Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices
© Kong et al.; licensee Springer. 2014
Received: 30 October 2013
Accepted: 26 December 2013
Published: 22 January 2014
In this paper, we extend and prove Ky Fan’s Theorem for discontinuous increasing maps f in a Banach lattice X when f has no compact conditions. The main tools of analysis are the variational characterization of the generalized projection operator and order-theoretic fixed-point theory. Moreover, we establish a sequence which converges strongly to the unique best approximation point. As an application of our best approximation theorems, a fixed-point theorem for non-self maps is established and proved under some conditions. Our results generalize and improve many recent results obtained by many authors.
MSC:06F30, 47H07, 41A50, 41A65.
Ky Fan’s approximation theorem (Theorem 2 in ) has attracted great attention worldwide over the last few decades. The normed space version of the theorem is as follows.
Theorem Let K be a non-empty compact convex set in a normed linear space X. If f is a continuous map from K into X, then there exists a point u in K such that . The point u in the theorem above is called a best approximation point of f in K.
Ky Fan’s Theorem is of great importance in nonlinear analysis, approximation theory, game theory and minimax theorems. In recent years, the theorem has been studied and generalized in various respects and applied in the analysis of many problems. Lin and Park , O’Regan and Shahzad  obtained a multivalued version of Ky Fan’s result for condensing maps. Tan and Yuan  and Liu [5, 6] extended the theorem to the more general continuous 1-set-contractive maps under some stronger hypothesis. In the last decade, the study of random approximations and random fixed points have been a very active area of research in probabilistic functional analysis. Some results have already been achieved in this line such as those by Lin , Seghal and Singh , Seghal and Water , Liu [10, 11], Tan and Yuan , Beg and Shahzad . Meanwhile, Lin  proved Fan’s theorem for a continuous condensing map defined on a closed ball in a Banach space. Subsequently, Lin and Yen  proved that Ky Fan’s Theorem is true for a semi-contractive map defined on a closed convex subset of a Hilbert space. Very recently, Liu  proved that Ky Fan’s Theorem is true for the 1-set-contractive maps defined on a bounded closed convex subset in a Banach space when is replaced by Minkowski’s function. For more results, the reader is referred to Shahzad , Markin and Shahzad , Amini-Harandi , Roux and Singh , Liu [19, 20], O’Regan , and so on.
However, so far, Ky Fan’s Theorem has not been well investigated for the cases where f is a discontinuous map and has no compact conditions. Partly motivated by this difficulty, Alber  introduced the notion of a generalized projection operator and noted that can be used instead of in Banach space. Based on this concept, Li and Ok  proved that the metric projection operator is order-preserving in partially ordered Banach spaces. Motivated and inspired by the above mentioned work, in this paper, we obtain two best approximation theorems through the order-theoretic fixed-point theorems by using instead of for reflexive, strictly convex and smooth Banach lattice. In the first best approximation theorem, we establish a sequence which converges strongly to the unique best approximation point; while in the second best approximation theorem, we obtain the existence of a minimum best approximation point and a maximum best approximation point in order intervals. As an application of our best approximation theorems, a fixed-point theorem for non-self maps is established under some conditions which do not need to require any continuous and compact conditions on f.
The rest of the paper is organized as follows. In Section 2, we review the definition of the generalized projection operator in Banach spaces and its basic properties, and also give some definitions in Banach lattice and some fundamental results as preliminaries for our theorems. In Section 3, we establish the properties of the generalized projection operator in Banach lattice under some assumptions. Then we combine these results with an order-theoretic fixed-point theorem to derive some best approximation theorems. Section 4 provides an application of these best approximation theorems to the fixed-point theory.
2.1 The generalized projection operator
for all , where denotes the generalized duality pairing between and X. It is well known that if X is reflexive, strictly convex and smooth, J is a surjective, injective, and single-valued map.
If X is a Hilbert space, then and .
