Existence of positive periodic solutions for abstract evolution equations
 Qiang Li^{1}Email author and
 Yongxiang Li^{1}
https://doi.org/10.1186/s1366201504355
© Li and Li; licensee Springer. 2015
Received: 8 October 2014
Accepted: 3 March 2015
Published: 30 April 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.
Keywords
MSC
1 Introduction
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].
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.
Our main results are as follows:
Theorem 1.1
 (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
 (H2):

there is a constant \(c\in(0,\nu_{0})\) such that for \(x,y\in K\) with \(y\leq x\),$$f(t,x)f(t,y)\leq c(xy),\quad t\in \mathbb{R}, $$
Corollary 1.3
 (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
 (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.
2 Preliminaries
In this section, we introduce some notions, definitions, and preliminary facts which are used through this paper.
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.
Lemma 2.1
([10])
Proof
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.
3 Proof of the main results
Proof of Theorem 1.1
Thus, the ArzelaAscoli theorem guarantees that \(Q:K_{C}\rightarrow K_{C}\) is a compact operator.
Proof of Theorem 1.2
4 Application
In this section, we present one example, which indicates how our abstract results can be applied to concrete problems.
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 \(\lambda_{1}>0\).
Therefore, by Corollary 1.3, we have the following result.
Theorem 4.1
 (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
 (H4):

there is a constant \(c\in(0,\lambda_{1})\) such that for \(0< v<w\)$$g(x,t,w)g(x,t,v)\leq c(wv),\quad (x,t)\in\Omega\times \mathbb{R}, $$
Declarations
Acknowledgements
Research supported by NNSF of China (11261053) and NSF of Gansu Province (1208RJZA129).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
References
 Hino, Y, Murakami, S: Periodic solutions of linear Voltera systems. In: Differential Equations. Lect. Notes Pure Appl. Math., vol. 118, pp. 319326. Dekker, New York (1987) Google Scholar
 Vrabie, I: Periodic solutions for nonlinear evolution equations in a Banach space. Proc. Am. Math. Soc. 109(3), 653661 (1990) View ArticleMATHMathSciNetGoogle Scholar
 Liu, J: Bounded and periodic solutions of differential equations in Banach space. Appl. Math. Comput. 65, 141150 (1994) View ArticleMATHMathSciNetGoogle Scholar
 Liu, J: Bounded and periodic solutions of semilinear evolution equations. Dyn. Syst. Appl. 4, 341350 (1995) MATHMathSciNetGoogle Scholar
 Li, Y: Periodic solutions of semilinear evolution equations in Banach spaces. Acta Math. Sin. 41, 629636 (1998) (in Chinese) MATHMathSciNetGoogle Scholar
 Li, Y, Cong, F, Lin, Z, Lin, W: Periodic solutions for evolution equations. Nonlinear Anal. 36, 275293 (1999) View ArticleMATHMathSciNetGoogle Scholar
 Akrid, T, Maniar, L, Ouhinou, A: Periodic solutions for some nonautonomous semilinear boundary evolution equations. Nonlinear Anal. 71, 10001011 (2009) View ArticleMATHMathSciNetGoogle Scholar
 Zitane, M, Bensoida, C: Massera problem for nonautonomous retarded differential equations. J. Math. Anal. Appl. 402, 453462 (2013) View ArticleMATHMathSciNetGoogle Scholar
 Li, Y: Existence and uniqueness of periodic solution for a class of semilinear evolution equations. J. Math. Anal. Appl. 349, 226234 (2009) View ArticleMATHMathSciNetGoogle Scholar
 Li, Y: Existence and uniqueness of positive periodic solution for abstract semilinear evolution equations. J. Syst. Sci. Math. Sci. 25, 720728 (2005) (in Chinese) MATHGoogle Scholar
 Li, Y: Existence and asymptotic stability of periodic solution for evolution equations with delays. J. Funct. Anal. 261, 13091324 (2011) View ArticleMATHMathSciNetGoogle Scholar
 Pazy, A: Semigroup of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983) View ArticleMATHGoogle Scholar
 Triggiani, R: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52, 383403 (1975) View ArticleMATHMathSciNetGoogle Scholar
 Li, Y: The positive solutions of abstract semilinear evolution equations and their applications. Acta Math. Sin. 39, 666672 (1996) (in Chinese) MATHMathSciNetGoogle Scholar
 Deimling, K: Nonlinear Functional Analysis. Springer, New York (1985) View ArticleMATHGoogle Scholar
 Amann, H: Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problem. In: Nonlinear Operators and the Calculus of Variations. Lecture Notes in Mathematics, pp. 155. Springer, Berlin (1976) View ArticleGoogle Scholar
 Amann, H: Periodic solutions of semilinear parabolic equations. In: Cesari, L, Kannan, R, Weinberger, R (eds.) Nonlinear Analysis: A Collection of Papers in Honor of Erich H Rothe, pp. 129. Academic Press, New York (1978) View ArticleGoogle Scholar