regular,
-
(ii)
T is non-decreasing,
-
(iii)
there exists such that ,
-
(iv)
T is an -contraction mapping of the first kind, where and .
Then T has a fixed point. Moreover, the sequence converges to the fixed point of T.
Proof Let for any . Since and T is non-decreasing, then we have
Now since T is an -contraction mapping of the first kind, we get
for all . Since and , so for all ,
Now, from (3) and (4), we have
for all . Since , so is a Cauchy sequence, hence there exists such that converges to . Now we show that is a fixed point of T. If T is continuous, then from the equality , we get . Now assume that is regular. Then, for each , we have . On the contrary, assume that . For any ,
Since and , then
and so from (5) we get . □
Corollary 2.7 Let be a complete metric space, and let be an -contraction mapping of the first kind, where and . Then
-
(i)
T has a unique fixed point. Moreover, for all , the sequence converges to the fixed point of T, that is, T is the Picard operator.
-
(ii)
T is continuous at .
Proof (i) Let . Then from Theorem 2.6 we deduce that T has a fixed point. To prove the uniqueness, on the contrary, assume that are distinct fixed points of T. So,
a contradiction. By the uniqueness of a fixed point and from Theorem 2.6, we get that the sequence converges to the fixed point of T for all .
(ii) Let and be a sequence in X such that . Since T is an -contraction mapping of the first kind, so for all we have
Since and , we have
Thus, for any ,
Thus , and so T is continuous at . □
Remark 2.8 Theorem 2.6 extends the main result of Babu et al. [3], Corollary 1 of Berinde et al. [4], Corollary 3.1 of Samet [29] and Theorem 2.1 of Agarwal et al. [30].
Theorem 2.9 Let be a complete preordered metric space, and let be a mapping. Suppose that the following conditions hold:
-
(i)
T is continuous or is regular,
-
(ii)
T is non-decreasing,
-
(iii)
there exists such that ,
-
(iv)
T is an -contraction mapping of the second kind, where and .
Then T has a fixed point. Moreover, the sequence converges to the fixed point of T.
Proof Let for any . If for some , then , and so is a fixed point of T, and we are finished. So, we may assume that for all . Now, since and T is non-decreasing, so
Since T is an -contraction of the second kind, so for all we have
Since and , for all ,
For all , we have
By the triangle inequality, we have
Hence, by (8), (9) and (10),
Now, if , then by (11) we have
a contradiction. So, for all , we have
As , so is a Cauchy sequence and so, by the completeness of , there exists such that converges to . Now we show that is a fixed point of T. If T is continuous, then from the equality , we get . Now, assume that is regular. Then, for each , we have . Now, on the contrary, assume that . So, for any ,
Since and , we have
Now let and choose such that for , we have , then
and
So, for , we have
Then, from (12) and (13), we get
a contradiction. □
Corollary 2.10 Let be a complete metric space, and let be an -contraction mapping of the second kind, where and . Then T has a unique fixed point. Moreover, for all , the sequence converges to the fixed point of T, that is, T is the Picard operator.
Proof By Theorem 2.9 it is sufficient to prove the uniqueness of the fixed point. On the contrary assume that are distinct fixed points of T. Then
a contradiction. □
Remark 2.11 Theorem 2.9 is a generalization of Theorem 2.2 and Theorem 2.3 of Agarwal et al. [30].
Remark 2.12 When for all we set where and where , in Corollary 2.10, we obtain Theorem 2.4 of Berinde [5].
3 Best proximity point theory
Let A and B be two nonempty subsets of a metric space . We denote by and the following sets:
where .
Definition 3.1 Let be a pair of nonempty subsets of the metric space with . Then the pair is said to have the P-property [31] if and only if
where and .
The following lemma is crucial in proving our best proximity point results.
Lemma 3.2 Let be a pair of nonempty closed subsets of a complete metric space such that and that satisfies the P-property. Then there exists a mapping satisfying
Furthermore, is closed.
Proof Let , then we show that there exists a unique such that . To prove the uniqueness, let us assume that there exists such that . Since has the P-property, we have and so . Let , then . Now, assume that , where . Then, by the P-property of , we get . Therefore, there exists a mapping such that
Now, we show that is closed. To prove the claim, let be a sequence in with (note that B is closed). Since A is a closed subset of a complete metric space, for each and is a Cauchy sequence, we deduce that . Since for each , we have
and so . Hence, is closed. □
Remark 3.3 It is clear that the mapping Q in Lemma 3.2 is a bijection and for any , we have .
Definition 3.4 Let be a preordered set. A non-self mapping is said to be proximally non-decreasing if and only if
where , .
The following lemma follows from Lemma 14 in [32].
Lemma 3.5 Let be a preordered metric space, and let be a non-self mapping such that . Let and Q be as in the statement of Lemma 3.2. Suppose that is proximally non-decreasing. Then the mapping defined by for each is non-decreasing.
The following lemma follows from Lemma 15 in [32].
