Characterization of the Best Approximation and Establishment of the Best Proximity Point Theorems in Lorentz Spaces

Abstract

Since the monotonicity of the best approximant is crucial to establish partial ordering methods, in this paper, we, respectively, characterize the best approximants in Banach function spaces and Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$, in which we especially focus on the monotonicity characterizations. We first study monotonicity characterizations of the metric projection operator onto sublattices in general Banach function spaces by the property $H_g$. The sufficient and necessary conditions for monotonicity of the metric projection onto cones and sublattices are then, respectively, established in ${\mathrm{\Gamma }}_{p,w}$. The Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$ are also shown to be reflexive under the condition $R{B}_p$, which is the basis for the existence of the best approximant. As applications, by establishing the partial ordering methods based on the obtained monotonicity characterizations, the solvability and approximation theorems for best proximity points are deduced without imposing any contractive and compact conditions in ${\mathrm{\Gamma }}_{p,w}$. Our results extend and improve many previous results in the field of the approximation and partial ordering theory.

1. Introduction

The best approximation theory is very important in nonlinear optimization theory, nonlinear analysis, game theory, image processing, and signal processing. Various aspects of the best approximation theory such as the existence, the Chebyshev problem, the continuity, the monotonicity, the derivability, etc., have attracted tremendous attention from many authors in recent years. In [1], Isac and Németh introduced the monotonicity of the metric projections onto cones and the new partial ordering approach in Euclidean spaces. In [2], by the lattice operator, the monotonicity of the set-valved metric projection was shown in Hilbert lattice. Later in [3,4], based on generalizing the lattice operator, the monotonicity of best approximants was extended in ordered Hilbert spaces. Recently, in [5,6], new extensions of the best monotone approximation property for general Banach spaces have been introduced and were used to resolve variational inequalities. In [7], the existence of the best monotone approximants on nonconvex sets in normed spaces was given. In [8], the best monotone approximants in the space $L^1[0,1]$ were introduced. In [9], Marano and Quesada researched the best monotone approximants in $L_{\varphi }[0,1]$ for a general function $\varphi$. More generally, in the space $L_{\varphi }$, Landers and Rogge showed the monotonicity of the best approximation operator in [10]. In [11], the extended best constant approximant operator over the Orlicz space $L^{{\phi }^{\prime }}(R^n)$ was constructed by Favier and Zo. In [12], in the space $L^p$ ($1\le p<\infty$), characterizing the best monotone approximations was established by a certain class of subcones. In [13,14], monotonicity was proposed in quasi-normed Orlicz and F-normed Musielak–Orlicz spaces, respectively.

Since the Banach function space theory was introduced by Luxemburg in [15], it has been an active research topic in mathematics and played an important role in operator theory, space theory, harmonic analysis, best approximation theory, and other branches of mathematical analysis. In [16], characterisations of separability for quasi-Banach function spaces over the Euclidean space were researched. As the special class of Banach function spaces, Lorentz introduced Lorentz spaces ${\mathrm{\Lambda }}_{p,w}$ in [17], while ${\mathrm{\Gamma }}_{p,w}$ was first shown in [18]. Lorentz spaces and their different generalizations such as ${\mathrm{\Gamma }}_{p,w}$, the Orlicz–Lorentz spaces ${\mathrm{\Lambda }}_{w,\phi }$, etc., play an important role in the Banach space theory. In [19,20], characterizations of the best constant approximants were deduced and the monotonicity of the best constant approximation operator in Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$ was constructed. In [21], the best monotone approximants were studied in general Banach spaces endowed with extended partial orders. In the same spirit as in [11], the extensions of the best constant approximants in Orlicz–Lorentz spaces ${\mathrm{\Lambda }}_{w,{\phi }^{\prime }}$ were studied. Refs. [22,23], Levis, Cuenya, and Priori investigated the best approximants in ${\mathrm{\Lambda }}_{w,\phi }$ with the Orlicz function $\phi$ and the weight function $w>0$. In [24], the variable exponent Lorentz spaces were introduced and their basic characterizations were researched. In [25,26], the best dominated approximation problem for $x⪯K$ was considered by the order continuity, where K is a sublattice. Very recently, in [27,28], the local geometry ($H_g$ points, $H_l$ points, strict K-monotonicity, and local uniform rotundity) of the Banach function space and ${\mathrm{\Gamma }}_{p,w}$ were discussed, and as an application, the solvability theorems of the best dominated approximation problem for $x⪯K$ were proved by property $H_g$.

In this paper, we aim to study the existence and monotonicity properties of the metric projection operator as the extension of the best constant approximant operator in Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$ and Banach function spaces and apply them to solve the best proximity point problems in ${\mathrm{\Gamma }}_{p,w}$ by partial ordering methods. The motivation is the difficulties in finding more points to satisfy the monotonicity of the metric projection and constructing the increasing sequence to converge to the best proximity point under some geometry and order assumptions. This paper shows the following four highlights. Firstly, using property $H_g$, we obtain the existence and monotonicity of the metric projections onto closed sublattices K in Banach function spaces for a wider range of problems than $f{⪯}_{E^{+}}K$. Then, we establish expressions of the metric projection onto subcones in $E^{+}$. By the expression of the Gâteaux derivative of ${\parallel ·\parallel }_{{\mathrm{\Gamma }}_{p,w}}$, we obtain the necessary and sufficient conditions for monotonicity of the metric projections onto cones and sublattices in Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$. Moreover, we show that the Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$ are reflexive under some assumptions, under the condition $R{B}_p$, which is basic for the existence of the best approximant. As applications of the monotonicity of the metric projections, solvability and approximation results for the best proximity points are established and proved. We should emphasize that we do not require any contractive and compact conditions in our best proximity point theorems.

The structure of this paper is as follows. In Section 2, we show the terminology, definitions, and notations, and present some basic results used in the paper. In Section 3, some existence and monotonicity characterizations of the metric projection operator are proved in Banach function spaces. In Section 4, based on the geometric characterizations of ${\mathrm{\Gamma }}_{p,w}$, we obtain the necessary and sufficient conditions for the monotonicity of the metric projection onto cones and sublattices. In Section 5, solvability and approximation theorems for best proximity points are obtained.

2. Preliminaries

In this section, we introduce some basic definitions, characterizations, notations, references on the Banach function space, Lorentz space, monotonicity, and the best approximation operator which will be used in the paper.

Let $\mathbb{R}$ be the set of all real numbers and $\mathbb{N}$ the set of all positive integers. The set of all extended real valued Lebesgue measurable functions on $[0,\tau )$ is denoted by $L^0[0,\tau )$ (or briefly, $L^0$), where $\tau \in (0,+\infty ]$. Let m be the Lebesgue measure on $[0,\tau )$ (see [29]) and denote by $\mathbb{S}(h)=\mathrm{supp}(h)$ the support of $h\in L^0$, which is the closure of the set where h is nonzero.

In the rest of the paper, we always denote by E a Banach lattice. The positive cone of E is $E^{+}=\{x\in E:x{⪰}_{E^{+}}\theta \}$, where $\theta$ is the zero element of E and ${⪰}_{E^{+}}$ denote the partial order with respect to $E^{+}$, i.e., $y{⪰}_{E^{+}}x$ if only if $y-x\in E^{+}$. We say that $E^{+}$ is regular if and only if any increasing sequence which is bounded from above is convergent (or equivalently, any decreasing sequence which is bounded from below is convergent). Some details of the lattice can be seen in [30,31,32].

Lemma 1

(see Theorem 2.2.2 in [33]). Let E be a Banach lattice and $E^{+}$ the positive cone of E. If E is reflexive, then $E^{+}$ is regular.

