Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

Abstract

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such that X becomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spaces and over nonatomic measure spaces in terms of the function . We will prove that the Fréchet differentiability of the farthest point map and the conditions and in reflexive Musielak-Orlicz function spaces are equivalent.

1. Introduction

The problem of farthest points in Banach spaces is studied with many authors (see [1–3]). An interesting question in this field is, under what conditions on the set does the point farthest from point in the spaces exist and is this unique? We recall that the mapping is called farthest point map and the mapping is called a metric antiprojection. Fitzpatrick in [2] gives some conditions such that farthest point map is Fréchet differentiable and the metric antiprojection is continuous; in fact he showed that these conditions are equivalent. Balashov and Ivanov in [1] proved that in Hilbert spaces, the set of conditions for the existence, uniqueness, and Lipschitz dependence (on ) of the metric antiprojection of on the set for points that are sufficiently far from the set is equivalent to the strong convexity of the set . Ivanov in [3] showed that the results of [1] generalized to uniformly convex Banach spaces with Fréchet differentiable norm.

Kurc in [4] introduces the dominated best approximation problem and examines the relations between monotonicity properties and the existence and uniqueness of the dominated best approximation problem. Hudzik and Kurc proved that strictly monotone and order continuity of the norm on is equivalent to unique solvability of the dominated best approximation problem (e.g., [5]).

In this paper we introduce the dominated farthest point problem in a Banach lattice and try to examine the relation between the dominated farthest point problem and monotonicity, Fréchet differentiability of farthest point map, and the continuity of antiprojection map in Banach lattices.

In preliminaries section we recall main definitions and some lemmas that will be used in this context. In Section 3, we introduce the dominated farthest points problem and state some conditions such that guaranteed, existence and uniqueness of the dominated farthest points problem. We give some criteria for strict monotonicity, lower locally uniformly monotone, upper locally uniformly monotone and uniformly monotone. Also we prove that for two equivalent norms such that becomes an STM space and LLUM space the dominated farthest point has the same solution. This note will prove that the Fréchet differentiability of farthest point map is equivalent to continuity of antiprojection map under some conditions. In fact these conditions are equivalent to strong solvability of the set . We give some conditions such that it is proved that if is a singleton set then is a singleton set.

Finally we will say some application of the stated results in Musielak-Orlicz function space and over nonatomic measure spaces in terms of the function . Equivalency of the Fréchet differentiability of the farthest point map and the conditions and in reflexive Musielak-Orlicz function spaces is the final result which will be proved.

2. Preliminaries

Let be a Banach lattice and a bounded sublattice in . Suppose that such that (i.e., for each ); we define as 1.1; we always refer to such problems as to the dominated Farthest points problem.

The dominated farthest points problem is called solvable if . The problem is said to be uniquely solvable if and is to be stable if for every maximizing sequence in , that is, a sequence in such that , there holds as . Finally, the problem is said to be strongly solvable if it is uniquely solvable and stable. A sequence in is a maximizing sequence for if .

In this section, we recall some definitions and lemmas which we need in main results.

Definition 1 (see [4]). A Banach lattice is said to be strictly monotone ( STM) if, for all , the conditions , , and imply .

Definition 2 (see [4]). A Banach lattice is said to be uniformly monotone () if, for all , such that , then .

Definition 3 (see [5]). A Banach lattice is said to be upper (lower) locally uniformly monotone, ULUM ( LLUM), if, for each , such that () and ,  then .

Definition 4 (see [6]). A Banach lattice is said to be decreasing (increasing) uniformly monotone, DUM (), if, for each , such that and ,  then .

Definition 5 (see [5]). A Banach lattice is said to be CWLLUM if for any nonnegative with and any , with , and any sequence in satisfying for all the condition implies .

Definition 6 (see [7]). A lattice seminorm on a Riesz space is said to be order continuous whenever implies . If the above condition holds for sequences, that is, implies , then is said to be -order continuous. If is a lattice norm then the norm is order continuous.

Definition 7 (see [7]). A Banach lattice is said to be a Kantorovich-Banach space (or briefly a KB-space) whenever every increasing norm bounded sequence of is norm convergent.

Definition 8 (see [8]). We say that the norm of the Banach space is Fréchet differentiable at whenever exists uniformly for . If the norm of is Fréchet differentiable at , then we say that has a Fréchet differentiable norm, where .

Definition 9 (see [9]). For a function from a Banach space into a Banach space the Gâteaux derivative at a point is by definition a bounded linear operator such that, for every , The operator is called the Fréchet derivative of at if it is a Gâteaux derivative of at and the limit in (4) holds uniformly in in the unit ball (or unit sphere) in .

Definition 10 (e.g., [10–12]). Let be a -finite, complete (nontrivial), positive measure space and a function such that for -a.e. , , and is nontrivial (continuous at zero with nonzero values), convex, and lsc.
Moreover, if is measurable, for all , then we call the Musielak-Orlicz function.

