# Proximinality in Banach space valued Musielak-Orlicz spaces

- Jingshi Xu
^{1}Email author

**2014**:146

https://doi.org/10.1186/1029-242X-2014-146

© Xu; licensee Springer. 2014

**Received: **22 December 2013

**Accepted: **27 March 2014

**Published: **9 April 2014

## Abstract

Let $(A,\mathcal{A},\mu )$ be a *σ*-finite complete measure space and let *Y* be a subspace of a Banach space *X*. Let *φ* be a generalized Φ-function on $(A,\mathcal{A},\mu )$. Denote by ${L}^{\phi}(A,Y)$ and ${L}^{\phi}(A,X)$ the Musielak-Orlicz spaces whose functions take values in *Y* and *X*, respectively. Firstly, let $f\in {L}^{\phi}(A,X)$, we characterize the distance of *f* from ${L}^{\phi}(A,Y)$. Then, if *Y* is weakly
-analytic and proximinal in *X*, we show that ${L}^{\phi}(A,Y)$ is proximinal in ${L}^{\phi}(A,X)$. Finally, we give the connection between the proximinality of ${L}^{\phi}(A,Y)$ in ${L}^{\phi}(A,X)$ and the proximinality of ${L}^{1}(A,Y)$ in ${L}^{1}(A,X)$.

## Keywords

## 1 Introduction

It is well known that Musielak-Orlicz spaces include many spaces as special spaces, such as Lebesgue spaces, weighted Lebesgue spaces, variable Lebesgue spaces and Orlicz spaces; see [1]. Especially, in recent decades, variable exponent function spaces, such as Lebesgue, Sobolev, Besov, Triebel-Lizorkin, Hardy, Morrey, and Herz spaces with variable exponents, have attracted much attention; see [2–16] and references therein. Cheng and the author discussed geometric properties of Banach space valued Bochner-Lebesgue and Bochner-Sobolev spaces with a variable exponent in [17]. Very recently, Musielak-Orlicz-Hardy spaces have been systemically developed; see, for example, [18–22]. These spaces have many applications in various fields such as PDE, electrorheological fluids, and image restoration; see [6, 23–25].

In recent years, proximinality in Banach space valued Bochner-Lebesgue spaces with constant exponent have been extensively studied; see [26–33]. Proximinality in Banach space valued Bochner-Lebesgue spaces with variable exponent was discussed by the author in [34]. In fact, we generalized those results in [29, 31] to Banach space valued Bochner-Lebesgue spaces with a variable exponent. Khandaqji, Khalil and Hussein considered proximinality in Orlicz-Bochner function spaces on the unit interval in [35], and Al-Minawi and Ayesh consider the same problem on finite measures in [36]. The best simultaneous approximation in Banach space valued Orlicz spaces was discussed in [37, 38]. Micherda discussed proximinality of subspaces of vector-valued Musielak-Orlicz spaces via modular in [39]. However, as usual, one considers the best approximation via the norm, so in this paper, we will discuss proximinality of subspaces of vector-valued Musielak-Orlicz spaces via the norm. To proceed, we need to recall some definitions. Our results will be given in the next section.

In what follows, $(A,\mathcal{A},\mu )$ will be a *σ*-finite complete measure space. Suppose *D* is a subset of *A*, let ${\chi}_{D}$ be the indicator function on *D*. Let $(X,\parallel \cdot \parallel )$ be a Banach space. The dual space of *X* is the vector space ${X}^{\ast}$ of all continuous linear mappings from *X* to ℝ or ℂ. To avoid a double definition we let
be either ℝ or ℂ.

**Definition 1** A convex, left-continuous function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}]$ with $\phi (0)=0$, ${lim}_{t\to {0}^{+}}\phi (t)=0$, ${lim}_{t\to \mathrm{\infty}}\phi (t)=\mathrm{\infty}$ is called a Φ-function. It is called positive if $\phi (t)>0$ for all $t>0$.

It is easy to see that if *φ* is a Φ-function, then it is nondecreasing on $[0,\mathrm{\infty})$.

**Definition 2**Let $(A,\mathcal{A},\mu )$ be a

*σ*-finite complete measure space. A real function $\phi :A\times [0,\mathrm{\infty})\to [0,\mathrm{\infty}]$ is called a generalized Φ-function on $(A,\mathcal{A},\mu )$ if

- (a)
$\phi (y,\cdot )$ is a Φ-function for all $y\in A$,

- (b)
$y\mapsto \phi (y,t)$ is measurable for all $t\u2a7e0$.

