- Research
- Open access
- Published:
Fixed points and endpoints of contractive set-valued maps in cone uniform spaces with generalized pseudodistances
Fixed Point Theory and Applications volume 2012, Article number: 176 (2012)
Abstract
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.
1 Introduction
Nadler [1] extended Banach’s fixed point theorem [2] for set-valued maps in complete metric spaces.
Theorem 1.1 ([[1], Th. 5])
Let be a complete metric space, let denote the class of all nonempty closed subsets of X, and let be defined by
where, for each and , . If a set-valued map is H-contractive, i.e., if T satisfies
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 [3], de Blasi et al. [4], Ćirić [5], Eldred et al. [6], Feng and Liu [7], Frigon [8], Al-Homidan et al. [9], Jachymski [10], Kaneko [11], Klim and Wardowski [12], Latif and Al-Mezel [13], Mizoguchi and Takahashi [14], Pathak and Shahzad [15], Quantina and Kamran [16], Reich [17, 18], Reich and Zaslavski [19, 20], Sintunavarat and Kumam [21–25], Suzuki [26], Suzuki and Takahashi [27], Takahashi [28] and Zhong et al. [29]. In particular, the significant fixed point existence results of Nadler type were obtained by Suzuki [[30], Th. 3.7] in metric spaces with τ-distances and by Wardowski [31] in cone metric spaces.
Recently, Włodarczyk and Plebaniak in [32] have studied among others the -families of generalized pseudodistances in cone uniform, uniform and metric spaces which generalize distances of Tataru [33], w-distances of Kada et al. [34], τ-distances of Suzuki [35] and τ-functions of Lin and Du [36] in metric spaces and distances of Vályi [37] 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 [[32], 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:
(H1) ;
(H2) ; and
(H3) .
It is clear that each cone defines, by virtue of
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 ([[38], Def. 2.2])
Let X be a nonempty set, and let L be an ordered normed space with a cone H.
-
(i)
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
() .
-
(ii)
If is a -family, then the pair is called a cone uniform space.
-
(iii)
A -family is said to be separating if
() .
-
(iv)
If a -family is separating, then the pair is called a Hausdorff cone uniform space.
Definition 2.2 ([[38], Def. 2.3])
Let L be an ordered normed space with a solid cone H, and let be a Hausdorff cone uniform space with a cone H.
-
(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
-
(iii)
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 ([[32], 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:
(P1) The sequence in X converges to iff
(P2) The sequence in X is a Cauchy sequence in X iff
Definition 2.3 Let L be an ordered Banach space with a cone H.
-
(i)
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.
-
(ii)
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.
Definition 2.4 Let L be an ordered normed space with a solid cone H. The cone H is called regular if for every increasing (decreasing) sequence in L which is bounded from above (below),
there exists such that . Every regular cone is normal.
Definition 2.5 ([[32], Def. 2.6])
Let L be an ordered normed space with a normal solid cone H, and let be a Hausdorff cone uniform space with a cone H.
-
(i)
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
() For any sequence in X such that
if there exists a sequence in X satisfying
then
-
(ii)
Each -family is a -family.
-
(iii)
If is a -family, then where
and
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.
Definition 2.6 Let L be an ordered Banach space with a normal solid cone H, let be a Hausdorff sequentially complete cone uniform space with a cone H, and let be a -family.
-
(i)
Let . We say that a pair is -admissible if:
-
(a)
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.,
and
respectively); and
-
(c)
For each , the elements and are comparable.
-
(ii)
Let , and let a pair be -admissible. For each , we define where
Here, for each , and by and since H is closed, .
-
(iii)
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
then we say that is -contractive.
-
(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 [3] and Yuan [47]).
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;
(A6) For each , the set is of the form:
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 , .
-
(ii)
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:
(A10) .
Then (A7) holds.
-
(ii)
Assume that one of the following conditions holds:
(A11) For each , the map
is -lsc;
(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.
Theorem 4.1 ([[32], Th. 4.5 ])
-
(i)
Assume that:
(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;
(C7) For each , the set is of the form:
(C8) For each , the set is a nonempty subset of X; and
(C9) For each , the set is a closed subset in X.
Then there exists such that
-
(c)
.
-
(ii)
Assume, in addition, that:
(C10) For each , each dynamic process starting at and satisfying satisfies .
Then assertion is of the form:
() .
Remark 4.1 ([[32], Remark 4.6 ])
-
(i)
A special case of condition (C9) is a condition () defined by:
() For each , the map
is -lsc.
-
(ii)
If , then a special case of condition (C9) is a condition () defined by:
() For each , the map
is -lsc.
The proof will be broken into seven steps.
Step 1. Let where
The following hold:
and
Indeed, by (A4), (A5) and (iii) and (iv) of Definition 2.6, we obtain
Hence, in particular, for , we get
This implies (4.1).
Note that
This, by (A6) (recall that ), implies (4.2).
By (4.1) and (4.2), we have
i.e., (4.3) holds.