If X is a reflexive, strictly convex, and smooth Banach space, then for , if and only if . It is sufficient to show that if then . From (2.4), we have . This implies that . From the definition of J, one has . Therefore, we have ; and for more details, the reader is referred to [24, 25].
The operator is fixed in each point , i.e., .
- (ii)The operator is d-accretive in X, i.e.,(2.5)
- (iii)The point is a generalized projection of x on if and only if the following inequality is satisfied:(2.6)
- (iv)The operator gives the absolutely best approximation of relative to the functional , i.e.,(2.7)
2.1.1 Banach lattices
Let be a real partially ordered Banach space with the dual and S be a subset of X. We say that an element x of X is an upper bound for S if , that is, for each (the notation is similarly understood). We say that S is bounded from above if for some , and bounded from below if for some . In turn, S is said to be bounded if it is bounded both from above and below. The supremum of S is the minimum of the set of all upper bounds for S, and is denoted by (the infimum of S is denoted as ). As is conventional, we denote as , and as , for any . If and exist for every x and y in X, we say that is a lattice. And if and exist for every non-empty (bounded) , we say that is a (Dedekind) complete lattice. If Y is a non-empty subset of X which contains and for every , then Y is said to be a sublattice of X.
where is defined by for each .
A Riesz space is a lattice where X is a (real) linear space whose linear structure is compatible with the partial order ⪯ in the sense that for all , implies for every and real number . The positive cone of is , which is a pointed convex cone in X. We will assume throughout the paper that the positive cones is closed.
Let be a Banach lattice, that is, is an ordered Riesz space with X being Banach space (if X is a Hilbert space here, is referred to as a Hilbert lattice). The cone is said to be solid if has a non-empty interior i.e. . The cone is said to be normal if there is a number such that for all , implies . The least positive number satisfying this inequality is called the normal constant of .
Definition 2.1 ()
Obviously, if is itself regular, then every sublattice of X is regular. We know every Hilbert normed lattice is regular and the positive cones of many Banach lattices are regular. For example, if , every sublattice S of with is regular; if , every sublattice S of with is regular.
Definition 2.2 ()
For any lattices and , we say that a map is order-preserving if implies .
It is well known that is a Banach lattice, which is called the dual of . As usual, we denote the positive cone of by , and recall that iff for every (see Meyer-Nieberg , Proposition 1.4.2).
Let be a Hilbert space and be a closed convex cone. Recall that is called the dual cone of K. The cone K is called subdual if and superdual if . Suppose be a Hilbert lattice and K be its positive cone, for any , we denote the minimum (supremum) with respect to K as () and the minimum (supremum) with respect to as ().
The following fixed-point theorem is fundamental for the proof of the best approximation theorem.
Theorem 2.1 ()
There exist and , such that , .
- (ii)For any and any bounded subset , there exists such that(2.10)
Then f has a unique fixed point in K such that . Moreover, if such that (), then for .
We denote , where and W is a Lyapunov functional in X.
3 Best approximation theorems
First we establish the following properties of the generalized projection operators.
Using (2.6), we obtain . □
Lemma 3.2 For a reflexive, strictly convex, and smooth Banach lattice , the following statements are equivalent:
(H1) The normalized duality mapping J is order-preserving;
(H2) , implies .
Proof (H1) ⇒ (H2) If J is order-preserving, for , we have , . It is thus obvious that (H2) holds.
This contradicts (H2). So J is order-preserving and the assertion is proved. □
Lemma 3.3 Let be a reflexive, strictly convex, smooth Banach lattice and satisfy condition (H2) and C a closed convex regular sublattice of X. Then, is increasing.
By Lemma 3.2, , and so does not belong to , which is a contradiction. This proves that is increasing. □
Lemma 3.4 Let H be a Hilbert normed lattice with its positive cone K and C a closed convex sublattice of H. Then, is increasing.