If K is a subset in E and $x\in E$, $x{⪯}_{E^{+}}K$ denotes $x{⪯}_{E^{+}}y$ for any $y\in K$. For convenience, we also adopt the following common notations:

$$\begin{array}{c}x{\vee }_{E^{+}}y=sup\{x,y\},x{\wedge }_{E^{+}}y=inf\{x,y\};\hfill \\ x_{E^{+}}^{+}=x{\vee }_{E^{+}}\theta ,x_{E^{+}}^{-}=(-x){\vee }_{E^{+}}\theta ;\hfill \\ {|x|}_{E^{+}}=(-x){\vee }_{E^{+}}x,\forall x,y\in E.\hfill \end{array}$$

(1)

Definition 1

(see [28]). If a Banach lattice $(E,\parallel ·\parallel ,{⪯}_{E^{+}})$ is a sublattice of $L^0$ and satisfies the following conditions:

(i) 

If $x\in L^0$, $y\in E$ and ${|x|}_{E^{+}}{⪯}_{E^{+}}{|y|}_{E^{+}}$ a.e., then $x\in E$ and $\parallel x\parallel \le \parallel y\parallel$;

(ii) 

There exists an element $x\in E$ such that $\theta {\prec }_{E^{+}}x$.

Then we say that $(E,\parallel ·\parallel ,{⪯}_{E^{+}})$ is a Banach function space or Köthe space.

If $x\in E$ and $\parallel x_n\parallel \to 0$ for each sequence $\left\{x_n\right\}\subset E$ with $\theta {⪯}_{E^{+}}{x}_n{⪯}_{E^{+}}{|x|}_{E^{+}}$ and $x_n\to \theta$ a.e., then we say that x has an order continuous norm. A Banach function space E is called order continuous if each $x\in E$ has an order continuous norm.

Definition 2

(see [28]). Let E be a Banach function space. Then

(i) 

E is said to have the Fatou property if $x\in E$ and $\underset{n\to \infty }{lim}\parallel x_n\parallel =\parallel x\parallel$ for $\theta {⪯}_{E^{+}}{x}_n\uparrow x\in L^0$ with ${\left\{x_n\right\}}_{n=1}^{\infty }$ in E and ${sup}_{n\in \mathbb{N}}\parallel x_n\parallel <\infty$;

(ii) 

E is mentioned to have the semi-Fatou property if $\parallel x_n\parallel \uparrow \parallel x\parallel$ for $\theta {⪯}_{E^{+}}{x}_n\uparrow x\in E$ with $x_n\in E$.

If $x\in E$ and $\underset{n\to \infty }{lim}\parallel x_n-x\parallel =0$ for every sequence $\left\{x_n\right\}$ in E with $x_n\to x$ globally in measure and $\underset{n\to \infty }{lim}\parallel x_n\parallel =\parallel x\parallel$, then we say that x is an $H_g$ point in E. The space $(E,\parallel ·\parallel ,{⪯}_{E^{+}})$ is said to have Kadec–Klee property globally in measure if every $x\in E$ is an $H_g$ point. Denote by $B_E$ and $S_E$ the closed unit ball and the unit sphere of E, respectively. A point $z\in B_E$ is an extreme point in $B_E$ if for any $x,y$ in $B_E$ with $z={\displaystyle \frac{x+y}{2}}$, we have $x=y$. If each element of $S_E$ is an extreme point of $B_E$, then E is said to be strictly convex. Other definitions on Banach function spaces can be found in [34,35,36].

Let x be any element of $L^0$. The distribution function of x is defined by

$${\mu }_x(\lambda ,\tau )=m\{s\in [0,\tau ):|x(s)|>\lambda \},\forall \lambda \ge 0.$$

(2)

The decreasing rearrangement of x is defined as

$$x^{*}(t)=inf\{\lambda >0:{\mu }_x(\lambda ,\tau )\le t\},\forall t\ge 0.$$

(3)

The maximal function of $x^{*}$ is defined by

$$x^{**}(t)={\displaystyle \frac{1}{t}}{\int }_0^t{x}^{*}(s)ds.$$

(4)

Note that $x^{*}(t)\le x^{**}(t)$ for all $t>0$ and $x^{**}$ is nonincreasing and subadditive, that is,

$${(x+y)}^{**}(t)\le x^{**}(t)+y^{**}(t),\forall x,y\in L^0,t>0.$$

More characterizations about ${\mu }_{x,\tau },x^{*}$ and $x^{**}$ can be found in [37,38,39].

Let $1\le p<\infty$ and a weight function w be in $L^0$, the Lorentz space $({\mathrm{\Gamma }}_{p,w},\parallel ·{\parallel }_{{\mathrm{\Gamma }}_{p,w}})$ (or briefly, ${\mathrm{\Gamma }}_{p,w}$) be a subspace of $L^0$ such that

$${\parallel x\parallel }_{{\mathrm{\Gamma }}_{p,w}}={\left({\int }_0^{\tau }{(x^{**})}^p(t)w(t)dt\right)}^{{\displaystyle \frac{1}{p}}}<\infty .$$

Throughout the paper, unless otherwise mentioned, we always assume that the weight function w is nonnegative. Aiming to ${\mathrm{\Gamma }}_{p,w}\ne \left\{\theta \right\}$, we also assume that w satisfies the condition ${\mathcal{D}}_p$, that is,

$$W(s)={\int }_0^{s}w(t)dt<\infty \mathrm{and}{W}_p(s)=s^p{\int }_s^{\tau }{t}^{-p}w(t)dt<\infty ,\forall 0<s\le \tau .$$

In ref. [40], it was proved that, for the case when $\tau =\infty$, the space $({\mathrm{\Gamma }}_{p,w},\parallel ·{\parallel }_{{\mathrm{\Gamma }}_{p,w}})$ is order continuous if and only if ${\int }_0^{\infty }w(t)dt=\infty$. Moreover, for the case when $\tau <\infty$, by Lebesgue’s dominated convergence theorem, it was proved that $({\mathrm{\Gamma }}_{p,w},\parallel ·{\parallel }_{{\mathrm{\Gamma }}_{p,w}})$ is order continuous. We now recall the well-known general characterizations on strictly convexity, Kadec–Klee property, and best approximants of space ${\mathrm{\Gamma }}_{p,w}(0,\tau )$.

Lemma 2

(see Theorem 3.1 in [41]). Suppose that $1\le p<\infty$ and w is positive. Then the space ${\mathrm{\Gamma }}_{p,w}(0,\tau )$ is strictly convex if and only if $1<p<\infty$ and $W(\infty )=\infty$ for the case when $\tau =\infty$.

Lemma 3

(see Theorem 4.1 in [28]). The space ${\mathrm{\Gamma }}_{p,w}(0,\tau )$ has the Kadec–Klee property with respect to global convergence in measure.

Lemma 4

(see Theorem 6.1 in [41]). Let K be a closed convex subset of ${\mathrm{\Gamma }}_{p,w}(0,\tau )$ and $x\in K^c=\{x\in {\mathrm{\Gamma }}_{p,w}(0,\tau ):x\notin K\}$. Then $\overline{x}\in K$ is a best approximant of x onto K if and only if $G_{+}(x-\overline{x},\overline{x}-z)\ge 0$ for all $z\in K$, where $G_{+}(x-\overline{x},\overline{x}-z)=\underset{ϵ\downarrow 0}{lim}{\displaystyle \frac{\parallel (x-\overline{x})+ϵ(\overline{x}-z)\parallel -\parallel x-\overline{x}\parallel }{ϵ}}$.

