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Existence of positive periodic solutions for abstract evolution equations
Advances in Difference Equations volume 2015, Article number: 135 (2015)
Abstract
In this paper, we discuss the existence of the positive time periodic mild solutions for the evolution equation in an ordered Banach space E, \(u'(t)+Au(t)=f(t,u(t))\), \(t\in \mathbb{R}\), where \(A:D(A)\subset E\rightarrow E\) is a closed linear operator and −A generates a positive compact semigroup \(T(t)\) (\(t\geq0\)) in E, the nonlinear function \(f:\mathbb{R}\times E\rightarrow E\) is continuous and \(f(t,x)\) is ωperiodic in t. We apply the operator semigroup theory and the LeraySchauder fixed point theorem to obtain the existence of a positive ωperiodic mild solution under the condition that the nonlinear function satisfies a linear growth condition concerning the growth exponent of the semigroup \(T(t)\) (\(t\geq0\)). In the end, an example is given to illustrate the applicability of our abstract results.
Introduction
Let E be an ordered Banach space, whose positive cone K is normal cone with normal constant N. In this paper, we discuss the existence of positive time periodic mild solutions for the evolution equation in the ordered Banach space E
where \(A:D(A)\subset E\rightarrow E\) is a closed linear operator and −A generates a positive compact semigroup \(T(t)\) (\(t\geq0\)) in E, the nonlinear mapping \(f:\mathbb{R}\times E\rightarrow E\) is continuous, and, for every \(x\in E\), \(f(t,x)\) is ωperiodic in t.
It is well known that a variety of partial differential equations with time t, such as the reactiondiffusion equation, the heat equation, the wave equation, the telegraph equation and so on, can be classified as a nonlinear evolution equation (1.1) in some Banach space E. In the nonlinear evolution equation (1.1), A corresponds to the linear partial differential operator with certain boundary conditions, f corresponds to the nonlinear term.
It is noteworthy at this point that the problem concerning periodic solutions of partial differential equations has become an important area of investigation in recent years. Specially, the periodic problems of abstract evolution equations in the form of (1.1) have been considered by several authors; see [1–11].
In [9], the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a Hilbert space H,
was studied, where \(A:D(A)\subset H\rightarrow H\) is a positive definite selfadjoint operator and A has compact resolvent. The author presented some spectral conditions for the nonlinearity \(f(t,x)\) to guarantee the existence and uniqueness by applying the Schauder fixed point theorem and the contraction mapping principle. Specially, in [10], the author established the existence and uniqueness results of periodic solutions for the linear evolution equation corresponding to (1.1) and accurately estimated the spectral radius of periodic resolvent operator. With the aid of the estimation, under the assumption that the nonlinearity \(f(t,x)\) satisfies some ordered conditions concerning the growth exponent of the semigroup \(T(t)\) (\(t\geq 0\)), the existence and uniqueness results of periodic mild solutions were obtained by applying the operator semigroup theorem and monotone iterative method.
Since in many practice models, only positive periodic solutions are significant, motivated by the papers mentioned above, we research the existence of positive ωperiodic mild solutions for (1.1) in the positive cone K. In this paper, we will use a completely different method to prove the existence of positive ωperiodic mild solutions for (1.1) under some new conditions by applying the LeraySchauder fixed point theorem in an ordered Banach space E. More precisely, the nonlinear term satisfies a linear growth condition concerning the growth exponent of the semigroup \(T(t)\) (\(t\geq0\)) or the first eigenvalue of the operator A.
For the \(C_{0}\)semigroup \(T(t)\) (\(t\geq0\)), there exist \(M>0\) and \(\gamma\in \mathbb{R}\) such that (see [12])
Let
then \(\nu_{0}\) is called the growth exponent of the semigroup \(T(t)\) (\(t\geq0\)). Furthermore, \(\nu_{0}\) can also be obtained by the following formula:
If \(C_{0}\)semigroup \(T(t)\) is continuous in the uniform operator topology for every \(t>0\) in E, it is well known that \(\nu_{0}\) can also be determined by \(\sigma(A)\) (see [13])
where −A is the infinitesimal generator of the \(C_{0}\)semigroup \(T(t)\) (\(t\geq0\)). We know that \(T(t)\) (\(t\geq0\)) is continuous in the uniform operator topology for \(t>0\) if \(T(t)\) (\(t\geq0\)) is a compact semigroup (see [1]).