Lemma 3.6 Let be a preordered metric space , and Q be as in Lemma 3.2 and be a non-self mapping such that . Suppose that there exist such that and . Let the mapping be defined by for each . Then .
Now, we are ready to establish our best proximity point theorems.
Theorem 3.7 Let be a pair of nonempty closed subsets of a complete preordered metric space such that . Let be a non-self mapping. Suppose that the following conditions hold:
-
(i)
and satisfy the P-property,
-
(ii)
T is continuous or is regular,
-
(iii)
T is proximally non-decreasing,
-
(iv)
there exist such that
-
(v)
For all such that , we have
(18)
where , and δ is non-decreasing in each of its variables.
Then T has a best proximity point in A.
Proof Since , so . By Lemma 3.2, is closed and there exists an isometry which satisfies (17). Let be defined by for each . Let and , then from (18) we have
but
and
So, from (19) we have
Thus S is an ordered -contraction mapping of the first kind. Now conditions (ii), (iii) and (iv) with Lemma 3.5 and Lemma 3.6 imply that S satisfies conditions (i), (ii) and (iii) of Theorem 2.6. Consequently, S has a fixed point such that and . That is, . Thus is the required best proximity point for T. □
Corollary 3.8 Let be a pair of nonempty closed subsets of a complete metric space such that and satisfies the P-property. Let such that for all ,
where , and δ is non-decreasing in each of its variables. Moreover, assume that . Then T has a best proximity point in A.
Theorem 3.9 Let be a pair of nonempty closed subsets of a complete preordered metric space such that . Let be a non-self mapping. Suppose that the following conditions hold:
-
(i)
and satisfy the P-property,
-
(ii)
T is continuous or is regular,
-
(iii)
T is proximally non-decreasing,
-
(iv)
there exist such that
-
(v)
For all such that , we have
(21)
where is non-decreasing, and δ is non-decreasing in each of its variables.
Then T has a best proximity point in A.
Proof Since , so . By Lemma 3.2, is closed and there exists an isometry which satisfies (17). Let be defined by for each . Let and , then from (21) we have
Since ω is non-decreasing and δ is non-decreasing in each of its variables, in view of the proof of Theorem 3.7, we get
for each , where . Thus S is an ordered -contraction mapping of the second kind. Now conditions (ii), (iii) and (iv) with Lemma 3.5 and Lemma 3.6 imply that S satisfies conditions (i), (ii) and (iii) of Theorem 2.9, so by Theorem 2.9 S has a fixed point such that and . Thus , as required. □
Corollary 3.10 Let be a pair of nonempty closed subsets of a complete metric space such that and satisfies the P-property. Let be such that for all ,
where is non-decreasing, and δ is non-decreasing in each of its variables. Moreover, assume that . Then T has a best proximity point in A.
Remark 3.11
-
(a)
Theorem 3.9 is a generalization of Theorem 20 of Jleli et al. [32].
-
(b)
From Lemma 2.2 and Theorem 3.7, we deduce the ordered version of Theorem 3.1 in [29].
From Lemma 2.2 and Corollary 3.8, we deduce the following result due to Samet [29].
Theorem 3.12 Let be a pair of nonempty closed subsets of a complete metric space such that , satisfies the P-property. Let such that for all ,
where , . Moreover, assume that . Then T has a best proximity point in A.
Now we provide the following example to show that Corollary 3.8 is an essential extension of the above mentioned theorem of Samet.
Example 3.13 Consider the complete metric space with the Euclidean metric. Let and . Then , , and has the P-property.
Let be defined by
Let for each , let and let for each . Then, for all , we have
and so the conditions of Corollary 3.8 are satisfied. Thus T has a best proximity point (indeed, is a best proximity point of T). But we cannot invoke the above mentioned theorem of Samet to show that the mapping T has a best proximity point in A because T is not an almost contraction. On the contrary, assume that there exist and such that for all ,
Letting , we get
Let for each . Then it is easy to see that f on is an increasing positive function. So, we have
Let , and let for each , then is a decreasing sequence of positive numbers. From the above, we have . Since φ is non-decreasing, then from the above, we get
Proceeding in this manner, we get for each , and so is convergent (note that is convergent). Let be a constant. Now we show that
Obviously, the inequality holds for . Now we proceed by induction. Assume that (24) holds for . Then we have
Then, from the above, we obtain (note that )
and so (24) holds for each . Since for each and , then we get , a contradiction.
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Acknowledgements
The first author was partially supported by a grant from IPM (No. 92470412). The second author was partially supported by a grant from IPM (No. 92550414). The first and the second author were also partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the fourth author acknowledges with thanks DSR, KAU for financial support.
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Amini-Harandi, A., Fakhar, M., Hajisharifi, H.R. et al. Some new results on fixed and best proximity points in preordered metric spaces. Fixed Point Theory Appl 2013, 263 (2013). https://doi.org/10.1186/1687-1812-2013-263
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DOI: https://doi.org/10.1186/1687-1812-2013-263