Abstract
In this paper, we establish a unified approach to study the existence of fixed points for common fixed point problems. Moreover, we introduce mixed systems of common fixed point problems, systems of common fixed point problems and common fixed point problems without convex assumptions. As applications, we establish a general type of some set-valued variational inequalities, and we obtain some existence theorems of solutions of set-valued variational inequalities. The results of this paper improve and generalize several known results on common fixed point problems.
MSC:47H10, 49J53, 49J40, 49J45.
Similar content being viewed by others
1 Introduction
In three recent papers [1–3], by using some new concepts of generalized KKM mappings, the authors established common fixed point theorems for families of set-valued mappings in Hausdorff topological vector spaces. Recently, Agarwal et al. [4] established a common fixed point theorem for a family of self set-valued mappings on a compact and convex set in a locally convex topological vector space. As applications, an existence theorem of solutions for a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities were obtained.
It is well known that the equilibrium problems are unified models of several problems, namely, optimization problems, saddle point problems, variational inequalities, fixed point problems, Nash equilibrium problems etc. Recently, Luc [5] introduced a more general model of equilibrium problems, which is called a variational relation problem (in short, VR). The stability of the solution set of variational relation problems was studied in [6, 7]. Some various types of variational relation problems or systems of variational relation problems have been investigated in many recent papers (see [8–12]). Recently, Agarwal et al. [13] presented a unified approach in studying the existence of solutions for two types of variational relation problems, and Balaj and Lin [14] established existence criteria for the solutions of two very general types of variational relation problems (see also [15–22] for further studies of variational relation problems).
Motivated and inspired by research works mentioned above, in this paper, we establish a unified approach to study the existence of fixed points for common fixed point problems. As generalizations, mixed systems of common fixed point problems, systems of common fixed point problems, and common fixed point problems without convex assumptions are obtained.
2 Preliminary
In this short section, we recall some definitions and known results concerning set-valued mappings which will be needed throughout this paper.
Let X, Y, Z be three Hausdorff topological spaces. We adopt the following notations: for a set U and point x, and, respectively, means , and, respectively, . Denote by and ; for two sets A, B, and, respectively, means , and, respectively, . Denote by and . A set-valued mapping is said to be: (1) upper semicontinuous at if, for any open subset O of Y with , there exists an open neighborhood of x such that for any ; (2) upper semicontinuous on X, if it is upper semicontinuous at each ; (3) lower semicontinuous at if, for any open subset O of Y with , there exists an open neighborhood of x such that for any ; (4 lower semicontinuous on X, if it is lower semicontinuous in each ; (5) closed if is a closed subset of .
Lemma 2.1 (Corollary 17.55 (Kakutani-Fan-Glicksberg fixed point theorem) of [23])
Let X be a nonempty, convex and compact subset of a locally convex topological linear space, and be an upper semicontinuous set-valued mapping with nonempty convex compact values. Then there exists such that .
Lemma 2.2 (Lemma 17.8, Theorem 17.10, Theorem 17.16, Theorem 17.19 of [23])
-
(i)
The image of a compact set under a compact-valued upper semicontinuous set-valued mapping is compact.
-
(ii)
If an upper semicontinuous set-valued mapping possess compact-valued, then it is closed.
-
(iii)
The correspondence φ is upper semicontinuous at x and is compact, if and only if, for every net in the graph of φ, that is, with for each α, if , then the net has a limit point in .
-
(iv)
If X and Y are topological spaces, a set-valued mapping is lower semicontinuous, if and only if, for any net in X, converging to , and each , there exists a net converging to y, with for all α.
3 Main results
Let , , and be set-valued mappings with nonempty values. A common fixed point problem of type α (CFP-α) consists in finding such that , for any ,
When , a common fixed point problem of type (CFP-) consists in finding such that , and
When , a common fixed point problem of type (CFP-) consists in finding such that and, for any , there exists for which
When and for any , the problems (CFP-) and (CFP-) reduce the common fixed point problem (CFP): finding such that , and
Theorem 3.1 Assume that the data of problem (CFP-α) satisfy the following conditions:
-
(i)
X is a nonempty, convex and compact subset of a locally convex topological linear space, Y is Hausdorff linear topological space, and Z is a Hausdorff topological space;
-
(ii)
S is upper semicontinuous with nonempty convex compact values;
-
(iii)
is open in X.
