Abstract
In this paper, we consider the existence of multiple solutions for discrete boundary value problems involving the mean curvature operator by means of Clark’s Theorem, where the nonlinear terms do not need any asymptotic and superlinear conditions at 0 or at infinity. Further, the existence of a positive solution has been considered by the strong comparison principle. As an application, some examples are given to illustrate the obtained results.
Similar content being viewed by others
1 Introduction
Denote the sets of integers and real numbers by , , respectively. For , denotes the discrete interval \(\{a,a+1\ldots,b\}\) if \(a\leq b\). Due to geometric and physical motivations, many authors [1–3] have studied the existence results for the prescribed mean curvature equations with Dirichlet boundary conditions
where \(f:[0,\infty )\rightarrow [0,\infty )\) is a continuous function, \(\phi _{c}\) is the mean curvature operator defined by \(\phi _{c}(\xi )=\frac{\xi}{\sqrt{1+ \kappa \xi ^{2}}}\) with \(\kappa >0\). For general background on the mean curvature operator, we refer to [4, 5].
Because of the wide applications of difference equations in various research fields such as computer science, economics, biology, and other fields [6–11], many authors have obtained excellent results for difference equations, for example, positive solutions [12–15], homoclinic solutions [16–21], and ground-state solutions [22, 23]. In particular, Guo and Yu [24] first used the critical-point theory to study the existence of a periodic solution for the following discrete problem
where Δ is the forward difference operator defined by \(\Delta u(t)=u(t+1)-u(t)\), \(\Delta ^{ 2}u(t)=\Delta (\Delta u(t))\), \(f(t,\cdot )\in C(R,R)\) for each . Also, the critical-point theory is an important tool to deal with the existence of solutions for the discrete boundary value problems [25–27]. However, few works have been done concerning the discrete problems (1). In [15], Zhou and Ling proved the existence results for the boundary value problem
where T is a given positive integer, \(f(t,\cdot )\in C(R,R)\) for each . Under some suitable oscillating assumption on the nonlinearity f at infinity, they investigated the existence of infinitely many positive solutions.
The aim of this paper is to study the existence of multiple solutions for the following nonlinear difference equations with mean curvature operator
where for each and \(\lambda >0\) is a positive parameter. Based on a version of Clark’s Theorem [28, 29], we investigate the existence of multiple solutions of (4).
Let f satisfy the following hypotheses:
- \((a_{1})\):
-
is a continuous function and \(f(t,0)=0\) for each ;
- \((a_{2})\):
-
\(\liminf_{\xi \rightarrow +\infty}f(t,\xi )<0\) and there exists a positive constant α such that
- \((a_{3})\):
-
\(f(t,\xi )\) is odd in ξ for any .
Example 1.1
It is easy to find some suitable functions satisfying assumptions \((a_{1})\)–\((a_{3})\). Let
We note that sinξ is a continuous odd function on . Thus, \((a_{1})\) and \((a_{3})\) hold. Lastly, if we set \(\alpha =\frac{\pi}{4}\), then
Therefore, \((a_{2})\) holds.
Obviously, if u is a solution of (4), then −u is also a solution of (4) by \((a_{3})\). We say that ±u is a pair of solutions.
2 Preliminaries
Consider the T-dimensional real space
endowed with the norm
From functional analysis theory, we know that E is a real Banach space. Moreover, we define the following two equivalent norms on E,
Let J be a \(C^{1}\) functional on E. A sequence \(\{u_{n}\}\subset E\) is called a Palais–Smale sequence (P.S. sequence for short) for J if \(\{J(u_{n})\}\) is bounded and \(J'(u_{n})\rightarrow 0\) as \(n\rightarrow \infty \). We say J satisfies the Palais–Smale condition (P.S. condition for short) if any P.S. sequence for J possesses a convergent subsequence in E.
Let θ be the zero element of Banach space E. Let Σ denote the family of sets \(A\subset E\backslash \{\theta \}\) such that A is closed in E and symmetric with respect to θ, i.e., \(u\in A\) implies \(-u\in A\).
The following Clark’s Theorem will be used to prove our main result.
