Positive periodic solutions in shifts ${\delta }_{\pm }$ for a nonlinear first-order functional dynamic equation on time scales

Abstract

Let $\mathbb{T}\subset \mathbb{R}$ be a periodic time scale in shifts ${\delta }_{\pm }$ with period $P\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$. We consider the existence of positive periodic solutions in shifts ${\delta }_{\pm }$ for the nonlinear functional dynamic equation of the form

$$x^{\mathrm{\Delta }}(t)=-a(t)x^{\sigma }(t)+\lambda b(t)f(t,x(h(t))),t\in \mathbb{T}$$

using the cone theory techniques. We extend and unify periodic differential, difference, h-difference and q-difference equations and more by a new periodicity concept on time scales.

MSC:34N05, 39A12, 35B10.

1 Introduction

Functional differential equations include many mathematical ecological and population models, such as the Lasota-Wazewska model [1–6], Nicholson’s blowflies model [1, 4, 7–10], the model for blood cell production [1, 4, 9, 11]etc. Particularly, since the periodic variation of the environment plays an important role in many biological and ecological systems, many researchers have been interested in studying the existence of periodic solutions of the above models. Chow [12], Freedman and Wy [13], Hadeler and Tomiuk [14], Kuang [8], Wang [15], Weng and Sun [16] and many others studied the existence of at least one and at least two positive periodic solutions of nonlinear first-order differential equations using the fixed point theorem of cone expansion and the cone compression method, the upper and lower solution method and iterative technique [17]. On the other hand, it has been observed that very few papers exist in the literature on the existence of at least three and the nonexistence of a nonnegative periodic solution for first-order differential equations. For example, see [1, 15, 18].

In fact, both continuous and discrete systems are very important in implementation and application. Therefore, the study of dynamic equations on time scales, which unifies differential, difference, h-difference, q-differences equations and more, has received much attention; see [19–23]. The theory of dynamic equations on times-scales was introduced by Stefan Hilger in 1988 [24]. There are only a few results concerning periodic solutions of dynamic equations on time scales such as in [20, 25]. In these papers, all periodic time scales must be unbounded above and below, but there are many time scales that do not satisfy this condition such as $\overline{q^{\mathbb{Z}}}=\{q^n:n\in \mathbb{Z}\}\cup \{0\}$ and $\sqrt{\mathbb{N}}=\{\sqrt{n}:n\in \mathbb{N}\}$. Adıvar introduced a new periodicity concept in [26] with the aid of shift operators ${\delta }_{\pm }$. With the new periodicity concept, the time scale need not be closed under the operation $t\pm \omega$ for a fixed $\omega >0$. There are only few existence results related with the new periodicity; see [19].

Let $\mathbb{T}$ be a periodic time scale in shifts ${\delta }_{\pm }$ with period $P\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$ and $t_0\in \mathbb{T}$. We are concerned with the existence, multiplicity and nonexistence of periodic solutions in shifts ${\delta }_{\pm }$ for the nonlinear dynamic equation

$$x^{\mathrm{\Delta }}(t)=-a(t)x^{\sigma }(t)+\lambda b(t)f(t,x(h(t))),t\in \mathbb{T},$$

(1)

where $a\in C(\mathbb{T},(0,\mathrm{\infty }))$ is Δ-periodic in shifts ${\delta }_{\pm }$ with period T and $a\in \mathcal{R}$, $\lambda >0$ is a positive parameter, $b\in C(\mathbb{T},[0,\mathrm{\infty }))$ is Δ-periodic in shifts with period T, ${\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\mathrm{\Delta }s>0$, $h\in C(\mathbb{T},[0,\mathrm{\infty }))$ is periodic in shifts with period T and $f\in C(\mathbb{T}\times (0,\mathrm{\infty }),(0,\mathrm{\infty }))$ is periodic in shifts ${\delta }_{\pm }$ with period T with respect to the first variable and $T\in {[P,\mathrm{\infty })}_{\mathbb{T}}$.

Hereafter, we use the notation ${[a,b]}_{\mathbb{T}}$ to indicate the time scale interval $[a,b]\cap \mathbb{T}$. The intervals ${[a,b)}_{\mathbb{T}}$, ${(a,b]}_{\mathbb{T}}$ and ${(a,b)}_{\mathbb{T}}$ are similarly defined.

In this study, we shall show that the number of positive periodic solutions in shifts ${\delta }_{\pm }$ of (1) can be determined by the asymptotic behaviors of the quotient of $\frac{f(t,x)}{x}$ at zero and infinity. We shall organize this paper as follows. In Section 2, we state some facts about exponential function on time scales, the new periodicity concept for time scales and some important theorems which will be needed to show the existence and nonexistence of periodic solutions in shifts ${\delta }_{\pm }$. Besides these, in Section 2, we give some lemmas about the exponential function and the graininess function with shift operators. We also present some lemmas to be used later. Finally, we state our main results and give their proofs in Section 3 by using the Krasnosel’skiĭ fixed point theorem.

2 Preliminaries

In this section, we mention some definitions, lemmas and theorems from calculus on time scales which can be found in [18, 27]. Next, we state some definitions, lemmas and theorems about the shift operators and the new periodicity concept for time scales which can be found in [26].

