Fixed points and endpoints of contractive set-valued maps in cone uniform spaces with generalized pseudodistances
© Włodarczyk and Plebaniak; licensee Springer 2012
Received: 23 April 2012
Accepted: 1 October 2012
Published: 17 October 2012
We introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type yields the fixed point and endpoint existence theorem for these contractive maps.
MSC:47H10, 54C60, 47H09, 54E15, 46A03, 54E50, 46B40.
Theorem 1.1 ([, Th. 5])
then T has a fixed point w in X, i.e., .
A number of authors introduce the new concepts of set-valued contractions of Nadler type and study the problem concerning the existence of fixed points for such contractions; see, e.g., Aubin and Siegel , de Blasi et al. , Ćirić , Eldred et al. , Feng and Liu , Frigon , Al-Homidan et al. , Jachymski , Kaneko , Klim and Wardowski , Latif and Al-Mezel , Mizoguchi and Takahashi , Pathak and Shahzad , Quantina and Kamran , Reich [17, 18], Reich and Zaslavski [19, 20], Sintunavarat and Kumam [21–25], Suzuki , Suzuki and Takahashi , Takahashi  and Zhong et al. . In particular, the significant fixed point existence results of Nadler type were obtained by Suzuki [, Th. 3.7] in metric spaces with τ-distances and by Wardowski  in cone metric spaces.
Recently, Włodarczyk and Plebaniak in  have studied among others the -families of generalized pseudodistances in cone uniform, uniform and metric spaces which generalize distances of Tataru , w-distances of Kada et al. , τ-distances of Suzuki  and τ-functions of Lin and Du  in metric spaces and distances of Vályi  in uniform spaces.
In the present paper, we introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances, and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type [, Th. 4.5] yields the fixed point and endpoint existence theorem for these contractive maps.
It is worth noticing that our fixed point and endpoint existence Theorem 3.1: has a simpler proof; is Nadler type; is new in cone uniform and cone locally convex spaces; is new even in cone metric and metric spaces; and is different from those given in the previous publications on this subject.
2 Definitions and notations
We define a real normed space to be a pair with the understanding that a vector space L over ℝ carries the topology generated by the metric , .
A nonempty closed convex set is called a cone in L if it satisfies:
(H2) ; and
an order of L under which L is an ordered normed space with a cone H.
We will write to indicate that , but . A cone H is said to be solid if ; denotes the interior of H. We will write to indicate that .
The cone H is normal if a real number such that for each , implies exists. The number M satisfying above is called the normal constant of H.
Let an element be such that for all .
Let denote the family of all nonempty subsets of a space X. Recall that a set-valued dynamic system is defined as a pair , where X is a certain space and T is a set-valued map ; in particular, a set-valued dynamic system includes the usual dynamic system where T is a single-valued map. We say that a map is proper if its effective domain, , is nonempty.
Definition 2.1 ([, Def. 2.2])
The family , -index set, is said to be a -family of cone pseudometrics on X (-family for short) if the following three conditions hold:
() ; and
If is a -family, then the pair is called a cone uniform space.
A -family is said to be separating if
If a -family is separating, then the pair is called a Hausdorff cone uniform space.
Definition 2.2 ([, Def. 2.3])
- (i)We say that a sequence in X is a -convergent in X (convergent in X for short) if there exists such that
- (ii)We say that a sequence in X is a -Cauchy sequence in X (Cauchy sequence in X, for short) if
If every Cauchy sequence in X is convergent in X, then is called a -sequentially complete cone uniform space (sequentially complete for short).
Theorem 2.1 ([, Th. 2.3])
Let L be an ordered Banach space with a normal solid cone H, and let be a Hausdorff cone uniform space with a cone H. The following hold:
A subset is said to have a minimal (maximal) element if there exists such that () for all , and we write then that (). It is clear that if D has a minimal (maximal) element, then the minimal (maximal) element is unique.
We say that is an infimum (supremum) for set if has the minimal (maximal) element and (), and we write then that (); here denotes the closure of D in L.
there exists such that . Every regular cone is normal.
Definition 2.5 ([, Def. 2.6])
The family is said to be a -family of cone pseudodistances on X (-family on X for short) if the following three conditions hold:
() ; and
Each -family is a -family.
- (iii)If is a -family, then where
Let be a sequentially complete cone uniform space. We say that a set is closed in X if where , the closure of Y in X, denotes the set of all for which there exists a sequence in Y which converges to w. If a set is closed in X, then is a sequentially complete cone uniform space with a cone H. Define ; that is, denotes the class of all nonempty closed subsets of X.
Let . We say that a pair is -admissible if:
For each , and , the set has a minimal element, say (i.e., ), and the set has a minimal element, say (i.e., );
- (b)The sets and have maximal elements, say and , respectively (i.e.,
For each , the elements and are comparable.