By (4.3) and (1),
and, by (4.5) and (4.1), we have
Consequently, (4.4) holds.
Step 2. The family Ω defined in Step 1 satisfies (C4).
Indeed, by (4.1),
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.
By (4.2) and (4.3), in particular,
and, by (A7) and (A8), the following property concerning intersection of these sets holds: for each ,
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).
Indeed, denote
and
By (4.2) and (4.3),
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).
Let be arbitrary and fixed, and let a sequence in X be convergent to , i.e., let (see Definition 2.2 and Theorem 2.1). If , and are arbitrary and fixed, then by (1),
Since and T satisfy (A5), this implies
that is,
Hence, by (A4), since H is closed, using the fact that is continuous and taking the limit as , we get
Therefore, for each ,
i.e., is lsc in X. Moreover, if , and are arbitrary and fixed, then by (1),
that is,
Since H is closed and is continuous, this implies
that is, for each , the map is lsc in X. Hence, in particular, we conclude that for each , the map
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 .
Example 5.1 If
, and, for each , is defined by the formula
then , is a cone metric space; let in the sequel .
Let be of the form:
Let , and let be of the form:
. Clearly, is a -family on X (see [[32], Ex. 5.1]).
We observe that .
We show that is -admissible and -contractive on X where
Indeed, let , and let be arbitrary and fixed.
We consider three cases:
Case 1. If , then by definition of T, we have that and
Case 2. If and , then by definition of T, and . Hence, by definition of J, we calculate:
-
(i)
, and
-
(ii)
, and
-
(iii)
By (i) and (ii),
Consequently,
for and .
Case 3. If and , then by analogous considerations as in Case 2, we get
Thus, T is -admissible and -contractive on X.
Let now . Then for each , we have and using the fact that , we obtain
This implies that
and
for .
Assumptions (A1)-(A8) of Theorem 3.1 hold, and .
Example 5.2 Let X, W, J, λ and γ be such as in Example 5.1, and let be of the form:
Then and
for since . Hence:
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.
Otherwise, , and
However, for and from X, we obtain:
-
(i)
and ;
-
(ii)
, and
-
(iii)
, and
-
(iv)
By (i)-(iii),
Consequently, for each ,
It is absurd.
-
(b)
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.
References
Nadler SB: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475
Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Aubin JP, Siegel J: Fixed points and stationary points of dissipative multivalued maps. Proc. Am. Math. Soc. 1980, 78: 391–398. 10.1090/S0002-9939-1980-0553382-1
de Blasi FS, Myjak J, Reich S, Zaslawski AJ: Generic existence and approximation of fixed points for nonexpansive set-valued maps. Set-Valued Var. Anal. 2009, 17: 97–112. 10.1007/s11228-009-0104-5
Ćirić L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 2009, 71: 2716–2723. 10.1016/j.na.2009.01.116
Eldred A, Anuradha J, Veeramani P: On the equivalence of the Mizoguchi-Takahashi fixed point theorem to Nadler’s theorem. Appl. Math. Lett. 2009, 22: 1539–1542. 10.1016/j.aml.2009.03.022
Feng Y, Liu S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 2006, 317: 103–112. 10.1016/j.jmaa.2005.12.004
Frigon M: Fixed point results for multivalued maps in metric spaces with generalized inwardness conditions. Fixed Point Theory Appl. 2010., 2010: Article ID 183217
Al-Homidan S, Ansari QH, Yao J-C: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. 2008, 69: 126–139. 10.1016/j.na.2007.05.004
Jachymski J: Caristi’s fixed point theorem and selections of set-valued contractions. J. Math. Anal. Appl. 1998, 227: 55–67. 10.1006/jmaa.1998.6074
Kaneko H: Generalized contractive multi-valued mappings and their fixed points. Math. Jpn. 1988, 33: 57–64.
Klim D, Wardowski A: Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 2007, 334: 132–139. 10.1016/j.jmaa.2006.12.012
Latif A, Al-Mezel SA: Fixed point results in quasimetric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 178306
Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 141: 177–188. 10.1016/0022-247X(89)90214-X
Pathak HK, Shahzad N: Fixed point results for set-valued contractions by altering distances in complete metric spaces. Nonlinear Anal. 2009, 70: 2634–2641. 10.1016/j.na.2008.03.050
Quantina K, Kamran T: Nadler’s type principle with hight order of convergence. Nonlinear Anal. 2008, 69: 4106–4120. 10.1016/j.na.2007.10.041
Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 4: 26–42.
Reich S: Some problems and results in fixed point theory. Contemp. Math. 21. In Topological Methods in Nonlinear Functional Analysis. Am. Math. Soc., Providence; 1983:179–187.
Reich S, Zaslavski AJ: Convergence of iterates of nonexpansive set-valued mappings. Ser. Math. Anal. Appl. 4. In Set-valued Mappings with Applications in Nonlinear Analysis. Taylor and Francis, London; 2002:411–420.
