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Some new results for single-valued and multi-valued mixed monotone operators of Rhoades type
Fixed Point Theory and Applications volume 2013, Article number: 73 (2013)
Abstract
In (2008), Zhang proved the existence of fixed points of mixed monotone operators along with certain convexity and concavity conditions. In this paper, mixed monotone single-valued and multi-valued operators of Rhoades type are defined and two fixed point theorems are proved.
MSC:47H10, 47H07.
1 Introduction and preliminaries
In (1987), mixed monotone operators were introduced by Guo and Lakshmikantham [1]. Then many authors studied them in Banach spaces and obtained lots of interesting results (see [2, 3] and [4–8]).
On the other hand, in (2001), Rhoades [9] introduced a new fixed point theorem as a generalization of Banach fixed point theorem.
Theorem 1.1 (Rhoades [9])
Let be a complete metric space. Suppose that is a single-valued mapping that satisfies
for each , where is continuous, nondecreasing and (i.e., weakly contractive mappings). Then T has a fixed point.
In this paper, a weak mixed monotone single-valued and multi-valued operator of Rhoades type is defined. Then two fixed point theorems for this kind of operators are proved.
Let E be a real Banach space. The zero element of E is denoted by θ. A subset P of E is called a cone if and only if:
-
P is closed, nonempty and ,
-
, and imply that ,
-
and imply that .
Given a cone , a partial ordering ≤ with respect to P is defined by if and only if . We write to indicate that but , while stands for , where intP denotes the interior of P. The cone P is called normal if there exists a number such that implies for every . The least positive number satisfying this is called the normal constant of P.
Assume that and . The set is denoted by .
Now, we recall the following definitions from [2, 3].
Definition 1.1 Let P be a cone of a real Banach space E. Suppose that and . An operator is said to be α-convex (α-concave) if it satisfies () for .
Definition 1.2 Let E be an ordered Banach space and . An operator is called mixed monotone on if and for any , where and . Also, is called a fixed point of A if .
Let be a collection of all closed subsets of E.
Definition 1.3 For two subsets X, Y of E, we write
-
if for all , there exists such that ,
-
if there exists such that ,
-
if for all , .
Definition 1.4 Let D be a nonempty subset of E. is called increasing (decreasing) upward if , and imply there exists such that (). Similarly, is called increasing (decreasing) downward if , and imply there exists such that (). T is called increasing (decreasing) if T is an increasing (decreasing) upward and downward.
Definition 1.5 Let D be a nonempty subset of E. A multi-valued operator is said to be mixed monotone upward if is increasing upward in x and decreasing upward in y, i.e.,
(A1) for each and any with , if , then there exists a such that ;
(A2) for each and any with , if , then there exists a such that .
Definition 1.6 is called a fixed point of T if .
Definition 1.7 [10]
A function is called an -function if , for , and for all .
In 2011, Khojasteh and Razani [10] extended the results given by Zhang [6]. Also, in 2011 Khojasteh and Razani [11] introduced the concept of integral with respect to a cone. We recall the following definitions and lemmas of cone integration and refer to [11, 12] for their proofs.
Definition 1.8 [11]
Suppose that P is a cone in E. Let and . Define
and
Definition 1.9 [11]
The set is called a partition for if and only if the intervals are pairwise disjoint and . Denote as the collection of all partitions of .
Definition 1.10 [12]
For each partition Q of and each increasing function , we define cone lower summation and cone upper summation as
and
respectively. Also, we denote .
Definition 1.11 [12]
Suppose that P is a cone in E. is called an integrable function on with respect to a cone P or, to put it simply, a cone integrable function if and only if for all partition Q of ,
where must be unique.
We show the common value by
We denote the set of all cone integrable functions by .
Lemma 1.1 [11]
Let M be a subset of P. The following conditions hold:
-
(1)
If , then for .
-
(2)
for and .
Remark 1.1 [[13], Remark 1.2]
Let P be a cone of E, and let . If for each , , then .
2 Main results
In this section, we introduce some new fixed point theorems in the class of mixed monotone operators. Due to this, the following definition is presented.
Definition 2.1 A mixed monotone operator is said to be a Weak Mixed Monotone single-valued operator of Rhoades type (WM2R property for short) if
for all , where is an -function.
Theorem 2.1 Let P be a cone of E, let S be a completely ordered closed subset of E with and let for all . Let , be a weak mixed monotone operator of Rhoades type with satisfying the following conditions:
-
(I)
there exists such that ,
-
(II)
,
-
(III)
for with , there exists such that ; similarly, for with , there exists such that .
