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 Open Access
On almost periodicity of solutions of secondorder differential equations involving reflection of the argument
 Peiguang Wang^{1},
 Dhaou Lassoued^{2},
 Syed Abbas^{3},
 Akbar Zada^{4} and
 Tongxing Li^{5, 6}Email author
https://doi.org/10.1186/s1366201819387
© The Author(s) 2019
 Received: 14 October 2018
 Accepted: 17 December 2018
 Published: 7 January 2019
Abstract
We study almost periodic solutions for a class of nonlinear secondorder differential equations involving reflection of the argument. We establish existence results of almost periodic solutions as critical points by a variational approach. We also prove structure results on the set of strong almost periodic solutions, existence results of weak almost periodic solutions, and a density result on the almost periodic forcing term for which the equation possesses usual almost periodic solutions.
Keywords
 Almost periodic solution
 Secondorder differential equation
 Reflection of the argument
 Variational principle
MSC
 34K14
 47H30
 58E30
1 Introduction
The study of existence, uniqueness, and stability of periodic and almost periodic solutions has become one of the most attractive topics in the qualitative theory of ordinary and functional differential equations for its significance in the physical sciences, mathematical biology, control theory, and other fields; see, for instance, [3, 8, 11, 19, 20, 28] and the references cited therein. Indeed, the almost periodic functions are closely connected with harmonic analysis, differential equations, and dynamical systems; cf. Corduneanu [12] and Fink [14]. These functions are basically generalizations of continuous periodic and quasiperiodic functions. Almost periodic functions are further generalized by many mathematicians in various ways; see Šarkovskii [26].
By a strong almost periodic solution of equation (1.1) we mean a function \(u:\mathbb{R}\rightarrow\mathbb{R}^{n}\) which is twice differentiable (in ordinary sense) such that u, \(u'\), and \(u''\) are almost periodic in the sense of Bohr [9] and u satisfies (1.1) for all \(t\in\mathbb{R}\). This solution is also called \({\mathcal {C}}^{2}\)almost periodic in some earlier work.
A weak almost periodic solution of equation (1.1) is a function \(u:\mathbb{R}\rightarrow\mathbb{R}^{n}\) which is almost periodic in the sense of Besicovitch [4] and possesses a firstorder and a secondorder generalized derivatives such that u satisfies (1.1) for all \(t\in\mathbb{R}\) and the difference between the two members of equation (1.1) has a quadratic mean value equal to zero. It is natural that a strong almost periodic solution is also a weak almost periodic solution.
This paper is organized as follows. Section 2 presents the considered notation for the various function spaces and auxiliary assumptions. In Sect. 3, we develop variational principles to study the almost periodic solutions of (1.1) and critical points of functionals defined on spaces of almost periodic functions. In Sect. 4, we establish some results about the structure of the set of strong almost periodic solutions of (1.1). Finally, in Sect. 5, we establish an existence result of weak almost periodic solutions of (1.1) by using the techniques in the spirit of the direct methods of calculus of variations, and a result on the density of the almost periodic forcing term for which (1.1) possesses a strong almost periodic solution.
2 Notation and preliminaries
First, we review some facts about Bohr almost periodic and Besicovitch almost periodic functions. For more details on almost periodic functions, we refer the reader to the monographs [4, 9, 12, 14, 21].
Remark 2.1
Remark 2.2
For \(p=2\), \(B^{2}(\mathbb{R}^{n})\) is a Hilbert space and its norm \(\\cdot \_{2}\) is associated to the inner product \((u\mid v):=\mathcal{M} \{ u\cdot v \}\). The elements of these spaces \(B^{p}(\mathbb{R}^{n})\) are called Besicovitch almost periodic functions, cf. [4].
 (H_{1}):

\(f\in\mathcal{C}^{1}(\mathbb{R}^{n}\times\mathbb {R}^{n},\mathbb{R})\);
 (H_{2}):

\(\vert Df(X)Df(Y) \vert \leq a\cdot \vert XY \vert \) for some constant \(a>0\) and for all \(X,Y\in\mathbb {R}^{n}\times\mathbb{R}^{n}\);
 (H_{3}):

f is a convex function on \(\mathbb{R}^{n}\times\mathbb{R}^{n}\);
 (H_{4}):

\(f(x,y)\geq c \vert \zeta \vert ^{2}+d\) for two numbers \(c>0\) and \(d\in\mathbb{R}\) and for all \((x,y)\in\mathbb {R}^{n}\times\mathbb{R}^{n}\), where \(\zeta=x\mbox{ or }y\).
3 Variational principles
We begin this section by establishing two lemmas which contain general properties of almost periodic functions.
