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Three positive periodic solutions of second order nonlinear neutral functional differential equations with delayed derivative
Advances in Difference Equations volume 2020, Article number: 164 (2020)
Abstract
This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity \((x(t)-cx(t-\delta)){''}+a(t)g(x(t))x(t)=\lambda b(t)f(t,x(t),x(t-\tau_{1}(t)),x'(t-\tau_{2}(t)))\). By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established.
1 Introduction
In the present work, we study the existence of three positive periodic solutions for the second order neutral functional differential equation of the form
where \(\lambda>0\) is a positive parameter, c, δ are constants, and \(|c|<1\). \(a(t)\), \(b(t)\) are nonnegative ω-periodic continuous functions, \(\tau_{i}(t)\), \(i=1,2\), are continuous ω-periodic functions, \(f: \mathbb{R}\times[0,+\infty)^{2}\times\mathbb {R}\rightarrow[0,+\infty)\) is a continuous function, and \(f(t,u,v,w)\) is ω-periodic with respect to t, \(g\in C([0,+\infty ),[0,+\infty))\).
Neutral functional differential equations have a wide range of applications in the field of physics, biology, economics, and so on, see [1–14] for more details. In [15], the authors pointed out that the growth of single or multiple species was mainly affected by seasonal changes (especially cyclical changes) and time lags. So it is important to study the periodic solutions of such models. The issues of the existence of positive periodic solutions of neutral functional differential equations have received more attention in recent years, see [7–14]. The existence of positive periodic solutions for first order neutral functional differential equations has been studied by many authors, see [7–12] and the references therein. But the research results on the case of second order are more seldom.
In [13], the authors studied the existence, multiplicity, and nonexistence of positive periodic solutions of second order neutral functional differential equations of the form
where \(\lambda>0\) is a positive parameter, c, δ are constants, and \(|c|<1\), \(a(t)\), \(b(t)\) are nonnegative ω-periodic continuous functions. But the nonlinear term does not contain the derivative term.
Recently, Li [14] discussed the existence and nonexistence of positive ω-periodic solutions of second order neutral functional differential equations with delayed derivative in nonlinear term by using the positive operator perturbation method and the fixed point index theory
where \(\delta>0\), \(|c|<1\), \(a\in C(\mathbb{R},\infty)\) is an ω periodic function, \(f:\mathbb{R}\times[0,\infty)^{2}\times\mathbb {R}\rightarrow[0,\infty)\) is continuous, and \(f(t,u,v,w)\) is ω-periodic with respect to t, \(\tau, \gamma\in C(\mathbb {R},[0,\infty))\) are ω-periodic functions. But he did not consider the multiplicity of the positive periodic solutions.
Motivated by the above mentioned results, in this work, by using a different method, we mainly study the existence and multiplicity of positive periodic solutions for a class of second order neutral nonlinear functional differential equations with delayed derivative of the form (1).
Let \(C_{\omega}(\mathbb{R})\) be the Banach space of all continuous ω-periodic functions endowed with the norm \(\|x\|_{C}=\max_{t\in[0,\omega]}|x( t)|\), \(C^{1}_{\omega}(\mathbb{R})\) be the Banach space of all continuous differentiable ω-periodic functions with the norm \(\|x\|_{C ^{1}}=\|x\|_{C}+\|x'\|_{C}\). In general, for \(n\in\mathbb{N}\), \(C^{n}_{\omega}(\mathbb{R})\) represents the Banach space of all nth order continuous differentiable ω-periodic functions. Let \(C^{+}_{\omega }(\mathbb{R})=C_{\omega}(\mathbb{R},[0,\infty))\) be a nonnegative function cone in \(C_{\omega}(\mathbb{R})\).
The main results of the present paper are summarized as follows:
- (i)
We establish the existence (and uniqueness) of ω-periodic solutions for the corresponding linear second order neutral functional differential equation
$$\bigl(x(t)-cx(t-\delta)\bigr)''+a(t)g\bigl(x(t) \bigr)x(t)=\lambda h(t),\quad t\in\mathbb{R}. $$See Lemma 5.
- (ii)
We provide the strong positive estimate and \(C^{1}\)-estimate of the periodic solution operator by using the positive operator perturbation method, see Lemma 6.