Our main results are as follows:
Theorem 1.1
Let E be an ordered Banach space, whose positive cone K is a normal cone, \(A : D(A) \subset E\rightarrow E\) be a closed linear operator, and −A generate an exponentially stable positive compact semigroup \(T(t)\) (\(t\geq0\)) in E, that is, \(\nu_{0}<0\). Assume that \(f:\mathbb{R}\times K\rightarrow K\) is continuous and \(f(t,x)\) is ωperiodic in t. If the following condition is satisfied:
 (H1):

there is a constant \(c\in(0,\nu_{0})\) and a function \(h_{0}\in C_{\omega}(\mathbb{R},K)\) such that
$$f(t,x)\leq cx+h_{0}(t), \quad t\in \mathbb{R}, x\in K, $$
Theorem 1.2
Let E be an ordered Banach space, whose positive cone K is a normal cone, \(A : D(A) \subset E\rightarrow E\) be a closed linear operator, and −A generate an exponentially stable positive compact semigroup \(T(t)\) (\(t\geq0\)) in E. Assume that \(f:\mathbb{R}\times K\rightarrow K\) is continuous and \(f(t,x)\) is ωperiodic in t. If the following condition is satisfied:
 (H2):

there is a constant \(c\in(0,\nu_{0})\) such that for \(x,y\in K\) with \(y\leq x\),
$$f(t,x)f(t,y)\leq c(xy),\quad t\in \mathbb{R}, $$
Furthermore, we assume that the positive cone K is a regeneration cone. By the characteristic of positive semigroups (see [14]), for sufficiently large \(\lambda_{0}>\inf\{\operatorname{Re} \lambda\mid \lambda\in \sigma(A)\}\), we see that \(\lambda_{0}I+A\) has a positive bounded inverse operator \((\lambda_{0}I+A)^{1}\). Since \(\sigma(A)\neq \emptyset\), the spectral radius \(r((\lambda_{0}I+A)^{1})=\frac {1}{\operatorname{dist}(\lambda_{0},\sigma(A))}>0\). By the famous KreinRutmann theorem, A has the first eigenvalue \(\lambda_{1}\), which has a positive eigenfunction \(e_{1}\), and
that is, \(\nu_{0}=\lambda_{1}\). Hence, by Theorem 1.1 and Theorem 1.2, we have the following.
Corollary 1.3
Let E be an ordered Banach space, whose positive cone K is a normal regeneration cone. −A generates an exponentially stable positive compact semigroup \(T(t)\) (\(t\geq0\)) in E. Assume that \(f:\mathbb{R}\times K\rightarrow K\) is continuous and \(f(t,x)\) is ωperiodic in t. If the following condition is satisfied:
 (H1)′:

there is a constant \(c\in(0,\lambda_{1})\) and a function \(h_{0}\in C_{\omega}(\mathbb{R},K)\) such that
$$f(t,x)\leq cx+h_{0}(t), \quad t\in \mathbb{R}, x\in K, $$
Corollary 1.4
Let E be an ordered Banach space, whose positive cone K is a normal regeneration cone. −A generates an exponentially stable positive compact semigroup \(T(t)\) (\(t\geq0\)) in E. Assume that \(f:\mathbb{R}\times K\rightarrow K\) is continuous and \(f(t,x)\) is ωperiodic in t. If the following condition is satisfied:
 (H2)′:

there is a constant \(c\in(0,\lambda_{1})\) such that for \(x,y\in K\) with \(y\leq x\),
$$f(t,x)f(t,y)\leq c(xy), \quad t\in \mathbb{R}, $$
Remark 1.5
In Corollary 1.3 and Corollary 1.4, since \(\lambda_{1}\) is the first eigenvalue of A, the condition \(c<\lambda_{1}\) in (H1)′ and (H2)′ cannot be extended to \(c\leq \lambda_{1}\). Otherwise, (1.1) does not always have a mild solution. For example, \(f(t,x)=\lambda_{1}x\).
