Abstract
In this paper, we consider two families and of mappings defined on satisfy some certain properties. Using the mentioned properties for and , we prove the analogous PPF dependent fixed point theorems for mappings as in Drici et al. (Nonlinear Anal. 67:641-647, 2007) in partially ordered Banach spaces where mappings satisfy the weaker contractive conditions without assuming the topological closedness with respect to the norm topology for the Razumikhin class .
MSC:54H25, 55M20.
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1 Introduction and preliminary results
The fixed point theorems for mappings satisfying certain contractive conditions have been continually studied for decade (see [1–9] and references contained therein). Bernfeld et al. [10] proved the existence of PPF (past, present and future) dependent fixed points in the Razumikhin class for mappings that have different domains and ranges. After that Dhage [11] extended the existence of PPF dependent fixed points to PPF common dependent fixed points for mappings satisfying the weaker contractive conditions. In 2007, Drici et al. [2] proved the fixed point theorems in partially ordered metric spaces for mappings with PPF dependence. In this paper, we consider two families and of mappings defined on satisfy some certain properties. Moreover, the PPF dependent fixed point theorems for mappings satisfying some generalized contractive conditions in partially ordered Banach spaces are proven using the mentioned properties for and .
Suppose that E is a real Banach space with the norm and I is a closed interval in ℝ. Let be the set of all continuous E-valued mappings on I equipped with the supremum norm defined by
for all . For a fixed element , the Razumikhin class of mappings in is defined by
Recall that a point is said to be a PPF dependent fixed point or a fixed point with PPF dependence of if for some .
Example 1.1 Let be defined by
Therefore T is a contraction with a constant . Suppose that for all . Since , we find that ϕ is a fixed point with dependence of T.
Definition 1.2 Let A be a subset of E. Then
-
(i)
A is said to be topologically closed with respect to the norm topology if for each sequence in A with as implies .
-
(ii)
A is said to be algebraically closed with respect to the difference if for all .
Recently, Dhage [11] proved the existence of PPF fixed points for mappings satisfying the condition of Cirić type generalized contraction assuming topological closedness with respect to the norm topology for a Razumikhin class.
Definition 1.3 (Dhage, [11])
A mapping is said to satisfy the condition of Cirić type generalized contraction if there exists a real number satisfying
for all and for some .
Theorem 1.4 (Dhage, [11])
Suppose that satisfies the condition of Cirić type generalized contraction. Assume that is topologically closed with respect to the norm topology and is algebraically closed with respect to the difference, then T has a unique PPF dependent fixed point in .
It is a natural question that the result of the previous theorem is still valid by omitting the topological closedness of . In the next result, we will answer the question.
Proposition 1.5 The Razumikhin class is topologically closed with respect to the norm topology.
Proof Let be a sequence in converging to ϕ. This implies that
Therefore
Since for all , we obtain . Therefore
By the uniqueness of the limit, we have . Hence and thus is topologically closed with respect to the norm topology. □
Hence, using Proposition 1.5, we can drop the topological closedness with respect to the norm topology for in Theorem 1.4.
The following example shows that the algebraic closedness with respect to the difference of Razumikhin class may fail.
Example 1.6 Let and . If we take and for all , then while .
Proposition 1.7 If the Razumikhin class is algebraically closed with respect to the difference, then is a convex set.
Proof Since is algebraically closed with respect to the difference, we have . Using the fact that , we obtain . Since for all , we get . Hence is a convex set. □
One can verify that Razumikhin class is a cone (i.e., , for each and ). Then by applying the previous theorem Razumikhin class is a convex cone (also closed).
In 2007, Drici et al. [2] proved the following the fixed point theorems in partially ordered complete metric spaces for mappings with PPF dependence.
Theorem 1.8 ([2])
Let be a partially ordered complete metric space and where and . Assume that
-
(i)
T is a nondecreasing mapping;
-
(ii)
for all with , where and ;
-
(iii)
there exists a lower solution such that ;
-
(iv)
T is a continuous mapping or if is a nondecreasing sequence in converging to , then for all .