Definition 2.1 [27]

A function $p:\mathbb{T}\to \mathbb{R}$ is said to be regressive provided $1+\mu (t)p(t)\ne 0$ for all $t\in {\mathbb{T}}^{\kappa }$, where $\mu (t)=\sigma (t)-t$. The set of all regressive rd-continuous functions $\phi :\mathbb{T}\to \mathbb{R}$ is denoted by ℛ, while the set ${\mathcal{R}}^{+}$ is given by ${\mathcal{R}}^{+}=\{\phi \in \mathcal{R}:1+\mu (t)\phi (t)>0\text{ for all }t\in \mathbb{T}\}$.

Let $\phi \in \mathcal{R}$ and $\mu (t)>0$ for all $t\in \mathbb{T}$. The exponential function on $\mathbb{T}$ is defined by

$$e_{\phi }(t,s)=exp\left({\int }_s^t{\zeta }_{\mu (r)}(\phi (r))\mathrm{\Delta }r\right),$$

(2)

where ${\zeta }_{\mu (s)}$ is the cylinder transformation given by

$${\zeta }_{\mu (r)}(\phi (r)):=\{\begin{array}{ll}\frac{1}{\mu (r)}Log(1+\mu (r)\phi (r))& \text{if }\mu (r)>0;\\ \phi (r)& \text{if }\mu (r)=0.\end{array}$$

(3)

Also, the exponential function $y(t)=e_p(t,s)$ is the solution to the initial value problem $y^{\mathrm{\Delta }}=p(t)y$, $y(s)=1$. Other properties of the exponential function are given in the following lemma ([27], Theorem 2.36).

Lemma 2.1 Let $p,q\in \mathcal{R}$. Then

• i. $e_0(t,s)\equiv 1$ and $e_p(t,t)\equiv 1$;

• ii. $e_p(\sigma (t),s)=(1+\mu (t)p(t))e_p(t,s)$;

• iii. $\frac{1}{e_p(t,s)}=e_{\ominus }(t,s)$, where $\ominus p(t)=-\frac{p(t)}{1+\mu (t)p(t)}$;

• iv. $e_p(t,s)=\frac{1}{e_p(s,t)}=e_{\ominus p}(s,t)$;

• v. $e_p(t,s)e_p(s,r)=e_p(t,r)$;

• vi. $e_p(t,s)e_q(t,s)=e_{p\oplus q}(t,s)$;

• vii. $\frac{e_p(t,s)}{e_q(t,s)}=e_{p\ominus q}(t,s)$;

• viii. ${(\frac{1}{e_p(\cdot ,s)})}^{\mathrm{\Delta }}=-\frac{p(t)}{e_p^{\sigma }(\cdot ,s)}$.

The following definitions, lemmas, corollaries and examples are about the shift operators and the new periodicity concept for time scales which can be found in [26].

Definition 2.2 [26]

Let ${\mathbb{T}}^{\ast }$ be a nonempty subset of the time scale $\mathbb{T}$ including a fixed number $t_0\in {\mathbb{T}}^{\ast }$ such that there exist operators ${\delta }_{\pm }:{[t_0,\mathrm{\infty })}_{\mathbb{T}}\times {\mathbb{T}}^{\ast }\to {\mathbb{T}}^{\ast }$ satisfying the following properties:

• (P.1) The functions ${\delta }_{\pm }$ are strictly increasing with respect to their second arguments, i.e., if

  $$(T_0,t),(T_0,u)\in {\mathcal{D}}_{\pm }:=\{(s,t)\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}\times {\mathbb{T}}^{\ast }:{\delta }_{\mp }(s,t)\in {\mathbb{T}}^{\ast }\},$$

  then

  $$T_0\le t<u\text{implies}{\delta }_{\pm }(T_0,t)<{\delta }_{\pm }(T_0,u);$$

• (P.2) If $(T_1,u),(T_2,u)\in {\mathcal{D}}_{-}$ with $T_1<T_2$, then ${\delta }_{-}(T_1,u)>{\delta }_{-}(T_2,u)$, and if $(T_1,u),(T_2,u)\in D_{+}$ with $T_1<T_2$, then ${\delta }_{+}(T_1,u)<{\delta }_{+}(T_2,u)$;

• (P.3) If $t\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$, then $(t,t_0)\in D_{+}$ and ${\delta }_{+}(t,t_0)=t$. Moreover, if $t\in {\mathbb{T}}^{\ast }$, then $(t_0,t)\in D_{+}$ and ${\delta }_{+}(t_0,t)=t$ holds;

• (P.4) If $(s,t)\in D_{\pm }$, then $(s,{\delta }_{\pm }(s,t))\in D_{\mp }$ and ${\delta }_{\mp }(s,{\delta }_{\pm }(s,t))=t$, respectively;

• (P.5) If $(s,t)\in D_{\pm }$ and $(u,{\delta }_{\pm }(s,t))\in D_{\mp }$, then $(s,{\delta }_{\mp }(u,t))\in D_{\pm }$ and ${\delta }_{\mp }(u,{\delta }_{\pm }(s,t))={\delta }_{\pm }(s,{\delta }_{\mp }(u,t))$, respectively.