Moreover, assume that there exists a set-valued mapping such that
-
(iv)
for any ;
-
(v)
is nonempty, convex and compact for any ;
-
(vi)
F is convex, i.e., for any and any with ;
-
(vii)
T is α--KKM, i.e., for any finite set of Y and any , there is such that , .
Then problem (CFP-α) has at least a solution.
Proof By way of contradiction suppose that, for any , , or, there is such that , . Denote , which is open in X by (ii). Therefore,
By (iii), there is a finite subset of Y such that
Let be the partition of unity subordinate to the open covering of X, i.e., is a set of continuous functions with following properties: , , , ; and if , for some , then , and if , then .
Now, we define the following set-valued mapping :
Clearly ϕ is upper semicontinuous on X. Moreover, since and are nonempty, convex and compact, is nonempty, convex and compact in X for any . By Lemma 2.1, there exists such that . Let . Then, for any , , it follows that for any . By (iv), for any . Therefore, , which implies that . It follows from the convexity of F that
By (vii), there is such that , , which implies that , i.e., . It contradicts the fact that , that is, . This completes the proof. □
Remark 3.1 By Proposition 3.3 of [14], when , condition (iii) in Theorem 3.1 can be replaced by (a1) is lower semicontinuous; (b1) is closed in X for any ; (c1) is open in X for any . Thus we have the following theorem.
Theorem 3.2 Assume that the data of problem (CFP-) satisfy the conditions (i), (ii), (iv)-(vi) of Theorem 3.1 and
(a1) is lower semicontinuous;
(b1) is closed in X for any ;
(c1) is open in X for any ;
(d1) T is --KKM, i.e., for any finite set of Y and any , there is such that for any .
Then problem (CFP-) has at least a solution.
Remark 3.2 By Proposition 3.1 of [14], when , condition (iii) in Theorem 3.1 can be replaced by (a2) is upper semicontinuous with nonempty compact values; (b2) is closed in X for any ; (c2) is open in X for any . Thus we have the following theorem.
Theorem 3.3 Assume that the data of problem (CFP-) satisfy the conditions (i), (ii), (iv)-(vi) of Theorem 3.1 and
(a2) is upper semicontinuous with nonempty compact values;
(b2) is closed in X for any ;
(c2) is open in X for any ;
(d2) T is --KKM, i.e., for any finite set of Y and any , there is such that there is for which .
Then problem (CFP-) has at least a solution.
If and for all , we have problem (CFP-α).
Theorem 3.4 Let X be a nonempty, convex and compact subset of a locally convex topological linear space, Z be a Hausdorff topological space, and , and be set-valued mappings with nonempty values. Assume that
-
(i)
S is upper semicontinuous with nonempty convex compact values, and for all ;
-
(ii)
the set is open in X for any ;
-
(iii)
T is α-P-KKM, i.e., for any finite set of X and any , there is such that , .
Then problem (CFP-α) has at least a solution.
When , and , for all , we obtain the following corollary.
Corollary 3.1 Let X be a nonempty, convex, and compact subset of a locally convex topological linear space, and , be set-valued mappings with nonempty values. Assume that
-
(i)
S is upper semicontinuous with nonempty convex compact values;
-
(ii)
is open in X for any , and for any ;
-
(iii)
the set is closed in X for any ;
-
(iv)
for any finite set of X and any , there is such that .
Then there exists such that and .
Remark 3.3 Theorems 3.1-3.4 generalize the results of [1–4]. When and for all , and for any , our Theorems 3.1-3.4 reduce to the results of [1–4].
