A function α: is said to be an -function (or ℛ-function) [12, 25–30] if for all . It is obvious that if α: is a nondecreasing function or a nonincreasing function, then α is an -function. So the set of -functions is a rich class.
Recently, Du [28] first proved the following characterizations of -functions which are quite useful for proving our main results.
Theorem D ([[28], Theorem 2.1])
Let be a function. Then the following statements are equivalent.
-
(a)
α is an -function.
-
(b)
For each , there exist and such that for all .
-
(c)
For each , there exist and such that for all .
-
(d)
For each , there exist and such that for all .
-
(e)
For each , there exist and such that for all .
-
(f)
For any nonincreasing sequence in , we have .
-
(g)
α is a function of contractive factor; that is, for any strictly decreasing sequence in , we have .
Let us recall the concept of -cyclic contractions introduced first by Du and Lakzian [12].
Definition 3.1 [12]
Let A and B be nonempty subsets of a metric space . If a map satisfies
-
(MT1)
and ;
-
(MT2)
there exists an -function α: such that
then T is called an -cyclic contraction with respect to α on .
The following example shows that there exists an -cyclic contraction which is not a cyclic contraction.
Example 3.1 [12]
Let be a countable set and be a strictly increasing convergent sequence of positive real numbers. Denote by . Then . Let be defined by for all and if . Then d is a metric on X. Set , . Define a map by
for and define by
Then T is an -cyclic contraction with respect to α, but not a cyclic contraction on .
The following result tells us the relation between an -cyclic contraction and a Caristi-type cyclic map.
Theorem 3.1 Let A and B be nonempty subsets of a metric space and be an -cyclic contraction with respect to α. Then there exist a bounded below function and a nondecreasing function such that T is a Caristi-type cyclic map dominated by f and φ.
Proof Denote (the identity mapping). Let be given. Define a sequence in by and for . Clearly, the condition (MT2) implies that T satisfies
So from the last inequality we deduce
Hence the sequence is nonincreasing in . Since α is an -function, by (g) of Theorem D, we obtain
Since is arbitrary, we can define a new function by
Clearly, for each , we have
and
(3.2)
Let be given. Without loss of generality, we may assume . Then . By (MT2), we get
and hence
(3.3)
By exploiting inequalities (3.1), (3.2) and (3.3), we obtain
Let and be defined by
and
respectively. Then φ is a nondecreasing function and f is a bounded below function. Clearly, for all . From (3.3), we obtain
which means that T is a Caristi-type cyclic map dominated by f and φ. □
Theorem 3.2 [12]
Let A and B be nonempty subsets of a metric space and be an -cyclic contraction with respect to α. Then there exists a sequence such that
Proof Applying Theorem 3.1, there exist a bounded below function and a nondecreasing function such that T is a Caristi-type cyclic map dominated by f and φ. Let be given. Let be defined by and for . Applying Theorem 2.1, we have
(3.4)
Since the condition (MT2) implies that T satisfies
we know that the sequence is nonincreasing in . By (3.4), we get
The proof is completed. □
Theorem 3.3 [12]
Let A and B be nonempty subsets of a metric space and be an -cyclic contraction with respect to α. For a given , define an iterative sequence by for . Suppose that has a convergent subsequence in A, then there exists such that .
Proof Applying Theorem 3.1, there exist a bounded below function and a nondecreasing function such that T is a Caristi-type cyclic map dominated by f and φ. Let be given. Let be defined by for . Since the condition (MT2) implies the condition (H2) as in Theorem 2.2, all the assumptions of Theorem 2.2 are satisfied. By applying (a) of Theorem 2.2, there exists such that . □
Remark 3.1 ([[3], Proposition 3.2])
(i.e., Theorem 1.1) is a special case of Theorem 3.3.
Finally, applying Theorem 2.1, we can establish a new Caristi-type fixed point theorem without assuming that the dominated functions possess the lower semicontinuity property.
Theorem 3.4 Let M be a nonempty subset of a metric space , be a proper and bounded below function, be a nondecreasing function and be a selfmap on X. Suppose that T is of Caristi type on M dominated by φ and f, that is,
(3.5)
Then there exists a sequence in M such that is Cauchy.
Moreover, if is complete and M is closed in X, and one of the following conditions is satisfied:
-
(D1)
T is continuous on M;
-
(D2)
T is closed, that is, , the graph of T, is closed in ;
-
(D3)
T he map defined by is l.s.c.
Then the mapping T admits a fixed point in X, and for any with , the sequence converges to a fixed point of T.
Proof Let . Then we have , , and . So (3.5) implies
Hence T a Caristi-type cyclic map dominated by f and φ on . Since f is proper, there exists such that . Let be defined by and for . By applying Theorem 2.1, we have
-
(a)
for each ,
-
(b)
for all ,
-
(c)
.
Since φ is nondecreasing, by (a), we have
(3.6)
Since f is bounded below and the sequence is nonincreasing in ,
(3.7)
For with , taking into account (3.5), (3.6) and (3.7), we get
Let , . Then
(3.8)
Since , . From (3.8) we obtain
which proves that is a Cauchy sequence in M.
Moreover, assume that is complete and M is closed in X. So is a complete metric space. By the completeness of M, there exists such that as . We claim . If (D1) holds, since T is continuous on M, for each and as , we get
If (D2) holds, since T is closed, for each and as , we have . Finally, assume that (D3) holds. Since , we obtain
we obtain and hence . This completes the proof. □