- (iii)
Let
$$K=\bigl\{ x\in C^{1}_{\omega}(\mathbb{R}): x(t)\geq\sigma \Vert x \Vert _{C}, \bigl\vert x'(\tau) \bigr\vert \leq C_{0} \bigl\vert x(t) \bigr\vert , \tau,t\in \mathbb{R}\bigr\} , $$where σ and \(C_{0}\) will be specified later. We define an operator \(Q_{\lambda}\) which maps K into itself and prove that the operator \(Q_{\lambda}\) has at least three positive fixed points by using Leggett–Williams fixed point theorem, see Theorem 1.
In this paper, we always assume that
- \((H1)\):
\(f\in C(\mathbb{R}\times[0,\infty)^{2}\times\mathbb {R},[0,\infty))\), \(f(t,u,v,w)\) is nondecreasing with respect to u, v, w and ω-periodic in t; \(g\in C([0,\infty),[0,\infty))\);
- \((H2)\):
\(a, b\in C^{+}_{\omega}(\mathbb{R})\), \(\overline{b}:=\frac {1}{\omega}\int_{0}^{\omega}b(s)\,ds>0\), and \(\tau_{i}\in C_{\omega }(\mathbb{R})\), \(i=1,2\);
- \((H3)\):
there exist two positive constants d and D satisfying \(0< d\leq a(t)g(x(t))\leq D<(\frac{\pi}{\omega})^{2}\) for any \(t\in [0,\omega]\), \(x\in C^{+}_{\omega}({\mathbb{R}})\).
2 Preliminaries
Firstly, let \(0< M<(\frac{\pi}{\omega})^{2}\). We consider the second order linear ordinary differential equation
By Lemma 2.1 of [14], the following lemma is obtained.
Lemma 1
For\(\forall h\in C_{\omega}(\mathbb{R})\), linear equation (2) has a uniqueω-periodic solution\(x\in C ^{2}_{\omega}(\mathbb {R})\)expressed by
where
And the operator\(T_{\lambda}: C_{\omega}(\mathbb{R})\rightarrow C^{1}_{\omega}(\mathbb{R})\)is a linear completely continuous operator.
For the sake of brevity, let \(\beta=\sqrt{M}\) and denote
Then
Clearly, if \(h\in C^{+}_{\omega}(\mathbb{R})\), the solution \(x\in C ^{2}_{\omega}(\mathbb{R})\) of (2) is positive.
Define a cone K in \(C^{1}_{\omega}(\mathbb{R})\) by
Lemma 2
\(T_{\lambda}(C^{+}_{\omega} (\mathbb{R}))\subset K\)and\(\| T_{\lambda}\|\leq\frac{\lambda}{M}\).
Proof
Let \(h\in C^{+}_{\omega}(\mathbb{R})\). By (3) and (4), we have
That is, \(\|x\|_{C}\leq\lambda L\int_{0}^{\omega}h(s)\,ds\). Hence we have
For \(\forall\tau\in\mathbb{R}\), noticing that \(x'(\tau)=\lambda \int_{\tau-\omega}^{\tau}U'(\tau-s)h(s)\,ds\), we have
Consequently, by (5), \(T_{\lambda}(C^{+}_{\omega}(\mathbb {R}))\subset K\). In addition, for \(h\in C^{+}_{\omega}(\mathbb{R})\), the inequality
implies that \(\|T_{\lambda}\|\leq\frac{\lambda}{M}\) and the proof is complete. □
In order to prove the existence of ω-periodic solutions of equation (1), we consider the corresponding linear neutral functional differential equation
Define a linear operator \(A: C_{\omega}(\mathbb{R})\rightarrow C_{\omega}(\mathbb{R})\) by
Then \(A: C_{\omega}(\mathbb{R})\rightarrow C_{\omega}(\mathbb{R})\) is bounded.