Now, we introduce a new class of problems, called mixed systems of common fixed point problems (MSCFP). Let X, Y be nonempty sets in two Hausdorff topological vector spaces, Z be a Hausdorff topological space, , , , , , , be set-valued mappings with nonempty values. A mixed system of common fixed point problems consists in finding such that , and
Theorem 3.5 Assume that
-
(i)
X, Y, Z are three nonempty, compact and convex subsets of three Hausdorff linear topological spaces;
-
(ii)
and are nonempty and closed in ;
-
(iii)
P is continuous with nonempty convex compact values;
-
(iv)
and are closed for any ;
-
(v)
, , , , and , are open for any ;
-
(vi)
for any fixed , any finite subset of X and any , there is such that, for any , ;
-
(vii)
for any fixed , any finite subset of Y and any , there is such that there is for which .
Then problem (MSCFP) has at least a solution.
Proof Define and as follows:
By (iii), (iv), and Remarks 3.1, 3.2, , are open in for any .
Suppose there exists such that , then there is a finite subset of such that . By (vi), there is such that for any , which contradicts the fact that for any . Hence for any .
Suppose there exists such that , then there is a finite subset of such that . By (vii), there is such that there exists for which , which contradicts the fact that for any . Hence for any .
Define the mappings and as follows:
For any ,
is open in . Similarly, is open for any . Hence, , are open for any , and , for any . By Theorem 3 of [24], there exists such that and , which implies that , and
As a generalization, we introduce the following system of common fixed point problems. Let I be any index set. For any , let , be Hausdorff topological spaces, and , , and be set-valued mappings with nonempty values. A system of common fixed point problems (SCFP) consists in finding such that, for each , and, for any ,
□
Theorem 3.6 Assume that, for each , the following conditions are satisfied:
-
(i)
is a nonempty, convex and compact subset of a locally convex topological linear space, and is a Hausdorff topological space;
-
(ii)
the set is nonempty and closed in X;
-
(iii)
the set-valued mapping , defined by , has open fibers;
-
(iv)
is open in X for any ;
-
(v)
for any finite set of and any with , there is such that , .
Then problem (SCFP) has at least a solution.
Proof For each , define the set
By (ii) and (iii), is open in X for each . Since is open in X for each , by a known continuous selection Theorem (see [[23], Theorem 17.63]), there is a continuous function such that . Thus, for each , we define the mapping as follows:
By Lemma 2.1, there exists such that , which implies that for each . If for some , . Then there exists a finite set such that . By (v), there is such that , , which contradicts the fact that . Therefore, for any , which implies that, for each , and, for any , , . This completes the proof. □
As a generalization of Theorem 3.1, we derive the following existence result for the solution of problem (CFP-α) without convex assumptions.
Theorem 3.7 Assume that
-
(i)
X is a nonempty and compact subset of a Hausdorff topological vector space E, and has the fixed point property, and Z is a Hausdorff topological space;
-
(ii)
is closed;
-
(iii)
for any ;
-
(iv)
and are open in X for any ;
-
(v)
for any finite set of X, there exists a continuous mapping such that
(v1) for any , there exists such that , ;
(v2) if for any , then , where
Then problem (CFP-α) has at least a solution, i.e., there exists such that and, for any ,
Proof Define the mapping as follows:
By (ii) and (iv), is closed for any .
By (v), for any finite subset of X, there exists a continuous mapping such that, for any , there exists such that , , then
-
(1)
if there exists such that , which implies that . Thus ;
-
(2)
if for any , then by (v). By (iii), . Then and, for any , there exists such that , . Thus .
Hence, by Theorem 2.1 of [17],
which implies that there is such that and, for any ,
□
By Remarks 3.1 and 3.2, we have the following results.
Theorem 3.8 If (iv) of Theorem 3.7 is replaced by
(a3) is lower semicontinuous;
(b3) is closed in X for any , and is open in X for any .
Then problem (CFP-) has at least a solution, i.e., there exists such that and
Theorem 3.9 If (iv) of Theorem 3.7 is replaced by
(a4) is upper semicontinuous with nonempty compact values;
(b4) is closed in X for any , and is open in X for any .
Then problem (CFP-) has at least a solution, i.e., there exists such that and, for any , there is for which .
Remark 3.4 In [1–4], convex assumptions or the KKM property played an important role in the proofs of common fixed points. In Theorems 3.7-3.9, the existence of common fixed points does not depend on convex assumptions.