If *φ* is a generalized Φ-function on $(A,\mathcal{A},\mu )$, we write $\phi \in \mathrm{\Phi}(A,\mu )$.

**Definition 3**Let $\phi \in \mathrm{\Phi}(A,\mu )$. Define

*μ*-measurable functions $f:A\to X$. Then the Bochner-Musielak-Orlicz space ${L}^{\phi}(A,X)$ is the collection of all strongly

*μ*-measurable functions $f:A\to X$ endowed with the norm:

**Definition 4**Let $\phi \in \mathrm{\Phi}(A,\mu )$. The function

*φ*is said to obey the ${\mathrm{\Delta}}_{2}$-condition if there exists $K\ge 2$ such that

for all $s\in A$ and all $t\ge 0$.

When *X* is ℝ or ℂ, we simply denote ${L}^{\phi}(A,X)$ by ${L}^{\phi}(A)$, and ${E}^{\phi}(A,X)$ by ${E}^{\phi}(A)$. Usually, ${E}^{\phi}(A,X)$ is a proper subspace of ${L}^{\phi}(A,X)$. But when the *φ* satisfies the ${\mathrm{\Delta}}_{2}$-condition, they are the same. It is easy to see that ${E}^{\phi}(A,X)={L}^{\phi}(A,X)$ is equivalent to ${E}^{\phi}(A)={L}^{\phi}(A)$, this means that the equality depends only on *φ*.

We remark that ${\rho}_{\phi}$ is a semimodular on the space of all *X*-valued strongly *μ*-measurable functions on *A*. For a semimodular, we recommend the reader reference [6]. Let *ρ* be a semimodular on vector space *E*, ${E}_{\rho}=\{x\in E:\rho (x/\lambda )<\mathrm{\infty}\text{for some}\lambda 0\}$, ${\parallel x\parallel}_{\rho}=inf\{\lambda >0:\rho (x/\lambda )\le 1\}$. We will use the following elementary result for a semimodular, which is Corollary 2.1.15 in [6].

**Lemma 1**

*Let*

*ρ*

*be a semimodular on*

*E*, $x\in {E}_{\rho}$.

- (i)
*If*${\parallel x\parallel}_{\rho}\le 1$,*then*$\rho (x)\le {\parallel x\parallel}_{\rho}$. - (ii)
*If*$1<{\parallel x\parallel}_{\rho}$,*then*${\parallel x\parallel}_{\rho}\le \rho (x)$.

*X*be a Banach space and let

*Y*be a closed subspace of

*X*. Then

*Y*is called proximinal in

*X*if for any $x\in X$ there exists $y\in Y$ such that

In this case *y* is called a best approximation of *x* in *Y*. If this best approximation is unique for any $x\in X$, then *Y* is said to be Chebyshev.

## 2 Main results

Firstly, we estimate ${dist}_{\phi}(f,{L}^{\phi}(A,Y))$.

**Theorem 1**

*Let*

*Y*

*be a subspace of Banach space*

*X*.

*Suppose*$\phi \in \mathrm{\Phi}(A,\mu )$.

*For*$f\in {L}^{\phi}(A,X)$,

*define*$\varphi :A\to \mathbb{R}$

*by*$\varphi (s):=dist(f(s),Y)$.

*Then*

- (i)
$\varphi \in {L}^{\phi}(A)$

*and*${dist}_{\phi}(f,{L}^{\phi}(A,Y))\ge {\parallel \varphi \parallel}_{\phi}$; - (ii)
${dist}_{\phi}(f,{L}^{\phi}(A,Y))={\parallel \varphi \parallel}_{\phi}$

*for*$f\in {E}^{\phi}(A,X)$.