Then T has a PPF dependent fixed point in .
It is a natural question if one can obtain the result of the aforementioned theorem for a generalized contraction (that is, condition (ii)). One of the aims of this paper is to answer the question by considering a general case of Cirić type generalized contractions.
In this paper, we consider two families and of mappings defined on satisfying some certain properties. Using the mentioned properties for and , we prove the analogous PPF dependent fixed point theorems for mappings as in [2] in partially ordered real Banach spaces where mappings satisfy the weaker contractive conditions.
2 PPF dependent fixed points in partially ordered Banach spaces
We begin this part with the consideration of the example of a partially ordered real Banach space. Recall that the set of all bounded linear operators from a normed space X into ℝ is a real Banach space with a norm defined by
We know that is a partially ordered real Banach space with a partial order defined as follows:
From now on, let be a partially ordered real Banach space. In this paper, we use the following notations:
Lemma 2.1 ([12])
Suppose that . If ψ is nondecreasing, then for each , implies .
Hence the difference between an element of and an element of is continuity.
Remark 2.2
-
(i)
It is easily seen that if is nondecreasing and for all , then .
-
(ii)
We can see that if , for all and , then ψ is continuous at 0.
We now prove the PPF dependent fixed point theorems for mappings satisfying the generalized contractive conditions concerning with without assuming the topological closedness with respect to the norm topology for the Razumikhin class .
Theorem 2.3 Suppose that , and satisfies the following conditions:
-
(i)
T is a nondecreasing mapping;
-
(ii)
for all with , we have
-
(iii)
there exists a lower solution such that ;
-
(iv)
T is a continuous mapping.
Assume that is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in .
Proof Since and , there exists such that . Choose such that . Since is a lower solution in such that , it follows that . Using the algebraic closedness with respect to the difference of , this yields
By the fact that T is nondecreasing, we obtain
By induction, we can construct the sequence such that
for all . Assume that for some . It follows that . Therefore T has a fixed point in . Suppose that for all . Therefore, for each , we obtain
If , then
which leads to a contradiction. Therefore, for each , we have
By induction, we obtain
Fix . This implies that there exists such that
For each with , we obtain
This implies that is a Cauchy sequence. By the completeness of , we have for some and
Since is algebraically closed with respect to the norm topology, we have . We next prove that ϕ is a PPF dependent fixed point of T. Using the continuity of T, we obtain . By the uniqueness of the limit, we have . □
Remark 2.4
-
(i)
From the proof of Theorem 2.3, we assume that the Razumikhin class is algebraically closed with respect to difference, that is, for all , in order to construct the sequence satisfying
-
(ii)
In the proof of Theorem 2.3, if we choose to be a constant mapping for each , then
Therefore the algebraic closedness with respect to the difference of can be dropped.
Example 2.5 Let with respect to the norm and . Define a mapping by
We see that ψ is a nondecreasing mapping with for all . Let . Therefore, for each , we obtain . Thus we can define mappings such that
Define a mapping by
where and . Suppose that with . For each , we obtain
Suppose that . We see that all assumptions in Theorem 2.3 are now satisfied and 0 is the PPF dependent fixed point of T in .
By applying Theorem 2.3, we obtain the following corollary.
Corollary 2.6 Suppose that and satisfies the following conditions:
-
(i)
T is a nondecreasing mapping;
-
(ii)
for all with , we have
-
(iii)
there exists a lower solution such that ;
-
(iv)
T is a continuous mapping.
Assume that is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in .
Proof Define a function by for all . Therefore ψ is a nondecreasing mapping and
This implies that all assumptions in Theorem 2.3 are satisfied. Hence we obtain the desired result. □
Theorem 2.7 Suppose that , and satisfies the following conditions:
-
(i)
T is a nondecreasing mapping;
-
(ii)
for all with , ;
-
(iii)
there exists a lower solution such that ;
-
(iv)
if is a nondecreasing sequence in converging to , then for all .