Definition 11. Musielak-Orlicz spaces consist of all -measurable functions such that for some (depending on ).

Musielak-Orlicz spaces under the natural ordering, when endowed with each of the following norms, become a Banach lattice (e.g., [12, 13]). Luxemburg norm is and Orlicz norm is where and denote the young conjugate of . The amemiya norm (see [14–16] for the Orlicz spaces and [17] for the general case) is All norms defined above are lattice monotone norms and they are equivalent: In the following we will write, for short, or , if for -a.e. the function is strictly positive (except zero) or assumes finite values only, respectively. In the case that is finitely valued and the -condition is satisfied then for all . We recall that is satisfied -condition if as for -a.e.  .

The function is said to satisfy a , condition (), if there exist a set of zero measure, a constant , and an integrable (nonnegative) function , such that, for all and , there holds Suppose that is a subspace of functions with order continuous norm Then as closed ideals (see [18], [13, p. 17], and [19]). If then is super order dense in and [13, p. 19], and has an order continuous norm precisely when . Clearly the norm in is order continuous.

Lemma 12 (see [20]). The following assertions are equivalent: (i)the norm on is order continuous,(ii) is Dedekind complete (-Dedekind complete) satisfying as for any decreasing sequence with ,(iii)every monotone order bounded sequence of is convergent,(iv)every disjoint order bounded sequence of is convergent to zero,(v) is an ideal in ,(vi)every order interval of is weakly compact,(vii).

Lemma 13 (see [20]). The following assertions are equivalent: (i) is reflexive,(ii) and are KB-spaces,(iii) does not contain any subspace isomorphic to or to ,(iv) does not contain any sublattice isomorphic to or to .

Lemma 14 (see [21]). A Banach lattice has an order-continuous norm if and only if it has an equivalent locally uniformly convex lattice norm.

Lemma 15 (see [22]). If is a Banach lattice, the following properties are equivalent: (i) and have order continuous norms,(ii)there exists an equivalent lattice norm on which is locally uniformly convexand Fréchet differentiable, such that its dual norm is also locally uniformly convex on .

Lemma 16 (see [22]). A Banach lattice is reflexive if and only if it can be given an equivalent lattice norm such that both the space and its dual are simultaneously locally uniformly convex and Fréchet differentiable.

Lemma 17 (see [2]). Suppose is a closed subset of a Banach space such that the norm of is Fréchet differentiable. If is bounded and is Fréchet differentiable at some , then every maximizing sequence for converges; also is continuous at .

Lemma 18 (see [2]). Suppose that is a Banach space such that the norms of and are both Fréchet differentiable. If is a closed bounded subset of and is a point of , then the following are equivalent: (i)the metric antiprojection is continuous at x,(ii)every maximizing sequence in for converges,(iii)the function is Fréchet differentiable at .

Lemma 19 (see [23]). Given a Banach lattice the following hold true: (i)if is rotund, then is strictly monotone;(ii)if is locally uniformly rotund, then is upper and lower locally uniformly monotone;(iii)if is uniformly rotund then is uniformly monotone,(iv)in the order intervals in the positive cone the inverse statement of each of the above is also true.

Lemma 20 (see [24]). The following statements are equivalent: (i),(ii) implies ,(iii) implies ,(iv) dose not contain an isometric copy of ,(v) dose not contain a lattice isometric copy of ,(vi); that is, the norm is order continuous on .

3. Dominated Farthest Points Problem in Banach Lattices

In this section we introduce the dominated farthest points problem in Banach lattices and give some criteria for strict monotonicity, lower locally uniformly monotone, upper locally uniformly monotone, and uniformly monotone. We prove that if and are two Banach lattices with the same order such that and are equivalent norms and is an STM space and is an LLUM space, then the problem of the dominated farthest points has the same solution in two spaces. We give some condition such that the Fréchet differentiability of farthest point map is equivalent to continuity of antiprojection map.

Lemma 21. Let be a Banach lattice, a sublattice, and such that . If and such that , then .

Proof. Since and , so and thus . Since we have . Therefore, .

Theorem 22. Let be a Banach lattice. Then is an STM space if and only if for every sublattice in and such that , .

Proof. Suppose that is an STM space and a sublattice in . Suppose that such that and . Since is a sublattice and , from Lemma 21, , so ; since is an STM space ; similarly ; therefore .
Conversely if is not an STM space, then there exist such that and . Define ; then is a sublattice and ; since for each , so for each , and thus , by the assumption ; so ; therefore .

Theorem 23. A Banach lattice is an STM space and has order continuous norm if and only if the dominated farthest points problem with respect to closed order bounded sublattices is uniquely solvable.