The paper is organized as follows. Section 2 provides the definitions and preliminary results to be used in theorems stated and proved in the paper. In Section 3, we apply the operator semigroup theory and the LeraySchauder fixed point theorem to prove Theorem 1.1 and Theorem 1.2. In the last section, we give an example to illustrate the applicability of the abstract results.
Preliminaries
In this section, we introduce some notions, definitions, and preliminary facts which are used through this paper.
Let J denote the infinite interval \([0,+\infty)\) and \(h:J\rightarrow E\), consider the initial value problem of the linear evolution equation
It is well known [12], Chapter 4, Theorem 2.9, when \(x_{0}\in D(A)\) and \(h\in C^{1}(J,E)\), the initial value problem (2.1) has a unique classical solution \(u\in C^{1}(J,E)\cap C(J,E_{1})\) expressed by
where \(E_{1}=D(A)\) is Banach space with the graph norm \(\\cdot\_{1} = \\cdot\+\A\cdot \\). Generally, for \(x_{0}\in E\) and \(h\in C(J,E)\), the function u given by (2.2) belongs to \(C(J,E)\) and it is called a mild solution of the linear evolution equation (2.1).
Let \(C_{\omega}(\mathbb{R},E)\) denote the Banach space \(\{u\in C(\mathbb{R},E)\mid u(t+\omega)=u(t), t\in \mathbb{R}\} \) endowed with the maximum norm \(\u\_{C}=\max_{t\in[0,\omega]}\ u(t)\\). Evidently, \(C_{\omega}(\mathbb{R},E)\) is also an ordered Banach space with the partial order ‘≤’ induced by the positive cone \(K_{C}=\{u\in C_{\omega}(\mathbb{R},E)\mid u(t)\geq\theta, t\in \mathbb{R}\}\) and \(K_{C}\) is also normal with the normal constant N.
Given \(h\in C_{\omega}(\mathbb{R},E)\), for the following linear evolution equation corresponding to (1.1):
we have the following result.
Lemma 2.1
([10])
If −A generates an exponentially stable positive \(C_{0}\)semigroup \(T(t)\) (\(t\geq0\)) in E, then for \(h\in C_{\omega}(\mathbb{R},E)\), for the linear evolution equation (2.3) there exists a unique ωperiodic mild solution u, which can be expressed by
and the solution operator \(P:C_{\omega}(\mathbb{R},E)\rightarrow C_{\omega }(\mathbb{R},E)\) is a positive bounded linear operator with the spectral radius \(r(P)\leq\frac{1}{\nu_{0}}\).
Proof
For any \(\nu\in(0,\nu_{0})\), there exists \(M>0\) such that
In E, define the equivalent norm \(\cdot\) by
then \(\x\\leqx\leq M\x\\). By \(T(t)\) we denote the norm of \(T(t)\) in \((E,\cdot)\), then for \(t\geq0\), it is easy to obtain \(T(t)< e^{\nu t}\). Hence, \((IT(\omega))\) has a bounded inverse operator
and its norm satisfies
Set
then the mild solution \(u(t)\) of the linear initial value problem (2.1) given by (2.2) satisfies the periodic boundary condition \(u(0)=u(\omega )=x_{0}\). For \(t\in\mathbb{R}^{+}\), by (2.2) and the properties of the semigroup \(T(t)\) (\(t\geq0\)), we have
Therefore, the ωperiodic extension of u on \(\mathbb{R}\) is a unique ωperiodic mild solution of (2.3). By (2.2) and (2.8), the ωperiodic mild solution can be expressed by
Evidently, by the positivity of the semigroup \(T(t)\) (\(t\geq0\)), we can see that \(P:C_{\omega}(\mathbb{R},E)\rightarrow C_{\omega}(\mathbb{R},E)\) is a positive bounded linear operator. By (2.7) and (2.9), we have
which implies that \(P\leq\frac{1}{\nu}\). Therefore, \(r(P)\leq P\leq\frac{1}{\nu}\). Hence, by the arbitrariness of \(\nu\in (0,\nu_{0})\), we have \(r(P)\leq\frac{1}{\nu_{0}}\). This completes the proof of Lemma 2.1. □
In the proof of our main results, we need the following fixed point theorem.
