Fixed points of set-valued Caristi-type mappings on semi-metric spaces via partial order relations
© Nimana et al.; licensee Springer. 2014
Received: 13 November 2013
Accepted: 13 January 2014
Published: 3 February 2014
In this paper, we present some fixed-point theorems that are related to a set-valued Caristi-type mapping. The main results extend the recent work which was presented by Jiang and Li (Fixed Point Theory Appl. 2013:74, 2013) from a single-valued setting to a set-valued case. Further, the presented results also improve essentially many results that have appeared, because we have removed some conditions from the auxiliary function. Meanwhile, we give some partial answers to an important problem which was raised by Kirk (Colloq. Math. 36:81-86, 1976).
for some a function and a function . He raised a problem relating to generalize Caristi’s fixed-point theorem stating whether a Caristi-type mapping T has a fixed point for a complete metric space .
for all . By assuming that η is nondecreasing, continuous, subadditive, and , they showed that the relation defined by (1.4) is a partial order on X and proved that each Caristi-type mapping has a fixed point, by inquiring the existence of maximal element of a partially ordered complete metric space provided that φ is bounded from below and lower semicontinuous. It was noticed that the subadditivity of η is necessarily assumed for insisting that the relation in (1.4) is a partial order. Thus if η is not a subadditive function, then the relationship ⪯ defined by (1.4) may not be a partial order in X, and consequently the technique used in [5, 6] becomes invalid. Observing this one, by introducing a partial order on a complete subset of X, Khamsi  removed the subadditivity of η and showed the existence theorem of fixed point for a Caristi type mapping. Recently, Li  noticed that is an essential condition for the existence of a fixed point for Caristi-type mapping.
Without the subadditivity of η and the continuity of both η and φ, they successfully obtained the following fixed-point theorem.
T has a maximal fixed point , i.e., let be a fixed point of T, then implies ;
T has a least fixed point , i.e., let be a fixed point of T, then .
Inspired by the above literature, in this work, we will present some extensions of results of Jiang and Li , say from a single-valued setting to a set-valued case. Further, meanwhile, our results also improve essentially many results. This is because we will remove some conditions, such as the assumption of an auxiliary function, as η, being nondecreasing. Also, some related results will be discussed. To complete our plan, the following useful basic concepts are needed.
A semi-metric for a nonempty set X is a nonnegative real-valued function d from into ℝ such that
for each . A pair is called a semi-metric space.
A subset A of a semi-metric space is said to be closed iff , where and . Let d be a semi-metric on X and be a sequence in X. A sequence is said to be convergent to if for any , there exists such that , for all . is called Cauchy if for any , there exists such that , for all . A semi-metric space is said to be complete if every Cauchy sequence is a convergent sequence.
Let be a partial order set and be denoted for a class of all nonempty subset of X. As the core of this work, we are interested in the following class of set-valued mappings: a set-valued mapping is called isotone if, whenever in X,
(I1) for each there is such that , and
(I2) dually, for each there is such that .
Isotone mappings were first studied by Smithson , and it is well known that one of these conditions may be satisfied while the other is not. In particular, a single-valued case of (I1) and (I2) must be similar, i.e., for all with we have .
2 Main results
In this section, we let be a semi-metric space, and be functions such that
Further if is an element in X, we assume that there is a partially ordered set such that the following conditions hold:
(H2) φ is bounded below function on .
(H4) is closed for each such that .
(H5) The set contains every limit points of increasing convergent sequence in P.
The following two examples show that (H4) and (H5) are different.
We see that . One can notice that there is an increasing convergent sequence in P which converges to 0, but .
For , we notice that the set P is nothing else but . It follows that (H5) must be satisfied. This notwithstanding, since is not a closed set, (H4) is not satisfied.
Now we show our main results.
If T is an isotone mapping (I1) and conditions (H1)-(H5) are satisfied, then T has a fixed point . Moreover, is a maximal fixed point in .
where ε is a positive real number as appeared in condition (H1).
Further, by (H5), we have .
Next, we will show that has an upper bound in P. We consider the following two cases.
Case I. For each , there exists such that .
Let be fixed. By the closedness of and passing to a subsequence of an increasing sequence , we know that . Since is arbitrary and , we reach the required result.
Case II. There exists such that for all .
This implies that , and this contradicts to (2.6). Hence, our claim is asserted.
By Cases I and II, we can conclude that has an upper bounded in P. Consequently, since is an arbitrary chain in P, Zorn’s lemma will therefore imply that has a maximal element, say .