Remark 3.5 This paper extends the research on common fixed point problems. The classical common fixed point problem (see [1–4]) is a special case of problem (CFP-α). Moreover, we introduce the system of common fixed points, and common fixed point problems without convex assumptions are obtained.
4 Applications
4.1 Variational inclusions
In this section, we fix our attention on variational inclusions described below:
Let X be a nonempty, convex and compact subset of a locally convex topological linear space, Z be a Hausdorff topological linear space, and , be set-valued mapping with nonempty values.
A variational inclusion of type γ (VI-γ) consists in finding such that and holds for any .
A variational inclusion of type (VI-) consists in finding such that and for any .
A variational inclusion of type (VI-) consists in finding such that and holds for any .
Theorem 4.1 Let X be a nonempty, convex and compact subset of a locally convex topological linear space, Z be a Hausdorff topological linear space. Assume that
-
(i)
S is upper semicontinuous with nonempty convex compact values;
-
(ii)
is open in X for any , and for any ;
-
(iii)
the set is closed in X for any ;
-
(iv)
for any finite set of X and any , there is such that holds.
Then problem (VI-γ) has at least a solution.
Proof Define the mapping as follows:
By (iii), is closed in X for any . By Corollary 3.1, there exists such that and , which implies and holds for any . This completes the proof. □
Theorem 4.2 Assume (ii) and (iii) of Theorem 4.1 are replaced by
(a5) for any , is lower semicontinuous, and is closed;
(b5) for any finite set of X and any , there is such that .
Then problem (VI-) has at least a solution.
Proof As soon as we show that the set is closed for any . Let be a net in X converging to x, such that for any α. For any , since is lower semicontinuous, by Lemma 2.2, there is such that . It follows from the closeness of that . Then . Thus, the set is closed for any . □
Theorem 4.3 Assume (ii) and (iii) of Theorem 4.1 are replaced by
(a6) for any , is upper semicontinuous with nonempty compact values, and is closed;
(b6) for any finite set of X and any , there is such that .
Then problem (VI-) has at least a solution.
Proof As soon as we show that the set is closed for any . Let be a net in X converging to x, such that for any α. Then there exists such that for any α. Since is upper semicontinuous with nonempty compact values, by (iii) of Lemma 2.2, there is a subnet of converging to some . Since , and is closed, . Thus . Hence, the set is closed for any . □
4.2 Generalized multiplied minimax inequality of Ky Fan type
Let X be a nonempty, convex and compact subset of a locally convex topological linear space, and , be a real-valued function. A generalized multiplied minimax inequality of Ky Fan type consists in finding such that and for any .
Theorem 4.4 Let X be a nonempty, convex and compact subset of a locally convex topological linear space. Assume that
-
(i)
S is upper semicontinuous with nonempty convex compact values;
-
(ii)
is open in X for any , and for any ;
-
(iii)
is lower semicontinuous on for any ;
-
(iv)
for any finite set of X and any , there is such that .
Then the generalized multiplied minimax inequality of Ky Fan type has at least a solution.
Proof Define the mapping as follows:
By (iii), is closed in X for any . By Corollary 3.1, there exists such that and , which implies and for any . This completes the proof. □
Remark 4.1 Theorem 4.4 is different from Theorems 4.2, 4.3 of [4]. (1) Our Theorem 4.4 with constraining mappings S, Q is a more general problem than Theorems 4.2, 4.3 of [4]. (2) The existence conditions are different between Theorem 4.4 and Theorems 4.2, 4.3 of [4].
From Theorem 4.4, when for any , we obtain a multiplied minimax inequality of Ky Fan type.
Theorem 4.5 Let X be a nonempty, convex and compact subset of a locally convex topological linear space. Assume that
-
(i)
is lower semicontinuous on for any ;
-
(ii)
for any finite set of X and any , there is such that .
Then the multiplied minimax inequality of Ky Fan type has at least a solution, i.e., for any .