Lemma 2.1
Let E be a real Banach space, with J even, bounded from below, and satisfying the P.S. condition. Suppose \(J(\theta )=0\), there is a set \(K\subset \Sigma \) such that K is homeomorphic to \(S^{j-1}\) by an odd map, and \(\sup_{K}J<0\). Then, J possesses at least j distinct pairs of critical points.
First, the following comparison principle is necessary for the positive solutions.
Lemma 2.2
Let \(u,v\in E\). If
then \(u\geq v\) in .
Proof
Arguing indirectly, if not, there exist some such that \(u(j_{0})< v(j_{0})\). Set that
If \(u(j-1)>v(j-1)\), by \(\phi _{c}(s)\) being increasing in s, we have
which contradicts (5).
If \(u(j-1)\leq v(j-1)\), we first consider the case for \(u(j)-u(j-1)< v(j)-v(j-1)\), which implies (6). This contradicts (5), since it remains to consider that \(u(j-1)\leq v(j-1)\) and \(u(j)-u(j-1)>v(j)-v(j-1)\). First, we assume that \(u(j-2)>v(j-2)\) holds, then
If \(u(j-2)\leq v(j-2)\) and \(u(j-1)-u(j-2)< v(j-1)-v(j-2)\), we obtain (7), again contradicting (5). Then, \(u(j)< v(j)\) can happen only if \(u(j-2)\leq v(j-2)\) and \(u(j-1)-u(j-2)>v(j-1)-v(j-2)\).
By repeating the above process, \(u(j)< v(j)\) can happen only if \(u(2)\leq v(2)\) and \(u(3)-u(2)>v(3)-v(2)\). In this case, if \(u(1)>v(1)\), or \(u(1)\leq v(1)\) and \(u(2)-u(1)< v(2)-v(1)\), we have
which contradicts (5). If \(u(1)\leq v(1)\) and \(u(2)-u(1)>v(2)-v(1)\), we have
which contradicts (5). Hence, \(u(j)\geq v(j)\) for all . The proof is completed. □
Lemma 2.3
Let \(u\in E\), if
then \(u(t\pm 1)=0\).
Proof
By the above assumptions, we have
Combining with the monotonicity of \(\phi _{c}\), we have \(u(t\pm 1)=0\). The proof is completed. □
In particular, let \(v=0\) in Lemma 2.2, the strong comparison principle is given by the two lemmas above.
Lemma 2.4
Let \(u\in E\), if \(u\neq \theta \) and
then \(u>0\) in .
Lemma 2.5
Fix \(u\in E\), if \(u(j)\leq 0\) for some and
then \(u>0\) in or \(u=0\).
Proof
Let u be a nontrivial function satisfying (10). We assume first that \(j=1\) and \(u(1)\leq 0\), by (10), then we obtain
Hence, \(u(2)\leq u(1)\leq 0\). Again, we apply (10) with \(j=2\) to conclude that \(u(3)\leq u(2)\leq 0\). Hence, we have \(u(T)\leq u(T-1)\leq \cdots \leq u(3)\leq u(2)\leq u(1)\leq 0\). If \(u(T)=0\), then u is a trivial function, contradicting the definition of u. If \(u(T)<0\), by (10) with \(j=T\), we obtain
This is absurd, hence \(u(1)>0\). By a similar argument, if \(u(2)<0\), then \(u(T)<0\), but we obtain \(u(T)=0\) from (11). Thus, we have \(u(2)\geq 0\). Repeating the above computation, we have \(u(3)\geq 0, u(4)\geq 0, \ldots,u(T-1)\geq 0\). Now, we show \(u(T)\geq 0\). If \(u(T)<0\), since \(u(T)<0\leq u(T-1)\), again we obtain (11) from (10). This implies \(u(T)=0\), showing \(u(T)\geq 0\). Hence, \(u(1)>0\) and \(u(j)\geq 0\) for all . If \(u(2)=0\), we have
contradicting \(u(1)>0\). By a similar argument, we obtain \(u(3)>0, u(4)>0,\ldots,u(T-1)>0\). If \(u(T)=0\), we have \(0>\frac{-u(T-1)}{\sqrt{1+ \kappa u(T-1)^{2}}}\geq 0\). Then, \(u(T-1)=0\), which is absurd, hence \(u>0\). □
3 Main results
Now, we state our main results.
