# On almost periodicity of solutions of second-order differential equations involving reflection of the argument

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## Abstract

We study almost periodic solutions for a class of nonlinear second-order 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 Second-order 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 quasi-periodic functions. Almost periodic functions are further generalized by many mathematicians in various ways; see Šarkovskii [26].

*X*and

*Y*, respectively, \(e:\mathbb{R}\rightarrow\mathbb{R}^{n}\) is an almost periodic forcing term. Equation (1.1) appears as an Euler–Lagrange equation.

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 first-order and a second-order 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].

*u*possesses a mean time

*u*associated to

*λ*. We denote by \(\varLambda(u):= \{\lambda\in \mathbb{R}: a(u,\lambda)\neq0 \}\) the set of exponents of

*u*. We use the notation \(\operatorname{mod}(u)\) for the module of

*u*which is the additive group generated by \(\varLambda(u)\).

### 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].

*u*exists in \(B^{2}(\mathbb {R}^{n})\), and the space \(B^{2,2}(\mathbb{R}^{n})\) is the space of \(u\in B^{1,2}(\mathbb {R}^{n})\) such that \(\nabla^{2} u=\nabla(\nabla u)\) exists in \(B^{2}(\mathbb {R}^{n})\). It is easy to verify that the above-mentioned spaces are Hilbert spaces with the respective norms

- (H
_{1}) -
\(f\in\mathcal{C}^{1}(\mathbb{R}^{n}\times\mathbb {R}^{n},\mathbb{R})\);

- (H
_{2}) -
\(\vert Df(X)-Df(Y) \vert \leq a\cdot \vert X-Y \vert \) 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**ϵ*-*translation of*\(u(t)\), *then**τ**is also an**ϵ*-*translation 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

*If*\(u\in B^{p}(\mathbb{R}^{n})\),

*then the following assertions hold*.

- (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

*Under condition*(

*H*

_{1}),

*the functional*\(J_{0}:AP^{1}(\mathbb{R}^{n})\rightarrow\mathbb{R}\)

*defined by*

*is of class*\(\mathcal{C}^{1}\),

*and for all*\(u,v\in AP^{0}(\mathbb{R}^{n})\),

### 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

_{1}) and (H

_{2}) hold, then, for all \(X\in\mathbb{R}^{n}\),

### Proposition 3.5

*Under condition*(H

_{1}),

*the following assertions are equivalent*.

- (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

*u*is a strong almost periodic solution of (1.1), then we have \(u''=q\). Hence, for all \(v\in AP^{1}(\mathbb{R}^{n})\), we obtain

### Lemma 3.6

*If conditions*(H

_{1})

*and*(H

_{2})

*are fulfilled*,

*then the functional*\(J_{1}:B^{1,2}(\mathbb{R}^{n})\rightarrow\mathbb{R}\)

*defined by*

*is of class*\(\mathcal{C}^{1}\).

*Moreover*,

*for all*\(u,v\in B^{1,2}(\mathbb{R}^{n})\),

### 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

*Under conditions*(H

_{1})

*and*(H

_{2}),

*the following assertions are equivalent*.

- (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

*u*is a weak almost periodic solution of (1.1).

## 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

*Under assumptions*(H

_{1})

*and*(H

_{3}),

*the following assertions hold*.

- (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

*f*is convex and is of class \(\mathcal{C}^{1}\), the operator \(J_{0}\) is convex and is of class \(\mathcal{C}^{1}\) on \(AP^{1}(\mathbb{R}^{n})\). Therefore,

### Theorem 4.2

*Under assumptions*(H

_{1})

*and*(H

_{3}),

*if*\(e=0\),

*then the following assertions hold*.

### Proof

*u*is a strong almost periodic solution of (1.1). According to the Besicovitch theorem [4, p. 144], there exists a

*T*periodic continuous function denoted by \(u_{T}\) such that

*T*periodic solution of (1.1). Now, using a straightforward calculation, we can easily observe that \(a (C_{T,\nu}(u),\frac{2\pi}{T} )=a (u,\frac{2\pi}{T} )\) and consequently \(a (u_{T},\frac{2\pi}{T} )=a (u,\frac {2\pi}{T} )\neq0\), then \(u_{T}\) is not constant which proves assertion (1).

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

_{1}) and (H

_{2}), the functional \(J_{1}\) is of class \(\mathcal{C}^{1}\). It follows from (H

_{3}) that \(J_{1}\) is a convex functional. Since the mean value is invariant by reflection, assumption (H

_{4}) implies that, for all \(u\in B^{1,2}(\mathbb{R}^{n})\),

*u*is a weak almost periodic solution of (1.1) by using Proposition 3.7. Hence, the existence is proved.

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

*Assume that*(H

_{1})

*–*(H

_{4})

*hold*.

*Then*,

*for each*\(e\in AP^{0}(\mathbb {R}^{n})\)

*and for each*\(\epsilon>0\),

*there exist an*\(e_{\epsilon}\in AP^{0}(\mathbb{R}^{n})\)

*and a*\(u_{\epsilon}\in AP^{2} (\mathbb {R}^{n})\)

*such that*\(\|e-e_{\epsilon}\|_{2}<\epsilon\)

*and*

### Proof

_{1}) and (H

_{2}), the operators

*Γ*is continuous.

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. □

## Notes

### 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.

### Authors’ contributions

All five authors contributed equally to this work. They all read and approved the final version of the manuscript.

### 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.

### Competing interests

The authors declare that they have no competing interests.