Then the operators ${\delta }_{-}$ and ${\delta }_{+}$ associated with $t_0\in {\mathbb{T}}^{\ast }$ (called the initial point) are said to be backward and forward shift operators on the set ${\mathbb{T}}^{\ast }$, respectively. The variable $s\in {[t_0,\mathrm{\infty })}_{\mathbb{T}}$ in ${\delta }_{\pm }(s,t)$ is called the shift size. The value ${\delta }_{+}(s,t)$ and ${\delta }_{-}(s,t)$ in ${\mathbb{T}}^{\ast }$ indicate s units translation of the term $t\in {\mathbb{T}}^{\ast }$ to the right and left, respectively. The sets ${\mathcal{D}}_{\pm }$ are the domains of the shift operator ${\delta }_{\pm }$, respectively. Hereafter, ${\mathbb{T}}^{\ast }$ is the largest subset of the time scale $\mathbb{T}$ such that the shift operators ${\delta }_{\pm }:{[t_0,\mathrm{\infty })}_{\mathbb{T}}\times {\mathbb{T}}^{\ast }\to {\mathbb{T}}^{\ast }$ exist.

Definition 2.3 [26] (Periodicity in shifts)

Let $\mathbb{T}$ be a time scale with the shift operators ${\delta }_{\pm }$ associated with the initial point $t_0\in {\mathbb{T}}^{\ast }$. The time scale $\mathbb{T}$ is said to be periodic in shift ${\delta }_{\pm }$ if there exists $p\in {(t_0,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$ such that $(p,t)\in D_{\pm }$ for all $t\in {\mathbb{T}}^{\ast }$. Furthermore, if

$$P:=inf\{p\in {(t_0,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}:(p,t)\in D_{\pm },\mathrm{\forall }t\in {\mathbb{T}}^{\ast }\}\ne t_0,$$

then P is called the period of the time scale $\mathbb{T}$.

Example 2.1 [26]

The following time scales are periodic in the sense of shift operators given in Definition 2.3.

• i. ${\mathbb{T}}_1=\{\pm n^2:n\in \mathbb{Z}\}$, ${\delta }_{\pm }(P,t)=\{\begin{array}{ll}{(\sqrt{t}\pm \sqrt{P})}^2,& t>0;\\ \pm P,& t=0;\\ -{(\sqrt{-t}\pm \sqrt{P})}^2,& t<0;\end{array}$ $P=1$, $t_0=0$.

• ii. ${\mathbb{T}}_2=\overline{q^{\mathbb{Z}}}$, ${\delta }_{\pm }(P,t)=P^{\pm 1}t$, $P=q$, $t_0=1$.

• iii. ${\mathbb{T}}_3=\overline{{\bigcup }_{n\in \mathbb{Z}}[2^{2n},2^{2n+1}]}$, ${\delta }_{\pm }(P,t)=P^{\pm 1}t$, $P=4$, $t_0=1$.

• iv. ${\mathbb{T}}_4=\{\frac{q^n}{1+q^n}:q>1\text{ is constant and }n\in \mathbb{Z}\}\cup \{0,1\}$,

  $${\delta }_{\pm }(P,t)=\frac{q^{(\frac{ln(\frac{t}{1-t})\pm ln(\frac{P}{1-P})}{lnq})}}{1+q^{(\frac{ln(\frac{t}{1-t})\pm ln(\frac{P}{1-P})}{lnq})}},P=\frac{q}{1+q}.$$

Notice that the time scale ${\mathbb{T}}_4$ in Example 2.1 is bounded above and below and ${\mathbb{T}}_4^{\ast }=\{\frac{q^n}{1+q^n}:q>1\text{ is constant and }n\in \mathbb{Z}\}$.

Remark 2.1 [26]

Let $\mathbb{T}$ be a time scale that is periodic in shifts with the period P. Thus, by (P.4) of Definition 2.2, the mapping ${\delta }_{+}^P:{\mathbb{T}}^{\ast }\to {\mathbb{T}}^{\ast }$ defined by ${\delta }_{+}^P(t)={\delta }_{+}(P,t)$ is surjective. On the other hand, by (P.1) of Definition 2.2, shift operators ${\delta }_{\pm }$ are strictly increasing in their second arguments. That is, the mapping ${\delta }_{+}^P(t)={\delta }_{+}(P,t)$ is injective. Hence, ${\delta }_{+}^P$ is an invertible mapping with the inverse ${({\delta }_{+}^P)}^{-1}={\delta }_{-}^P$ defined by ${\delta }_{-}^P(t):={\delta }_{-}(P,t)$.

We assume that $\mathbb{T}$ is a periodic time scale in shift ${\delta }_{\pm }$ with period P. The operators ${\delta }_{\pm }^P:{\mathbb{T}}^{\ast }\to {\mathbb{T}}^{\ast }$ are commutative with the forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ given by $\sigma (t):=inf\{s\in \mathbb{T}:s>t\}$. That is, $({\delta }_{\pm }^P\circ \sigma )(t)=(\sigma \circ {\delta }_{\pm }^P)(t)$ for all $t\in {\mathbb{T}}^{\ast }$.

Lemma 2.2 [26]

The mapping ${\delta }_{+}^P:{\mathbb{T}}^{\ast }\to {\mathbb{T}}^{\ast }$ preserves the structure of the points in ${\mathbb{T}}^{\ast }$. That is,

$$\begin{array}{c}\sigma (t)=t\mathit{\text{implies}}\sigma ({\delta }_{+}(P,t))={\delta }_{+}(P,t)\mathit{\text{and}}\hfill \\ \sigma (t)>t\mathit{\text{implies}}\sigma ({\delta }_{+}(P,t))>{\delta }_{+}(P,t).\hfill \end{array}$$

Corollary 2.1 [26]

${\delta }_{+}(P,\sigma (t))=\sigma ({\delta }_{+}(P,t))$ and ${\delta }_{-}(P,\sigma (t))=\sigma ({\delta }_{-}(P,t))$ for all $t\in {\mathbb{T}}^{\ast }$.