Lemma 3
If\(|c|\neq1\), then the operatorA, defined by (7), has a linear bounded inverse operator\(A^{-1}\)on\(C_{\omega}({\mathbb{R}})\)given by
and
Lemma 4
If\(|c|<\sigma\), then for any\(y\in K\), we have
Proof
For any \(y\in K\), by virtue of Lemma 3, we have
Then the proof of Lemma 4 is complete. □
Let \(y=Ax\). Then by (7), equation (6) can be rewritten as
It is available from Lemma 3 that when \(y\in C^{1}_{\omega}(\mathbb {R})\), \(A^{-1}y\in C^{1}_{\omega}(\mathbb{R})\), and \((A^{-1}y)'=A^{-1}y'\), when \(y\in C ^{2}_{\omega}(\mathbb{R})\), \(A^{-1}y\in C^{2}_{\omega}(\mathbb{R})\), and \((A^{-1}y)''=A^{-1}y''\). Therefore, \(x\in C^{2}_{\omega}(\mathbb{R})\) is an ω-periodic solution of equation (6) if and only if \(y=Ax\in C^{2}_{\omega }(\mathbb{ R})\) is an ω-periodic solution of equation (8).
Lemma 5
If\(|c|<\frac{d}{D+d}\), equation (8) has a uniqueω-periodic solution\(y\in C_{\omega}^{2}(\mathbb{R})\)for any\(h\in C_{\omega }(\mathbb{R})\). When\(h\in C_{\omega}^{+}(\mathbb{R})\)and\(|c|<\min \{\frac{d}{D+d},\frac{\sigma(M-D)}{\sigma(M-D)+D}\}\), theω-periodic solution\(y\in K\).
Proof
Define an operator \(B_{\lambda}: C_{\omega}(\mathbb{R})\rightarrow C_{\omega}(\mathbb{R})\) by
where \(G(y(t))=ca(t)g((A^{-1}y)(t))(A^{-1}y)(t-\delta)\). Then equation (8) can be rewritten as
By Lemma 1, we have
It follows from (9) that
Consequently,
Combining this fact with \(\|T_{\lambda}\|\leq\frac{\lambda}{M}\), we have
Then \(\|T_{\lambda}B_{\lambda}\|<1\) because of \(|c|<\frac{d}{D+d}\). Hence the operator \(I-T_{\lambda}B_{\lambda}\) has a bounded inverse operator \((I-T_{\lambda}B_{\lambda})^{-1}\) which can be expressed by
Therefore, operator equation (11) has a unique ω-periodic solution \(y\in C_{\omega}^{2}(\mathbb{R})\) expressed by
Let \(z=T_{\lambda}h\) for any \(h\in C^{+}_{\omega}(\mathbb{R})\). By Lemma 2, we get \(z\in K\). Then we have
Since \(|c|<\frac{\sigma(M-D)}{\sigma(M-D)+D}\), it follows that
Hence \(B_{\lambda}z(t)\geq0\) for any \(t\in\mathbb{R}\), that is, \(B_{\lambda}z\in C^{+}_{\omega}(\mathbb{R})\). Then applying Lemma 2 again, \((T_{\lambda}B_{\lambda})z=T_{\lambda}B_{\lambda}z\in K\). Consequently, \((T_{\lambda}B_{\lambda})^{n}z\in K\) for \(\forall n\in \mathbb{N}\). By boundedness of the linear operator \(T_{\lambda}: C_{\omega}(\mathbb{R})\rightarrow C^{1}_{\omega}(\mathbb{R})\), \(\sum_{n=1}^{\infty} (T_{\lambda}B_{\lambda})^{n}z\) is convergence in \(C^{1}_{\omega}(\mathbb{R})\). Since the cone \(K\subset C^{1}_{\omega}(\mathbb{R})\) is closed, then by (12), we have
This completes the proof of Lemma 5. □
Lemma 6
Let\(h\in C^{+}_{\omega}({\mathbb{R}})\)and\(|c|<\min\{\frac{d}{D+d},\frac {\sigma(M-D)}{\sigma(M-D)+D}\}\). Then\(h^{*}(\cdot):=b(\cdot)h(\cdot )\)belongs to\(C^{+}_{\omega}({\mathbb{R}})\), and the operator\(S_{\lambda}: C^{+}_{\omega}({\mathbb{R}})\rightarrow C^{+}_{\omega}({\mathbb{R}})\)defined by
maps\(C^{+}_{\omega}({\mathbb{R}})\)toKand it is completely continuous.
