Steady states of a nonlinear elliptic system arising from population dynamics

Abstract

In this paper, we study the existence and multiplicity of coexistence states for an elliptic system modeling two subpopulations of the same species competing for resources. Our results generalize and complement the work of Bouguima et al. (Nonlinear Anal., Real World Appl. 9:1184-1201, 2008). In particular, an open problem proposed by Bouguima et al. is partially solved.

MSC: 34B10, 34B18.

1 Introduction

This paper studies the coexistence states of the nonlinear elliptic system

$$\{\begin{array}{cc}-\mathrm{\Delta }u=a(x)v-e(x)u-c(x)u(u+v),\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v=b(x)u-f(x)v-d(x)v(u+v),\hfill & x\in \mathrm{\Omega },\hfill \\ u=v=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$$

(1.1)

where Ω is a bounded regular domain of ${\mathbb{R}}^n$ ($n\ge 1$), i.e., $\overline{\mathrm{\Omega }}$ is an n-dimensional compact connected smooth submanifold of ${\mathbb{R}}^n$ with boundary ∂ Ω. Throughout the paper, we suppose that a, b, c, d, e, and f are positive functions in $C^{\alpha }(\overline{\mathrm{\Omega }})$ for a certain $\alpha \in (0,1)$.

System (1.1) arises from population dynamics where it models the steady-state solutions of the corresponding nonlinear evolution problem [1], where u and v represent, respectively, the concentrations of the adult and juvenile populations. The function a gives the rate at which juveniles become adults and as adults give birth to juveniles, the function b corresponds to the birth rate of the population, e and f reflect the result of harvesting a portion of the population (fishing effort for marine population), c and d measure the competition between u and v. Both populations are living in the same region Ω, and the boundary conditions in (1.1) may be interpreted as the condition that the populations u and v may not stay on ∂ Ω. The Laplacian operator shows the diffusive character of u and v within Ω.

Obviously, system (1.1) can be rewritten as

$$\{\begin{array}{cc}\mathfrak{L}U=A(x)U-F(U),\hfill & x\in \mathrm{\Omega },\hfill \\ U=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$$

where $U={(u,v)}^T$, $\mathfrak{L}=diag(-\mathrm{\Delta },-\mathrm{\Delta })$,

$$A(x)=\left(\begin{array}{cc}-e(x)& a(x)\\ b(x)& -f(x)\end{array}\right)$$

and F the nonlinear term such that $F(U)(x)=\left(\begin{array}{c}c(x)u(u+v)\\ d(x)v(u+v)\end{array}\right)$. In the following, for $q\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, we denote by $\underline{q}$ and $\overline{q}$ the essential infimum and supremum of q, respectively. Let ${\rho }_1(q)$ be the principal eigenvalue of the linear problem

$$\{\begin{array}{cc}-\mathrm{\Delta }u+q(x)u=\rho u,\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$$

and ${\phi }_1(q)$ the principal eigenfunction associated to ${\rho }_1(q)$. Then it is not difficult to check that ${\rho }_1(q)=q+{\lambda }_1$ and ${\phi }_1(q)={\phi }_1$ if $q(\cdot )\equiv q$ is a constant. Here $({\lambda }_1,{\phi }_1)$ is the principal eigen-pair of −Δ under a homogeneous Dirichlet boundary condition such that ${\phi }_1(x)>0$, $x\in \mathrm{\Omega }$, and

$${\parallel {\phi }_1\parallel }_{\mathrm{\infty }}=\underset{x\in \overline{\mathrm{\Omega }}}{max}|{\phi }_1(x)|=1.$$