For more details about the properties of ${\mathrm{\Gamma }}_{p,w}$, the reader is referred to [40,42].

In [18], by an analogous way, Calderón defined the spaces $({\mathrm{\Gamma }}_{p,w},\parallel ·{\parallel }_{{\mathrm{\Gamma }}_{p,w}})$ as the famous Lorentz spaces.

$${\mathrm{\Lambda }}_{p,w}=\left\{x\in L^0:{\parallel x\parallel }_{{\mathrm{\Lambda }}_{p,w}}={\left({\int }_0^{\tau }{(x^{*}(t))}^{p}w(t)dt\right)}^{{\displaystyle \frac{1}{p}}}<\infty \right\},$$

where w is nonnegative and nonincreasing and $p\ge 1$. It is easy to obtain ${\mathrm{\Gamma }}_{p,w}\subset {\mathrm{\Lambda }}_{p,w}$. Conversely, ${\mathrm{\Lambda }}_{p,w}\subset {\mathrm{\Gamma }}_{p,w}$ holds if and only if w satisfies condition $B_p$, that is, there exists $\alpha >0$ such that for all $s>0$,

$${\int }_s^{\infty }{t}^{-p}w(t)dt\le \alpha s^{-p}{\int }_0^{s}w(t)dt.$$

If $\infty >p>0$, and for all $s>0$, there exists $\beta >0$ such that

$$s^{-p}{\int }_0^{s}w(t)dt\le \beta {\int }_s^{\infty }{t}^{-p}w(t)dt,$$

then $w\in {\mathcal{D}}_p$ is said to satisfy the reverse condition of $B_p$, denoted also by $R{B}_p$. By the Sawyer’s result (see Theorem 1 in [43]), the Köthe dual of ${\mathrm{\Gamma }}_{p,w}$ and the Köthe dual of ${\mathrm{\Lambda }}_{p,w}$ are also related for $1<p<\infty$ and ${\int }_0^{\infty }w(t)dt=\infty$, that is, the Köthe dual of ${\mathrm{\Lambda }}_{p,w}$ coincides with ${\mathrm{\Gamma }}_{p^{\prime },\tilde{w}}$, where $\tilde{w}(t)={(t/{\int }_0^{t}w(s)ds)}^{p^{\prime }}w(t)$ and ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p^{\prime }}}=1$.

Let B be a subset in any Banach space $(E,\parallel ·{\parallel }_E)$. The set-valued mapping $P_B:E\rightrightarrows B$,

$$P_B(x)={\{z\in B:\parallel x-z\parallel }_E=\underset{y\in B}{inf}{\parallel x-y\parallel }_E\},\forall x\in E$$

is called the metric projection operator from E onto B. It is known that, for every closed convex subset B of a Banach space E, $P_B(x)\ne \varnothing$ for any $x\in E$ if and only if E is reflexive. If E is strict convex, then $P_B(x)$ is single-valued. Then, set-valued mapping is $f:E\rightrightarrows E$ if $x{⪯}_{E^{+}}y$ implies that there exist $x^{\prime }\in f(x)$ and $y^{\prime }\in f(y)$ such that $x^{\prime }{⪯}_{E^{+}}{y}^{\prime }$; in this case, we say that f is monotone. If f is single valued, then f is monotone if and only if $f(x){⪯}_{E^{+}}f(y)$ for all $x,y\in E$, satisfying $x{⪯}_{E^{+}}y$.

Lemma 5

(see Lemma 2.11 in [44]). Let $(E,{⪯}_K,\parallel ·\parallel )$ be a partially ordered Banach lattice induced by the positive cone K. Suppose that $\parallel ·\parallel$ is $p-$additive, $u_0,v_0\in E$ with $u_0{\prec }_K{v}_0$. Then $u_0+{(x-u_0)}_K^{+}{\wedge }_K(v_0-u_0)\in P_{[u_0,v_0]}(x)$ for all $x\in E$.

3. Characterizing Best Approximations in Banach Function Spaces

In the section, we consider characterizations of the metric projection operator $P_K$, including the existence and monotonicity results of $P_K$ onto general sublattices and special lattices. The following basic existence relation between the best approximant of x and $x^{*}$ is first proposed.

Proposition 1.

Let $(E,\parallel ·\parallel ,{⪯}_{E^{+}})$ be a Banach function space and K a subset of E. If $x\in E^{+}$ and $P_K(x^{*})\ne \varnothing$, then $P_{K\circ \gamma }(x)\ne \varnothing$, where $\gamma :\mathbb{S}(x)\to \mathbb{S}(x^{*})$ is a measure preserving transformation from $\mathbb{S}(x)$ onto $\mathbb{S}(x^{*})$, satisfying ${|x|}_{E^{+}}=x^{*}\circ \gamma a.e.$ on $\mathbb{S}(x)$ and $K\circ \gamma =\{f:f(t)=y(\gamma (t)),y\in K,t\in \mathbb{S}(x)\}$.

Proof. 

Since $P_K(x^{*})\ne \varnothing$, there exists an ${\tilde{x}}^{*}\in K$ such that

$$\parallel x^{*}-{\tilde{x}}^{*}\parallel \le \parallel x^{*}-y\parallel ,\forall y\in K.$$

(5)

As $x\in E^{+}$, we get ${|x|}_{E^{+}}=x$ and the left of (5) becomes

$$\parallel x^{*}-{\tilde{x}}^{*}\parallel =\parallel (x^{*}-{\tilde{x}}^{*})\circ \gamma \parallel =\parallel x-{\tilde{x}}^{*}\circ \gamma \parallel .$$

Similarly, the right of (5) becomes

$$\parallel x^{*}-y\parallel =\parallel (x^{*}-y)\circ \gamma \parallel =\parallel x-y\circ \gamma \parallel .$$

Therefore,

$$\parallel x-{\tilde{x}}^{*}\circ \gamma \parallel \le \parallel x-y\circ \gamma \parallel ,\forall y\in K.$$

It follows that ${\tilde{x}}^{*}\circ \gamma \in P_{K\circ \gamma }(x)$. Thus, the assertion is proved. □

In the following proposition, based on the above semi-Fatou property and definition of the $H_g$ point, we introduce the existence and monotonicity proposition of the metric projection operator onto general closed sublattices in Banach function spaces.

Proposition 2.

Let $(E,\parallel ·\parallel ,{⪯}_{E^{+}})$ be a Banach function space satisfying the semi-Fatou property. Let $K{⪰}_{E^{+}}\theta$ be a closed sublattice of E. Suppose that, for each point h in K, there exists a $v\in K$ with $v{⪰}_{E^{+}}x$, such that $v-x{⪯}_{E^{+}}{|x-h|}_{E^{+}}$, and there exists a $v_0\in K$ with $v_0^{*}(\infty )=0$, such that $v_0-x\in E^{+}$ is an $H_g$ point of E. Then, $P_K(x)\ne \varnothing$. Moreover, $P_K(x)$ is monotone.

Proof. 