*Proof*(i) Given $f\in {L}^{\phi}(A,X)$, we see that there exists a sequence of simple functions $\{{f}_{n}\}$ which converges to

*f*almost everywhere and in ${L}^{\phi}(A,X)$. Since the distance function $d(x,Y)$ is a continuous function of $x\in X$, $\parallel {f}_{n}(s)-f(s)\parallel \to 0$ implies that $|dist({f}_{n}(s),Y)-dist(f(s),Y)|\to 0$. Moreover, each function ${\varphi}_{n}:A\to \mathbb{R}$ defined by ${\varphi}_{n}(s):=dist({f}_{n}(s),Y)$ is a simple function; therefore we conclude that

*ϕ*is measurable. Now, for any $g\in {L}^{\phi}(A,Y)$ and any $\lambda >0$,

*f*is a simple function. Let $f(s):={\sum}_{i=1}^{m}{\chi}_{{A}_{i}}{x}_{i}$ where ${\{{A}_{i}\}}_{i=1}^{m}$ are disjoint measurable subsets in

*A*such that $0<\mu ({A}_{i})<\mathrm{\infty}$ and $0\ne {x}_{i}\in X$ for $i\in \{1,\dots ,m\}$. Without loss of generality, we suppose that ${dist}_{\phi}(f,{L}^{\phi}(A,Y))=1$. Let $0<\u03f5<1$. Since $\varphi (s)\le \parallel f(s)\parallel $, we have ${\rho}_{\phi}(\lambda \varphi )\le {\rho}_{\phi}(\lambda f)<\mathrm{\infty}$ for any $\lambda >0$. Then, by the dominated convergence theorem, we find that there exists $\delta >0$ such that

Thus, ${\rho}_{\phi}(\varphi )\ge 1-\u03f5$. Since *ϵ* is arbitrary, we have ${\rho}_{\phi}(\varphi )\ge 1$. By Lemma 1 again, we have ${\parallel \varphi \parallel}_{\phi}\ge 1$. This means that ${\parallel dist(f(\cdot ),Y)\parallel}_{\phi}\ge {dist}_{\phi}(f,{L}^{\phi}(A,Y))$. Therefore, we have proved that ${\parallel dist(f(\cdot ),Y)\parallel}_{\phi}={dist}_{\phi}(f,{L}^{\phi}(A,Y))$ for simple functions.

*f*

*μ*-almost everywhere, $\parallel {g}_{n}(s)\parallel \le \parallel f(s)\parallel $

*μ*-almost everywhere and ${\parallel f-{g}_{n}\parallel}_{\phi}\to 0$ as

*n*tends to ∞. Let ${\varphi}_{n}(s)=dist({g}_{n}(s),Y)$. From the previous proof, we have

*μ*-almost everywhere, and ${\varphi}_{n}(s)\to \varphi (s)$

*μ*-almost everywhere as

*n*tends to ∞, by Lemma 2.3.16(c) in [6], we conclude that ${\varphi}_{n}\to \varphi $ in ${L}^{\phi}(A)$. Hence, letting $n\to \mathrm{\infty}$, we see that

which completes the proof of Theorem 1. □

**Corollary 1** *Let* *Y* *be a closed subspace of a Banach space* *X*. *Suppose* $\phi \in \mathrm{\Phi}(A,\mu )$. *An element* *g* *of* ${L}^{\phi}(A,Y)$ *is a best approximation to an element* *f* *in* ${E}^{\phi}(A,X)$ *if and only if* $g(s)$ *is a best approximation in* *Y* *to* $f(s)$ *for almost every* $s\in A$. *Furthermore*, *if* *φ* *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition*, *then an element* *g* *of* ${L}^{\phi}(A,Y)$ *is a best approximation to an element* *f* *in* ${L}^{\phi}(A,X)$ *if and only if* $g(s)$ *is a best approximation in* *Y* *to* $f(s)$ *for almost every* $s\in A$.

**Corollary 2** *Let* *Y* *be a Chebyshev subspace of a Banach space* *X*. *Suppose* $\phi \in \mathrm{\Phi}(A,\mu )$ *satisfies the* ${\mathrm{\Delta}}_{2}$-*condition*. *If* ${L}^{\phi}(A,Y)$ *is proximinal in* ${L}^{\phi}(A,X)$, *then it is a Chebyshev subspace of* ${L}^{\phi}(A,X)$.

**Remark 1** Our results Theorem 1, Corollaries 1 and 2 cover the results for vector Orlicz spaces in [36]. Indeed, in [36] the authors only considered vector Orlicz spaces on finite measures. Analogous results for best simultaneous approximation were obtained in [37, 38] for vector Orlicz spaces on finite measures.

Next, we transfer the proximinality of *Y* in *X* to ${L}^{\phi}(A,Y)$ in ${L}^{\phi}(A,X)$. To do so, we need some preliminaries.