In recent years, system (1.1) has been studied by several authors, see for example, [1]–[5] and the references listed therein. In [1] and [2], system (1.1) was discussed as a problem in optimal control. Existence and uniqueness results are given in terms of hypotheses which are appropriate for control problems, that is, the coefficients are required to satisfy certain uniform bounds. Brown and Zhang [3] studied system (1.1) subject to Neumann boundary condition, and obtained more precise existence results in terms of the principal eigenvalue of

$$\{\begin{array}{cc}\mathfrak{L}U-A(x)U=\mu U,\hfill & x\in \mathrm{\Omega },\hfill \\ \frac{\partial U}{\partial \mathbf{n}}=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$$

where n is the outward unit normal vector on ∂ Ω; see [3], Theorem 3.2]. Recently, Bouguima et al.[4] investigated system (1.1) and showed that it has a positive solution if and only if ${\lambda }_1(\mathfrak{L}-A(x))<0$ via fixed point theory, where ${\lambda }_1(\mathfrak{L}-A(x))$ is the principal eigenvalue of the linear eigenvalue problem

$$\{\begin{array}{cc}\mathfrak{L}U-A(x)U=\lambda U,\hfill & x\in \mathrm{\Omega },\hfill \\ U=0,\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}$$

(1.2)

We note that ${\lambda }_1(\mathfrak{L}-A(x))$ exists if $a(x)>0$, $b(x)>0$ for $x\in \mathrm{\Omega }$; see Lemma 12 and Corollary 18 in [4] for the details. In addition, implicit function theorem was adopted in [4] to discuss the limit case of system (1.1), where the authors supposed that $a(x)={\rho }_1(e)+\epsilon a_1(x)$, $b(x)={\rho }_1(e)+\epsilon b_1(x)$ with $\epsilon >0$ sufficiently small and $a_1,b_1\in L_{+}^{\mathrm{\infty }}(\mathrm{\Omega })$; the strength of the crowding effect and the competition between u and v are negligible, i.e., $c(x)=\epsilon c_1(x)$, $d(x)=\epsilon d_1(x)$ for $c_1,d_1\in L_{+}^{\mathrm{\infty }}(\mathrm{\Omega })$, and small $\epsilon >0$; the functions e and f are assumed to have the same effect, that is, $f(x)=e(x)+\epsilon f_1(x)$ with $f_1\in L_{+}^{\mathrm{\infty }}(\mathrm{\Omega })$ and $\epsilon >0$ is small enough. Under the above assumptions, Bouguima et al. obtained the following multiplicity results.

Theorem A

[4], Theorem 24]

Assume that

$$\alpha :=\frac{1}{4}{\int }_{\mathrm{\Omega }}(a_1+b_1-f_1){\phi }_1^2(e)dx>0,\beta :=\frac{1}{2}{\int }_{\mathrm{\Omega }}(c_1+d_1){\phi }_1^3(e)dx>0.$$

Then there exists${\epsilon }_0>0$, such that system (1.1) has two nontrivial solutions of the form

$${\xi }_i(\epsilon )=S^i(\epsilon )({\phi }_1(e),{\phi }_1(e))+U(S^i(\epsilon ),\epsilon ),i=1,2,$$

(1.3)

where$S^1:(-{\epsilon }_0,{\epsilon }_0)\to V_0$and$S^2:(-{\epsilon }_0,{\epsilon }_0)\to V_{\ast }$are some functions w.r.t. ε, and$V_0$, $V_{\ast }$are neighborhoods of 0 and$\frac{\alpha }{\beta }$, respectively. The function$U(S^i(\epsilon ),\epsilon )$ ($i=1,2$) appearing in (1.3) is given as in[4], Lemma  23].

By an inspection of the arguments in [4], Section 5], it is easy to see that the second nontrivial solution ${\xi }_2$ obtained in Theorem A is positive and satisfies

$$(u(\epsilon ),v(\epsilon ))\to \frac{\alpha }{\beta }({\phi }_1(e),{\phi }_1(e))\mathit{\text{as }}\epsilon \to 0,$$

see also [4], Theorem 2]. However, Bouguima et al. pointed out that the positiveness of the small solution ${\xi }_1$ is not obvious and the question remains open. In this paper, we shall give a partial answer to this open problem.

Our main result can be stated as below.

Theorem 1.1

Let$a_1$, $b_1$, $f_1$, and e be positive constants such that$a_1+b_1-f_1>0$. Then the small solution${\xi }_1$is positive.