Definition 2.4 [26] (Periodic function in shift ${\delta }_{\pm }$)

Let $\mathbb{T}$ be a time scale that is periodic in shifts ${\delta }_{\pm }$ with the period P. We say that a real-valued function f defined on ${\mathbb{T}}^{\ast }$ is periodic in shifts ${\delta }_{\pm }$ if there exists $T\in {[P,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$ such that

$$(T,t)\in D_{\pm }\text{and}f({\delta }_{\pm }^T(t))=f(t)\text{for all }t\in {\mathbb{T}}^{\ast },$$

(4)

where ${\delta }_{\pm }^T:={\delta }_{\pm }(T,t)$. The smallest number $T\in {[P,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$ such that (4) holds is called the period of f.

Definition 2.5 [26] (Δ-periodic function in shifts ${\delta }_{\pm }$)

Let $\mathbb{T}$ be a time scale that is periodic in shifts ${\delta }_{\pm }$ with the period P. We say that a real-valued function f defined on ${\mathbb{T}}^{\ast }$ is Δ-periodic in shifts ${\delta }_{\pm }$ if there exists $T\in {[P,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$ such that

$$(T,t)\in D_{\pm }\text{for all }t\in {\mathbb{T}}^{\ast },$$

(5)

$$\text{the shifts }{\delta }_{\pm }^T\text{ are }\mathrm{\Delta }\text{-differentiable with rd-continuous derivatives}$$

(6)

and

$$f({\delta }_{\pm }^T(t)){\delta }_{\pm }^{\mathrm{\Delta }T}=f(t)\text{for all }t\in {\mathbb{T}}^{\ast },$$

(7)

where ${\delta }_{\pm }^T:={\delta }_{\pm }(T,t)$. The smallest number $T\in {[P,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$ such that (5)-(7) hold is called the period of f.

Notice that Definition 2.4 and Definition 2.5 give the classic periodicity definition on time scales whenever ${\delta }_{\pm }^T:=t\pm T$ are the shifts satisfying the assumptions of Definition 2.4 and Definition 2.5.

Now, we give a theorem which is the substitution rule on periodic time scales in shifts ${\delta }_{\pm }$ which can be found in [26].

Theorem 2.1 Let $\mathbb{T}$ be a time scale that is periodic in shifts ${\delta }_{\pm }$ with period $P\in {[t_0,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$, and let f be a Δ-periodic function in shifts ${\delta }_{\pm }$ with the period $T\in {[P,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$. Suppose that $f\in {\mathcal{C}}_{\mathrm{rd}}(\mathbb{T})$, then

$${\int }_{t_0}^{t}f(s)\mathrm{\Delta }s={\int }_{{\delta }_{\pm }^T(t_0)}^{{\delta }_{\pm }^T(t)}f(s)\mathrm{\Delta }s.$$

(8)

We give some interesting properties of the exponential functions $e_p(t,t_0)$ and shift operators on time scales which can be found in [19].

Lemma 2.3 Let $\mathbb{T}$ be a time scale that is periodic in shifts ${\delta }_{\pm }$ with the period P. Suppose that the shifts ${\delta }_{\pm }^T$ are Δ-differentiable on $t\in {\mathbb{T}}^{\ast }$ where $T\in {[P,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$. Then the graininess function $\mu :\mathbb{T}\to [0,\mathrm{\infty })$ satisfies

$$\mu ({\delta }_{\pm }^T(t))={\delta }_{\pm }^{\mathrm{\Delta }T}(t)\mu (t).$$

Lemma 2.4 Let $\mathbb{T}$ be a time scale that is periodic in shifts ${\delta }_{\pm }$ with the period P. Suppose that the shifts ${\delta }_{\pm }^T$ are Δ-differentiable on $t\in {\mathbb{T}}^{\ast }$ where $T\in {[P,\mathrm{\infty })}_{{\mathbb{T}}^{\ast }}$ and $p\in \mathcal{R}$ is Δ-periodic in shifts ${\delta }_{\pm }$ with the period T. Then

• i. $e_p({\delta }_{\pm }^T(t),{\delta }_{\pm }^T(t_0))=e_p(t,t_0)$ for $t,t_0\in {\mathbb{T}}^{\ast }$,

• ii. $e_p({\delta }_{\pm }^T(t),\sigma ({\delta }_{\pm }^T(s)))=e_p(t,\sigma (s))=\frac{e_p(t,s)}{1+\mu (t)p(t)}$ for $t,s\in {\mathbb{T}}^{\ast }$.

We define

$$P_T=\{x\in \mathcal{C}(\mathbb{T},\mathbb{R}):x({\delta }_{+}^T(t))=x(t)\},$$

where $\mathcal{C}(\mathbb{T},\mathbb{R})$ is the space of all real-valued continuous functions endowed with the norm

$$\parallel x\parallel =\underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}|x(t)|,$$

then $P_T$ is a Banach space.

Lemma 2.5 [19]

Let $x\in P_T$. Then $\parallel x^{\sigma }\parallel$ exists and $\parallel x^{\sigma }\parallel =\parallel x\parallel$.

Lemma 2.6 $x(t)\in P_T$ is a solution of (1) if and only if

$$x(t)=\lambda {\int }_t^{{\delta }_{+}^T(t)}G(t,s)b(s)f(s,x(h(s)))\mathrm{\Delta }s,$$

where $G(t,s)=\frac{e_{a(s)}(s,t)}{e_{a(t)}({\delta }_{+}^T(t),t)-1}$ is the Green’s function.