**Lemma 2** *Let* $(A,\mathcal{A},\mu )$ *be a* *σ*-*finite complete measure space*. *Suppose* $\phi \in \mathrm{\Phi}(A,\mu )$. *Let* *Y* *be a proximinal subspace of a Banach space* *X*. *Suppose* $f\in {L}^{\phi}(A,X)$ *and* *g* *is a strongly* *μ*-*measurable function such that* $g(s)$ *is a best approximation to* $f(s)$ *from* *Y* *for almost everywhere* $s\in A$. *Then* *g* *is a best approximation to* *f* *from* ${L}^{\phi}(A,Y)$.

*Proof* Since $0\in Y$, it follows that $\parallel g(s)\parallel \le 2\parallel f(s)\parallel $ *μ*-almost everywhere. Thus, $g\in {L}^{\phi}(A,Y)$. For each $h\in {L}^{\phi}(A,Y)$, by assumption we know that $\parallel f(s)-g(s)\parallel \le \parallel f(s)-h(s)\parallel $ *μ*-almost everywhere. So ${\rho}_{\phi}(\lambda (f-g))\le {\rho}_{\phi}(\lambda (f-h))$ for any $\lambda >0$. Thus, ${\parallel f-g\parallel}_{\phi}\le {\parallel f-h\parallel}_{\phi}$. This ends the proof. □

**Definition 5** Let $(T,\tau )$ be a Polish space (*i.e.* a topological space which is separable and completely metrizable). A set $Q\subset T$ is analytic if it is empty or if there exists a continuous mapping $f:{\mathbb{N}}^{\mathbb{N}}\to T$ satisfying $f({\mathbb{N}}^{\mathbb{N}})=Q$, where ${\mathbb{N}}^{\mathbb{N}}$ denotes the space of all infinite sequences of natural numbers endowed with the Tychonoff topology.

**Definition 6** Let $(T,\tau )$ be a Polish space, *H* a topological space and denote by $\sigma (\mathcal{A})$ the smallest *σ*-algebra containing all analytic subsets of *T*. Then a mapping $f:T\to H$ is said to be analytic measurable if ${f}^{-1}(C)\in \sigma (\mathcal{A})$ for every $C\in \mathcal{B}(H)$, where $\mathcal{B}(H)$ is for the Borel sets of *H*.

**Definition 7** Let *H*, *T* be topological spaces. Then a multifunction $F:H\to {2}^{T}$ is said to be upper semi-continuous if for every $x\in H$ and for every open set *U* satisfying $F(x)\subset U$, there exists an open neighborhood *V* of *x* such that $F(V)\subset U$.

**Definition 8** A subset *C* of a topological space *T* is
-analytic if it can be written as $C={\bigcup}_{\sigma \in {\mathbb{N}}^{\mathbb{N}}}F(\sigma )$ for some upper semi-continuous mapping $F:{\mathbb{N}}^{\mathbb{N}}\to {2}^{T}$ with compact values. In the case when *T* is a Banach space endowed with its weak topology, *C* is said to be weakly
-analytic.

For the theory of -analytic sets, we recommend [40]. Specially, all reflexive and all separable Banach spaces are weakly -analytic. The following lemma is just Theorem 3.3 in [31].

**Lemma 3** *Let* $(X,\parallel \cdot \parallel )$ *be a real Banach space and let* *Y* *be a proximinal*, *weakly*
-*analytic convex subset of* *X*. *Then*, *for each closed and separable set* $M\subset X$, *there exists an analytic measurable mapping* $h:M\to Y$ *such that* $h(M)$ *is separable in* *Y* *and* $h(x)$ *is a best approximation of* *x* *in* *Y* *for any* $x\in M$.

Thus, following the argument of the proof of (i) → (ii) in [[39], p.185], we have the following conclusion, the details being omitted.

**Theorem 2** *Let* $(A,\mathcal{A},\mu )$ *be a* *σ*-*finite complete measure space*. *Suppose* $\phi \in \mathrm{\Phi}(A,\mu )$, *and* *Y* *is a weakly*
-*analytic linear subspace of a real Banach space* *X*. *If* *Y* *is proximinal in* *X*, *then* ${L}^{\phi}(A,Y)$ *is proximinal in* ${L}^{\phi}(A,X)$.