Proof. Suppose that is an STM space and has order continuous norm and is a closed sublattice. We assume that and is a maximizing sequence; that is, . Put . Since is a sublattice and , is a decreasing maximizing sequence. Since there exists , we have , by the order continuity of the norm . On the other hand is a closed sublattice so and, since , therefore .
Suppose that the norm on is not order continuous; then there exists a sequence such that but . We can assume that for , otherwise replacing by , for . Suppose that such that and put ; then is a sublattice and . On the other hand by Dini’s theorem is norm closed. Indeed if , and , then we use the fact that if is downward directed sequence which is weakly convergent to then , and thus ; hence ; this is a contradiction and so . By the assumption this contradiction completes the proof.

Remark 24. In Theorem 23, if is an LLUM, CWLLUM, IUM, or DUM space then the theorem is also true.

Theorem 25. Let be an order continuous Banach lattice with the ULUM property; then the dominated farthest points problem with respect to closed order bounded sublattices is strongly solvable.

Proof. From Theorem 23 the dominated farthest points problem with respect to closed sublattices is uniquely solvable. The proof of stability is the same as the proof of Theorem  4.4 in [5].

Theorem 26. Let be an STM space and a sublattice in . If such that (or ) and , then .

Proof. Suppose that such that and . Since is a sublattice so ; from Lemma 21, . Similarly ; therefore .

Corollary 27. Let be an STM space with order continuous norm. If is a sublattice in , such that or ; then the metric antiprojection is in this form: for some .

Proof. It is a consequence of Theorems 23 and 26.

Theorem 28. Let be an STM space and a sublattice in . If such that and and , then is singleton.

Proof. Suppose that and . Since is a sublattice so , from Lemma 21, , thus , thus , and again, from Lemma 21, we have . Therefore .

Lemma 29. Let and be two Banach lattices with the same order such that and are equivalent norm. If is an order continuous Banach lattice, then is also a Banach lattice with order continuous norm.

Proof. It is a consequence of equivalency of two norms.

Example 30. (a) Every equivalent norm on with for is order continuous.
(b) Every equivalent norm on ( is not a finite set) with sup-norm is not order continuous.

Theorem 31. Let and be two Banach lattices with the same order such that and are equivalent norms. Suppose that is an STM space and is an LLUM space. If is a sublattice of and such that (or ), then , where is the set of farthest points with respect to for .

Proof. Since is order continuous and and are equivalent, from Lemma 29, is order continuous. From Theorem 23 and Remark 24, the dominated farthest points problem is uniquely solvable for two norms. Suppose that ; if , there exists such that . Since is a sublattice ; since , we have either or .
If , then and since is an LLUM space that is a contradiction.
If , then and so ; also and so ; since , we have and so that is a contradiction; therefore ; thus .

Remark 32. If we assume that is an STM space with order continuous norm, then Lemmas 14 and 19 guaranteed the existence of , such that is a LLUM space.

Theorem 33. Let be a Banach lattice and a closed order bounded subset of . If , , the following statements are equivalent: (i)any sequence such that is convergent;(ii)the metric antiprojection is uniquely defined on and, for any vector , any maximizing sequences have a convergent subsequence;(iii)dominated farthest point problem with respect to closed bounded sublattices is strongly solvable.

Proof. (i) (ii). Suppose that there exists an , such that . We assume that , , and are two maximizing sequences in convergent to and , respectively.
We define then is not convergent, that is, a contradiction, so . Since every maximizing sequence is convergent the proof is complete.
(ii) (i). Suppose that and is a maximizing sequence. If the condition (i) is not true, then has a subsequent such that , for any . Since , the sequence has a limit point of  , we have and that is a contradiction; therefore (i) is true.
(ii) (iii). By the assumption , suppose that is a maximizing sequence in for . From (ii), has a convergent subsequence to , so .
(iii) (ii). From the definition of strongly solvable  , put . Suppose that is a maximizing sequence in for . So so is a limit point of ; thus there exists a subsequence convergent to ; this completes the proof.

Corollary 34. Let be a Banach lattice with ULUM property and order continuous norm and a closed order bounded sublattice of ; then (i)any sequence such that is convergent;(ii)the metric antiprojection is uniquely defined on and, for any vector , any maximizing sequences have a convergence subsequence,(iii)dominated farthest point problem with respect to closed order bounded sublattices is strongly solvable.

Proof. It is a compound of Theorems 25 and 33.

Theorem 35. Let be an STM space such that and have order continuous norm, a closed order bounded sublattice in , and , such that . If is Fréchet differentiable at then (i),(ii)any sequence such that is convergent,(iii)if and for then ,(iv)the dominated farthest points problem is strongly solvable.