Proof Let $x(t)\in P_T$ be a solution of (1). We can rewrite equation (1) as

$$x^{\mathrm{\Delta }}(t)+a(t)x^{\sigma }(t)=\lambda b(t)f(t,x(h(t))).$$

Multiply both sides of the above equation by $e_{a(t)}(t,t_0)$ and then integrate from t to ${\delta }_{+}^T(t)$ to obtain

$${\int }_t^{{\delta }_{+}^T(t)}{[x(s)e_{a(s)}(s,t_0)]}^{\mathrm{\Delta }}\mathrm{\Delta }s=\lambda {\int }_t^{{\delta }_{+}^T(t)}{e}_{a(s)}(s,t_0)b(s)f(s,x(h(s)))\mathrm{\Delta }s.$$

We arrive at

$$[e_{a(t)}({\delta }_{+}^T(t),t_0)-e_{a(t)}(t,t_0)]x(t)=\lambda {\int }_t^{{\delta }_{+}^T(t)}{e}_{a(s)}(s,t_0)b(s)f(s,x(h(s)))\mathrm{\Delta }s.$$

Dividing both sides of the above equation by $e_{a(t)}(t,t_0)$ and using Lemma 2.1, we have

$$x(t)[e_{a(t)}({\delta }_{+}^T(t),t)-1]=\lambda {\int }_t^{{\delta }_{+}^T(t)}{e}_{a(s)}(s,t)b(s)f(s,x(h(s)))\mathrm{\Delta }s.$$

(9)

We get

$$x(t)=\lambda {\int }_t^{{\delta }_{+}^T(t)}\frac{e_{a(s)}(s,t)}{e_{a(t)}({\delta }_{+}^T(t),t)-1}b(s)f(s,x(h(s)))\mathrm{\Delta }s.$$

Thus, the proof is complete. □

It is easy to verify that the Green’s function $G(t,s)$ satisfies the property

$$0<\frac{1}{\xi -1}\le G(t,s)\le \frac{\xi }{\xi -1}\text{for }s\in {[t,{\delta }_{+}^T(t)]}_{\mathbb{T}},$$

(10)

where $\xi =e_{a(t_0)}({\delta }_{+}^T(t_0),t_0)$ and

$$G({\delta }_{+}^T(t),{\delta }_{+}^T(s))=G(t,s)\text{for }t\in {\mathbb{T}}^{\ast },s\in {[t,{\delta }_{+}^T(t)]}_{\mathbb{T}}.$$

(11)

Define K, a cone in $P_T$, by

$$K=\{x\in P_T:x(t)\ge \frac{1}{\xi }\parallel x\parallel ,\mathrm{\forall }t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}\}$$

and an operator $A_{\lambda }:K\to P_T$ by

$$(A_{\lambda }x)(t)=\lambda {\int }_t^{{\delta }_{+}^T(t)}G(t,s)b(s)f(s,x(h(s)))\mathrm{\Delta }s.$$

Define

$$B:={\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\mathrm{\Delta }s,C:=\underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{sup}a(t).$$

(12)

Lemma 2.7 $A_{\lambda }(K)\subset K$ and $A_{\lambda }:K\to K$ is compact and continuous.

Proof By using Theorem 2.1, for $x\in K$, we have

$$\begin{array}{rcl}(A_{\lambda }x)({\delta }_{+}^T(t))& =& \lambda {\int }_{{\delta }_{+}^T(t)}^{{\delta }_{+}^T({\delta }_{+}^T(t))}G({\delta }_{+}^T(t),s)b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ =& \lambda {\int }_t^{{\delta }_{+}^T(t)}G({\delta }_{+}^T(t),{\delta }_{+}^T(s))b({\delta }_{+}^T(s)){\delta }_{+}^{T\mathrm{\Delta }}(s)f({\delta }_{+}^T(s),x\left(h({\delta }_{+}^T(s))\right))\mathrm{\Delta }s\\ =& \lambda {\int }_t^{{\delta }_{+}^T(t)}G(t,s)b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ =& (A_{\lambda }x)(t).\end{array}$$

One can show that for $x\in K$, we have

$$\begin{array}{rcl}{A}_{\lambda }x(t)& \ge & \lambda {\int }_t^{{\delta }_{+}^T(t)}\frac{1}{\xi -1}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ =& \frac{1}{\xi }\lambda {\int }_{t_0}^{{\delta }_{+}^T(t_0)}\frac{\xi }{\xi -1}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \ge & \frac{1}{\xi }\parallel A_{\lambda }x\parallel .\end{array}$$

Therefore, $A_{\lambda }(K)\subset K$. We will prove that $A_{\lambda }$ is continuous and compact. Firstly, we will consider the continuity of $A_{\lambda }$. Let $x_n\in K$ and $\parallel x_n-x\parallel \to 0$ as $n\to \mathrm{\infty }$, then $x\in K$ and $|x_n(t)-x(t)|\to 0$ as $n\to \mathrm{\infty }$ for any $t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$. Because of the continuity of f, for any $t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$ and ${ϵ}^{\ast }>0$, we have

$$\begin{array}{rcl}\parallel A_{\lambda }{x}_n-A_{\lambda }x\parallel & =& \underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}|A_{\lambda }{x}_n-A_{\lambda }x|\\ \le & \underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}\lambda {\int }_t^{{\delta }_{+}^T(t)}G(t,s)b(s)|f(s,x_n(h(s)))-f(s,x(h(s)))|\mathrm{\Delta }s\\ \le & \lambda {ϵ}^{\ast }\frac{\xi }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\mathrm{\Delta }s=ϵ,\end{array}$$

where ${ϵ}^{\ast }={[\lambda B\frac{\xi }{\xi -1}]}^{-1}ϵ$. Thus $A_{\lambda }$ is continuous on K.