Theorem 3.1
Assume that \((a_{1})\)–\((a_{3})\) hold, then there exists a positive constant λ̄, when \(\lambda >\bar{\lambda}\), and problem (4) admits at least T distinct pairs of nontrivial solutions. Furthermore, there exists a positive constant M such that each solution u satisfies \(\|u\|_{\infty}\leq M\).
Proof
Using the condition \((a_{2})\), there exists a positive real sequence \(\{d_{n}\}\) with \(\lim_{n\rightarrow \infty}d_{n}=+\infty \) such that
We can find a positive integer \(n_{0}\) such that \(M=d_{n_{0}}>\alpha \) and \(f(t,M)<0\), where α comes from \((a_{2})\). First, we consider the following boundary value problem
where \(\hat{f}(t,\xi )\) is a truncation function defined by
We show that if u satisfies problem (12), then \(\|u\|_{\infty}\leq M\) and u is a solution of problem (4). Arguing indirectly, there exists a such that \(|u(k_{0})|>M\) and \(|u(t)|\leq M\) for . If \(u(k_{0})>M\), then \(\hat{f}(t,u(k_{0}))=f(t,M)<0\). We have
or
which implies that \(u(k_{0}+1)>u(k_{0})>M\). By repeating the above process, we obtain
Further,
which is a contradiction. If \(u(k_{0})<-M\), we can similarly obtain a contradiction. Thus, \(\|u\|_{\infty}\leq M\) holds.
Define the functional Ĵ on E as follows:
where \(\hat{F}(t,\xi )=\int _{0}^{\xi}\hat{f}(t,s)\,ds\), . It is easy to verify that and is even. By using \(u(0)=u(T+1)=0\), we can compute the Frećhet derivative,
for all \(u, v\in E\). It is clear that the critical points of Ĵ are the solutions of problem (12). In what follows, we will prove that Ĵ has at least T distinct pairs of nonzero critical points by Lemma 2.1.
For any sequence \(\{u_{n}\}\subset E\), if \(\{\hat{J}(u_{n})\}\) is bounded and \(\hat{J}'(u_{n})\rightarrow 0\) as \(n\rightarrow +\infty \), we claim that \(\{u_{n}\}\) is bounded. In fact, there exists a positive constant such that \(|\hat{J}(u_{n})|\leq C\). Since E is a finite-dimensional real Banach space, there is \(\|u\|_{2}\leq \|u\|\leq 2\|u\|_{2}\) for all \(u\in E\) (see [30]). Assume that \(\|u_{n}\|\rightarrow +\infty \) as \(n\rightarrow +\infty \), then
where and \(D=\max |f(t,u)|\) for . This is impossible, since C is a fixed constant. Thus, \(\{u_{n}\}\) is bounded in E. This implies that \(\{u_{n}\}\) has a convergent subsequence. Then, the functional Ĵ satisfies the P.S. condition.
Moreover, the coerciveness of Ĵ,
implies that Ĵ is bounded from below.
Let \(\{e_{i}\}_{i=1}^{T}\) be a base of E and \(\|e_{i}\|=1\) for each . We define
Obviously, \(\theta \notin A(\rho )\), \(A(\rho )\) is closed in E and symmetric with respect to θ. We note that \(A(\rho )\) is homeomorphic to \(S^{T-1}\) for any \(\rho >0\). For \(u\in A(\rho )\), we see that
Take \(\rho =\frac{\alpha}{T+1}\), thus
For \(u\in A(\frac{\alpha}{T+1})\), we note that \(u\neq \theta \) and \(\hat{f}(t,u(t))=f(t,u(t))\). By \((a_{2})\) and \((a_{3})\), then
Let \(\tau =\inf_{u\in A(\frac{\alpha}{T+1}) }\sum^{T}_{t=1} \hat{F}(t,u(t))\) and \(\bar{\lambda}=\frac{(2+q^{\ast})\alpha ^{2}}{2T\tau}\). By \((a_{2})\), we know \(\tau >0\). If \(\lambda >\bar{\lambda}\), then
where . Since all conditions of Lemma 2.1 hold, problem (4) admits at least T distinct pairs of nontrivial solutions. The proof is completed. □
At the end of this section, we try to prove a pair of constant-sign solutions of the original problem. We first introduce the following assumptions:
- \((a_{1}')\):
-
\(f(t,\cdot )\) is a continuous functional on for each ;
- \((a_{2}')\):
-
for some \(\alpha >0\), where ;
- \((a_{3}')\):
-
\(f(t,\xi )=-f(t,-\xi )\) in \(\xi \neq 0\) for any .