## References

- 1.Aftabizadeh, A.R., Huang, Y.K., Wiener, J.: Bounded solutions for differential equations with reflection of the argument. J. Math. Anal. Appl.
**135**, 31–37 (1988) MathSciNetCrossRefGoogle Scholar - 2.Alexeev, V.M., Tihomirov, V.M., Fomin, S.V.: Commande Optimale, French edn. MIR, Moscow (1982) Google Scholar
- 3.Ayachi, M., Lassoued, D.: On the existence of Besicovitch almost periodic solutions for a class of neutral delay differential equations. Facta Univ., Ser. Math. Inform.
**29**, 131–144 (2014) MathSciNetzbMATHGoogle Scholar - 4.Besicovitch, A.S.: Almost Periodic Functions. Cambridge University Press, Cambridge (1932) zbMATHGoogle Scholar
- 5.Blot, J.: Calculus of variations in mean and convex Lagrangians. J. Math. Anal. Appl.
**134**, 312–321 (1988) MathSciNetCrossRefGoogle Scholar - 6.Blot, J.: Une approche variationnelle des orbites quasi-périodiques des systèmes hamiltoniens. Ann. Sci. Math. Qué.
**13**, 7–32 (1990) (French) MathSciNetzbMATHGoogle Scholar - 7.Blot, J.: Oscillations presque-périodiques forcées d’équations d’Euler–Lagrange. Bull. Soc. Math. Fr.
**122**, 285–304 (1994) (French) CrossRefGoogle Scholar - 8.Blot, J., Lassoued, D.: Bumps of potentials and almost periodic oscillations. Afr. Diaspora J. Math.
**12**, 122–133 (2011) MathSciNetzbMATHGoogle Scholar - 9.Bohr, H.: Almost Periodic Functions. Chelsea, New York (1956) zbMATHGoogle Scholar
- 10.Brézis, H.: Analyse Fonctionnelle. Théorie et Applications. Masson, Paris (1983) (French) zbMATHGoogle Scholar
- 11.Bu̧se, C., Lassoued, D., Nguyen, T.L., Saierli, O.: Exponential stability and uniform boundedness of solutions for nonautonomous periodic abstract Cauchy problems. An evolution semigroup approach. Integral Equ. Oper. Theory
**74**, 345–362 (2012) MathSciNetCrossRefGoogle Scholar - 12.Corduneanu, C.: Almost Periodic Functions, 2nd English edn. Chelsea, New York (1989) zbMATHGoogle Scholar
- 13.de Figueiredo, D.G.: The Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research, Bombay. Springer, Berlin (1989) zbMATHGoogle Scholar
- 14.Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (1974) CrossRefGoogle Scholar
- 15.Gupta, C.P.: Boundary value problems for differential equations in Hilbert spaces involving reflection of the argument. J. Math. Anal. Appl.
**128**, 375–388 (1987) MathSciNetCrossRefGoogle Scholar - 16.Gupta, C.P.: Existence and uniqueness theorems for boundary value problems involving reflection of the argument. Nonlinear Anal.
**11**, 1075–1083 (1987) MathSciNetCrossRefGoogle Scholar - 17.Gupta, C.P.: Two-point boundary value problems involving reflection of the argument. Int. J. Math. Math. Sci.
**10**, 361–371 (1987) MathSciNetCrossRefGoogle Scholar - 18.Hai, D.D.: Two point boundary value problem for differential equations with reflection of argument. J. Math. Anal. Appl.
**144**, 313–321 (1989) MathSciNetCrossRefGoogle Scholar - 19.Lassoued, D.: Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems. Electron. J. Differ. Equ.
**2013**, 89 (2013) MathSciNetCrossRefGoogle Scholar - 20.Lassoued, D.: New aspects of nonautonomous discrete systems stability. Appl. Math. Inf. Sci.
**9**, 1693–1698 (2015) MathSciNetGoogle Scholar - 21.Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Cambridge University Press, Cambridge (1982) zbMATHGoogle Scholar
- 22.O’Regan, D.: Existence results for differential equations with reflection of the argument. J. Aust. Math. Soc. A
**57**, 237–260 (1994) MathSciNetCrossRefGoogle Scholar - 23.Piao, D.: Periodic and almost periodic solutions for differential equations with reflection of the argument. Nonlinear Anal.
**57**, 633–637 (2004) MathSciNetCrossRefGoogle Scholar - 24.Piao, D.: Pseudo almost periodic solutions for differential equations involving reflection of the argument. J. Korean Math. Soc.
**41**, 747–754 (2004) MathSciNetCrossRefGoogle Scholar - 25.Piao, D., Sun, J.: Besicovitch almost periodic solutions for a class of second order differential equations involving reflection of the argument. Electron. J. Qual. Theory Differ. Equ.
**2014**, 41 (2014) MathSciNetCrossRefGoogle Scholar - 26.Šarkovskii, A.N.: Functional-differential equations with a finite group of argument transformations. In: Asymptotic Behavior of Solutions of Functional-Differential Equations, pp. 118–142, 157. Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev (1978) zbMATHGoogle Scholar
- 27.Vo-Khac, K.: Étude des fonctions quasi-stationnaires et de leurs applications aux équations différentielles opérationnelles. Bull. Soc. Math. France Mém.
**6**(1966) (French) Google Scholar - 28.Wang, Y., Zada, A., Ahmad, N., Lassoued, D., Li, T.: Uniform exponential stability of discrete evolution families on space of
*p*-periodic sequences. Abstr. Appl. Anal.**2014**, Article ID 784289 (2014) MathSciNetGoogle Scholar - 29.Wiener, J., Aftabizadeh, A.R.: Boundary value problems for differential equations with reflection of the argument. Int. J. Math. Math. Sci.
**8**, 151–163 (1985) MathSciNetCrossRefGoogle Scholar - 30.Zima, M.: On positive solutions of functional-differential equations in Banach spaces. J. Inequal. Appl.
**6**, 359–371 (2001) MathSciNetzbMATHGoogle Scholar

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