Assume that $x\in E$, $h\in K$ and $\left\{h_n\right\}\subset K$ satisfies

$$d(x,h)=\underset{h\in K}{inf}\parallel x-h\parallel =\underset{n\to \infty }{lim}\parallel x-h_n\parallel .$$

(6)

It follows that there exists $\left\{v_n\right\}{⪰}_{E^{+}}x$ such that $|x-h_n{|}_{E^{+}}{⪰}_{E^{+}}{v}_n-x$. Without loss of generality, we may suppose that $v_n{⪯}_{E^{+}}{v}_0$. Otherwise, it is enough to replace $v_n$ with $v_n{\wedge }_{E^{+}}{v}_0$. Take $u_n={\wedge }_{E+\stackrel{n}{k}=1}{v}_k$. As K is a sublattice, we have $u_n\in K$ and $\theta {⪯}_{E^{+}}{u}_n-x{⪯}_{E^{+}}{v}_n-x{⪯}_{E^{+}}{|x-h_n|}_{E^{+}}$. It follows from (6) that

$$d(x,h)\le \parallel x-u_n\parallel \le \parallel x-h_n\parallel ,\forall n\in \mathbb{N}.$$

If we take $u={\wedge }_{E+\stackrel{\infty }{k}=1}{v}_k$, then $x{⪯}_{E^{+}}u{⪯}_{E^{+}}{u}_n$, which implies that

$$u\in E \mathrm{and} \theta {⪯}_{E^{+}}{u}_n-u\downarrow \theta ,$$

pointwisely. If $\tau =\infty$, using the fact that $v_0^{*}(\infty )=0$ and $\theta {⪯}_{E^{+}}{u}_n{⪯}_{E^{+}}{v}_0$, then we get $u_n^{*}(\infty )=0$ for any n in $\mathbb{N}$. Let $\epsilon >0$ be random. As $u^{*}(\infty )=0$, there exists $t_{\epsilon }>0$, such that

$${(u_1-u)}^{*}(t_{\epsilon })\le \epsilon .$$

Since $\theta {⪯}_{E^{+}}{u}_n-u{⪯}_{E^{+}}{u}_1-u$, we have

$$\begin{array}{cc}m\{t:|u_n(t)-u(t)|>\epsilon \}\hfill & =m\{t:{(u_n-u)}^{*}(t)>\epsilon \}\hfill \\ \hfill & =m\{t\in [0,t_{\epsilon }):{(u_n-u)}^{*}(t)>\epsilon \}.\hfill \end{array}$$

(7)

As $\theta {⪯}_{E^{+}}{u}_n-u\downarrow \theta$ pointwisely, by the Proposition 12° in [39], we get $\left\{{(u_n-u)}^{*}\right\}$, which pointwisely converges to $\theta$. Without loss of generality, we may assume that

$$m\{t\in [0,t_{\epsilon }):{(u_n-u)}^{*}(t)>\epsilon \}\to 0.$$

Otherwise, it is enough to consider a subsequence of $\left\{u_n\right\}$. It follows that $u_n\to u$ globally in measure. Hence

$$v_0-u_n+u-x\to v_0-x\in E$$

(8)

globally in measure. As $v_0{⪰}_{E^{+}}{u}_n$, we have $\theta {⪯}_{E^{+}}{v}_0-(u_n-u)-x{⪯}_{E^{+}}{v}_0-x$. Further, adopting the semi-Fatou property of E, we have that

$$\parallel v_0-(u_n-u)-x\parallel \uparrow \parallel v_0-x\parallel .$$

Since $v_0-x$ is an $H_g$ point, it follows from (8) that

$$\parallel u_n-u\parallel \to 0.$$

By the conditon that K is closed, we obtain that $u\in K$ and

$$d(x,h)\le \parallel u-x\parallel \le \parallel u_n-x\parallel \to d(x,h).$$

(9)

Hence, $u\in P_K(x)$. Similarly, for the case when $\tau <\infty$, we get $P_K(x)\ne \varnothing$.

Now we prove that $P_K(x)$ is monotone. Take $C=\{x\in E:$ for any $h\in K$; there is a $v\in K$ with $v{⪰}_{E^{+}}x$ such that $v-x{⪯}_{E^{+}}{|x-h|}_{E^{+}}$ and $v_0-x\in E^{+}$ is an $H_g$ point in E for some $v_0\in K$ with $v_0(\infty )=0\}$. Let $x,y\in C$ with $x{⪯}_{E^{+}}y$. Adopting the similar proof as that of the previous stage, we have the $P_K(y)\ne \varnothing$ and there exists $y_0\in P_K(y)$ such that $y_0{⪰}_{E^{+}}y{⪰}_{E^{+}}x$. In the process of the above proof that $P_K(x)\ne \varnothing$, we take $u_n=({\wedge }_{E+\stackrel{n}{k}=1}{v}_k){\wedge }_{E^{+}}{y}_0$. Similarly, if we take $x_0=({\wedge }_{E+\stackrel{\infty }{k}=1}{v}_k){\wedge }_{E^{+}}{y}_0$, then it can be proved that $x_0\in P_K(x)$ such that $x_0{⪯}_{E^{+}}{y}_0$. Hence $P_K(x)$ is monotone. □

Remark 1.

Lemma 5.1 in [28] and Propostions 3.1 and 3.3 in [25] show $P_K(x)\ne \varnothing$ for $x{⪯}_{E^{+}}K$. Note that we obtain the such existence result for a wider range than $x{⪯}_{E^{+}}K$ in Proposition 2. Indeed, rather than $x{⪯}_{E^{+}}y$ for all $y\in K$, it just needs to satisfy that there exists a $v\in K$ with $v{⪰}_{E^{+}}x$ such that $v-x{⪯}_{E^{+}}{|x-h|}_{E^{+}}$ in Proposition 2. Moreover, we prove the monotonicity of $P_K(x)$.

Remark 2.

If K is a closed sublattice in E satisfying $K{⪰}_{E^{+}}{u}_0$ and $u_0\in E$, by using $P_{u_0+K}(x)=u_0+P_K(x-u_0)$, the corresponding existence and monotonicity results can be obtained.

We now propose the following example in which all the conditions in Proposition 2 are satisfied and the existence and monotonicity results hold.

Example 1.

Let $(E,{⪯}_{E^{+}})=(L^p(\mathrm{\Omega }),{⪯}_{E^{+}})$, which is the space of all measurable functions being pth power summable on Ω, where $m(\mathrm{\Omega })\in (0,\infty ),p\in (1,\infty )$. The space $(L^p(\mathrm{\Omega }),{⪯}_{E^{+}})$ is endowed with the following positive cone and norm:

$${\parallel x\parallel }_{L^p(\mathrm{\Omega })}={\left({\int }_{\mathrm{\Omega }}{|x(s)|}^{p}d\mu \right)}^{{\displaystyle \frac{1}{p}}},$$

$$E^{+}=\{x\in L^p(\mathrm{\Omega }):x(s)\ge 0,\forall s\in \mathrm{\Omega },a.e.\}.$$

Taking $u_0,v_0\in L^p(\mathrm{\Omega })$ such that $\theta {⪯}_{E^{+}}{u}_0{\prec }_{E^{+}}{v}_0$, it is easy to prove that the order interval ${[u_0,v_0]}_{E^{+}}$ is a sublattice and all the conditions in Proposition 2 are satisfied. From Lemma 5, for any $x\in E$, $u_0+(v_0-u_0){\wedge }_{E^{+}}{(x-u_0)}^{+}\in P_{{[u_0,v_0]}_{E^{+}}}(x)$ and it is monotone.

In the two results, by the positive operator, we propose the explicit expression of the metric projection operator, which is also monotone.

Proposition 3.

Let $(E,\parallel ·\parallel ,{⪯}_{E^{+}})$ be a Banach function space and B any closed subset of $[0,\tau )$. Let $B_0=\{x\in E:x(s)\ge 0,s\in B \mathrm{and} x=0,s\in [0,\tau )\setminus B\}$. Then, for any $x\in E$, $P_{B_0}(x)\ne \varnothing$ and $x_{E^{+}}^{+}{\chi }_B\in P_{B_0}(x)$.

Proof. 