Proof It is well known that J is an identity function in Hilbert space. , we have . As H is a normed lattice, we get , that is, . Furthermore, . Thus, . Conclusion: K is subdual cone. Let be such that , we have , that is, . Thus and . We have . So (H2) holds. From Lemma 3.2.1 in , we know that C is regular. By Lemma 3.3, is increasing. □
From Theorem 2.1 and the properties of the generalized projection operator, we obtain the following best approximation theorems.
Theorem 3.1 Let be a reflexive, strictly convex, smooth Banach lattice satisfying condition (H2), and be normal, solid and regular. Suppose that is increasing and satisfies the following conditions:
(H3) There exist and , such that , and .
Then f has a unique point in , satisfying , such that . Moreover, if and (), then for .
The assertion is proved. □
Remark 3.1 In Theorem 3.1, f is a discontinuous map and has no compact conditions.
Corollary 3.1 Let H be a Hilbert normed lattice and its positive cone K be solid. Suppose that is increasing and satisfies (H3), (H4) in K. Then f has a unique point in K, satisfying , such that . Moreover, if and (), then for .
Proof The assertion follows from the above Lemma 3.4 and Theorem 3.1. □
It is easy to see that is a sublattice of X.
where , ().
Proof Define by . From Lemma 3.3, we see that F is increasing. It is easy to see that and . Thus, F satisfies all conditions of Theorem 2.1.2 in . Then, F has a minimum fixed point and a maximum fixed point and satisfies (3.18). By the definition of , the assertion is proved. □
Remark 3.2 In Theorem 3.2, f is a discontinuous map and has no compact conditions.
Example 3.1 Let . Here ⪯ stands for the coordinatewise ordering. Given such that . Then, by Theorem 3.2, every increasing has a minimum best approximation point and a maximum best approximation point with respect to in .
It is easy to see that is a Hilbert normed lattice. Given such that ; then, by Theorem 3.2, every increasing has a minimum best approximation point and a maximum best approximation point with respect to in .
4 Fixed-point theorems
From the above best approximation theorems, we can obtain the following fixed-point theorems.
Theorem 4.1 Suppose that all conditions in Theorem 3.1 are satisfied. Moreover, one of the following conditions holds:
Then, f has a unique fixed point in , which satisfies . Moreover, if and (), then for .
Proof It suffices to show that is the fixed point of f. Indeed, if (H5) holds, using , we get . Thus and . Hence .
If (H6) holds, using , we get . From (H5), we have . The assertion is proved. □
If is self-projective, then , and Theorem 3.2 reduces to the following fixed-point theorem:
Then, f has a minimum fixed point and a maximum fixed point in . Moreover, if and (), then equation (3.18) holds.
The first and second authors were supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
- Fan K: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 1969, 112: 234–240. 10.1007/BF01110225View ArticleMathSciNetGoogle Scholar
- Lin TC, Park S: Approximation and fixed point theorems for condensing composites of multifunctions. J. Math. Anal. Appl. 1998, 233: 1–8.View ArticleMathSciNetGoogle Scholar
- O’Regan D, Shahzad N: Approximation and fixed point theorems for countable condensing composite maps. Bull. Aust. Math. Soc. 2003, 68: 161–168. 10.1017/S0004972700037515View ArticleMathSciNetGoogle Scholar
- Tan KK, Yuan XZ: Random fixed-point theorems and approximation in cones. J. Math. Anal. Appl. 1994, 85: 378–390.View ArticleMathSciNetGoogle Scholar
- Liu LS: Approximation theorems and fixed point theorems for various classes of 1-set-contractive mappings in Banach spaces. Acta Math. Sin. Engl. Ser. 2001, 17: 103–112. 10.1007/s101140000095View ArticleMathSciNetGoogle Scholar