Lemma 3.1
If \(u\in AP^{0}(\mathbb{R}^{n})\), then \([t\mapsto u(t) ]\in AP^{0}(\mathbb{R}^{n})\). Furthermore, if τ is an ϵtranslation of \(u(t)\), then τ is also an ϵtranslation number of \(u(t)\) and \(\operatorname{mod}(u(t))=\operatorname{mod}(u(t))\).
Proof
The proof can be completed by using Bohr’s definition [9, p. 32]. □
Lemma 3.2
 (1)
\(\mathcal{M} \{u(t) \}_{t}=\mathcal{M} \{u(t) \}_{t}\).
 (2)
\([t\mapsto u(t) ]\in B^{p}(\mathbb{R}^{n})\).
Proof
Lemma 3.3
Proof
We consider the operator \(Q_{0}:AP^{1}(\mathbb{R}^{n})\rightarrow\mathbb{R}\) defined by \(Q_{0}(u):=\mathcal{M} \{\frac{1}{2} \vert u' \vert ^{2} \}\). The mapping \(q: \mathbb{R}^{n} \rightarrow\mathbb{R}\), \(q(x)=\frac{1}{2} \vert x \vert ^{2}\), is of class \({\mathcal {C}}^{1}\), so the Nemytskiĭ operator \({\mathcal {N}}^{0}_{q} :AP^{0}(\mathbb{R}^{n}) \rightarrow AP^{0}(\mathbb {R})\), \({\mathcal {N}}^{0}_{q}(\phi):= [t\mapsto\frac{1}{2} \vert \phi (t) \vert ^{2}]\), is of class \({\mathcal {C}}^{1}\), cf. [5]. The operator \(\frac{d}{dt}: AP^{1}(\mathbb{R}^{n}) \rightarrow AP^{0}(\mathbb {R}^{n})\) defined by \(\frac{d}{dt}(u):= u'\) is linear continuous, therefore, it is of class \({\mathcal {C}}^{1}\). The functional \({\mathcal {M}}^{0}: AP^{0}(\mathbb{R}) \rightarrow\mathbb {R}\) defined by \({\mathcal {M}}^{0}(\phi):={\mathcal {M}}^{0}_{t}\{\phi(t)\}\) is linear continuous, and hence it is of class \({\mathcal {C}}^{1}\).
Since \(Q_{0}= {\mathcal {M}}^{0} \circ{\mathcal {N}}^{0}_{q} \circ\frac{d}{dt}\), \(Q_{0}\) is of class \({\mathcal {C}}^{1}\) as composition of \({\mathcal {C}}^{1}\)mappings. Hence, by the chain rule, we have \(DQ_{0}(u)v=\mathcal{M} \{u' \cdot v' \}\).
Furthermore, the operator \(\varTheta_{0}:AP^{1}(\mathbb{R}^{n})\rightarrow \mathbb{R}\) defined by \(\varTheta_{0}(u):=\mathcal{M} \{e \cdot u \} \) is linear continuous, so it is of class \({\mathcal {C}}^{1}\) and its differential is given by \(D\varTheta _{0}(u)v=\mathcal{M} \{e \cdot v \}\).
We consider the operator \(\varPhi_{0}:AP^{1}(\mathbb{R}^{n})\rightarrow\mathbb {R}\) defined by \(\varPhi_{0}(u):=\mathcal{M} \{f (u (t ),u (t ) ) \}_{t}\). It is not difficult to observe that the operator \(L_{0}:AP^{0}(\mathbb {R}^{n})\rightarrow AP^{0}(\mathbb{R}^{n})\times AP^{0}(\mathbb{R}^{n})\) defined by \(L_{0}(u)(t):=(u(t),u(t))\) is linear. Both components of \(L_{0}\) are continuous and hence \(L_{0}\) is continuous. Therefore, \(L_{0}\) is of class \(\mathcal{C}^{1}\) and \(DL_{0}(u)v=L_{0}(v)\) for all \(u,v\in AP^{0}(\mathbb{R}^{n})\).
Now, under assumption (H_{1}), the Nemytskiĭ operator \(\mathcal {N}_{f}^{0}:AP^{0}(\mathbb{R}^{n}\times\mathbb{R}^{n})\rightarrow AP^{0}(\mathbb {R})\) defined by \(\mathcal{N}_{f}^{0}(U)(t):=f (U(t) )\) is of class \(\mathcal{C}^{1}\) (see [6] for details). Moreover, for all \(U,V\in (AP^{0}(\mathbb{R}^{n}) )^{2}\), \(D\mathcal{N}_{f}^{0}(U)\cdot V=Df(U)\cdot V\).