Remark 1.1

We note that the constants α and β defined in [4] were wrong, and this mistake is corrected in the statement of Theorem A. Moreover, we would like to point out that under the assumptions (which are similar to the corresponding ones of Theorem A) of Theorem 1.1, the discussions in [4], Section 5] can also be carried out, and thus the positiveness of ${\xi }_1$ remains unclear. However, in our case the positiveness of ${\xi }_1$ could be obtained. Consequently, the open problem proposed by Bouguima et al.[4] is partially solved, and it is the first time to obtain the multiplicity of positive solutions for system (1.1). For other related results on this topic, we refer the readers to [6]–[8] and the references therein.

2 Proof of Theorem 1.1

To obtain the positiveness of ${\xi }_1$, we introduce the following nonlinear elliptic system:

$$\{\begin{array}{cc}-\mathrm{\Delta }u=\tilde{a}u+\tilde{b}v+f(x,u,v),\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v=\tilde{c}u+\tilde{d}v+g(x,u,v),\hfill & x\in \mathrm{\Omega },\hfill \\ u=v=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$$

(2.1)

where $\tilde{a}$, $\tilde{b}$, $\tilde{c}$, $\tilde{d}$ are real parameters. For a matrix $\tilde{A}=\left(\begin{array}{cc}\tilde{a}& \tilde{b}\\ \tilde{c}& \tilde{d}\end{array}\right)$, we denote by $\sigma (\tilde{A})$ the spectrum of $\tilde{A}$. Let $\sigma (-\mathrm{\Delta })$ be the spectrum of −Δ under a homogeneous Dirichlet boundary condition. We shall also use the notation ${\tilde{A}}_0=\left(\begin{array}{cc}{\tilde{a}}_0& {\tilde{b}}_0\\ {\tilde{c}}_0& {\tilde{d}}_0\end{array}\right)$. Let $\mathcal{S}$ be the closure of the set of nontrivial solutions of system (2.1). We define the Banach space

$$E=\{u\in C^{1+\alpha }(\overline{\mathrm{\Omega }}):u=0\text{ on }\partial \mathrm{\Omega }\}$$

equipped with the usual norm. Denoting by $\partial /\partial \mathbf{n}$ the outward normal derivative, we introduce

$$K^{+}:=\{u\in E:u>0\text{ in }\mathrm{\Omega },\frac{\partial u}{\partial \mathbf{n}}<0\text{ on }\partial \mathrm{\Omega }\},K^{-}=-K^{+}.$$

The subsets $K^{+}$, $K^{-}$, and $K=K^{+}\cup K^{-}$ are open in E.

Lemma 2.1

[9], Lemma 3.1]

Assume that f and g are continuous functions in$\mathrm{\Omega }\times {\mathbb{R}}^2$satisfying the following conditions:

(H1)$f(x,r)$and$g(x,r)$are continuous and locally Lipschitz with respect to$r=(r_1,r_2)\in {\mathbb{R}}^2$.

(H2)$f(\cdot ,r)=o(|r|)$, $g(\cdot ,r)=o(|r|)$at 0, uniformly in$x\in \overline{\mathrm{\Omega }}$.

Suppose$\sigma ({\tilde{A}}_0)\cap \sigma (-\mathrm{\Delta })=\{{\lambda }_1\}$and either${\tilde{b}}_0({\lambda }_1-{\tilde{a}}_0)>0$or${\tilde{c}}_0({\lambda }_1-{\tilde{d}}_0)>0$. Then there exists a neighborhood${\mathcal{U}}_0$of$({\tilde{A}}_0,(0,0))$, such that if$(A,(u,v))\in {\mathcal{U}}_0\cap \mathcal{S}$for some matrix A and$(u,v)\ne (0,0)$, then either$u,v\in K^{+}$or$u,v\in K^{-}$.