References
Balaj M: A common fixed point theorem with applications to vector equilibrium problems. Appl. Math. Lett. 2010, 23: 241–245. 10.1016/j.aml.2009.09.019
Agarwal RP, Balaj M, O’Regan D: Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces. Appl. Anal. 2009, 88: 1691–1699. 10.1080/00036810903331874
Lin LJ, Chuang CS, Yu ZT: Generalized KKM theorems and common fixed point theorems. Nonlinear Anal. 2011, 74: 5591–5599. 10.1016/j.na.2011.05.044
Agarwal RP, Balaj M, O’Regan D: A common fixed point theorem with applications. J. Optim. Theory Appl. 2013. 10.1007/s10957-013-0490-6
Luc DT: An abstract problem in variational analysis. J. Optim. Theory Appl. 2008, 138: 65–76. 10.1007/s10957-008-9371-9
Khanh PQ, Luc DT: Stability of solutions in parametric variational relation problems. Set-Valued Anal. 2008, 16: 1015–1035. 10.1007/s11228-008-0101-0
Pu YJ, Yang Z: Stability of solutions for variational relation problems with applications. Nonlinear Anal. 2012, 75: 1758–1767. 10.1016/j.na.2011.09.007
Lin LJ, Wang SY: Simultaneous variational relation problems and related applications. Comput. Math. Appl. 2009, 58: 1711–1721. 10.1016/j.camwa.2009.07.095
Balaj M, Luc DT: On mixed variational relation problems. Comput. Math. Appl. 2010, 60: 2712–2722. 10.1016/j.camwa.2010.09.026
Balaj M, Lin LJ: Equivalent forms of a generalized KKM theorem and their applications. Nonlinear Anal. 2010, 73: 673–682. 10.1016/j.na.2010.03.055
Balaj M, Lin LJ: Generalized variational relation problems with applications. J. Optim. Theory Appl. 2011, 148: 1–13. 10.1007/s10957-010-9741-y
Lin LJ, Ansari QH: Systems of quasi-variational relations with applications. Nonlinear Anal. 2010, 72: 1210–1220. 10.1016/j.na.2009.08.005
Agarwal RP, Balaj M, O’Regan D: A unifying approach to variational relation problems. J. Optim. Theory Appl. 2012, 155: 417–429. 10.1007/s10957-012-0090-x
Balaj M, Lin LJ: Existence criteria for the solutions of two types of variational relation problems. J. Optim. Theory Appl. 2013, 156: 232–246. 10.1007/s10957-012-0136-0
Luc DT, Sarabi E, Soubeyran A: Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 2010, 364: 544–555. 10.1016/j.jmaa.2009.10.040
Pu YJ, Yang Z: Variational relation problem without the KKM property with applications. J. Math. Anal. Appl. 2012, 393: 256–264. 10.1016/j.jmaa.2012.04.015
Yang Z, Pu YJ: Generalized Knaster-Kuratowski-Mazurkiewicz theorem without convex hull. J. Optim. Theory Appl. 2012, 154: 17–29. 10.1007/s10957-012-9994-8
Yang Z, Pu YJ: Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications. Nonlinear Anal. 2012, 75: 516–525. 10.1016/j.na.2011.08.053
Yang Z: On existence and essential stability of solutions of symmetric variational relation problems. J. Inequal. Appl. 2014., 2014: Article ID 5
Hung NV: Sensitivity analysis for generalized quasi-variational relation problems in locally G -convex spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 158
Hung NV: Continuity of solutions for parametric generalized quasi-variational relation problems. Fixed Point Theory Appl. 2012., 2012: Article ID 102
Latif A, Luc DT: Variational relation problems: existence of solutions and fixed points of contraction mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 315
Aliprantis CD, Border KC: Infinite Dimensional Analysis: A Hitchhiker’s Guide. 3rd edition. Springer, Berlin; 2006.
Deguire P, Tan KK, Yuan GXZ:The study of maximal elements, fixed points for-majorized mapping and their applications to minimax and variational inequalities in the product topological spaces. Nonlinear Anal. 1999, 37: 933–951. 10.1016/S0362-546X(98)00084-4
Acknowledgements
This research is supported by the Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (no. 13CG35), and open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (no. 201309KF02).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Yang, Z. Some generalizations of common fixed point problems with applications. Fixed Point Theory Appl 2014, 189 (2014). https://doi.org/10.1186/1687-1812-2014-189
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-189