Lemma 2.2
(LeraySchauder fixed point theorem [15])
Let Ω be a convex subset of Banach space E with \(\theta\in\Omega\), and let \(Q: \Omega\rightarrow\Omega\) be a compact operator. If the set \(\{u\in\Omega\mid u=\eta Qu, 0<\eta<1\}\) is bounded, then Q has a fixed point in Ω.
Proof of the main results
Proof of Theorem 1.1
Evidently, the normal cone \(K_{C}\) is a convex subset of the Banach space \(C_{\omega}(\mathbb{R},E)\) and \(\theta\in K_{C}\). Consider the operator Q defined by
where
From the positivity of semigroup of \(T(t)\) (\(t\geq0\)) and the conditions of Theorem 1.1, it is easy to see that \(Q:K_{C}\rightarrow K_{C} \) is well defined. It is clear that
By the definition of P, the positive ωperiodic mild solution of (1.1) is equivalent to the fixed point of the operator Q. In the following, we will prove Q has a fixed point by applying the famous LeraySchauder fixed point theorem.
At first, we prove that Q is continuous on \(K_{C}\). To this end, let \(\{u_{m}\}\subset K_{C}\) be a sequence such that \(u_{m}\rightarrow u\in K_{C} \) as \(m\rightarrow\infty\), so for every \(t\in \mathbb{R}\), \(\lim_{m\rightarrow\infty}u_{m}(t)=u(t)\). Since \(f:\mathbb{R}\times K\rightarrow K\) is continuous, for every \(t\in \mathbb{R}\), we get
By (3.3) and the Lebesgue dominated convergence theorem, for every \(t\in \mathbb{R}\), we have
where \(C=\(IT(\omega))^{1}\\). Therefore, we can conclude that
Thus, \(Q:K_{C}\rightarrow K_{C}\) is continuous.
Subsequently, we show that Q maps every bounded set in \(K_{C}\) into a bounded set. For any \(R>0\), let
For each \(u\in\Omega_{R}\), from the continuity of f, we know that there exists \(M_{1}>0\) such that
hence, we get
Therefore, \(Q(\Omega_{R})\) is bounded.
Next, we demonstrate that \(Q(\Omega_{R})\) is equicontinuous. For every \(u\in\Omega_{R}\), by the periodicity of u, we only consider it on \([0,\omega]\). Set \(0\leq t_{1}< t_{2}\leq\omega\), we get
where
It is clear that
Now, we only need to check that the \(\I_{i}\\) tend to 0 independently of \(u\in\Omega_{R}\) when \(t_{2}t_{1}\rightarrow0\), \(i=1,2,3\). From the definition of the \(I_{i}\), we can easily see
As a result, \(\ Qu(t_{2}) Qu(t_{1})\\) tends to 0 independently of \(u\in\Omega_{R}\) as \(t_{2} t_{1}\rightarrow0\), which means that \(Q(\Omega_{R})\) is equicontinuous.