We note that the function \(f(t,\cdot )\) is locally bounded from below for each in the right-hand side of 0 from \((a_{2}')\) and problem (4) has no trivial solution. When θ is not the solution of the problem, many problems become more complicated [31, 32]. For example, we put \(f(t,\xi )=\frac{1}{\sqrt[3]{\xi}}\), \(\alpha =1\) and \(q^{\ast}=\frac{1}{2}\), then , which satisfies the conditions \((a_{1}')\)–\((a_{3}')\).
Let
We observe that \(\frac{\sum^{T}_{t=0}\frac{(\Delta u(t))^{2}}{\sqrt{1+\kappa (\Delta u(t))^{2}}}}{\|u\|_{2}^{2}}\) is positive in \(E\backslash \{\theta \}\). Thus, \(\mu _{1}\geq 0\).
Theorem 3.2
Assume that \((a_{1}')\)–\((a_{3}')\) hold and
Then, problem (4) has a positive solution and a negative solution for each \(\lambda \in (0,\frac{\mu _{1}+q_{\ast}}{\mu})\), where \(\mu \in (0,\mu _{1}+q_{\ast})\).
Proof
We consider the following problem
Define the variational framework of problem (15)
then we have
Hence, \(J_{q^{\ast}}\) is coercive on E and has a global minimum point \(u_{0}\) that is its critical point. Combining with Lemma 2.4, \(u_{0}\) is a positive solution of problem (15). We take \(\varepsilon >0\) so small that \(u_{1}(t)=\varepsilon u_{0}(t)<\alpha \).
Define a continuous function as follows:
By (14), there exist a \(\mu \in [0,\mu _{1}+q_{\ast})\) and \(M_{1}>u_{1}(t)\) such that
Thus,
and
Next, we claim that the following problem
admits a positive solution u and \(u>u_{1}>0\).
We define the following variational framework corresponding to problem (19)
where \(F_{u_{1}}(t,\xi )=\int _{0}^{\xi}f_{u_{1}}(t,s)\,ds\), .
Using (18), there is one positive constant M̅ such that
Let \(\eta =\frac{\mu _{1}+q_{\ast}}{\mu}\). For \(\eta >\lambda >0\), we have
This shows that Ĵ is also coercive. As the functional is coercive and continuous, it has a global minimum point \(u\in E\), which is a critical point. By (17) and Lemma 2.5, u is a positive solution. Moreover, if we can show \(u>u_{1}\), then u must be one positive solution of problem (4). First, we assume that \(u\leq u_{1}\) for every . Since
by Lemma 2.2, we obtain \(u\geq u_{0}>u_{1}\), contradicting the assumption above. Secondly, we consider that u and \(u_{1}\) are not ordered vectors. There exist some such that \(u(j_{0})< u_{1}(j_{0})\). Let
From the proof of Lemma 2.2, if \(u(j)< u_{1}(j)\) holds, we have the following inequality
where \(i=1,2,j\). Since \(q^{\ast}\leq f(i,u_{1}(i))\) from the proof of Lemma 2.2 and \((a_{2}')\), this implies
We see from (22) that
By the strict monotonicity of \(\phi _{c}\), we find that if \(\Delta u_{0}(i-1)>0\), then \(\Delta u_{0}(i)>0\). That is, if \(\Delta u_{1}(i-1)>0\), then \(\Delta u_{1}(i)>0\).
