Before giving our main results, we first introduce the weak P-monotone property.
Weak P-monotone property Let be a pair of nonempty subsets of a partially ordered metric space with . Then the pair is said to have the weak P-monotone property if for any and ,
furthermore, implies .
Now we are in a position to give our main results.
Theorem 2.1 Let be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be an increasing mapping with , and let there exist such that
Assume that either f is continuous or that is such that if an increasing sequence , then , ∀n. Suppose that the pair has the weak P-monotone property. And for some , there exists such that and . Besides, if for each , there exists which is comparable to x and y, then there exists an in A such that .
Proof We first prove that is closed. Let be a sequence such that . It follows from the weak P-monotone property that
as , where and , . Then is a Cauchy sequence so that converges strongly to a point . By the continuity of a metric d, we have , that is, , and hence is closed.
Let be the closure of , we claim that . In fact, if , then there exists a sequence such that . By the continuity of f and the closeness of , we have . That is, .
Define an operator , by . Since the pair has the weak P-monotone property and f is increasing, we have
for any . Obviously, is increasing. Let , , , when f is continuous, then we have
Then is continuous. When is such that if an increasing sequence , then (∀n), we need not prove the continuity of . For some , there exists such that and . That is,
By the weak P-monotone property, we have .
This shows that is a contraction satisfying all the conditions in Theorem 1.3. Using Theorem 1.3, we can get has a unique fixed point . That is, , which implies that
Therefore, is the unique one in such that . □
Theorem 2.2 Let be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be a continuous and nondecreasing mapping with , and let f satisfy
where is a continuous and nondecreasing function such that ψ is positive in , and . Suppose that the pair has the weak P-monotone property. If for some , there exists such that and , then there exists an in A such that .
Proof In Theorem 2.1, we have proved that is closed and . Now we define an operator by . Since the pair has the weak P-monotone property, by the definition of f, we have
for any . Obviously, is continuous and nondecreasing. For some , there exists such that and . That is,
By the weak P-monotone property, we have .
This shows that is a contraction satisfying all the conditions in Theorem 1.4. Using Theorem 1.4, we can get has a fixed point . That is, , which implies that
Therefore, is the one in such that . □
Theorem 2.3 Let X be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be a continuous and nondecreasing mapping with , and let T satisfy
where ψ is an altering distance function and is a continuous function with the condition for all . Suppose that the pair has the weak P-monotone property. If for some , there exists such that and , then there exists an in A such that .
Proof In Theorem 2.1, we have proved that is closed and . Define an operator by . Since the pair has the weak P-monotone property and ψ is nondecreasing, by the definition of T, we have
for any . Since
this shows that is continuous and nondecreasing. Because there exist and such that and , by the weak P-monotone property, we have .
This shows that is a contraction from a complete metric subspace into itself and satisfies all the conditions in Theorem 1.6. Using Theorem 1.6, we can get has a fixed point . That is, , which implies that
Therefore, is the one in such that . □
Theorem 2.4 Let be a partially ordered set, and let be a complete metric space. Let be a pair of nonempty closed subsets of X such that . Let be a continuous and nondecreasing mapping with , and let T satisfy
where ψ and ϕ are altering distance functions. Suppose that the pair has the weak P-monotone property. If for some , there exists such that and , then there exists an in A such that .
Proof In Theorem 2.1, we have proved that is closed and . Define an operator by . Since the pair has the weak P-monotone property and ψ is nondecreasing, by the definition of T, we have
for any . Since
This shows that is continuous and nondecreasing. Because there exist and such that and , by the weak P-monotone property, we have .
This shows that is a contraction from a complete metric subspace into itself and satisfies all the conditions in Theorem 1.7. Using Theorem 1.7, we can get has a fixed point . That is, , which implies that
Therefore, is the one in such that . □