Proof. Part (i). From Theorem 23, the dominated farthest points problem is uniquely solvable so .
Part (ii). From (i) , suppose that is a maximizing sequence in , . is a limit point of and so there is a subsequence such that from Lemma 15, there exists an equivalent norm such that and are locally uniformly convex space and has Fréchet differentiable norm, from Theorem 31, and   is unique farthest point in from with . Since has a subsequence convergent to , is also a maximizing sequence with and hence . From Lemma 17, converges with , so it is convergent with and this completes the proof.
Part (iii). From Lemma 17, is continuous so if ; then or equivalently .
Part (iv). By part (i), . Suppose that is a maximizing sequence in ; that is, . Since we have ; therefore the dominated farthest points problem is strongly solvable.

Theorem 36. Let be a reflexive, STM space , a closed order bounded sublattice in , and , such that . The following statements are equivalent: (i)the dominated farthest points problem is strongly solvable;(ii)any sequence such that is convergent;(iii)if and for , then ;(iv) is Fréchet differentiable at .

Proof. (i) (ii). From Theorem 33 thus is true.
(iv) (ii), (iv) (iii). From Lemma 13, and are KB-spaces and so from Lemma 12 they have order continuous norm. Therefore it is a part of Theorem 35.
(ii) (iv), (iii) (iv). From Lemma 16, there exists an equivalent norm such that and are locally uniformly convex space and , has Fréchet differentiable norm, and from Lemma 18, (ii), (iii), and (iv) are equivalent to norm , so is Fréchet differentiable with norm since is equivalent to from [25, p. 3]; is Fréchet differentiable with norm ; this completes the proof.

Theorem 37. Let be a uniformly convex Banach lattice space, a closed order bounded sublattice in , and , such that . The following statements are equivalent: (i)the dominated farthest points problem is strongly solvable;(ii)any sequence such that is convergent;(iii)if and for , then ;(iv) is Fréchet differentiable at .

Proof. Since is uniformly convex, so it is a reflexive Banach lattice and from Lemma 19, it is a UM space; from Theorem 36 the proof is complete.

Theorem 38. Let be a UM space with an order unit , a closed order bounded sublattice in , and , such that . The following statements are equivalent: (i)the dominated farthest points problem is strongly solvable;(ii)any sequence such that is convergent;(iii)if and for , then ;(iv) is Fréchet differentiable at .

Proof. Since is a UM space with order unit , . So from Lemma 19, rotundity properties are equivalent to monotonicity properties on , so is a uniformly convex Banach lattice from Theorem 37; the proof is complete.

4. Some Applications of the Dominated Farthest Points Problem in Musielak-Orlicz Spaces

In this section we give some applications of the theorems that were proved in Section 3 in Musielak-Orlicz function space. The most important result that will be proved in this section is the equivalency of the Fréchet differentiability of farthest point map and the conditions and in reflexive Musielak-Orlicz function spaces.

Theorem 39. For the Musielak-Orlicz space the following statements are equivalent: (i);(ii)for each closed order bounded sublattice in and such that (or ), .
Moreover (ii) is true for with .

Proof. It is a consequence of Theorem 23.

Remark 40. In Musielak-Orlicz space with and nonatomic measure , if and with any norm the sets of farthest points in dominated farthest points problem are coinciding; this means that , for every and sublattice with , and that for denote the set of farthest points with respect to Luxemburg norm, Orlicz norm, and Ammemyia norm, respectively.

Corollary 41. The dominated farthest points problem in ( with ) with respect to closed order bounded sublattices is unique if and only if and (resp., ).

Proof. From [4, Theorem 2.7] (resp., [4, Theorem 2.8]) has order continuous norm and it is an STM space (resp., ); so from Theorem 22 the proof is complete.

Theorem 42. In the Musielak-Orlicz spaces the following statements are equivalent: (i);(ii)for all closed order bounded linear sublattices , for each .
Moreover (ii) is true for with .

Proof. (ii) (i). It is an immediate consequence of (ii) (i), in Theorem 39.
(i) (ii). Suppose that is a maximizing sequence in ; for example,
Since is a bounded subset of , so . is a KB-space [4, Theorem 3.5] so is convergent to some . Since is a closed linear sublattice, so ; this complete the proof.

Remark 43. Let be a KB-space; with the same argument as Theorem 44, we can show that for all closed bounded linear sublattices , for each .

Theorem 44. Let be a closed order bounded sublattice in the reflexive Musielak-Orlicz spaces with the Luxemburg norm; the following statements are equivalent: (i)the dominated farthest points problem is strongly solvable;(ii)any sequence such that is convergent;(iii)if and for , then ;(iv) is Fréchet differentiable at ;(v) and ;(vi) (resp. with ) is a UM (resp. STM) space.

Proof. From Theorem 36 (i)–(iv) are equivalent. From [4, Theorem 2.7, 2.8] (v) and (vi) are equivalent. From Theorems 38 and 25 (i) and (v) are equivalent.

Conflict of Interests

The authors declare that they have no conflict of interests.