- Research
- Open Access
Compact almost automorphic solutions for some nonlinear integral equations with time-dependent and state-dependent delay
- El Hadi Ait Dads1, 2,
- Fatima Boudchich1 and
- Brahim Es-sebbar1Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1364-2
© The Author(s) 2017
- Received: 4 May 2017
- Accepted: 14 September 2017
- Published: 3 October 2017
Abstract
We study the existence of compact almost automorphic solutions for a class of integral equations with time-dependent and state-dependent delay. An application to a blowflies model and a transmission lines model is carried out to support the theoretical finding.
Keywords
- almost periodic and almost automorphic solutions
- integral equations; neutral equation
- state-dependent delay
- Nicholson’s model
- lossless transmission lines
1 Introduction
The existence of periodic and almost periodic solutions of differential equations has an important theoretical and practical significance and is a problem of great interest. The existence of such solutions for ordinary as well as abstract differential equations has been intensively studied [1–7]. Such dynamics can be found in celestial mechanics, electronic circuits, problems of ecology and many other physical and biological systems. The parameters of such nonautonomous models are usually assumed to be periodic with respect to time due to periodic time-fluctuating environment. For example in epidemiology, the periodic aspect comes from the periodic seasonal effects. Even if the parameters of the system are periodic in time, the overall time dependence may not be periodic; i.e., if the quotient of periods of these functions is not rational, the overall time dependence will not be periodic but almost periodic in the sense of Bohr.
In reality the parameters of a system may be outputs of other almost periodic dynamical systems. However, it is well known in general that almost periodic systems do not carry necessarily almost periodic dynamics [4, 6, 8]. Although these systems may have bounded oscillating solutions, these oscillations belong to a class larger than the class of almost periodic functions, we are talking about almost automorphic functions. Bochner introduced the concept of almost automorphy in the literature in [9] as a generalization of almost periodicity. This concept was then deeply investigated by Veech [10] and many other authors. That is why it is natural to assume that the parameters of such systems are almost automorphic. Since most of such systems give rise to differential equations with solutions having bounded derivatives, a stronger concept of almost automorphy comes into play, that is, the notion of uniformly continuous almost automorphic functions. It turns out that this notion coincides with the notion of compact almost automorphy (see Lemma 6).
2 Almost periodic and almost automorphic functions
Definition 1
[16]
A useful characterization of almost periodic functions was given by Bochner.
Theorem 3
[9]
In [17], Bochner introduced the concept of almost automorphy which is a generalization of the almost periodicity.
Definition 4
[17]
Let \(\operatorname{AA}(\mathbb{R},X)\) and \(\operatorname {KAA}(\mathbb{R},X)\) denote respectively the space of almost automorphic and compact almost automorphic X-valued functions.
Remark
By the pointwise convergence, the function f̃ in Definition 4 is only measurable and not necessarily continuous. If one of the two convergences in Definition 4 is uniform on \(\mathbb{R}\), then f becomes almost periodic. For more details about this topic, we refer the reader to the books [18, 19].
Definition 5
- (i)
for all \(x\in X\), \(f(\cdot,x)\in\operatorname {AA}(\mathbb{R},X)\) (resp. \(f(\cdot,x)\in\operatorname{KAA}(\mathbb{R},X)\));
- (ii)f is uniformly continuous on each compact set K in X with respect to the second variable x, namely, for each compact set K in X, for all \(\varepsilon>0\), there exists \(\delta>0\) such that for all \(x_{1},x_{2}\in K\) one haswhenever \(\vert x_{1}-x_{2} \vert \leq\delta\).$$\sup_{t\in\mathbb{R}} \bigl\Vert f(t,x_{1})-f(t,x_{2}) \bigr\Vert \leq \varepsilon $$
Denote by \(\operatorname{AAU}(\mathbb{R}\times X,X)\) (resp. \(\operatorname{KAAU}(\mathbb {R}\times X,X)\)) the set of all such functions.
3 Some preliminary lemmas
In this section we introduce some results concerning compact almost automorphic functions which will be used to establish the main results.
The following lemma is essential for the rest of this work. It gives a characterization of compact almost automorphic functions.
Lemma 6
[20], Lemma 3.7
A function f is compact almost automorphic if and only if it is almost automorphic and uniformly continuous.