Reich S, Zaslavski AJ: Generic existence of fixed points for set-valued mappings. Set-Valued Anal. 2002, 10: 287–296. 10.1023/A:1020602030873
Sintunavarat W, Kumam P:Weak condition for generalized multi-valued -weak contraction mappings. Appl. Math. Lett. 2011, 24: 460–465. 10.1016/j.aml.2010.10.042
Sintunavarat W, Kumam P: Gregus-type common fixed point theorems for tangential multivalued mappings of integral type in metric spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 923458
Sintunavarat W, Kumam P: Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2011., 2011: Article ID 3
Sintunavarat W, Kumam P: Common fixed point theorems for hybrid generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25: 52–57. 10.1016/j.aml.2011.05.047
Sintunavarat W, Kumam P: Common fixed point theorem for cyclic generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25: 1849–1855. 10.1016/j.aml.2012.02.045
Suzuki T: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. J. Math. Anal. Appl. 2008, 340: 752–755. 10.1016/j.jmaa.2007.08.022
Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. 1997, 8: 371–382.
Takahashi W: Existence theorems generalizing fixed point theorems for multivalued mappings. Pitman Res. Notes Math. Ser. 252. In Fixed Point Theory and Applications (Marseille, 1989). Edited by: Baillon JB, Théra M. Longman Sci. Tech., Harlow; 1991:397–406.
Zhong C-H, Zhu J, Zhao P-H: An extension of multi-valued contraction mappings and fixed points. Proc. Am. Math. Soc. 1999, 128: 2439–2444.
Suzuki T: Several fixed point theorems concerning τ -distance. Fixed Point Theory Appl. 2004, 2004: 195–209.
Wardowski D: On set-valued contractions of Nadler type in cone metric spaces. Appl. Math. Lett. 2011, 24: 275–278. 10.1016/j.aml.2010.10.003
Włodarczyk K, Plebaniak R: Maximality principle and general results of Ekeland and Caristi types without lower semicontinuity assumptions in cone uniform spaces with generalized pseudodistances. Fixed Point Theory Appl. 2010., 2010: Article ID 175453
Tataru D: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 1992, 163: 345–392. 10.1016/0022-247X(92)90256-D
Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 1996, 44: 381–391.
Suzuki T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 2001, 253: 440–458. 10.1006/jmaa.2000.7151
Lin L-J, Du W-S: Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 2006, 323: 360–370. 10.1016/j.jmaa.2005.10.005
Vályi I: A general maximality principle and a fixed point theorem in uniform spaces. Period. Math. Hung. 1985, 16: 127–134. 10.1007/BF01857592
Włodarczyk K, Plebaniak R, Doliński M: Cone uniform, cone locally convex and cone metric spaces, endpoints, set-valued dynamic systems and quasi-asymptotic contractions. Nonlinear Anal. 2009, 7(1):5022–5031.
Włodarczyk K, Plebaniak R: Periodic point, endpoint, and convergence theorems for dissipative set-valued dynamic systems with generalized pseudodistances in cone uniform and uniform spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 864536
Włodarczyk K, Plebaniak R, Obczyński C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 2010, 72: 794–805. 10.1016/j.na.2009.07.024
Włodarczyk K, Plebaniak R: A fixed point theorem of Subrahmanyam type in uniform spaces with generalized pseudodistances. Appl. Math. Lett. 2011, 24: 325–328. 10.1016/j.aml.2010.10.015
Włodarczyk K, Plebaniak R: Quasi-gauge spaces with generalized quasi-pseudodistances and periodic points of dissipative set-valued dynamic systems. Fixed Point Theory Appl. 2011., 2011: Article ID 712706
Włodarczyk K, Plebaniak R: Kannan-type contractions and fixed points in uniform spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 90. doi:10.1186/1687–1812–2011–90
Włodarczyk K, Plebaniak R: Contractivity of Leader type and fixed points in uniform spaces with generalized pseudodistances. J. Math. Anal. Appl. 2012, 387: 533–541. 10.1016/j.jmaa.2011.09.006
Włodarczyk K, Plebaniak R: Generalized uniform spaces, uniformly locally contractive set-valued dynamic systems and fixed points. Fixed Point Theory Appl. 2012., 2012: Article ID 104. doi:10.1186/1687–1812–2012–104
Włodarczyk K, Plebaniak R: Leader type contractions, periodic and fixed points and new completivity in quasi-gauge spaces with generalized quasi-pseudodistances. Topol. Appl. 2012, 159: 3504–3512. 10.1016/j.topol.2012.08.013
Yuan GX-Z: KKM Theory and Applications in Nonlinear Analysis. Marcel Dekker, New York; 1999.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have equitably contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Włodarczyk, K., Plebaniak, R. Fixed points and endpoints of contractive set-valued maps in cone uniform spaces with generalized pseudodistances. Fixed Point Theory Appl 2012, 176 (2012). https://doi.org/10.1186/1687-1812-2012-176
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-176