Assume that is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in .
Proof By the analogous proof as in Theorem 2.3, we can construct a nondecreasing sequence in converging to . This implies that for all . Therefore, for each , we have
Since ψ is continuous at 0, we get . Taking the limit of the above inequality, this yields and so ϕ is a PPF dependent fixed point of T in . □
We next ensure the result on PPF dependent fixed points for mappings concerning with .
Theorem 2.8 Suppose that , and satisfies the following conditions:
-
(i)
T is a nondecreasing mapping;
-
(ii)
for all with , we have
-
(iii)
there exists a lower solution such that ;
-
(iv)
T is a continuous mapping.
Assume that is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in .
Proof Since and , there exists such that . As in the proof of Theorem 2.3, we can construct the sequence such that
for all . Assume that for some . It follows that . Therefore T has a fixed point in . Suppose that for all . For each , we obtain
If , then
This leads to a contradiction. Therefore
It follows that for all . Since the sequence is a nonincreasing sequence of nonnegative real numbers, we see that it is a convergent sequence. Suppose that
for some nonnegative real number α. We will prove that . Suppose that . Since
for all and the continuity of ψ, we have which leads to a contradiction. This implies that . We next prove that the sequence is a Cauchy sequence in . Assume that is not a Cauchy sequence. It follows that there exist and two sequences of positive integers and satisfying for each and
Let be the sequence of the least positive integers exceeding which satisfies (2.1) and
We will prove that . Since for all , we have
For each , we obtain
This implies that . Therefore
Similarly, we can prove that
and
Since is algebraically closed with respect to the difference, for each , we obtain
By taking the limit of both sides, we have
This leads to a contradiction. It follows that the sequence is a Cauchy sequence. By the completeness of , we have for some and
Since is algebraically closed with respect to the norm topology, we have . We will prove that ϕ is a PPF dependent fixed point of T. Using the continuity of T, we obtain . By the uniqueness of the limit, we can conclude that . □
Example 2.9 Assume that and . Define a mapping by
We see that ψ is a continuous nondecreasing mapping with . Define a mapping by
Suppose that with and . Therefore
Suppose that . We find that all assumptions in Theorem 2.8 are now satisfied and 0 is the PPF dependent fixed point of T in .
For the next result, we drop the continuity of T.
Theorem 2.10 Suppose that , , and that satisfies the following conditions:
-
(i)
T is a nondecreasing mapping;
-
(ii)
for all with , we have
-
(iii)
there exists a lower solution such that ;
-
(iv)
if is a nondecreasing sequence in converging to , then for all .
Assume that is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in .
Proof As in the proof of Theorem 2.8, we can construct a nondecreasing sequence converging to and this yields
Using (iv), we have for all . Therefore, for each , we obtain
Letting , we obtain . If , then
This leads to a contradiction. Therefore . This implies that ϕ is a PPF dependent fixed point of T. □
By applying Theorem 2.8 and Theorem 2.10, we obtain the following corollary.
Corollary 2.11 Suppose that and that satisfies the following conditions:
-
(i)
T is a nondecreasing mapping;
-
(ii)
for all with , we have
-
(iii)
there exists a lower solution such that ;
-
(iv)
T is a continuous mapping or if is a nondecreasing sequence in converging to , then for all .
Assume that is algebraically closed with respect to the difference. Then T has a PPF dependent fixed point in .
Proof Define a function by for all . Therefore ψ is a continuous nondecreasing mapping and
This implies that all assumptions in Theorem 2.8 or Theorem 2.10 are satisfied. Hence the proof is complete. □
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The second author would like to express her deep thanks to the Centre of Excellence in Mathematics, the Commission of Higher Education and Naresuan University, Thailand for the support.
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Farajzadeh, A., Kaewcharoen, A. On fixed point theorems for mappings with PPF dependence. J Inequal Appl 2014, 372 (2014). https://doi.org/10.1186/1029-242X-2014-372
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DOI: https://doi.org/10.1186/1029-242X-2014-372