Take any $x\in E$. As $x=x_{E^{+}}^{+}-x_{E^{+}}^{-}$; we have

$$\begin{array}{cc}|x-x_{E^{+}}^{+}{\chi }_{B|E^{+}}\hfill & =|(x-x_{E^{+}}^{+}){\chi }_B+x{\chi }_{[0,\tau )\setminus B|E+}=\{\begin{array}{cc}{x}_{E^{+}}^{-}(s),\hfill & s\in B,\hfill \\ {|x|}_{E^{+}}(s)\hfill & s\in [0,\tau )\setminus B\hfill \end{array}\hfill \\ \hfill & =x_{E^{+}}^{-}{\chi }_B+{|x|}_{E^{+}}{\chi }_{[0,\tau )\setminus B}.\hfill \end{array}$$

(10)

Similarly, for any $y\in B_0$, we have $y=0$ on $[0,\tau )\setminus B$ and

$${|x-y|}_{E^{+}}=|(x-y){\chi }_B+x{\chi }_{[0,\tau )\setminus B}{|}_{E^{+}}={|x-y|}_{E^{+}}{\chi }_B+{|x|}_{E^{+}}{\chi }_{[0,\tau )\setminus B}.$$

(11)

Since $x_{E^{+}}^{-}{⪯}_{E^{+}}{|x-y|}_{E^{+}}$, from (10) and (11), we get

$$|x-x_{E^{+}}^{+}{\chi }_B{|}_{E^{+}}{⪯}_{E^{+}}{|x-y|}_{E^{+}}.$$

Therefore,

$$\parallel x-x_{E^{+}}^{+}{\chi }_B\parallel \le \parallel x-y\parallel ,\forall y\in B_0.$$

Thus, the assertion is proved. □

Corollary 1.

Let $(E,\parallel ·\parallel ,{⪯}_{E^{+}})$ be a Banach function space and ${\left\{A_i\right\}}_{i=1}^n$ a closed subset sequence of $[0,\tau )$ with $A_i\cap A_j=\varnothing$ for $i\ne j$. Let

$$K=\{x\in E:x=k_1{\chi }_{A_1}+k_2{\chi }_{A_2}+\cdots +k_n{\chi }_{A_n} \mathrm{with} k_1,k_2,\cdots ,k_n\ge 0\}.$$

Then, for any $x\in span\{{\chi }_{A_1},{\chi }_{A_2}\cdots ,{\chi }_{A_n}\}$, $x_{E^{+}}^{+}{\chi }_{{\displaystyle \bigcup _{i=1}^n}{A}_i}\in P_K(x)$.

Proof. 

Take any $l_i\in \mathbb{R}$ ($i=1,2,\cdots ,n$) and

$$x=l_1{\chi }_{A_1}+l_2{\chi }_{A_2}+\cdots +l_n{\chi }_{A_n}\in \mathrm{span} \{{\chi }_{A_1},{\chi }_{A_2}\cdots ,{\chi }_{A_n}\}.$$

Then

$$x_{E^{+}}^{+}{\chi }_{{\displaystyle \bigcup _{i=1}^n}{A}_i}=l_1^{+}{\chi }_{A_1}+l_2^{+}{\chi }_{A_2}+\cdots +l_n^{+}{\chi }_{A_n},$$

where $l_i^{+}=max\{0,l_i\}$ ($i=1,2,\cdots ,n$). By the proof of Proposition 3,

$$\parallel x-x_{E^{+}}^{+}{\chi }_{{\displaystyle \bigcup _{i=1}^n}{A}_i}\parallel \le \parallel x-z\parallel ,\forall z\in K.$$

The assertion is proved. □

4. Characterizing Best Approximations in Lorentz Spaces ${\mathrm{\Gamma }}_{\mathit{p},\mathit{w}}$

In the section, based on the above existence and monotonicity results in general Banach function spaces, using the better characterizations of Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$, we continue to study more specific existence and monotonicity propositions of the metric projection operators. Firstly, based on Proposition 2, the corresponding proposition in ${\mathrm{\Gamma }}_{p,w}$ is shown.

Proposition 4.

Let $\theta {⪯}_{E^{+}}K$ be a closed sublattice of ${\mathrm{\Gamma }}_{p,w}$ with $W(\infty )=\infty$. If for every $h\in K$, there exists $v\in K$ with $v{⪰}_{E^{+}}x$ such that $v-x{⪯}_{E^{+}}{|x-h|}_{E^{+}}$ and $v_0^{*}(\infty )=0$ for some $x{⪯}_{E^{+}}{v}_0\in K$. Then $P_K(x)\ne \varnothing$. Moreover, $P_K(x)$ is monotone.

Proof. 

From Lemma 3, we get that each point in ${\mathrm{\Gamma }}_{p,w}$ is an $H_g$ point. Since $W(\infty )=\infty$, ${\mathrm{\Gamma }}_{p,w}$ is order continuous. Then

$$\theta {⪯}_{E^{+}}{x}_n\uparrow x\in {\mathrm{\Gamma }}_{p,w} \mathrm{implies} \underset{n\to \infty }{lim}{(x_n-x)}^{**}(s)=0,\forall s\in [0,\tau )$$

and

$$\underset{n\to \infty }{lim}{\parallel x_n-x\parallel }_{{\mathrm{\Gamma }}_{p,w}}=0.$$

It follows that ${\mathrm{\Gamma }}_{p,w}$ has the semi-Fatou property. Following the same proof as that of Proposition 2, $P_K(x)\ne \varnothing$ and $P_K(x)$ is monotone. □

By the Gâteaux derivative of the norm and the boundary conditions, we establish the necessary and sufficient conditions for the best approximants onto the cone in the following proposition.

Proposition 5.

Let K be the closed convex positive cone of ${\mathrm{\Gamma }}_{p,w}$. Suppose that $x\in {\mathrm{\Gamma }}_{p,w}$ is such that $m(\mathbb{S}(x-y))=\tau$ for any $y\in \mathsf{\partial }K$ and $m\{u:|x-y{|}_K(u)=v\}=0$ for all $v>0$ and $y\in \mathsf{\partial }K$. Then, $\widehat{x}\in K$ is a best approximant of $x\notin K$ onto K if and only if

(i) 

for any $y\in K$,

$$\begin{array}{c}{\int }_0^{\tau }\left({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x})}sign(x(s)-\widehat{x}(s))y(s){\chi }_{(0,t)}({\mu }_{(x-\widehat{x})}(|x-\widehat{x}{|}_K(s)))ds\right)\hfill \\ {\left({\displaystyle \frac{{(x-\widehat{x})}^{**}(t)}{\parallel x-\widehat{x}\parallel }}\right)}^{p-1}w(t)dt\le 0;\hfill \end{array}$$

(ii) 

$$\begin{array}{c}{\int }_0^{\tau }\left({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x})}sign(x(s)-\widehat{x}(s))\widehat{x}(s){\chi }_{(0,t)}({\mu }_{(x-\widehat{x})}(|x-\widehat{x}{|}_K(s)))ds\right)\hfill \\ {\left({\displaystyle \frac{{(x-\widehat{x})}^{**}(t)}{\parallel x-\widehat{x}\parallel }}\right)}^{p-1}w(t)dt=0.\hfill \end{array}$$

Proof. 