Lemma 3.4
Assume that assumptions (H_{1}) and (H_{2}) are satisfied. Then the Nemytskiĭ operator \(\mathcal{N}^{1}_{f}:B^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})\rightarrow B^{1}(\mathbb{R})\) defined by \(\mathcal{N}^{1}_{f}(U)(t):=f (U(t) )\) is well defined and is of class \(\mathcal{C}^{1}\), and \(D\mathcal{N}^{1}_{f}(U) \cdot V=Df(U)\cdot V\) for all \(U, V \in B^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})\).
Proof
Proposition 3.5
 (1)
u is a critical point of \(J_{0}\) on \(AP^{1}(\mathbb{R}^{n})\).
 (2)
u is a strong almost periodic solution of (1.1).
Proof
Lemma 3.6
Proof
We consider the operator \(Q_{1}:B^{1,2}(\mathbb{R}^{n})\rightarrow\mathbb {R}\) defined by \(Q_{1}(u):=\mathcal{M} \{\frac{1}{2} \vert \nabla u \vert ^{2} \}\). The mapping \(q: \mathbb{R}^{n} \rightarrow\mathbb{R}\), \(q(x)=\frac{1}{2} \vert x \vert ^{2}\), is of class \({\mathcal {C}}^{1}\). Since \(Dq(x)=x\) satisfies conditions of [13, Theorem 2.6], the Nemytskiĭ operator \({\mathcal {N}}_{q}: B^{2}(\mathbb{R}^{n}) \rightarrow B^{1}(\mathbb {R})\) defined by \({\mathcal {N}}_{q}(v):= [t\mapsto\frac{1}{2} \vert v(t) \vert ^{2}]\) is of class \({\mathcal {C}}^{1}\) and \(D{\mathcal {N}}_{q}(v) \cdot h=[t\mapsto v(t) \cdot h(t)]\) for all \(v, h \in B^{2}(\mathbb{R}^{n})\).
Since the derivation operator \(\nabla: B^{1,2}(\mathbb{R}^{n})\rightarrow B^{2}(\mathbb{R})\) and the operator \({\mathcal {M}} : B^{1}(\mathbb{R}) \rightarrow\mathbb{R}\) are linear continuous, ∇ and \({\mathcal {M}}\) are of class \({\mathcal {C}}^{1}\). Therefore, \(Q_{1}={\mathcal {M}} \circ {\mathcal {N}}_{q} \circ\nabla\) is of class \({\mathcal {C}}^{1}\) as a composition of \({\mathcal {C}}^{1}\)mappings. Moreover, using the chain rule, we have \(DQ_{1}(u)\cdot v=\mathcal{M} \{\nabla u \cdot\nabla v \}\) for all \(u, v \in B^{1,2}(\mathbb{R}^{n})\).
Now, the operator \(\varTheta_{1} :B^{1,2}(\mathbb{R}^{n})\rightarrow\mathbb {R}\) defined by \(\varTheta_{1}(u):=\mathcal{M} \{e \cdot u \}\) is linear continuous, and thus it is of class \(\mathcal{C}^{1}\) and its differential is given by \(D\varTheta_{1}(u)v=\mathcal{M} \{e \cdot v \}\).
Let us consider the operator \(\varPhi_{1}:B^{1,2}(\mathbb{R}^{n})\rightarrow \mathbb{R}\) defined by \(\varPhi_{1}(u):=\mathcal{M} \{f (u (t ),u (t ) ) \}_{t}\). Note that the linear operator \(L_{1}:B^{2}(\mathbb{R}^{n})\rightarrow B^{2}(\mathbb{R}^{n})\times B^{2}(\mathbb{R}^{n})\) defined by \(L_{1}(u)(t):=(u(t),u(t))\) is continuous and so it is of class \(\mathcal{C}^{1}\). Moreover, for all \(u,v\in B^{2}(\mathbb{R}^{n})\), we have \(DL_{1}(u)v=L_{1}(v)\).
Under assumptions (H_{1}) and (H_{2}), by virtue of Lemma 3.4, the Nemytskiĭ operator \(\mathcal{N}^{1}_{f}:B^{2}(\mathbb{R}^{n}\times \mathbb{R}^{n})\rightarrow B^{1}(\mathbb{R})\) defined by \(\mathcal {N}^{1}_{f}(U)(t):=f (U(t) )\) is of class \(\mathcal{C}^{1}\) and for all \(U,V\in B^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})\), \(D\mathcal {N}^{1}_{f}(U) \cdot V=Df(U) \cdot V\).
The continuous linear operator \(\mathcal{M}_{1}:B^{1}(\mathbb{R})\rightarrow \mathbb{R}\) defined by \(\mathcal{M}_{1}(u):=\mathcal{M} \{u(t) \} _{t}\) is of class \(\mathcal{C}^{1}\) and for all \(\phi,\psi\in B^{1}(\mathbb {R})\), \(D\mathcal{M}_{1}(\phi)\psi=\mathcal{M}(\psi)\). Besides, the linear operator \(in_{1}:B^{1,2}(\mathbb{R}^{n})\rightarrow B^{2}(\mathbb {R}^{n})\), \(in_{1}(u)=u\) is of class \(\mathcal{C}^{1}\) and \(Din_{1}(u)v=in_{1}(v)\).