Further, we estimate the inequality (22) from the following three cases. First, if \(\Delta u_{1}(i-1)\leq \Delta u_{1}(i)\), we have
For the second case, if \(\Delta u_{1}(i-1)>\Delta u_{1}(i)>0\), then
For the last case, if \(0>\Delta u_{1}(i-1)>\Delta u_{1}(i)\), then
We note that \(\Delta u_{1}(i-1)=0\) or \(\Delta u_{1}(i)=0\) still satisfies the above cases. Obviously, when ε is taken sufficiently small, (23), (24), and (25) cannot hold. These are contradictions. Thus, \(u\geq u_{1}\). u is one positive solution of problem (4). Moreover, we see that −u is a negative solution of problem (4) because of \((a_{3}')\). This completes the proof. □
Example 3.1
Let \(\lambda =1\), we consider the problem
we note that the conditions \((a_{1})\)–\((a_{3})\) hold from Example 1.1, and λ̄ can be less than 1 by the definition, thus the problem (26) admits at least T distinct pairs of nontrivial solutions by Theorem 3.1.
In fact, in [30], such a problem can be found when \(\kappa =0\) and \(q(t)=0\) for each in Corollary 5.1. We see that the conditions of Theorem 3.1 are different from the conditions of Corollary 5.1 of [30], and we find more solutions of problem (26).
Example 3.2
Let \(\kappa =0\) and \(T=3\). We consider the problem
Put \(\alpha =1\), \(q_{\ast}=\frac{1}{4}\) and \(q^{\ast}=\frac{1}{2}\). The conditions \((a_{1}')\)–\((a_{3}')\) hold from the previous example. We have \(\mu _{1}=2-\sqrt{2}\) from [30]. Clearly,
All conditions of Theorem 3.2 are verified. If we take \(\mu >0\) sufficiently small, then the problem (27) has a positive solution and a negative solution for each \(\lambda \in (0,+\infty )\).
Availability of data and materials
Not applicable.
References
Manxásevich, R., Mawhin, J.: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equ. 145(2), 367–393 (1998)
Benevieri, P., Marcos, J., Medeiros, E.: Periodic solutions for nonlinear equations with mean curvature-like operators. Appl. Math. Lett. 20(5), 484–492 (2007)
Lu, Y.Q., Ma, R.Y., Gao, H.L.: Existence and multiplicity of positive solutions for one-dimensional prescribed mean curvature equations. Bound. Value Probl. 2014, Article ID 120 (2014)
Bonanno, G., Livrea, R., Mawhin, J.: Existence results for parametric boundary value problems involving the mean curvature operator. Nonlinear Differ. Equ. Appl. 22(3), 411–426 (2015)
Xu, M., Ma, R.Y., He, Z.Q.: Positive solutions of the periodic problems for quasilinear difference equation with sign-changing weight. Adv. Differ. Equ. 2018, 393 (2018)
Kelly, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications. Academic Press, San Diego (1991)
Agarwal, R.P.: Equations and Inequalities. Theory, Methods, and Applications. Dekker, New York (2000)
Yu, J.S., Zheng, B.: Modeling Wolbachia infection in mosquito population via discrete dynamical models. J. Differ. Equ. Appl. 25(11), 1549–1567 (2019)
Zheng, B., Li, J., Yu, J.S.: One discrete dynamical model on the Wolbachia infection frequency in mosquito populations. Sci. China Math. (2021). https://doi.org/10.1007/s11425-021-1891-7
Yu, J.S., Li, J.: Discrete-time models for interactive wild and sterile mosquitoes with general time steps. Math. Biosci. 346, Article ID 108797 (2022)
Zheng, B., Yu, J.S.: Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency. Adv. Nonlinear Anal. 11(1), 212–224 (2022)
D’Aguì, G., Mawhin, J., Sciammetta, A.: Positive solutions for a discrete two point nonlinear boundary value problem with p-Laplacian. J. Math. Anal. Appl. 447(1), 383–397 (2017)
Agarwal, R.P., Luca, R.: Positive solutions for a system of second order discrete boundary value problem. Adv. Differ. Equ. 2018, 470 (2018)