Since K is a closed convex cone, it is easy to get $\widehat{x}\in \mathsf{\partial }K$. By the assumptions, we have

$$m(\mathbb{S}(x-\widehat{x}))=\tau \mathrm{and} m\{u:|x-\widehat{x}{|}_K(u)=v\}=0,\forall v>0.$$

From Theorem 5.3 in [41], we have that x$-\widehat{x}$ is a smooth point and the Gâteaux derivative of $\parallel ·\parallel$ in ${\mathrm{\Gamma }}_{p,w}$ at $x-\widehat{x}$ is

$$\begin{array}{cc}G(x-\widehat{x},z)=\hfill & {\int }_0^{\tau }\left({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x})}\mathrm{sign}(x(s)-\widehat{x}(s))z(s){\chi }_{(0,t)}(\gamma (s))ds\right)\hfill \\ \hfill & {\left({\displaystyle \frac{{(x-\widehat{x})}^{**}(t)}{\parallel x-\widehat{x}\parallel }}\right)}^{p-1}w(t)dt\hfill \end{array}$$

for any $z\in {\mathrm{\Gamma }}_{p,w}$, where $\gamma :\mathbb{S}(x-\widehat{x})\to \mathbb{S}({(x-\widehat{x})}^{*})$ is a measure preserving transformation satisfying $|x-\widehat{x}{|}_{E^{+}}={(x-\widehat{x})}^{*}\circ \gamma$. From Definition 1.2 in [41], by using the conditions that $m(\mathbb{S}(x-\widehat{x}))=\tau$ and $m\{u:|x-\widehat{x}{|}_{E^{+}}(u)=v\}=0$ for every $v>0$, we get

$$\gamma (s)={\mu }_{(x-\widehat{x})}(|x-\widehat{x}{|}_K(s)).$$

It follows from Lemma 4 that $\widehat{x}\in K$ is a best approximant of $x\notin K$ onto K if and only if $G(x-\widehat{x},\widehat{x}-y)\ge 0$ for all $y\in K$; that is,

$$\begin{array}{cc}{\int }_0^{\tau }\hfill & ({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x})}\mathrm{sign}(x(s)-\widehat{x}(s))(\widehat{x}(s)-y(s)){\chi }_{(0,t)}({\mu }_{(x-\widehat{x})}\hfill \\ \hfill & (|x-\widehat{x}{|}_K(s)))ds){\left({\displaystyle \frac{{(x-\widehat{x})}^{**}(t)}{\parallel x-\widehat{x}\parallel }}\right)}^{p-1}w(t)dt\ge 0,\forall y\in K.\hfill \end{array}$$

(12)

As K is a closed convex cone, we get $\theta ,2\widehat{x}\in K$. It follows that

$$\begin{array}{cc}{\int }_0^{\tau }\hfill & ({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x})}\mathrm{sign}(x(s)-\widehat{x}(s))\widehat{x}(s){\chi }_{(0,t)}({\mu }_{(x-\widehat{x})}(|x-\widehat{x}{|}_K(s)))ds)\hfill \\ \hfill & {\left({\displaystyle \frac{{(x-\widehat{x})}^{**}(t)}{\parallel x-\widehat{x}\parallel }}\right)}^{p-1}w(t)dt=0.\hfill \end{array}$$

The assertion is proved. □

In turn, we consider the necessary and sufficient conditions for the monotonicity of the metric projection onto random closed convex sublattices in the strictly convex spaces ${\mathrm{\Gamma }}_{p,w}$ in the following proposition.

Proposition 6.

Let $1\le p<\infty$ and $W(\infty )=\infty$ for the case when $\tau =\infty$. Let $E^{+}$ be the closed convex positive cone, and K a closed convex sublattice of ${\mathrm{\Gamma }}_{p,w}$. Let the set $C=\{x:x\in {\mathrm{\Gamma }}_{p,w}$, such that $m(\mathbb{S}(x-u{\wedge }_{E^{+}}v))=\tau$ for all $u,v\in \mathsf{\partial }K$ and $m\{s:|x-u{\wedge }_{E^{+}}v{|}_{E^{+}}(s)=t\}=0$ for any $t>0$ and $u,v\in \mathsf{\partial }K\}$. Then $P_K(x)$ is monotone in C if and only if for any $x,y\in C$ with $x{⪯}_{E^{+}}y$ and $\widehat{x}\in P_K(x),\widehat{y}\in P_K(y)$,

$$\begin{array}{cc}{\int }_0^{\tau }\hfill & ({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}\mathit{sign}((x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})(s))(\widehat{x}{\wedge }_{E^{+}}\widehat{y}-\widehat{x})(s){\chi }_{(0,t)}({\mu }_{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}\hfill \\ \hfill & (|x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}{|}_{E^{+}}(s)))ds){\left({\displaystyle \frac{{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}^{**}(t)}{\parallel x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}\parallel }}\right)}^{p-1}w(t)dt\ge 0.\hfill \end{array}$$

(13)

Proof. 

By assumptions and Lemma 2, we get that ${\mathrm{\Gamma }}_{p,w}$ is strictly convex, which yields that $P_K(x)$ is single-valued.

$[\Rightarrow ]$ If $P_K(x)$ is monotone, for $x{⪯}_{E^{+}}y$ and $P_K(x)=\widehat{x},P_K(y)=\widehat{y}$, then $\widehat{x}{⪯}_{E^{+}}\widehat{y}$ and $\widehat{x}{\wedge }_{E^{+}}\widehat{y}=\widehat{x}$, which implies that (13) holds.

$[\Leftarrow ]$ Inversely, if we suppose that $P_K$ is not monotone, then there exist $x,y\in C$ with $x{⪯}_{E^{+}}y$ such that $\widehat{x}{/⪯}_{E^{+}}\widehat{y}$; that is, $\widehat{x}\ne \widehat{x}{\wedge }_{E^{+}}\widehat{y}$. Therefore,

$$\parallel x-\widehat{x}{\parallel }_{{\mathrm{\Gamma }}_{p,w}}<{\parallel x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}\parallel }_{{\mathrm{\Gamma }}_{p,w}}.$$

(14)

Since K is convex and closed, we get $\widehat{x},\widehat{y}\in \mathsf{\partial }K$. Moreover, through Theorem 5.3 in [41], $x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}$ is a smooth point of ${\mathrm{\Gamma }}_{p,w}$ and

$$\begin{array}{c}{\int }_0^{\tau }({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}\mathrm{sign}((x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})(s))(\widehat{x}{\wedge }_{E^{+}}\widehat{y}-\widehat{x})(s){\chi }_{(0,t)}\hfill \\ ({\mu }_{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}(|x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}{|}_{E^{+}}(s)))ds){\left({\displaystyle \frac{{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}^{**}(t)}{\parallel x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}\parallel }}\right)}^{p-1}w(t)dt\hfill \end{array}$$

is the Gâteaux derivative of ${\parallel ·\parallel }_{{\mathrm{\Gamma }}_{p,w}}$ at $x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}$. It follows that

$$\begin{array}{c}{\int }_0^{\tau }({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}\mathrm{sign}((x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})(s))(\widehat{x}{\wedge }_{E^{+}}\widehat{y}-\widehat{x})(s){\chi }_{(0,t)}\hfill \\ ({\mu }_{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}(|x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}{|}_{E^{+}}(s)))ds){\left({\displaystyle \frac{{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}^{**}(t)}{\parallel x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}\parallel }}\right)}^{p-1}w(t)dt\hfill \\ \le \parallel x-\widehat{x}{\parallel }_{{\mathrm{\Gamma }}_{p,w}}-{\parallel x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}\parallel }_{{\mathrm{\Gamma }}_{p,w}}.\hfill \end{array}$$

By (14), we get

$$\begin{array}{c}{\int }_0^{\tau }({\displaystyle \frac{1}{t}}{\int }_{\mathbb{S}(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}\mathrm{sign}((x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})(s))(\widehat{x}{\wedge }_{E^{+}}\widehat{y}-\widehat{x})(s){\chi }_{(0,t)}\hfill \\ ({\mu }_{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}(|x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}{|}_{E^{+}}(s)))ds){\left({\displaystyle \frac{{(x-\widehat{x}{\wedge }_{E^{+}}\widehat{y})}^{**}(t)}{\parallel x-\widehat{x}{\wedge }_{E^{+}}\widehat{y}\parallel }}\right)}^{p-1}w(t)dt<0,\hfill \end{array}$$

which contradicts to (13). Thus, $P_K(x)$ is monotone. □

In the following proposition, by the partial ordering relation between the metric projection operator and identity operator, we establish the sufficient condition for the monotonicity of the metric projection operator onto closed convex sublattices.