- Liu LS: Random approximations and random fixed point theorems for random 1-set-contractive non-self-maps in abstract cones. Stoch. Anal. Appl. 2000, 18: 125–144. 10.1080/07362990008809659View ArticleGoogle Scholar
- Lin TC: Random approximations and random fixed point theorems for non-self maps. Proc. Am. Math. Soc. 1988, 103: 1129–1135. 10.1090/S0002-9939-1988-0954994-0View ArticleGoogle Scholar
- Sehgal VM, Singh SP: On random approximations and a random fixed point theorem for set-valued mappings. Proc. Am. Math. Soc. 1985, 95: 91–94. 10.1090/S0002-9939-1985-0796453-1View ArticleMathSciNetGoogle Scholar
- Sehgal VM, Waters C: Some random fixed point theorems for condensing operators. Proc. Am. Math. Soc. 1984, 90: 425–429. 10.1090/S0002-9939-1984-0728362-7View ArticleMathSciNetGoogle Scholar
- Liu LS: Some random approximations and random fixed point theorems for 1-set-contractive random operators. Proc. Am. Math. Soc. 1997, 125: 515–521. 10.1090/S0002-9939-97-03589-2View ArticleGoogle Scholar
- Liu LS: Random approximations and random fixed point theorems for random 1-set-contractive non-self-maps in infinite dimensional Banach spaces. Indian J. Pure Appl. Math. 1997, 28(2):139–150.MathSciNetGoogle Scholar
- Beg I, Shahzad N: Random fixed points of random multivalued operators on Polish spaces. Nonlinear Anal. 1993, 20: 835–847. 10.1016/0362-546X(93)90072-ZView ArticleMathSciNetGoogle Scholar
- Lin TC: A note on a theorem of Ky Fan. Can. Math. Bull. 1979, 22: 513–515. 10.4153/CMB-1979-067-xView ArticleGoogle Scholar
- Lin TC, Yen CL: Applications of the proximity map to fixed point theorems in Hilbert space. J. Approx. Theory 1988, 52: 141–148. 10.1016/0021-9045(88)90053-6View ArticleMathSciNetGoogle Scholar
- Shahzad N: Fixed point and approximation results for multimaps in S-KKM class. Nonlinear Anal. 2004, 56: 905–918. 10.1016/j.na.2003.10.019View ArticleMathSciNetGoogle Scholar
- Markin J, Shahzad N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. Nonlinear Anal. 2009, 70: 2435–2441. 10.1016/j.na.2008.03.045View ArticleMathSciNetGoogle Scholar
- Amini-Harandi A: Best and coupled best approximation theorems in abstract convex metric spaces. Nonlinear Anal. 2011, 74: 922–926. 10.1016/j.na.2010.09.045View ArticleMathSciNetGoogle Scholar
- Roux D, Singh SP: On some fixed point theorems. Int. J. Math. Math. Sci. 1989, 12: 61–64. 10.1155/S0161171289000074View ArticleMathSciNetGoogle Scholar
- Liu LS: On approximation theorems and fixed point theorems for non-self-mapping in infinite dimensional Banach spaces. J. Math. Anal. Appl. 1994, 188(2):541–551. 10.1006/jmaa.1994.1444View ArticleMathSciNetGoogle Scholar
- Liu LS, Li XK: On approximation theorems and fixed point theorems for non-self-mappings in uniformly convex Banach spaces. Banyan Math. J. 1997, 4: 11–20.Google Scholar
- O’Regan D: Existence and approximation of fixed points for multivalued maps. Appl. Math. Lett. 1999, 12: 37–43.View ArticleGoogle Scholar
- Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996:15–50.Google Scholar
- Li J, Ok EA: Optimal solutions to variational inequalities on Banach lattices. J. Math. Anal. Appl. 2012, 388: 1157–1165. 10.1016/j.jmaa.2011.11.009View ArticleMathSciNetGoogle Scholar
- Cioranescu I: Geometry of Banach spaces. In Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.View ArticleGoogle Scholar
- Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.Google Scholar
- Meyer-Nieberg P Universitext. In Banach Lattices. Springer, Berlin; 1991.View ArticleGoogle Scholar
- Guo D: Existence and uniqueness of positive fixed points for mixed monotone operators and applications. Appl. Anal. 1992, 46: 91–100. 10.1080/00036819208840113View ArticleMathSciNetGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.Google Scholar
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