Next, we prove that $A_{\lambda }$ is a compact operator. It is equal to proving that $A_{\lambda }$ maps bounded sets into relatively compact sets.

Let $S\subset K$ be an arbitrary bounded set in K, then there exists a number $R>0$ such that $\parallel x\parallel <R$ for any $x\in K$. We prove that $\overline{A_{\lambda }S}$ is compact. In fact, for any ${\{x_n\}}_{n\in \mathbb{N}}\in K$ and $t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$, we have

$$\begin{array}{rcl}\parallel A_{\lambda }{x}_n\parallel & \le & \underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}\lambda {\int }_t^{{\delta }_{+}^T(t)}G(t,s)b(s)\left|f(s,x_n(h(s)))\right|\mathrm{\Delta }s\\ \le & \lambda \frac{\xi }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\left|f(s,x_n(h(s)))\right|\mathrm{\Delta }s:=D\end{array}$$

and

$$\begin{array}{rcl}\parallel A_{\lambda }^{\mathrm{\Delta }}{x}_n\parallel & \le & \underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}|a(t)A_{\lambda }{x}_n(t)+\lambda (a(t)+\frac{a(t)}{1+\mu (t)a(t)})\\ \times {\int }_t^{{\delta }_{+}^T(t)}G(t,s)b(s)f(s,x_n(h(s)))\mathrm{\Delta }s+\frac{\lambda b(t)f(t,x_n(h(t)))}{1+\mu (t)a(t)}|\\ \le & C\parallel A_{\lambda }{x}_n\parallel +2\lambda \frac{\xi }{\xi -1}C{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\left|f(s,x_n(h(s)))\right|\mathrm{\Delta }s\\ +\lambda \underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}|b(t)|\left|f(t,x_n(h(t)))\right|\\ =& 3CD+\lambda \underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}|b(t)|\left|f(t,x_n(h(t)))\right|:=L,\end{array}$$

which imply that ${\{A_{\lambda }{x}_n\}}_{n\in \mathbb{N}}$ and ${\{A_{\lambda }^{\mathrm{\Delta }}{x}_n\}}_{n\in \mathbb{N}}$ are uniformly bounded on ${[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$. There exists a subsequence of ${\{A_{\lambda }{x}_n\}}_{n\in \mathbb{N}}$ converging uniformly on ${[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$, namely, $\overline{A_{\lambda }S}$ is compact. The proof is complete. □

Lemma 2.8 The existence of positive periodic solutions in shifts ${\delta }_{\pm }$ of (1) is equivalent to the existence of fixed point problem of $A_{\lambda }$ in K.

The proof of Lemma 2.8 is straightforward and hence omitted.

3 Main result

In this section, we consider the existence of one or two positive T-periodic solutions in ${\delta }_{\pm }$ of (1). Let us define

$$\begin{array}{c}\underline{f_0}=\underset{x\to 0^{+}}{lim}\underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{min}\frac{f(t,x)}{x},\overline{f_0}=\underset{x\to 0^{+}}{lim}\underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}\frac{f(t,x)}{x},\hfill \\ \underline{f_{\mathrm{\infty }}}=\underset{x\to \mathrm{\infty }}{lim}\underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{min}\frac{f(t,x)}{x},\overline{f_{\mathrm{\infty }}}=\underset{x\to \mathrm{\infty }}{lim}\underset{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}{max}\frac{f(t,x)}{x}.\hfill \end{array}$$

To prove the results, we will use the following theorem which can be found in Krasnosel’skiĭ’s book [28].

Theorem 3.1 (Guo-Krasnoselskiĭ fixed point theorem)

Let X be a Banach space, $K\subset X$ be a cone, and suppose that ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are open, bounded subsets of X with $0\in {\mathrm{\Omega }}_1$ and $\overline{{\mathrm{\Omega }}_1}\subset {\mathrm{\Omega }}_2$. Suppose further that $A:K\cap (\overline{{\mathrm{\Omega }}_2}\setminus {\mathrm{\Omega }}_1)\to K$ is a completely continuous operator such that either

• (i) $\parallel Au\parallel \le \parallel u\parallel$ for $u\in K\cap \partial {\mathrm{\Omega }}_1$, $\parallel Au\parallel \ge \parallel u\parallel$ for $u\in K\cap \partial {\mathrm{\Omega }}_2$, or

• (ii) $\parallel Au\parallel \ge \parallel u\parallel$ for $u\in K\cap \partial {\mathrm{\Omega }}_1$, $\parallel Au\parallel \le \parallel u\parallel$ for $u\in K\cap \partial {\mathrm{\Omega }}_2$

holds. Then A has a fixed point in $K\cap (\overline{{\mathrm{\Omega }}_2}\setminus {\mathrm{\Omega }}_1)$.