Proposition 7.

Let $1<p<\infty$, and in the case when $\tau =\infty$, it holds $W(\infty )=\infty$. Let $E^{+}$ be the closed convex positive cone and K a closed convex sublattice of ${\mathrm{\Gamma }}_{p,w}$. Suppose that the set $C=\{x:x\in {\mathrm{\Gamma }}_{p,w}$ is such that $m(\mathbb{S}(x-u{\wedge }_{E^{+}}v))=\tau$ for any $u,v\in \mathsf{\partial }K$ and $m\{s:|x-u{\wedge }_{E^{+}}v{|}_{E^{+}}(s)=t\}=0$ for all $t>0$ and $u,v\in \mathsf{\partial }K\}$. If $x{⪯}_{E^{+}}{P}_K(x)$ for any $x\in C$, then $P_K(x)$ is monotone in C.

Proof. 

Take any $x,y\in C$ with $x{⪯}_{E^{+}}y$. Since $x{⪯}_{E^{+}}{P}_K(x),y{⪯}_{E^{+}}{P}_K(y)$, we have $x{⪯}_{E^{+}}{P}_K(x){\wedge }_{E^{+}}{P}_K(y)$. It follows that (13) holds. By Proposition 6, $P_K(x)$ is monotone. □

Example 2.

Let $K=\{x\in {\mathrm{\Gamma }}_{p,w}:x is increasing on [0,\tau ) such that x(0)=0\}$. It is easy to see that K is a cone in ${\mathrm{\Gamma }}_{p,w}$. Take $C=\{x\in {\mathrm{\Gamma }}_{p,w}:x{⪯}_{E^{+}}K\}$. It is easy to see that $x{⪯}_{E^{+}}{P}_K(x)$ for any $x\in C$. From Proposition 7, we have $P_K(x)\ne \varnothing$ and $P_K(x)$ is monotone in C.

Since the reflexivity of Banach spaces is crucial to the existence of the best approximant, by the condition $R{B}_p$, we introduce the sufficient condition for the reflexivity of ${\mathrm{\Gamma }}_{p,w}$.

Proposition 8.

Let $1<p<\infty$. If $\tau <\infty$ and w satisfies the condition $R{B}_p$ and ${\int }_0^1{s}^{-p}w(s)ds={\int }_1^{\infty }w(s)ds=V(\infty )=\infty$ for the case when $\tau =\infty$, then ${\mathrm{\Gamma }}_{p,w}$ is reflexive, where $V(x)={\left({\int }_x^{\infty }{s}^{-p}w(s)ds\right)}^{-1/(p-1)},x>0$.

Proof. 

If $\tau =\infty$, by $W(\infty )=\infty$ and Propostion 1.4 in [40], then ${\mathrm{\Gamma }}_{p,w}$ is order continuous. It follows from Theorem 1.8 in [40] that ${\mathrm{\Lambda }}_{p^{\prime },V^{\prime }}$ is an associated space of ${\mathrm{\Gamma }}_{p,w}$, where ${\displaystyle \frac{1}{p^{\prime }}}+{\displaystyle \frac{1}{p}}=1$. Since $V(\infty )=\infty$, ${\mathrm{\Lambda }}_{p^{\prime },V^{\prime }}$ is order continuous. As w satisfies the conditions $R{B}_p$ and ${\int }_0^1{s}^{-p}w(s)ds={\int }_1^{\infty }w(s)ds=\infty$, by Proposition 3.5 and Corollary 4.4 in [37], we get that ${\mathrm{\Gamma }}_{p,w}$ is reflexive. If $\tau <\infty$, by the Lebesgue dominated convergence theorem, ${\mathrm{\Gamma }}_{p,w}$, and the order continuity of its associated space, then ${\mathrm{\Gamma }}_{p,w}$ is reflexive. □

5. Best Proximity Point Theorems

In this section, based on the above characterizations of the monotonicity of the best approximants and partial ordering fixed point theory, we establish the existence and approximation results for the following proximity problems.

Assume that A, B are two nonempty subsets of ${\mathrm{\Gamma }}_{p,w}$ and $f:A\to B$. The best proximity problem is to find a point $x^{*}\in A$ satisfying

$$\parallel x^{*}-f(x^{*}){\parallel }_{{\mathrm{\Gamma }}_{p,w}}=d(A,B)=\underset{x\in A,y\in B}{inf}{\parallel x-y\parallel }_{{\mathrm{\Gamma }}_{p,w}}.$$

We denote

$$A_0{=\{x\in A:d(A,B)=\parallel x-y\parallel }_{{\mathrm{\Gamma }}_{p,w}}, \mathrm{for\; some} y\in B\}$$

and

$$B_0{=\{y\in B:d(A,B)=\parallel x-y\parallel }_{{\mathrm{\Gamma }}_{p,w}}, \mathrm{for\; some} x\in A\}.$$

In ref. [45], Kirk et al. proved that $A_0$ and $B_0$ are nonempty in reflexive Banach spaces. Also, in ref. [46], Basha and Veeramani showed that $A_0\subset \mathsf{\partial }A$. It is well known that if A and B are closed and convex, then $A_0$ and $B_0$ are also convex closed subsets of A and B, respectively. More results on the best proximity point problems can refer to [44,45,46]. However, partial ordering methods are seldom considered in the previous literature. We first show the partial ordering method based on the Proposition 4 to resolve the best proximity point problem in the following theorem.

Theorem 1.

Let A be a convex closed and bounded subset in ${\mathrm{\Gamma }}_{p,w}$ with $1<p<\infty$. Let $B{⪰}_{E^{+}}\theta$ be a convex closed sublattice in ${\mathrm{\Gamma }}_{p,w}$. Suppose that $\tau <\infty$ and w satisfies condition $R{B}_p$ and ${\int }_0^1{t}^{-p}w(t)dt={\int }_1^{\infty }w(t)dt=V(\infty )=\infty$ for the case when $\tau =\infty$. Assume that the following conditions are satisfied:

(i) 

$f:B\to A$ is a monotone and continuous mapping such that $f(B_0)\subseteq A_0$;

(ii) 

There exists a point $x_0\in B_0$ such that $x_0{⪯}_{E^{+}}{P}_B(f(x_0))$;

(iii) 

For all $x\in A,h\in B$, there exists $v\in B$ with $v{⪰}_{E^{+}}x$ such that $v-x{⪯}_{E^{+}}{|x-h|}_{E^{+}}$;

(iv) 

$v_0^{*}(\infty )=0$ for some $A{⪯}_{E^{+}}{v}_0\in B$.

Then there exists a point $x^{*}\in B$ such that $\parallel x^{*}-f(x^{*}){\parallel }_{{\mathrm{\Gamma }}_{p,w}}=d(A,B)$. Moreover, if $x_n={(P_B\circ f)}^n(x_0)$, then

$$\parallel x_n-x^{*}{\parallel }_{{\mathrm{\Gamma }}_{p,w}}\to 0, for n\to \infty .$$

Proof. 