Proposition 3.7
 (1)
u is a critical point of \(J_{1}\) on \(B^{1,2}(\mathbb{R}^{n})\).
 (2)
u is a weak almost periodic solution of (1.1).
Proof
4 Structure results on \(AP^{0}(\mathbb{R}^{n})\)
In this section, we give some structure results on the set of strong almost periodic solutions of equation (1.1). The main tool is the variational structure of the problem.
Theorem 4.1
 (1)
The set of the strong almost periodic solutions of (1.1) is a convex closed subset of \(AP^{1}(\mathbb{R}^{n})\).
 (2)
If \(u_{1}\) is a \(T_{1}\) periodic solution of (1.1), \(u_{2}\) is a \(T_{2}\) periodic solution of (1.1), and \(T_{1}/T_{2}\) is not rational, then \((1\theta)u_{1}+\theta u_{2}\) is a strong almost periodic but nonperiodic solution of (1.1) for all \(\theta\in(0,1)\).
Proof
Theorem 4.2
Proof
To prove assertion (2), it suffices to choose \(T\in(0,\infty)\) such that \(\frac{2\pi}{T} (\mathbb{Z} \{0 \} )\cap\varLambda (u)=\emptyset\). So all the Fourier–Bohr coefficients of \(u_{T}\) are zero except (perhaps) the mean value of \(u_{T}\) which is equal to \(\mathcal {M} \{u \}\). This completes the proof. □
5 Existence results
In this section, we study the weak almost periodic solutions of equation (1.1). In the previous section, we use a variational viewpoint but here the Hilbert structure of \(B^{2}(\mathbb{R}^{n})\) permits us to obtain an existence theorem by using direct methods of calculus of variations. Finally, in Theorem 5.2, we give a result of density of the almost periodic forcing term for which equation (1.1) possesses usual almost periodic solutions.
Theorem 5.1
Under assumptions (H_{1})–(H_{4}), for each \(e\in B^{2}(\mathbb{R}^{n})\), there exists a \(u\in B^{2,2}(\mathbb{R}^{n})\) which is a weak almost periodic solution of (1.1). Moreover, the set of the weak almost periodic solutions of (1.1) is a convex set.
Proof
On the basis of Lemma 3.6, the set of the weak almost periodic solutions of (1.1) is equal to the set \(\{u\in B^{1,2}(\mathbb {R}^{n}) : DJ_{1}(u)=0 \}\). Since \(J_{1}\) is convex, this set is also equal to \(\{u\in B^{1,2}(\mathbb{R}^{n}) : J_{1}(u)= \inf J_{1} (B^{1,2}(\mathbb{R}^{n}) ) \}\) which is a convex set. Thus, the set of the weak almost periodic solutions of (1.1) is convex. The proof is complete. □
Theorem 5.2
Proof
From Theorem 5.1, we know that \(\varGamma (B^{2,2}(\mathbb {R}^{n}) )=B^{2}(\mathbb{R}^{n})\), and so \(AP^{0}(\mathbb {R}^{n})\subset\varGamma (B^{2,2}(\mathbb{R}^{n}) )\). Let \(e\in AP^{0}(\mathbb{R}^{n})\). Then \(e\in\varGamma ( B^{2,2}(\mathbb {R}^{n}) )\), and thus there exists a \(u\in B^{2,2}(\mathbb {R}^{n})\) such that \(\varGamma(u)=e\). Since \(AP^{2}(\mathbb{R}^{n})\) is dense in \(B^{2,2}(\mathbb{R}^{n})\), for each \(\epsilon\in(0,\infty)\), there exists a \(u_{\epsilon}\in AP^{2}(\mathbb{R}^{n})\) such that \(\ u_{\epsilon}u\_{2,2}<\epsilon\). An application of continuity of Γ implies that \(\\varGamma(u_{\epsilon})e\_{2}<\epsilon\). Taking into account that \(\varGamma(u_{\epsilon})\in AP^{0}(\mathbb{R}^{n})\), let \(e_{\epsilon}:=\varGamma(u_{\epsilon})\). Then \(e_{\epsilon}\) and \(u_{\epsilon}\) satisfy the desired results. This completes the proof. □
Declarations
Acknowledgements
The authors express their sincere gratitude to the editors for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
This research is supported by NNSF of P.R. China (Grant Nos. 11771115, 11271106, and 61503171), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2016JL021), KRDP of Shandong Province (Grant No. 2017CXGC0701), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China.
Authors’ contributions
All five authors contributed equally to this work. They all read and approved the final version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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