Example
Lemma 7
Let \(y(\cdot)\in\operatorname{KAA}(\mathbb{R},\mathbb{R})\) and \(\sigma(\cdot)\in \operatorname{KAA}(\mathbb{R},\mathbb{R})\). Then \(t\mapsto y(t-\sigma(t))\in\operatorname{KAA}(\mathbb {R},\mathbb{R})\).
Proof
Lemma 8
Let f and g be both in \(\operatorname{KAA}(\mathbb{R},\mathbb {R})\), then the product \(f.g\) is also in \(\operatorname{KAA}(\mathbb{R},\mathbb{R})\).
Proof
Lemma 9
Proof
The next lemma is a composition result for compact almost automorphic functions.
Lemma 10
[22], Lemma 4.36
Let \(f\in\operatorname{KAAU}(\mathbb{R}\times X,X)\) and \(x\in \operatorname{KAA}(\mathbb{R},X)\). Then \([t\mapsto f(t,x(t))]\in\operatorname{KAA}(\mathbb{R},X)\).
Lemma 11
Let \(y(\cdot)\in\operatorname{KAA}(\mathbb{R},\mathbb{R})\). Then \(t\mapsto y_{t}\in\operatorname{KAA}(\mathbb{R},C)\).
Proof
Remark
If \(y\in\operatorname{AA}(\mathbb{R},\mathbb{R})\), then \(t\mapsto y_{t}\) does not belong necessarily to \(\operatorname{AA}(\mathbb{R},C)\).
Corollary 13
The space \(\operatorname{KAA}(\mathbb{R},X)\) is a Banach space.
Proof
Let \((f_{n} )_{n}\) be a Cauchy sequence in \(\operatorname{KAA}(\mathbb{R},X)\), then by Proposition 12, \((f_{n} )_{n}\) converges uniformly to an almost automorphic function f. Since by Lemma 6 \(f_{n}\) is uniformly continuous for each \(n\in\mathbb{N}\), then f is also uniformly continuous. It follows again by Lemma 6 that \(f\in\operatorname{KAA}(\mathbb{R},X)\). □
4 Compact almost automorphic solutions of integral equations
4.1 Time-dependent delay integral equations
- (\(H_{1}\)):
-
\(\alpha,\sigma \in\operatorname{KAA}(\mathbb {R},\mathbb{R})\);
- (\(H_{2}\)):
-
\(\beta:\mathbb{R}\times\mathbb{R}^{+}\to\mathbb {R}\) satisfies \(t\mapsto\beta(t,\cdot)\in\operatorname{KAA}(\mathbb {R},L^{1}(\mathbb{R}^{+}))\);
- (\(H_{3}\)):
-
\(f\in\operatorname{KAAU}(\mathbb{R}\times C,\mathbb{R})\). Moreover, there exists \(L_{f}>0\) such that for all \(t\in\mathbb{R}\) and \(\phi,\psi\in C\)$$\bigl\vert f(t,\phi)-f(t,\psi) \bigr\vert \leq L_{f} \vert \phi-\psi \vert _{C}. $$
Theorem 14
Proof
- (\(H^{\prime}_{1}\)):
-
\(\alpha,\sigma \in\operatorname {KAA}^{+}(\mathbb {R},\mathbb{R})\).
- (\(H^{\prime}_{2}\)):
-
\(\beta:\mathbb{R}\times\mathbb {R}^{+}\to\mathbb{R}^{+}\) satisfies \(t\mapsto\beta(t,\cdot)\in\operatorname{KAA}(\mathbb {R},L^{1}(\mathbb{R}^{+}))\).
- (\(H^{\prime}_{3}\)):
-
\(f\in\operatorname{KAAU}(\mathbb {R}\times C,\mathbb{R})\) and, for all \(\varphi\in C^{+}\), \(t\in\mathbb{R}\), \(f(t,\varphi)\geq0\). Moreover, there exists \(L_{f}>0\) such that for all \(t\in\mathbb{R}\) and \(\phi,\psi\in C\)$$\bigl\vert f(t,\phi)-f(t,\psi) \bigr\vert \leq L_{f} \vert \phi-\psi \vert _{C}. $$
Then we have the following result.