Proof of Theorem 1.1

We shall consider the system

$$\{\begin{array}{cc}-\mathrm{\Delta }u=-eu+({\lambda }_1+e)v-\epsilon c_1(x)u(u+v),\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v=({\lambda }_1+e)u-ev-\epsilon d_1(x)v(u+v),\hfill & x\in \mathrm{\Omega },\hfill \\ u=v=0,\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}$$

(2.2)

Let

$$A_0=\left(\begin{array}{cc}-e& {\lambda }_1+e\\ {\lambda }_1+e& -e\end{array}\right).$$

Then it is not hard to check that $A_0$ has two eigenvalues ${\mu }_1={\lambda }_1$ and ${\mu }_2=-{\lambda }_1-2e$, and therefore $\sigma (A_0)\cap \sigma (-\mathrm{\Delta })=\{{\lambda }_1\}$. Moreover,

$$\begin{array}{c}{\tilde{b}}_0({\lambda }_1-{\tilde{a}}_0)=({\lambda }_1+e)({\lambda }_1-(-e))={({\lambda }_1+e)}^2>0,\hfill \\ {\tilde{c}}_0({\lambda }_1-{\tilde{d}}_0)=({\lambda }_1+e)({\lambda }_1-(-e))={({\lambda }_1+e)}^2>0.\hfill \end{array}$$

Clearly, the functions $f(x,u,v):=-\epsilon c_1(x)u(u+v)$ and $g(x,u,v):=-\epsilon d_1(x)v(u+v)$ satisfy (H1) and (H2). Consequently, all of the assumptions of Lemma 2.1 are satisfied.

On the other hand, by Theorem A, we know that the system

$$\{\begin{array}{cc}-\mathrm{\Delta }u=-eu+({\lambda }_1+e+\epsilon a_1)v-\epsilon c_1(x)u(u+v),\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }v=({\lambda }_1+e+\epsilon b_1)u-(e+\epsilon f_1)v-\epsilon d_1(x)v(u+v),\hfill & x\in \mathrm{\Omega },\hfill \\ u=v=0,\hfill & x\in \partial \mathrm{\Omega }\hfill \end{array}$$

(2.3)

admits the nontrivial small solution ${\xi }_1$, i.e., the nontrivial solution $(A_{\epsilon },(u_{\epsilon },v_{\epsilon }))$ with $u_{\epsilon }$ and $v_{\epsilon }$ small, where

$$A_{\epsilon }=A_0+\epsilon \left(\begin{array}{cc}0& a_1\\ b_1& -f_1\end{array}\right).$$

In addition, since $\epsilon >0$ is sufficiently small, it follows from Lemma 2.1 that the nontrivial small solution $(A_{\epsilon },(u_{\epsilon },v_{\epsilon }))$ lies in the neighborhood ${\mathcal{U}}_0$ of $(A_0,(0,0))$, and either $u_{\epsilon },v_{\epsilon }\in K^{+}$ or $u_{\epsilon },v_{\epsilon }\in K^{-}$.

We claim that $u_{\epsilon },v_{\epsilon }\in K^{+}$.

Suppose on the contrary that $u_{\epsilon },v_{\epsilon }\in K^{-}$. Then we have

$$u_{\epsilon }(x)<0,v_{\epsilon }(x)<0,x\in \mathrm{\Omega },$$

(2.4)

and they satisfy

$$\{\begin{array}{cc}-\mathrm{\Delta }{u}_{\epsilon }=-e{u}_{\epsilon }+({\lambda }_1+e+\epsilon a_1)v_{\epsilon }-\epsilon c_1(x)u_{\epsilon }(u_{\epsilon }+v_{\epsilon }),\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }{v}_{\epsilon }=({\lambda }_1+e+\epsilon b_1)u_{\epsilon }-(e+\epsilon f_1)v_{\epsilon }-\epsilon d_1(x)v_{\epsilon }(u_{\epsilon }+v_{\epsilon }),\hfill & x\in \mathrm{\Omega },\hfill \\ u_{\epsilon }=v_{\epsilon }=0,\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}$$

(2.5)

