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Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices
Fixed Point Theory and Applications volume 2014, Article number: 18 (2014)
Abstract
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.
1 Introduction
Ky Fan’s approximation theorem (Theorem 2 in [1]) 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 [2], O’Regan and Shahzad [3] obtained a multivalued version of Ky Fan’s result for condensing maps. Tan and Yuan [4] 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 [7], Seghal and Singh [8], Seghal and Water [9], Liu [10, 11], Tan and Yuan [4], Beg and Shahzad [12]. Meanwhile, Lin [13] proved Fan’s theorem for a continuous condensing map defined on a closed ball in a Banach space. Subsequently, Lin and Yen [14] 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 [5] 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 [15], Markin and Shahzad [16], Amini-Harandi [17], Roux and Singh [18], Liu [19, 20], O’Regan [21], 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 [22] 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 [23] 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 Preliminaries
2.1 The generalized projection operator
Let X be a real Banach space with the dual . We denote by J the normalized duality mapping from X to defined by
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.
Let X be a reflexive, strictly convex and smooth Banach space and C a non-empty closed convex subset of X. Consider the Lyapunov functional defined by
Following Alber [22], the generalized projection operator is a map that assigns to an arbitrary point the minimum point of the functional , that is, , where is the solution to the minimization problem
Existence and uniqueness of the operator follows from the properties of the functional and the strict monotonicity of the mapping J. It is obvious from the definition of the functional W that
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].
As shown in [22], the generalized projection operator on a convex closed set C satisfies the following properties:
-
(i)
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.
A normed lattice X is a vector lattice with a norm such that the following condition is satisfied:
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 ([23])
Let be a Banach lattice, a sublattice Y of X is said to be regular if is submodular on Y with respect to ⪯, that is,
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 ([23])
For any lattices and , we say that a map is order-preserving if implies .
2.1.2 Order-dual
Let be a Banach lattice. The dual of ⪯ is the partial order ⪯∗ on defined as follows:
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 [26], 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 ([27])
Let K be a normal and solid cone. Suppose that is increasing and satisfies the following conditions:
-
(i)
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.
Lemma 3.1 Let be a partially ordered space, and let be its positive cone, then
Proof For every , , we take . It is obvious that , and so by equation (2.6) we have
and further, by the positive homogeneity of J, we get
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.
(H2) ⇒ (H1) Assume that J is not order-preserving, then there exist , such that , . Since X is a reflexive Banach lattice, J is surjective and is a Banach lattice, which implies that there exist , , such that , . Indeed we have
Since X is strictly convex, which implies only in the case , , the relation holds. Moreover, only in the case , the relation holds. This obviously implies if and only if . From the assumption, it is impossible that . Thus
In a similar way, we get
Adding equations (3.4) and (3.5), we have
and
Using , we obtain
and thus
Since and , we have
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.
Proof To derive a contradiction, assume that is not increasing. Then, there exist , such that , . Because C is a sublattice of X, we have . It follows from the definition of that , that is,
On the other hand, as , we trivially have , that is,
Using the fact that , we can write the inequality (3.10) as
Combining equations (3.11) and (3.12) yields
Thus, by the regularity of C, we get
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 [23], 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 .
(H4) For any and any bounded subset , there exists such that
Then f has a unique point in , satisfying , such that . Moreover, if and (), then for .
Proof Define by . It is obvious that is a sublattice of X. By Lemma 3.3, it is easy to see that F is increasing. Since is increasing and , , we get
Using Lemma 3.1, we have
From , , , we obtain
Thus F satisfies all conditions of Theorem 2.1, and so F has a unique fixed point in , such that , and for . Now we consider , i.e. . By the definition of , we get
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. □
Let be a Banach lattice. Given such that , we denote by the set:
It is easy to see that is a sublattice of X.
Theorem 3.2 Let be a reflexive, strictly convex, smooth Banach and Dedekind complete lattice satisfying condition (H2). Suppose that is regular and is increasing. Then, f has a minimum best approximation point and a maximum best approximation point with respect to in , such that
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 [28]. 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 .
Example 3.2 Let , the space of measurable functions which are 2nd power summable on Ω. Endow with the following norm and ⪯:
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:
(H5) ;
(H6) .
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:
Corollary 4.1 Let be a Dedekind complete lattice. Suppose that is increasing and satisfies:
Then, f has a minimum fixed point and a maximum fixed point in . Moreover, if and (), then equation (3.18) holds.
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Acknowledgements
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.
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Kong, D., Liu, L. & Wu, Y. Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices. Fixed Point Theory Appl 2014, 18 (2014). https://doi.org/10.1186/1687-1812-2014-18
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DOI: https://doi.org/10.1186/1687-1812-2014-18