Theorem 15
Proof
Remark just that under (\(H^{\prime}_{1}\))-(\(H^{\prime}_{3}\)) one can establish easily that Lemmas 7, 8, 9, 10 and 11 preserve non-negativity. It follows that P maps \(\operatorname{KAA}^{+}(\mathbb {R},\mathbb {R})\) into itself. Moreover, as \(\operatorname{KAA}^{+}(\mathbb{R},\mathbb{R})\) is a closed subset of \(\operatorname{KAA}(\mathbb{R},\mathbb{R})\), which is a Banach space, then \(\operatorname{KAA}^{+}(\mathbb{R},\mathbb {R})\) is complete. The rest of the proof is similar to that of Theorem 14. □
4.2 State-dependent delay integral equations
Lemma 16
Let \(y(\cdot)\in\operatorname{KAA}(\mathbb{R},\mathbb{R})\). If \(\gamma :C\to\mathbb{R}^{+}\) is uniformly continuous, then \([t\mapsto y (t-\gamma (y_{t}) ) ]\in\operatorname{KAA}(\mathbb{R},\mathbb{R})\).
Proof
Since \(y(\cdot)\in\operatorname{KAA}(\mathbb{R},\mathbb {R})\), then from Lemma 11, \([t\mapsto y_{t} ]\in \operatorname{KAA}(\mathbb{R},C)\). The function \(\sigma :t\mapsto\gamma(y_{t})\) is then almost automorphic and uniformly continuous. It follows by Lemma 6 that \(\sigma \in\operatorname{KAA}(\mathbb{R},\mathbb{R})\). The proof ends by applying Lemma 7. □
- (\(H_{4}\)):
-
(i) \(\alpha,\gamma\) are Lipschitz and are in \(\operatorname{KAA}(\mathbb{R},\mathbb{R})\).
(ii) \(\beta:\mathbb{R}\times\mathbb{R}^{+}\to\mathbb{R}\) satisfies \(t\mapsto\beta (t,\cdot)\in \operatorname{KAA}(\mathbb{R},L^{1}(\mathbb{R}^{+}))\) and β is Lipschitz in the following sense:$$\operatorname{Lip}(\beta):=\sup_{t\neq s}\frac{ \vert \beta (t,\cdot)-\beta (s,\cdot) \vert _{L^{1}(\mathbb{R}^{+})}}{ \vert t-s \vert }< \infty. $$
- (\(H_{5}\)):
-
\(f\in\operatorname{KAAU}(\mathbb{R}\times C,\mathbb{R})\). Moreover, there exist \(L_{f}>0, \widetilde{L}_{f}>0\) such that for all \(t,s\in\mathbb{R}\) and \(\phi,\psi\in C\)$$\begin{aligned}& \bigl\vert f(t,\phi)-f(t,\psi) \bigr\vert \leq L_{f} \vert \phi-\psi \vert _{C}. \\& \bigl\vert f(t,\phi)-f(s,\phi) \bigr\vert \leq\widetilde{L}_{f} \vert t-s \vert . \end{aligned}$$
Theorem 17
Remark
Proof
4.3 Case of a separated kernel
Remark
\(\beta_{1} \in\operatorname{KAA}(\mathbb{R},\mathbb{R})\) and \(\beta _{2} \in L^{1} (\mathbb{R}^{+},\mathbb{R})\) is equivalent to the fact that \(\beta:\mathbb{R}\times\mathbb {R}^{+}\to\mathbb{R}\) satisfies \(t\mapsto\beta(t,\cdot)\in\operatorname{KAA}(\mathbb{R},L^{1} (\mathbb{R}^{+}))\).
- (\(H_{2}^{\prime\prime}\)):
-
\(\beta:\mathbb{R}\times\mathbb {R}^{+}\to\mathbb{R}\) satisfies \(\beta(t,s) = \beta_{1}(t)\beta_{2}(s) \) such that \(\beta_{1} \in \operatorname{KAA}(\mathbb{R},\mathbb{R} )\) and \(\beta_{2} \in L^{1} (\mathbb{R}_{+},\mathbb{R})\).