**Theorem 3** *Let* $(A,\mathcal{A},\mu )$ *be a* *σ*-*finite measure space*. *Suppose* $\phi \in \mathrm{\Phi}(A,\mu )$ *such that* ${E}^{\phi}(A)={L}^{\phi}(A)$. *Let* *Y* *be a linear subspace of a real Banach space* *X*. *If* ${L}^{\phi}(A,Y)$ *is proximinal in* ${L}^{\phi}(A,X)$, *then* *Y* *is proximinal in* *X*.

*Proof* Since the measure *μ* is *σ*-finite, let us choose positive measure set *Q* such that ${\chi}_{Q}\in {L}^{\phi}(A)$. For any $x\in X$, let $f(t):={\chi}_{Q}(t)\cdot x$, $t\in A$. Then $f\in {L}^{\phi}(A,X)$. By the assumption, we know that there is a *g* in ${L}^{\phi}(A,Y)$ which is a best approximation element of *f*. Consequently, $g(s)$ is a best approximation to $f(s)$ in *Y* for almost every $s\in A$ by Corollary 1. Therefore, there is a best approximation element of *x* in *Y*. Thus, *Y* is proximinal in *X*. □

From Theorems 2 and 3, we deduce the following corollary.

**Corollary 3**

*Let*$(A,\mathcal{A},\mu )$

*be a*

*σ*-

*finite complete measure space*.

*Suppose*$\phi \in \mathrm{\Phi}(A,\mu )$

*such that*${E}^{\phi}(A)={L}^{\phi}(A)$.

*Let*

*Y*

*be a weakly*-

*analytic linear subspace of a real Banach space X*.

*Then the following conditions are equivalent*:

- (i)
*Y**is proximinal in**X*; - (ii)
${L}^{\phi}(A,Y)$

*is proximinal in*${L}^{\phi}(A,X)$.

**Remark 2** An analog to Corollary 3 in terms of a modular was obtained in [39].

Finally, we give a characterization of proximinity of ${L}^{\phi}(A,Y)$ in ${L}^{\phi}(A,X)$ via the proximinity of ${L}^{1}(A,Y)$ in ${L}^{1}(A,X)$. When ${L}^{\phi}(A,X)$ is a Bochner-Lebesgue space, which was obtained in [29] and [32, 33] on finite measure spaces and *σ* finite measure spaces, respectively. ${L}^{\phi}(A,X)$ is a Bochner-Orlicz space, which was discussed in [35].

**Theorem 4** *Let* $(A,\mathcal{A},\mu )$ *be a* *σ*-*finite complete measure space*. *Suppose* $\phi \in \mathrm{\Phi}(A,\mu )$ *such that the set of simple functions*, $S(A,\mu )$, *satisfies* $S(A,\mu )\subset {L}^{{\phi}^{\ast}}(A,\mu )$, *where* ${\phi}^{\ast}$ *is the conjugate function of* *φ* (*see* [6]). *Let* *Y* *be a closed subspace of a Banach space* *X*. *If* ${L}^{1}(A,Y)$ *is proximinal in* ${L}^{1}(A,X)$, *then* ${L}^{\phi}(A,Y)$ *is proximinal in* ${L}^{\phi}(A,X)$.

*Proof*Since

*A*is

*σ*-finite, we may write $A={\bigcup}_{i=1}^{\mathrm{\infty}}{A}_{i}$, where $\{{A}_{i}\}$ is a sequence of disjoint measurable sets each of finite measure. Let $f\in {L}^{\phi}(A,X)$. For any $n\in \mathbb{N}$, since $\mu ({A}_{n})<\mathrm{\infty}$, then ${\chi}_{{A}_{n}}\in {L}^{{\phi}^{\ast}}(A)$. Thus, by the norm conjugate formula (see Corollary 2.7.5 in [6]), we find that $f{\chi}_{{A}_{n}}\in {L}^{1}(A,X)$. By assumption, we know that there exists ${g}_{n}\in {L}^{1}(A,Y)$ such that

*μ*-almost everywhere. Therefore ${g}_{n}(t)=0$

*μ*-almost every $t\in {A}_{n}^{c}$. Let $g={\sum}_{n=1}^{\mathrm{\infty}}{g}_{n}$. Since $f={\sum}_{n=1}^{\mathrm{\infty}}f{\chi}_{{A}_{n}}$, it follows that for all $h\in {L}^{\phi}(A,Y)$,