Consequently, in view of the isotonicity (I1) of T, we can find such that . This, alternatively, implies that . Thus, by (2.7) together with the maximality of , we obtain . This means is a maximal fixed point of T, and the proof is completed. □
Remark 2.5 Assume that conditions (H1)-(H5) are satisfied and T is linked to X by the following stronger relationships: there exists a point such that for all , and
(SI) if with , then for each , .
This implies that . Now, by the maximality of , we see that . Since y is arbitrary, we conclude that .
By using Theorem 2.4, we also have the following results.
Assume that T is an isotone mapping (I2) on and the conditions (H2), (H3) and the following are satisfied:
(H1)′ φ is a bounded above functional on ;
(H4)′ is closed for each such that ;
(H5)′ the set contains every limit points of decreasing convergent sequence in .
Then T has a fixed point . Moreover, is a minimal fixed point in .
Proof Let ⪯1 be the inverse partial order of ⪯ and . Clearly, for some and is a bounded below functional on . Set . Then, by (H4)′, we find that is a closed set for each such that . It is easy to check that (H3) is satisfied for ⪯1 and T is an isotone mapping (I1) on . Moreover, any decreasing sequence in is an increasing sequence in P. Applying Theorem 2.4 with respect to ⪯1, we have T has a maximal fixed point . This gives the result that T has a minimal fixed point in . □
Remark 2.7 Similarly, as we have mentioned in Remark 2.5, if all assumptions of Theorem 2.6 are satisfied and T is linked to X by the stronger relationship (SI), and there exists a point such that such that for all , then one can show that there is a minimal end point of T.
Now we give an example to demonstrate Theorem 2.4.
We know that is a complete semi-metric space, see .
for all , and for all , respectively.
Note that , () are closed sets. Then for each , is a closed set. This means (H4) is satisfied. Therefore, all assumptions of Theorem 2.4 are satisfied. In fact, we see that 0 is a maximal fixed point in .
3 Further results
By using Theorems 2.4 and 2.6, in this section, we will provide some largest and least fixed-point theorems for a single-valued mapping on semi-metric space.
Theorem 3.1 Let be a complete semi-metric space and be an isotone mapping. Assume that there exist a partial ordering ⪯ on X and a point such that . If conditions (H1)-(H4) hold and the following condition is satisfied:
(H6) is a closed set for each such that ,
T has a maximal fixed point ;
T has a least fixed point , that is, if is a fixed point of T, then .
Proof (i) Set ; in this case, it suffices to show that the condition (H5) in Theorem 2.4 can be removed. Indeed, let an increasing sequence in P be such that , for some . Since , we have , for all . Furthermore, for each , we have , for each . It follows from is a closed set that . So, we have . Since is arbitrary, we obtain , for each . This implies that , for each . By taking n approaching infinity and the closedness of , we have and hence . Therefore, the result can be obtained immediately from Theorem 2.4.
It is easy to check that the relation is a partial order on S.
By the fact that T is an isotone mapping, we have and for each . Set . Clearly, and by (3.1). Thus, we have by (3.2). By the minimality of in S, we can conclude that . This implies that . Therefore is a least fixed point of T in . □
By Theorem 3.1 and using a technique as in Theorem 2.6, we can obtain the following result.
T has a minimal fixed point ;
T has a largest fixed point , that is, if be a fixed point of T, then .
Remark 3.3 In Theorems 3.1 and 3.2, we replace the conditions that η is a nondecreasing function on and , which have proposed in Theorem 1.1, by the condition (H1).
The following example is inspired by Example 2 in .
respectively. We know that T is an isotone mapping. Moreover, for each , and are closed sets, see also . Further, with respect to the given function η and as showed in Example 2.8, we know that the condition (H3) is also satisfied. In fact, 0 is the largest fixed point and is the least fixed point in .
Remark 3.5 In Example 3.4, the considered function η is not a decreasing function and . Thus, Theorem 1.1 cannot be applied in this situation.
In this work, the set-valued Caristi-type mapping in the setting of generalized metric space, as a semimetric space, is considered. Evidently, the presented results improve essentially many results because we also removed some conditions, such as a nondecreasingness assumption from an auxiliary function, which have been imposed in the literature. In fact, we would like to point out that this paper gives some partial answers to an important problem which was raised by Kirk .
The authors would like to thank the anonymous referees for their careful reading, suggestions, and appropriate questions, which permitted us to improve the first version of this paper. This research was partly supported by Naresuan University, Thailand (grant no. R2556C022).
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