Let $A_{\epsilon }^T$ be the transpose of $A_{\epsilon }$. Then it is easy to verify that $\mathfrak{L}-A_{\epsilon }^T$ is the adjoint operator of $\mathfrak{L}-A_{\epsilon }$. By the well-known functional analytic techniques (see [10], [11] and [12], Lemma 2.3]), we know that the principal eigenvalue ${\lambda }_1(\mathfrak{L}-A_{\epsilon })$ of $\mathfrak{L}-A_{\epsilon }$ is also the principal eigenvalue of $\mathfrak{L}-A_{\epsilon }^T$. In the following, we denote by ${({\psi }_1,{\psi }_2)}^T$ the unique eigenfunction of $\mathfrak{L}-A_{\epsilon }^T$ (up a constant multiple) associated to ${\lambda }_1(\mathfrak{L}-A_{\epsilon })$, such that ${\psi }_1>0$, ${\psi }_2>0$ in Ω. Then ${({\psi }_1,{\psi }_2)}^T$ verifies that

$$\{\begin{array}{cc}-\mathrm{\Delta }{\psi }_1=-e{\psi }_1+({\lambda }_1+e+\epsilon b_1){\psi }_2+{\lambda }_1(\mathfrak{L}-A_{\epsilon }^T){\psi }_1,\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }{\psi }_2=({\lambda }_1+e+\epsilon a_1){\psi }_1-(e+\epsilon f_1){\psi }_2+{\lambda }_1(\mathfrak{L}-A_{\epsilon }^T){\psi }_2,\hfill & x\in \mathrm{\Omega },\hfill \\ {\psi }_1={\psi }_2=0,\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}$$

(2.6)

Multiplying the system (2.5) by ${({\psi }_1,{\psi }_2)}^T$, the system (2.6) by ${(u_{\epsilon },v_{\epsilon })}^T$, integrating over Ω, and subtracting, we can get

$$\begin{array}{c}{\lambda }_1(\mathfrak{L}-A_{\epsilon }^T)\cdot {\int }_{\mathrm{\Omega }}({\psi }_1{u}_{\epsilon }+{\psi }_2{v}_{\epsilon })dx\hfill \\ =-\epsilon {\int }_{\mathrm{\Omega }}(c_1(x)u_{\epsilon }(u_{\epsilon }+v_{\epsilon }){\psi }_1+d_1(x)v_{\epsilon }(u_{\epsilon }+v_{\epsilon }){\psi }_2)dx,\hfill \end{array}$$

which together with (2.4) implies ${\lambda }_1(\mathfrak{L}-A_{\epsilon }^T)>0$, and hence ${\lambda }_1(\mathfrak{L}-A_{\epsilon })>0$. Consequently, by [4], Corollary 18], we know that system (1.1) has no positive solutions. But this is impossible since (1.1) admits a positive solution ${\xi }_2$. □

3 Related results

In this section, we shall investigate the effect of death rates and birth rates on the coexistence states. Bouguima et al.[4] established the following existence result.

Theorem B

[4], Theorem 20]

Assume$f=e+k$with$k\in \mathbb{R}$. If$\underline{a}\underline{b}>{\rho }_1(e){\rho }_1(f)$, then system (1.1) admits at least one positive solution.

Obviously, the assumption $f=e+k$, $k\in \mathbb{R}$ used in Theorem B is too restrictive, we shall weaken it below. Since a, b, c, d, e, and f are $C^{\alpha }(\overline{\mathrm{\Omega }})$ continuous positive functions, system (1.2) has a principal eigenvalue ${\lambda }_1(\mathfrak{L}-A(x))$, which corresponds to a unique (up to a constant multiple) eigenfunction

$$U_0={(u_0,v_0)}^T\in C^{2,\alpha }(\overline{\mathrm{\Omega }})\times C^{2,\alpha }(\overline{\mathrm{\Omega }})$$

satisfying $u_0>0$, $v_0>0$ in Ω.