Proof
By Lemma 1, \(y=S_{\lambda}h^{*}\in C^{2}_{\omega}({\mathbb{R}})\) is an ω-periodic solution of equation (10) since equation (10) is equivalent to equation (8). By Lemma 5, equation (8) has a unique ω-periodic solution \(y\in C^{2}_{\omega}({\mathbb{R}})\) expressed by
By (13), we know that \(y\in K\). Hence \(S_{\lambda}: C^{+}_{\omega}({\mathbb{R}})\rightarrow K\) and it is completely continuous. This completes the proof of Lemma 6. □
At the end of this section, we introduce a fixed point theorem, which will be used in the proof of our main result.
Let \((X, \|\cdot\|)\) be a real Banach space and K be a cone in X. A map ρ is called a nonnegative continuous concave function on K if \(\rho: K\rightarrow[0,+\infty)\) is continuous and
for all \(x, y\in K\) and \(t\in[0, 1]\).
Let \(0< r< R\) and ρ be a nonnegative continuous concave function on K, set
Lemma 7
Let\(Q:\overline{K}_{R}\rightarrow\overline{K}_{R}\)be a completely continuous mapping andρbe a nonnegative continuous concave function onKwith\(\rho(x)\leq\|x\|\)for all\(x\in\overline {K}_{R}\). Suppose that there exist positive constantsr, \(r_{1}\), \(r_{2}\), Rwith\(0< r< r_{1}< r_{2}< R\)such that
- (a)
\(\{x\in K(\rho,r_{1},r_{2}):\rho(x)>r_{1}\}\neq\emptyset\)and\(\rho(Qx)>r_{1}\)for\(x\in K(\rho,r_{1},r_{2})\);
- (b)
\(\|Qx\|< r\)for\(x\in\overline{K}_{r}\);
- (c)
\(\rho(Qx)>r_{1}\)for\(x\in K(\rho,r_{1},R)\)with\(\|Qx\| >r_{2}\).
ThenQhas at least three fixed points\(x_{1}\), \(x_{2}\), \(x_{3}\)satisfying
3 Existence theorem
Theorem 1
Let assumptions\((H1)\)–\((H3)\)hold. In addition, we suppose that
- \((H4)\):
\(|c|<\min\{\sigma, \frac{d}{D+d}, \frac{\sigma(M-D)}{\sigma (M-D)+D}\}\);
- \((H5)\):
\(1-|c|>L\omega(M-d)(1-|c|)+LD\omega|c|\);
- \((H6)\):
there exist positive constantsr, \(r_{1}\), andRwith\(0 < r< r_{1} < R\)such that
$$\begin{aligned}[b] \frac{\sup_{t\in[0,\omega]}f(t,\frac{r}{1- \vert c \vert },\frac {r}{1- \vert c \vert },\frac{C_{0}r}{1- \vert c \vert })}{\frac{A_{0}r}{L(1- \vert c \vert )}}&< \frac{\sup_{t\in[0,\omega]}f(t,\frac{R}{1- \vert c \vert },\frac{R}{1- \vert c \vert },\frac {C_{0}R}{1- \vert c \vert })}{\frac{A_{0}R}{L(1- \vert c \vert )}}\\& < \frac{\inf_{t\in[0,\omega]}f(t,\frac{\sigma- \vert c \vert }{1-c^{2}}r_{1},\frac {\sigma- \vert c \vert }{1-c^{2}}r_{1},-\frac{\sigma- \vert c \vert }{\sigma (1-c^{2})}C_{0}r_{1})}{\frac{B_{0}r_{1}}{l(1- \vert c \vert )}},\hspace{-12pt}\end{aligned} $$(14)where
$$\begin{gathered} A_{0}=1- \vert c \vert -L\omega(M-d) \bigl(1- \vert c \vert \bigr)-LD\omega \vert c \vert , \\B_{0}=1- \vert c \vert -l\omega\sigma(M-D) \bigl(1- \vert c \vert \bigr)+lD\omega \vert c \vert .\end{gathered} $$
Then equation (1) associated with\(\lambda\in(\lambda_{1},\lambda _{2})\)has at least three positiveω-periodic solutions, where
Proof
By \((H5)\), we obtain \(A_{0}>0\) and
Furthermore, in view of (14), we get \(0<\lambda_{1}<\lambda_{2}\).