- (ii)Let , and let a pair be -admissible. For each , we define where
Let a set-valued dynamic system satisfy . We say that is -admissible if for each , a pair is -admissible.
- (iv)Let satisfy , and let be -admissible. If there exists the family such that
- (v)Let , . The map is lower semicontinuous on E with respect to X (written: F is -lsc when and F is lsc when ) if the set is a closed subset in X for each . Equivalently, for each ,
- (vi)We say that the family is continuous in X if for each and for each sequence in X converging to , we have
If , then is continuous in X.
3 Statement of result
Let be a set-valued dynamic system. By and we denote the sets of all fixed points and endpoints of T, respectively, i.e., and . A dynamic process or a trajectory starting at or a motion of the system at is a sequence defined by for (see, Aubin-Siegel  and Yuan ).
The aim of this paper is to prove the following fixed point and endpoint existence general result of Nadler type.
Theorem 3.1 (i) Assume that:
(A1) L is an ordered Banach space with a regular solid cone H;
(A2) is a Hausdorff sequentially complete cone uniform space with a cone H;
(A3) is a -family on X such that ;
(A4) The set-valued dynamic system satisfies and is -admissible;
(A5) There exists the family such that is -contractive;
where the family satisfies ;
(A7) For each , the set is a nonempty subset in X; and
(A8) For each , the set is a closed subset in X.
Then the following hold:
() ; and
() For each , .
Assume, in addition, that:
(A9) For each , each dynamic process starting at and satisfying satisfies .
Then the assertions and are of the forms:
() ; and
() For each , .
Remark 3.1 (i) Assume that:
Then (A7) holds.
Assume that one of the following conditions holds:
(A12) The family is continuous in X.
Then (A8) holds.
4 Proof of Theorem 3.1
We will use the following fixed point and endpoint existence general result of Caristi type.
(C1) L is an ordered Banach space with a regular solid cone H;
(C2) is a Hausdorff sequentially complete cone uniform space with a cone H;
(C3) The family is a -family on X such that ;
(C4) The family satisfies ≠∅;
(C5) is a set-valued dynamic system;
(C6) is a family of finite positive numbers;
(C8) For each , the set is a nonempty subset of X; and
(C9) For each , the set is a closed subset in X.
Assume, in addition, that:
(C10) For each , each dynamic process starting at and satisfying satisfies .
Then assertion is of the form:
A special case of condition (C9) is a condition () defined by:
If , then a special case of condition (C9) is a condition () defined by:
The proof will be broken into seven steps.
This implies (4.1).
This, by (A6) (recall that ), implies (4.2).
i.e., (4.3) holds.
Consequently, (4.4) holds.
Step 2. The family Ω defined in Step 1 satisfies (C4).
Also, by Definition 2.5, , (1) holds and, by (A4), . Hence, we conclude that .
Step 3. Assumptions (C8) and (C9) hold where and Ω is defined in Step 1.
is a nonempty closed subset in X.
Step 4. The assertions of Theorem 3.1 hold.
This follows from Assumptions (A1)-(A9), Steps 1-3, definition of and Theorem 4.1.
Step 5. Assumption (A10) implies (A7).
Hence, we conclude that for each , the set is nonempty whenever .
Step 6. Assumption (A11) implies (A8).
This follows from Remark 4.1(i)
Step 7. Assumption (A12) implies (A8).
is -lsc, that is, () holds.
5 Remarks, examples and comparisons
Remark 5.1 Examples 5.1 and 5.2 illustrate a fixed point version and an endpoint version of Theorem 3.1, respectively, in cone metric spaces with -family where and .
then , is a cone metric space; let in the sequel .
. Clearly, is a -family on X (see [, Ex. 5.1]).
We observe that .
Indeed, let , and let be arbitrary and fixed.
We consider three cases:
- (i), and
- (ii), and
- (iii)By (i) and (ii),
for and .
Thus, T is -admissible and -contractive on X.
Assumptions (A1)-(A8) of Theorem 3.1 hold, and .
Assumptions (A1)-(A9) of Theorem 3.1 hold, and .
Remark 5.2 In Example 5.3, we show that in our concept of -contractive set-valued dynamic systems, the existence of -family such that is essential; from Example 5.3, it follows that for maps defined in Examples 5.1 and 5.2, we cannot use Theorem 3.1 when .
Example 5.3 (a) Let X and T be such as in Example 5.1. We observe that for each , T is not -contractive on X.
- (ii), and
- (iii), and
- (iv)By (i)-(iii),
Let X and T be such as in Example 5.2. By similar argumentation as in , we observe that for each , T is not -contractive on X.
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