Then A has at least one fixed point .
Proof By the above condition (III), there exists such that . Then there exists such that . Likewise, there exists such that . Then there exists such that . In general, there exists such that . Then there exists such that ().
Take , thus and . Since , thus . In addition, S is completely ordered and for all , then . Now, one can prove . Otherwise, .
Since and , hence , and we have
which is a contradiction. Thus, . Let be given. Choose such that , where . Since , one can choose a natural number such that for all . Therefore . Also, and
By Remark 1.1, .
For all , applying the same argument, we have
Also,
Hence, and are Cauchy sequences in E, then there exist such that , () and . Write .
It is easy to see for all . In addition, S is closed, then ().
Finally, by the mixed monotone property of A,
On taking limit on both sides of (11), when , we have . This means is a fixed point of A in . □
Corollary 2.1 Let P be a cone of E, let S be a completely ordered closed subset of E with and let for all . Let , satisfy
for all , where is an -function, and let be a non-vanishing, cone integrable mapping on each such that for each , and the mapping for has a continuous inverse at zero. Also, satisfies the following conditions:
-
(I)
there exists such that ,
-
(II)
,
-
(III)
for with , there exists such that ; similarly, for with , there exists such that .
Then A has at least one fixed point .
Proof
Define
A is a mixed monotone operator, and one can easily see that all conditions of Theorem 2.1 hold. Thus we obtain the desired result. □
3 M3R property
In this section, we introduce a new fixed point theorem in the class of multi-valued mixed monotone operators. Due to this, the following definition is given.
Definition 3.1 A mixed monotone operator is said to be a Mixed Monotone Multi-valued operator of Rhoades type (M3R property for short) if
for each , where is an -function.
Theorem 3.1 Let P be a cone of E, let S be a completely ordered closed subset of E with and let for all . Let , be a mixed monotone multi-valued operator of Rhoades type with satisfying the following conditions:
-
(I)
there exists such that ,
-
(II)
,
-
(III)
for with , there exists such that ; similarly, for with , there exists such that .
Then T has at least one fixed point .
Proof By the above condition (III), there exists such that . Then there exists such that . Likewise, there exists such that . Then there exists such that . In general, there exists such that . Then there exists such that ().
Take , thus , and . Since , thus . In addition, S is completely ordered and for all , then . Now, one can prove . Otherwise, . We claim
Suppose that is arbitrary. We have . If , and , then by (A1) of Definition 1.5, there exists such that . Thus, .
Also, if , and , then for , there exists such that . It means that
Thus,
and this is a contradiction. Therefore, . Let be given. Choose such that , where . Since , one can choose a natural number such that for all . Therefore . Also, and
By Remark 1.1, .
For all , applying the same argument, we have
Also,
Hence, and are Cauchy sequences in E, then there exist such that , () and . Write .
It is easy to see that for all . Thus, there exists such that . By taking limit on both sides of (17),
So, . Since T has closed values, then and
□
Remark 3.1 One can see easily that Theorem 2.1 should be included as a corollary of Theorem 3.1.
Example 3.1 Let , and . Then .
Define as
A is a mixed monotone operator. Now suppose that is as . Then Ψ is an -function. Moreover,
for each . Also, by taking , and , we have
-
(I)
,
-
(II)
,
-
(III)
for with , there exists such that ; similarly, for with , there exists such that .
For further explanation on (III), since , by (III) there exists such that . It means that . Thus must be greater than . Therefore we can set . Similarly, since , thus by (III) there exists such that . It means that must be less than . We can set . By the continuity of such ways, we can consider the following reflexive sequences:
which satisfy (I), (II) and (III) (see Figure 1). Moreover, and and .
4 Application
The following result is given by Zhang [6] and is obtained by our main result.
Corollary 4.1 Let P be a normal cone of E, let S be a completely ordered closed subset of E with and let for all . Let , be a mixed monotone operator with and . Assume that there exists a function such that , where for all . Suppose that
-
(I)
for with , there exists such that ; similarly, for with , there exists such that .
-
(II)
there exists an element such that for all , and for all .
Then A has at least one fixed point .
Proof Set . Then Ψ is an -function, and we have
Thus, by Theorem 2.1 the desired result is obtained. □
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Khojasteh, F. Some new results for single-valued and multi-valued mixed monotone operators of Rhoades type. Fixed Point Theory Appl 2013, 73 (2013). https://doi.org/10.1186/1687-1812-2013-73
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DOI: https://doi.org/10.1186/1687-1812-2013-73