Theorem 3.2 If either $\underline{f_{\mathrm{\infty }}}=\mathrm{\infty }$, $\overline{f_0}=0$ or $\underline{f_0}=\mathrm{\infty }$, $\overline{f_{\mathrm{\infty }}}=0$ holds, then equation (1) has a positive T-periodic solution x in shifts ${\delta }_{\pm }$ for

$$\frac{\xi -1}{\xi B}\le \lambda \le \frac{\xi -1}{B}.$$

(13)

Proof At first, in view of $\underline{f_{\mathrm{\infty }}}={lim}_{x\to \mathrm{\infty }}{min}_{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}\frac{f(t,x)}{x}=\mathrm{\infty }$ uniformly on ${[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$, there exists $r>0$ such that $f(t,x)\ge \mu x$ for $t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$, $x\ge r$, where $\mu \ge {\xi }^2$. We define ${\mathrm{\Omega }}_1=\{x\in P_T:\parallel x\parallel <\xi r\}$ and if $x\in \partial {\mathrm{\Omega }}_1\cap K$, then $\parallel x\parallel =\xi r$ and $r\le x\le \xi r$ for all t. We get

$$\begin{array}{rcl}\parallel A_{\lambda }x\parallel & \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\mu x(h(s))\mathrm{\Delta }s\\ \ge & \frac{\xi r}{B}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\mathrm{\Delta }s\\ =& \xi r=\parallel x\parallel \end{array}$$

and so $\parallel A_{\lambda }x\parallel \ge \parallel x\parallel$ for all $x\in K\cap \partial {\mathrm{\Omega }}_1$.

Next we use the assumption $\overline{f_0}={lim}_{x\to 0^{+}}{max}_{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}\frac{f(t,x)}{x}=0$ uniformly on ${[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$. We can choose $R>\xi r>0$ large enough such that

$$f(t,x)\le \eta x,0\le x\le R,t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}},$$

where $\eta \in (0,\frac{1}{\xi }]$. Then if ${\mathrm{\Omega }}_2$ is the ball in K centered at the origin with radius R and if $x\in K\cap \partial {\mathrm{\Omega }}_2$, then we have

$$\begin{array}{rcl}\parallel A_{\lambda }x\parallel & \le & \frac{\lambda \xi }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \le & \frac{\lambda \xi }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\eta x(h(s))\mathrm{\Delta }s\\ \le & \frac{1}{B}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)R\mathrm{\Delta }s\\ =& R=\parallel x\parallel \end{array}$$

and so $\parallel A_{\lambda }x\parallel \ge \parallel x\parallel$ for all $x\in K\cap \partial {\mathrm{\Omega }}_2$. Consequently, Theorem 3.1 yields the existence of a positive T-periodic solution $x\in K\cap (\overline{{\mathrm{\Omega }}_2}\setminus {\mathrm{\Omega }}_1)$ of (1) in shifts ${\delta }_{\pm }$, that is,

$$\xi r<x(t)\le R,t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}.$$

Next, let $\underline{f_0}=\mathrm{\infty }$, $\overline{f_{\mathrm{\infty }}}=0$ hold. In view of $\underline{f_0}={lim}_{x\to 0^{+}}{min}_{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}\frac{f(t,x)}{x}=\mathrm{\infty }$, there exists $r_1>0$ such that

$$f(t,x)\ge \overline{\mu }x\text{for }t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}},0<x\le r_1,$$

where $\overline{\mu }\ge {\xi }^2$. We define ${\mathrm{\Omega }}_1=\{x\in P_T:\parallel x\parallel <r_1\}$ and if $x\in \partial {\mathrm{\Omega }}_1\cap K$, then $\parallel x\parallel =r_1$ and $\frac{r_1}{\xi }\le x\le r_1$ for all t. We get

$$\begin{array}{rcl}\parallel A_{\lambda }x\parallel & \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\overline{\mu }x(h(s))\mathrm{\Delta }s\\ \ge & \frac{r_1}{B}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\mathrm{\Delta }s\\ =& r_1=\parallel x\parallel \end{array}$$

and so $\parallel A_{\lambda }x\parallel \ge \parallel x\parallel$ for all $x\in K\cap \partial {\mathrm{\Omega }}_1$.

Next, since $\overline{f_{\mathrm{\infty }}}={lim}_{x\to \mathrm{\infty }}{max}_{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}\frac{f(t,x)}{x}=0$, there exists $r_2>0$ such that

$$f(t,x)\le \overline{\eta }x\text{for }x\ge r_2,t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}},$$

where $\overline{\eta }\in (0,\frac{1}{\xi }]$. Let $\overline{r_2}\ge max\{2r_1,\xi r_2\}$ and it follows that $x(t)\ge \frac{1}{\xi }\parallel x\parallel \ge r_2$ for $x\in \partial {\mathrm{\Omega }}_2$ and $t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}$, where ${\mathrm{\Omega }}_2=\{x\in P_T:\parallel x\parallel <\overline{r_2}\}$. For $x\in K\cap \partial {\mathrm{\Omega }}_2$, we have

$$\begin{array}{rcl}\parallel A_{\lambda }x\parallel & \le & \frac{\lambda \xi }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \le & \frac{\lambda \xi }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\overline{\eta }x(h(s))\mathrm{\Delta }s\\ \le & \frac{1}{B}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\overline{r_2}\mathrm{\Delta }s\\ =& \overline{r_2}=\parallel x\parallel ,\end{array}$$

and again we have $\parallel A_{\lambda }x\parallel \le \parallel x\parallel$ for $x\in K\cap \partial {\mathrm{\Omega }}_2$. It follows from part (ii) of Theorem 3.1 that $A_{\lambda }$ has a fixed point in $K\cap (\overline{{\mathrm{\Omega }}_2}\setminus {\mathrm{\Omega }}_1)$ and this implies that our given equation (1) has a positive T-periodic solution x in shifts ${\delta }_{\pm }$, that is,

$$r_1<x(t)\le \overline{r_2},t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}.$$