*μ*-almost everywhere. Because $0\in Y$, it follows that $\parallel g(t)\parallel \le 2\parallel f(t)\parallel $. Thus, $g\in {L}^{\phi}(A,Y)$ and

for all $h\in {L}^{\phi}(A,Y)$. This finishes the proof. □

**Theorem 5** *Let* $(A,\mathcal{A},\mu )$ *be a* *σ*-*finite complete measure space*. *Suppose* $\phi \in \mathrm{\Phi}(A,\mu )$ *satisfies* ${E}^{\phi}(A)={L}^{\phi}(A)$ *and*, *for each* $t\in A$, $\phi (t,\cdot )$ *is strictly increasing*. *Let* *Y* *be a closed subspace of a Banach space* *X*. *If* ${L}^{\phi}(A,Y)$ *is proximinal in* ${L}^{\phi}(A,X)$, *then* ${L}^{1}(A,Y)$ *is proximinal in* ${L}^{1}(A,X)$.

*Proof*We use the idea from [35]. Indeed in [35] the authors only considered Banach space valued Orlicz spaces on the unit interval. Since, for each $t\in A$, $\phi (t,\cdot )$ is strictly increasing, let ${\phi}^{-1}(t,\cdot )$ be its inverse function, which means, for each $s\in [0,\mathrm{\infty})$, $\phi (t,{\phi}^{-1}(t,s))=s$. Define the map $J:{L}^{1}(A,X)\to {L}^{\phi}(A,X)$ by setting

*J*is injective. Moreover, if $g\in {L}^{\phi}(A,X)$, let

If $g(t)=0$, then $f(t)=0$ also, thus $J(f)(t)=0=g(t)$. Hence *J* is surjective and $J({L}^{1}(A,Y))={L}^{\phi}(A,Y)$ also.

*μ*-almost everywhere, for otherwise we can restrict our measure to the support of

*f*. Since $J(f)\in {L}^{\phi}(A,X)$, by the assumption, we know that there exists some $g\in {L}^{1}(A,Y)$ such that

*μ*-almost everywhere. Multiplying both sides of the last inequality by $\frac{\parallel f(t)\parallel}{{\phi}^{-1}(t,\parallel f(t)\parallel )}$, we obtain, for all $y\in Y$,

*Y*, and $0\in Y$, it follows that $\parallel h(t)\parallel \le 2\parallel f(t)\parallel $. Therefore, $h\in {L}^{1}(A,Y)$. Thus, for all $w\in {L}^{1}(A,Y)$,

*μ*-almost everywhere. Thus, by Corollary 1 *h* is a best approximation of *f* in ${L}^{1}(A,Y)$. This finishes the proof. □

From Theorems 4 and 5, we deduce the following corollary.

**Corollary 4**

*Let*$(A,\mathcal{A},\mu )$

*be a*

*σ*-

*finite complete measure space*.

*Suppose*$\phi \in \mathrm{\Phi}(A,\mu )$

*such that*${E}^{\phi}(A)={L}^{\phi}(A)$,

*for each*$t\in A$, $\phi (t,\cdot )$

*is strictly increasing and the set of simple functions*$S(A,\mu )$

*satisfies*$S(A,\mu )\subset {L}^{{\phi}^{\ast}}(A,\mu )$.

*Let*

*Y*

*be a closed subspace of a Banach space*

*X*.

*Then the following conditions are equivalent*:

- (i)
${L}^{1}(A,Y)$

*is proximinal in*${L}^{1}(A,X)$; - (ii)
${L}^{\phi}(A,Y)$

*is proximinal in*${L}^{\phi}(A,X)$.

**Remark 3** When $(A,\mu )$ is a finite measure and *φ* is a Orlicz function that satisfies the ${\mathrm{\Delta}}_{2}$-condition, the result of Corollary 4 was obtained in [36]. While $(A,\mu )$ is the unit interval and *φ* is a Young function that satisfies the ${\mathrm{\Delta}}_{2}$-condition, the result of Corollary 4 was obtained in [35].

## Declarations

### Acknowledgements

The author would like to thank the referee for carefully reading which made the presentation more readable and for his or her suggestion for references [36–38]. The author was supported by the National Natural Science Foundation of China (Grant No. 11361020) and the National Natural Science Foundation of Hainan Providence (113004).

## Authors’ Affiliations

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