From Proposition 8, we have that the space ${\mathrm{\Gamma }}_{p,w}$ is reflexive. By Lemma 3.2 in [45], $A_0$ and $B_0$ are two nonempty, convex and closed subsets. Therefore,

$$P_B(x)=P_{B_0}(x),\forall x\in A_0.$$

By assumptions, we get that ${\mathrm{\Gamma }}_{p,w}$ is strictly monotone, which implies that $P_B(x)$ is single valued. As $x_0\in B_0$, $f(x_0)\in A_0$, let $y_n=f({(P_B\circ f)}^n(x_0))$ and $x_n={(P_B\circ f)}^n(x_0)$. Then

$$\parallel x_{n+1}-y_n{\parallel }_{{\mathrm{\Gamma }}_{p,w}}=d(A,B)$$

(15)

and $y_n\in A_0$, $x_n\in B_0$ ($n=0,1,2,\cdots$). By Proposition 4, $P_B(x)$ is monotone. Since $x_0{⪯}_{E^{+}}{P}_B(f(x_0))$ and f is monotone, we have

$$f(x_0)=y_0{⪯}_{E^{+}}{y}_1{⪯}_{E^{+}}{y}_2{⪯}_{E^{+}}\cdots {⪯}_{E^{+}}{y}_n{⪯}_{E^{+}}\cdots$$

and

$$x_0{⪯}_{E^{+}}{x}_1{⪯}_{E^{+}}{x}_2{⪯}_{E^{+}}\cdots {⪯}_{E^{+}}{x}_n{⪯}_{E^{+}}\cdots .$$

As A is bounded, there exists an element $u_0\in {\mathrm{\Gamma }}_{p,w}$, such that

$$f(x_0)=y_0{⪯}_{E^{+}}{y}_1{⪯}_{E^{+}}{y}_2{⪯}_{E^{+}}\cdots {⪯}_{E^{+}}{y}_n{⪯}_{E^{+}}\cdots {⪯}_{E^{+}}{u}_0.$$

(16)

It follows that

$$x_0{⪯}_{E^{+}}{x}_1{⪯}_{E^{+}}{x}_2{⪯}_{E^{+}}\cdots {⪯}_{E^{+}}{x}_n{⪯}_{E^{+}}\cdots {⪯}_{E^{+}}{P}_B(u_0).$$

(17)

Since ${\mathrm{\Gamma }}_{p,w}$ is reflexive, by Lemma 1, we have that the positive cone of ${\mathrm{\Gamma }}_{p,w}$ is regular, which deduces that $\left\{x_n\right\}$ is convergent. Let $x_n\to x^{*}$ for $n\to \infty$. As f is continuous, we have

$$y_n=f(x_n)\to f(x^{*}),$$

for $n\to \infty$. It follows from (15) that $\parallel x^{*}-f(x^{*}){\parallel }_{{\mathrm{\Gamma }}_{p,w}}=d(A,B)$. The proof ends. □

We now give a corresponding example of the above Theorem 1.

Example 3.

Suppose that $p>2$ and $w(t)=t^{p-2}$. Let K be the positive cone of ${\mathrm{\Gamma }}_{p,w}$. Let $u_0,v_0\in {\mathrm{\Gamma }}_{p,w}$ with $u_0{\prec }_K{v}_0$ and $A={[u_0,v_0]}_K$. We have

$$x^{-p}{\int }_0^x{t}^{p-2}dt={\displaystyle \frac{x^{-p}}{p-1}}·x^{p-1}={\displaystyle \frac{1}{(p-1)x}}={\displaystyle \frac{1}{p-1}}{\int }_x^{\infty }{t}^{-p}w(t)dt,$$

which implies that the condition $R{B}_p$ of w holds. One can easily prove that $W(\infty )=\infty$. Since

$$V(x)={\left({\int }_x^{\infty }{t}^{-p}{t}^{p-2}dt\right)}^{1/(1-p)}={\left({\int }_x^{\infty }{t}^{-2}dt\right)}^{1/(1-p)}=x^{1/(p-1)},$$

we have $V(\infty )=\infty$. For each $x\in A,h\in K$, $x_K^{+}-x{⪯}_K{|x-h|}_K$. Define $f:K\to A$ as

$$f(x)=\{\begin{array}{cc}{v}_0,\hfill & x{⪰}_K{v}_{0K}^{+},\hfill \\ u_0+x{\wedge }_K(v_0-u_0),\hfill & x\in K\setminus \{x\in K:x{⪰}_K{v}_{0K}^{+}\}.\hfill \end{array}$$

Then f is continuous and monotone. It is easy to see that $v_{0K}^{+}\in K_0$ and $v_{0K}^{+}{⪯}_K{P}_K(f(v_{0K}^{+}))$. Therefore, all the conditions of Theorem 1 are satisfied. Hence f has the best proximity points and $\parallel v_{0K}^{+}-f(v_{0K}^{+}){\parallel }_{{\mathrm{\Gamma }}_{p,w}}=d(A,K).$

In the following theorem, by Proposition 6, we show the existence and approximation of the best proximity points.

Theorem 2.

Let $1<p<\infty$, $\tau <\infty$ and w satisfy the condition $R{B}_p$ and ${\int }_0^1{t}^{-p}w(t)dt={\int }_1^{\infty }w(t)dt=V(\infty )=\infty$ for the case when $\tau =\infty$. Let B be a convex closed sublattice of ${\mathrm{\Gamma }}_{p,w}$ and $C=\{x:x\in {\mathrm{\Gamma }}_{p,w}$ satisfies $m(\mathbb{S}(x-u{\wedge }_{E^{+}}v))=\tau$ for any $u,v\in \mathsf{\partial }B$ and $m\{s:|x-u{\wedge }_{E^{+}}v{|}_{E^{+}}(s)=t\}=0$ for all $t>0$ and $u,v\in \mathsf{\partial }B\}$. Let $A\subset C$ be convex closed and bounded. Suppose that the following conditions are satisfied:

(i) 

$f:B\to A$ is monotone and continuous and $f(B_0)\subseteq A_0$;

(ii) 

There exists $x_0\in B_0$ such that $x_0{⪯}_{E^{+}}{P}_B(f(x_0))$ and (13) holds.

Then there exists $x^{*}\in B$ such that $\parallel x^{*}-f(x^{*}){\parallel }_{{\mathrm{\Gamma }}_{p,w}}=d(A,B).$ Moreover, if $x_n={(P_B\circ f)}^n(x_0)$, then

$$\parallel x_n-x^{*}{\parallel }_{{\mathrm{\Gamma }}_{p,w}}\to 0, for n\to \infty .$$

Proof. 

By Proposition 6, $P_B(x)$ is monotone. By the same proof as that of Theorem 1, one can deduce the assertion. □

6. Conclusions

In this paper, based on the works on Banach space theory and the best constant approximant operator, we studied the existence and monotonicity characterizations of the metric projection operator in Lorentz spaces ${\mathrm{\Gamma }}_{p,w}$ and Banach function spaces, respectively. Based on these monotonicity results and partial ordering iterative theory, solvability and approximation theorems for the best proximity points in ${\mathrm{\Gamma }}_{p,w}$ were established by partial ordering methods, in which the mappings need not to satisfy contractive and compact conditions. In further research, we shall deduce more monotonicity, generalized isotonicity and subadditivity characterizations of the metric projection operator in partial ordering spaces endowed with different and general cones. Specially, we shall focus on the monotonicity and general monotonicity of similar projection operators, such as the Bregman projection operator, the proximity operator, the Bregman proximity operator, etc. As applications, we shall solve more optimization problems without continuous conditions and establish iterative algorithms without contractive and compact conditions by different order methods.