Now, we prove that \((Q\Omega_{R})(t)\) is relatively compact in K for all \(t\in \mathbb{R}\). To this end, we define a set \((Q_{\varepsilon}\Omega_{R})(t)\) by
where
Then the set \((Q_{\varepsilon}\Omega_{R})(t)\) is relatively compact in K since the operator \(T(\varepsilon)\) is compact in K. For any \(u\in\Omega_{R}\) and \(t\in \mathbb{R}\), from the following inequality:
one see that the set \((Q\Omega_{R})(t)\) is relatively compact in K for all \(t\in \mathbb{R}\).
Thus, the ArzelaAscoli theorem guarantees that \(Q:K_{C}\rightarrow K_{C}\) is a compact operator.
Finally, we prove the set \(\Lambda(Q):=\{u\in K_{C}\mid u=\eta Qu, \forall 0<\eta<1\}\) is bounded. For every \(u\in K_{C}\), by (3.2) and the condition (H1), we have
Let \(u\in\Lambda(Q)\), then there is a constant \(\eta\in(0,1)\) such that \(u=\eta Qu\). Therefore, by the definition of Q, Lemma 2.1, and (3.12), we have
inductively, we can see
where \(\mathcal{P}=c^{n1}P^{n}+c^{n2}P^{n1}+\cdots+cP^{2}+P\) is a bounded linear operator, and there exists a constant \(M_{2}>0\) such that \(\\mathcal{P}\\leq M_{2}\). Hence, by the normality of the cone \(K_{C}\), we can see
From the spectral radius of the Gelfand formula \(\lim_{n\rightarrow\infty}\sqrt[n]{\P^{n}\}=r(P)=\frac{1}{\nu _{0}}\), and the condition (H1), when n is large enough, we get \(c^{n}\P^{n}\<\frac{1}{N}\), then
which implies that \(\Lambda(Q)\) is bounded. By the LeraySchauder fixed point theorem of a compact operator, the operator Q has at least one fixed point u in \(K_{C}\), which is a positive ωperiodic mild solution of (1.1). This completes the proof of Theorem 1.1. □
Proof of Theorem 1.2
From the condition (H2), it is easy to see that the condition (H1) holds. Hence by Theorem 1.1, (1.1) has positive ωperiodic mild solutions. Let \(u_{1},u_{2}\in K_{C}\) be the positive ωperiodic solutions of (1.1), then they are the fixed points of the operator \(Q=P\circ F\). Let us assume \(u_{1}\leq u_{2}\), by the definition of F and the condition (H2), we see that \(F(u_{2})(t)F(u_{1})(t)\leq c(u_{2}(t)u_{1}(t))\). So by Lemma 2.1, we obtain
By the normality of the cone \(K_{C}\), we can see that
From the proof of Theorem 1.1, when n is large enough, \(Nc^{n}\ P^{n}\<1\), so \(\u_{2}u_{1}\_{C}=0\), it follows that \(u_{2}\equiv u_{1}\). Thus, (1.1) has only one positive ωperiodic mild solution. □
Application
In this section, we present one example, which indicates how our abstract results can be applied to concrete problems.
Let \(\Omega\in \mathbb{R}^{N}\) be a bounded domain with a sufficiently smooth boundary ∂Ω. Let
be a uniformly elliptic differential operator in \(\overline{\Omega}\), whose coefficients \(a_{ij}(x)\), \(a_{j}(x)\) (\(i,j=1,\ldots,N\)) and \(a_{0}(x)\) are Höldercontinuous on \(\overline{\Omega}\) and \(a_{0}(x)\geq0\). We let \(B=B(x,D)\) be a boundary operator on ∂Ω of the form
where either \(\delta=0\) and \(b_{0}(x)\equiv1\) (Dirichlet boundary operator), or \(\delta=1\) and \(b_{0}(x)\geq0\) (regular oblique derivative boundary operator; at this point, we further assume that \(a_{0}(x)\not\equiv0\) or \(b_{0}(x)\not\equiv0\)), β is an outward pointing, nowhere tangent vector field on ∂Ω.
Let \(\lambda_{1}\) be the first eigenvalue of elliptic operator \(A(x,D)\) under the boundary condition \(Bu=0\). It is well known ([16], Theorem 1.16) that \(\lambda_{1}>0\).