Theorem 3.1

System (1.1) has at least one positive solution provided that

$$max\{\overline{e}+{\lambda }_1,\overline{f}+{\lambda }_1\}<min\{\underline{a}\frac{{\int }_{\mathrm{\Omega }}{\phi }_1(e)v_{0}dx}{{\int }_{\mathrm{\Omega }}{\phi }_1(e)u_{0}dx},\underline{b}\frac{{\int }_{\mathrm{\Omega }}{\phi }_1(f)u_{0}dx}{{\int }_{\mathrm{\Omega }}{\phi }_1(f)v_{0}dx}\}.$$

(3.1)

Proof

By [4], Corollary 18], to prove the theorem we need only to show that ${\lambda }_1(\mathfrak{L}-A(x))<0$. Recall that $U_0={(u_0,v_0)}^T$ satisfies

$$\{\begin{array}{cc}-\mathrm{\Delta }{u}_0+e(x)u_0-a(x)v_0={\lambda }_1(\mathfrak{L}-A(x))u_0,\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }{v}_0+f(x)v_0-b(x)u_0={\lambda }_1(\mathfrak{L}-A(x))v_0,\hfill & x\in \mathrm{\Omega },\hfill \\ u_0=v_0=0,\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}$$

(3.2)

Multiplying (3.2) with $\left(\begin{array}{c}{\phi }_1(e)\\ {\phi }_1(f)\end{array}\right)$, multiplying the system

$$\{\begin{array}{cc}-\mathrm{\Delta }{\phi }_1(e)+e(x){\phi }_1(e)={\rho }_1(e){\phi }_1(e),\hfill & x\in \mathrm{\Omega },\hfill \\ -\mathrm{\Delta }{\phi }_1(f)+f(x){\phi }_1(f)={\rho }_1(f){\phi }_1(f),\hfill & x\in \mathrm{\Omega },\hfill \end{array}$$

(3.3)

with ${(u_0,v_0)}^T$, integrating over Ω, and subtracting, we can obtain

$$\begin{array}{c}{\lambda }_1(\mathfrak{L}-A(x))({\int }_{\mathrm{\Omega }}{\phi }_1(e)u_{0}dx+{\int }_{\mathrm{\Omega }}{\phi }_1(f)v_{0}dx)\hfill \\ ={\rho }_1(e){\int }_{\mathrm{\Omega }}{\phi }_1(e)u_{0}dx-{\int }_{\mathrm{\Omega }}a(x){\phi }_1(e)v_{0}dx+{\rho }_1(f){\int }_{\mathrm{\Omega }}{\phi }_1(f)v_{0}dx-{\int }_{\mathrm{\Omega }}b(x){\phi }_1(f)u_{0}dx\hfill \\ \le (\overline{e}+{\lambda }_1){\int }_{\mathrm{\Omega }}{\phi }_1(e)u_{0}dx-\underline{a}{\int }_{\mathrm{\Omega }}{\phi }_1(e)v_{0}dx\hfill \\ +(\overline{f}+{\lambda }_1){\int }_{\mathrm{\Omega }}{\phi }_1(f)v_{0}dx-\underline{b}{\int }_{\mathrm{\Omega }}{\phi }_1(f)u_{0}dx,\hfill \end{array}$$

(3.4)

which together with (3.1) implies ${\lambda }_1(\mathfrak{L}-A(x))<0$. □

Remark 3.1

Theorem B was established in the special case $f=e+k$, $k\in \mathbb{R}$. In such a case, we have ${\rho }_1(f)={\rho }_1(e)+k$ and ${\phi }_1(e)={\phi }_1(f)$. By (3.1) we can easily get

$$\underline{a}\frac{{\int }_{\mathrm{\Omega }}{v}_0{\phi }_1(e)dx}{{\int }_{\mathrm{\Omega }}{u}_0{\phi }_1(e)dx}>\overline{e}+{\lambda }_1\ge {\rho }_1(e),\underline{b}\frac{{\int }_{\mathrm{\Omega }}{u}_0{\phi }_1(e)dx}{{\int }_{\mathrm{\Omega }}{v}_0{\phi }_1(e)dx}>\overline{f}+{\lambda }_1\ge {\rho }_1(f),$$

and therefore $\underline{a}\underline{b}>{\rho }_1(e){\rho }_1(f)$, which is just the crucial condition used in Theorem B. Although (3.1) seems to be stronger than the assumption of Theorem B, the restrictive condition $f=e+k$, $k\in \mathbb{R}$ is weakened, and Theorem 3.1 is applicable to more general classes of f and e rather than $f=e+k$, $k\in \mathbb{R}$.