For each \(\lambda\in(\lambda_{1},\lambda_{2})\) and \(y\in K\), denote F by
then \(F: K\rightarrow C^{+}_{\omega}({\mathbb{R}})\) is continuous. We define a mapping \(Q_{\lambda}\) by
By Lemma 6, \(Q_{\lambda}: K\rightarrow K\) is completely continuous.
Define a function \(\rho: K\rightarrow[0,\infty)\) by
Then ρ is a nonnegative continuous concave function on K and
For any \(y\in\overline{K}_{R}\) and \(\lambda\in(\lambda_{1},\lambda _{2})\), by (4), (14), (15), Lemmas 4 and 5, we have
Hence \(\|Q_{\lambda}y\|_{C}\leq R\) and \(Q_{\lambda}\) is completely continuous on \(\overline{K}_{R}\).
We now verify that condition (b) of Lemma 7 holds. Indeed, if \(y\in \overline{K}_{r}\), we have
Hence, \(\|Q_{\lambda}y\|_{C}< r\).
Choose a positive constant \(r_{2}\) such that \(0 < r_{1} =\sigma r_{2} <r_{2}\leq R\). In the next discussion, we prove that condition (a) of Lemma 7 holds. Obviously, ρ is a concave continuous function on K with \(\rho(y)\leq\|y\|_{C}\) for \(y\in\overline{K}_{R}\). Noticing that if \(y(t)=\frac{1}{3}r_{1}+\frac{2}{3}r_{2}\) for any \(t\in [0,\omega]\), then \(y\in K(\rho,r_{1},r_{2})\) and \(\rho(y)>r_{1}\), which means that \(\{y\in K(\rho,r_{1},r_{2}): \rho(y)>r_{1}\}\neq \emptyset\). So, for any \(y\in K(\rho,r_{1},r_{2})\), we have
Hence, for any \(y\in K(\rho,r_{1},r_{2})\), by Lemma 4, we have
Consequently, condition (a) of Lemma 7 holds.
In the end, we prove that condition (c) of Lemma 7 holds.
Let \(y\in K(\rho,r_{1},R)\) and \(\|Q_{\lambda}y\|_{C}>r_{2}\). We prove \(\rho(Q_{\lambda}y)>r_{1}\). It follows from (15) that
Therefore,
Now, all the conditions of Lemma 7 are satisfied. By Lemma 7, \(Q_{\lambda}\) has at least three positive fixed points \(y_{1}\), \(y_{2}\), and \(y_{3}\) satisfying
Then equation (1) has at least three positive ω-periodic solutions:
This completes the proof. □
Example 1
We consider the positive 2π-periodic solutions for the second order neutral differential equation
where \(\lambda>0\) is a constant. Corresponding to equation (1), we choose
Let \(M=\frac{1}{16}\). Then
Therefore, it is easy to verify that conditions \((H2)\)–\((H5)\) are satisfied.
Let \(F\in C(\mathbb{R}\times[0,\infty)^{2}\times\mathbb{R},[0,\infty ))\), \(F(t,u,v,w)\) is nondecreasing with respect to u, v, w and ω-periodic with respect to t. If there exist positive constants r, \(r_{1}\), and R with \(0 < r< r_{1} < R\) such that the function F satisfies the following inequalities:
where \(A_{0}\approx0.095\), \(B_{0}\approx0.602\), then equation (16) has at least three positive 2π-periodic solutions provided that
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The authors thank the referees and editors for their careful reading of the manuscript and valuable comments, which helped to improve the quality of the paper.
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The research is supported by the National Natural Science Function of China (No. 11701457) and Gansu Science and Technology Plan (No. 17JR5RA071).
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The main idea of this paper was proposed by HY. HY and LZ prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Yang, H., Zhang, L. Three positive periodic solutions of second order nonlinear neutral functional differential equations with delayed derivative. Adv Differ Equ 2020, 164 (2020). https://doi.org/10.1186/s13662-020-02630-z
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DOI: https://doi.org/10.1186/s13662-020-02630-z