Zhou, Z., Ling, J.X.: Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with \(\phi _{c}\)-Laplacian. Appl. Math. Lett. 91, 28–34 (2019)
Ling, J.X., Zhou, Z.: Positive solutions of the discrete Dirichlet problem involving the mean curvature operator. Open Math. 17(1), 1055–1064 (2019)
Lin, G.H., Zhou, Z.: Homoclinic solutions in periodic difference equations with mixed nonlinearities. Math. Methods Appl. Sci. 39(2), 245–260 (2016)
Lin, G.H., Zhou, Z.: Homoclinic solutions in non-periodic discrete ϕ-Laplacian equations with mixed nonlinearities. Appl. Math. Lett. 64, 15–20 (2017)
Lin, G.H., Zhou, Z.: Homoclinic solutions of discrete ϕ-Laplacian equations with mixed nonlinearities. Commun. Pure Appl. Anal. 17(5), 1723–1747 (2018)
Zhang, Q.Q.: Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Commun. Pure Appl. Anal. 18(1), 425–434 (2019)
Lin, G.H., Yu, J.S., Zhou, Z.: Homoclinic solutions of discrete nonlinear Schrödinger equations with partially sublinear nonlinearities. Electron. J. Differ. Equ. 2019(96), 1 (2019)
Lin, G.H., Yu, J.S.: Homoclinic solutions of periodic discrete Schrödinger equations with local superquadratic conditions. SIAM J. Math. Anal. 54(2), 1966–2005 (2022)
Lin, G.H., Zhou, Z., Yu, J.S.: Ground state solutions of discrete asymptotically linear Schrödinger equations with bounded and non-periodic potentials. J. Dyn. Differ. Equ. 32(2), 527–555 (2020)
Lin, G.H., Yu, J.S.: Existence of a ground-state and infinitely many homoclinic solutions for a periodic discrete system with sign-changing mixed nonlinearities. J. Geom. Anal. 32, Article ID 127 (2022)
Guo, Z.M., Yu, J.S.: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China Ser. A 46(4), 506–515 (2003)
Bereanu, C., Mawhin, J.: Boundary value problems for second-order nonlinear difference equations with discrete ϕ-Laplacian and singular ϕ. J. Differ. Equ. Appl. 14(10–11), 1099–1118 (2008)
Xiong, F., Zhou, Z.: Three solutions to Dirichlet problems for second-order self-adjoint difference equations involving p-Laplacian. Adv. Differ. Equ. 2021, 192 (2021)
Wang, Z.G., Zhou, Z.: Multiple solutions for boundary value problems of p-Laplacian difference equations containing both advance and retardation. Math. Probl. Eng. 2020, 1–8 (2020)
Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol. 65. Am. Math. Soc., Providence (1986)
Bai, D.Y., Xu, Y.T.: Nontrivial solutions of boundary value problems of second-order difference equations. J. Math. Anal. Appl. 326(1), 297–302 (2007)
Bonanno, G., Candito, P., D’Aguì, G.: Variational methods on finite dimensional Banach space and discrete problems. Adv. Nonlinear Stud. 14(4), 915–939 (2014)
Bonanno, G.: Relations between the mountain pass theorem and local minima. Adv. Nonlinear Anal. 1(3), 205–220 (2012)
Wang, Z.G., Zhou, Z.: Boundary value problem for a second order difference equation with resonance. Complexity 2020, 1–10 (2020)
Acknowledgements
The authors wish to thank the editor and the anonymous reviewer for their valuable comments and suggestions.
Funding
This work is supported by the Scientific Research Project of Lüliang City (Grant No. Rc2020213).
Author information
Authors and Affiliations
Contributions
ZW proposed the idea of this paper and performed all the steps of the proofs. QX wrote the whole paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wang, Z., Xie, Q. Boundary value problems for a second-order difference equation involving the mean curvature operator. Bound Value Probl 2022, 55 (2022). https://doi.org/10.1186/s13661-022-01637-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-022-01637-7