 □

Theorem 3.3 Let $\underline{f_0}=\underline{f_{\mathrm{\infty }}}=\mathrm{\infty }$ hold. Further, assume that there is a constant $R_1>0$ such that

$$f(t,x(t))\le \frac{R_1}{\xi }\mathit{\text{for }}x\in [0,R_1],t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}.$$

(14)

Then equation (1) has two positive T-periodic solutions in shifts ${\delta }_{\pm }$ for

$$\frac{\xi -1}{\xi B}\le \lambda \le \frac{\xi -1}{B}.$$

(15)

Proof At first, in view of $\underline{f_0}={lim}_{x\to 0^{+}}{min}_{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}\frac{f(t,x)}{x}=\mathrm{\infty }$, there exists $R_1>R^{\ast }>0$ such that

$$f(t,x)\ge ϵx\text{for }t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}},0<x\le R^{\ast },$$

where $ϵ\ge {\xi }^2$. Set ${\mathrm{\Omega }}_1=\{x\in P_T:\parallel x\parallel <R^{\ast }\}$. Then, for $x\in K\cap \partial {\mathrm{\Omega }}_1$, we have

$$\begin{array}{rcl}\parallel A_{\lambda }x\parallel & \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)ϵx(h(s))\mathrm{\Delta }s\\ \ge & \frac{\xi }{B}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\frac{R^{\ast }}{\xi }\mathrm{\Delta }s\\ =& R^{\ast }=\parallel x\parallel ,\end{array}$$

which implies

$$\parallel A_{\lambda }x\parallel \ge \parallel x\parallel \text{for }x\in K\cap \partial {\mathrm{\Omega }}_1.$$

(16)

Next, since $\underline{f_{\mathrm{\infty }}}={lim}_{x\to \mathrm{\infty }}{min}_{t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}}\frac{f(t,x)}{x}=\mathrm{\infty }$, then for any $\alpha \ge {\xi }^2$ there exists $R_{\ast }>R_1$ such that $f(t,x)\ge \alpha x$ for $x\ge R_{\ast }$. Set ${\mathrm{\Omega }}_2=\{x\in P_T:\parallel x\parallel <R_{\ast }\}$. For $x\in K\cap \partial {\mathrm{\Omega }}_2$, since $x\in K$, $x(t)\ge \frac{1}{\xi }\parallel x\parallel =\frac{1}{\xi }{R}_{\ast }$, we have

$$\begin{array}{rcl}\parallel A_{\lambda }x\parallel & \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \ge & \frac{\lambda }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\alpha x(h(s))\mathrm{\Delta }s\\ \ge & \frac{\xi }{B}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\frac{R_{\ast }}{\xi }\mathrm{\Delta }s\\ =& R_{\ast }=\parallel x\parallel ,\end{array}$$

which implies

$$\parallel A_{\lambda }x\parallel \ge \parallel x\parallel \text{for }x\in K\cap \partial {\mathrm{\Omega }}_2.$$

(17)

Finally, let ${\mathrm{\Omega }}_3=\{x\in P_T:\parallel x\parallel <R_1\}$. For $x\in K\cap \partial {\mathrm{\Omega }}_3$, then from (14), we have

$$\begin{array}{rcl}\parallel A_{\lambda }x\parallel & \le & \frac{\lambda \xi }{\xi -1}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)f(s,x(h(s)))\mathrm{\Delta }s\\ \le & \frac{\xi }{B}{\int }_{t_0}^{{\delta }_{+}^T(t_0)}b(s)\frac{R_1}{\xi }\mathrm{\Delta }s\\ =& R_1=\parallel x\parallel ,\end{array}$$

which implies

$$\parallel A_{\lambda }x\parallel \le \parallel x\parallel \text{for }x\in K\cap \partial {\mathrm{\Omega }}_3.$$

(18)

Hence, since $R^{\ast }<R_1<R_{\ast }$ and (16), (17) and (18), it follows from Theorem 3.1 that $A_{\lambda }$ has a fixed point $x_1$ in $K\cap (\overline{{\mathrm{\Omega }}_3}\setminus {\mathrm{\Omega }}_1)$ and a fixed point $x_2$ in $K\cap (\overline{{\mathrm{\Omega }}_2}\setminus {\mathrm{\Omega }}_3)$. Both are positive T-periodic solutions in shifts ${\delta }_{\pm }$ of equation (1) and $0<\parallel x_1\parallel <R_1<\parallel x_2\parallel$. The proof is therefore complete. □

Theorem 3.4 Let $\overline{f_0}=\overline{f_{\mathrm{\infty }}}=0$ hold. There exists a constant $R_2>0$ such that

$$f(t,x(t))\ge \xi R_2\mathit{\text{for }}x\in [\frac{1}{\xi }{R}_2,R_2],t\in {[t_0,{\delta }_{+}^T(t_0)]}_{\mathbb{T}}.$$

Then equation (1) has two positive T-periodic solutions in shifts ${\delta }_{\pm }$ for

$$\frac{\xi -1}{\xi B}\le \lambda \le \frac{\xi -1}{B}.$$

Proof It can be proved similarly to the second part of Theorem 3.2 and Theorem 3.3. □