Under the above assumptions, we discuss the existence and uniqueness of positive time ωperiodic solutions of the semilinear parabolic boundary value problem
where \(g:\overline{\Omega}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) is a local Höldercontinuous function which is ωperiodic in t.
Let \(E=L^{p}(\Omega)\) (\(p>1\)), \(K=\{u\in E\mid u(x)\geq0 \mbox{ a.e. } x\in \Omega\}\), then E is an ordered Banach space, whose positive cone K is a normal regeneration cone. Define an operator \(A:D(A)\subset E\rightarrow E\) by
If \(a_{0}(x)\geq0\), then −A generates an exponentially stable analytic semigroup \(T_{p}(t)\) (\(t\geq0\)) in E (see [17]). By the maximum principle of elliptic operators, we know that \((\lambda I+A)\) has a positive bounded inverse operator \((\lambda I+A)^{1}\) for \(\lambda>0\), hence \(T_{p}(t)\) (\(t\geq0\)) is a positive semigroup (see [14]). From the operator A has compact resolvent in \(L^{p}(\Omega)\), we obtain \(T_{p}(t)\) (\(t\geq0\)) is also a compact semigroup (see [12]).
Therefore, by Corollary 1.3, we have the following result.
Theorem 4.1
Assume that \(g:\overline {\Omega}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) is a local Höldercontinuous function which is ωperiodic in t and satisfies \(g(x,t,u)\geq0\) for \((x,t,u)\in(\overline{\Omega}\times \mathbb{R}\times \mathbb{R}^{+})\). If the following condition holds:
 (H3):

there is a constant \(c\in(0,\lambda_{1})\) and a function \(h_{0}\in C_{\omega}(\Omega\times \mathbb{R})\) satisfying \(h_{0}(x,t)\geq0\) such that
$$g(x,t,u)\leq cu+h_{0}(x,t),\quad (x,t)\in\Omega\times \mathbb{R}, u\geq0, $$
Proof
Let \(u(t)=u(\cdot,t)\), \(f(t,u(t))=g(\cdot ,t,u(\cdot,t))\), then the parabolic boundary value problem (4.3) can be reformulated as the abstract evolution (1.1) in E. It is easy to see that the conditions of Corollary 1.3 are satisfied. By Corollary 1.3, the parabolic boundary value problem (4.3) has a time positive ωperiodic mild solution \(u\in C_{\omega}(\mathbb{R},E)\). By the analyticity of the semigroup \(T_{p}(t)\) (\(t\geq0\)) and the regularization method used in [17], we can see that \(u\in C^{2,1}(\overline{\Omega}\times \mathbb{R})\) is a classical time ωperiodic solution of the problem (4.3). This completes the proof of the theorem. □
From Corollary 1.4 and Theorem 4.1, we obtain the uniqueness result.
Theorem 4.2
Assume that \(g:\overline {\Omega}\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}\) is a local Höldercontinuous function which is ωperiodic in t and satisfies \(g(x,t,u)\geq0\) for \((x,t,u)\in(\overline{\Omega}\times \mathbb{R}\times \mathbb{R}^{+})\). If the following condition holds:
 (H4):

there is a constant \(c\in(0,\lambda_{1})\) such that for \(0< v<w\)
$$g(x,t,w)g(x,t,v)\leq c(wv),\quad (x,t)\in\Omega\times \mathbb{R}, $$
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Acknowledgements
Research supported by NNSF of China (11261053) and NSF of Gansu Province (1208RJZA129).
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Li, Q., Li, Y. Existence of positive periodic solutions for abstract evolution equations. Adv Differ Equ 2015, 135 (2015). https://doi.org/10.1186/s1366201504355
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MSC
 34K30
 47H07
 47H08
Keywords
 abstract evolution equation
 positive periodic mild solutions
 positive compact semigroup
 the growth exponent of the semigroup
 fixed point theorem