- (\(H^{\prime\prime}_{3}\)):
-
\(f\in\operatorname{AAU}(\mathbb {R}\times C,\mathbb{R})\). Moreover, there exists \(L_{f}>0\) such that for all \(t\in\mathbb{R}\) and \(\phi,\psi\in C\)$$\bigl\vert f(t,\phi)-f(t,\psi) \bigr\vert \leq L_{f} \vert \phi-\psi \vert _{C}. $$
Theorem 18
Remark
Note here that the function f is just in \(\operatorname {AAU}(\mathbb{R}\times C,\mathbb{R})\), whereas the obtained solution x is more regular, namely, x is in \(\operatorname{KAA}(\mathbb{R},\mathbb{R})\).
To prove Theorem 18, we need the following lemma.
Lemma 19
Proof of Lemma 19
Proof of Theorem 18
5 Applications
5.1 A neutral Nicholson’s blowflies model with time-dependent delay
- (i)
\(a, g, \delta, \eta, p, r , \dot{r} \in \operatorname{KAA}^{+}(\mathbb{R},\mathbb{R})\),
- (ii)
\(\underline{\delta}= \inf_{t\in \mathbb{R}}\delta(t)>0\), \(\inf_{t\in\mathbb{R}}(1-\dot {r}(t))>0\), and \(g(t)>0\) for all \(t\in\mathbb{R}\),
- (iii)the following condition holds:$$ p(t) \bigl(1-2a(t)\bigr)e^{-2a(t)} \geq\delta(t)\alpha(t) + \dot{\alpha}(t), $$(12)
Lemma 20
\(t\mapsto\beta(t,\cdot) \in \operatorname{KAA}(\mathbb{R},L^{1}(\mathbb{R}^{+}))\).
Proof
Remark
The obtained solution is not trivial since \(g(t)\) is positive.
5.2 A lossless transmission lines model
System with lossless transmission line [ 27 ].
Assume that the resistance R is affected by local environment conditions (principally temperature which depends on time in an oscillating way), then it can be given as a time-dependent function \(R(t)\) having an oscillating behavior. Let L and C denote the inductance and capacitance of a long line, respectively, and assume that the line is lossless.
- (i)
g and R are in \(\operatorname{KAA}(\mathbb{R}, \mathbb{R})\),
- (ii)
\(\underline{R}= \inf_{t\in\mathbb{R}}R(t)>0, \underline{K}= \inf_{t\in\mathbb{R}}K(t)>0\),
- (iii)
g is Lipschitz continuous on \(\mathcal{C}:=C([-\tau ,0];\mathbb{R})\).
Declarations
Authors’ contributions
All authors read and approved the final 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
References
- Coronel, A, Maulén, C, Pinto, M, Sepúlveda, D: Almost automorphic delayed differential equations and Lasota-Wazewska model. Discrete Contin. Dyn. Syst., Ser. A 37(4), 1959-1977 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Ait Dads, EH, Cieutat, P, Lhachimi, L: Existence of positive almost periodic or ergodic solutions for some neutral nonlinear integral equations. Differ. Integral Equ. 22(11), 1075-1096 (2009) MathSciNetMATHGoogle Scholar
- Ding, HS, Chen, YY, N’Guérékata, GM: Existence of positive pseudo almost periodic solutions to a class of neutral integral equations. Nonlinear Anal. 74(18), 7356-7364 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Ortega, R, Tarallo, M: Almost periodic linear differential equations with non-separated solutions. J. Funct. Anal. 237, 402-426 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Pinto, M: Pseudo-almost periodic solutions of neutral integral and differential equations with applications. Nonlinear Anal. 72(12), 4377-4383 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Shen, W, Yi, Y: Almost automorphic and almost periodic dynamics in skew-product semiflows. Mem. Am. Math. Soc. 136(647), 1-93 (1998) MathSciNetMATHGoogle Scholar
- Torrejón, R: Positive almost periodic solutions of a state-dependent delay nonlinear integral equation. Nonlinear Anal. 20(12), 1383-1416 (1993) MathSciNetView ArticleMATHGoogle Scholar
- Johnson, RA: A linear, almost periodic equation with an almost automorphic solution. Proc. Am. Math. Soc. 82, 199-205 (1981) MathSciNetView ArticleMATHGoogle Scholar
- Bochner, S: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA 48(12), 2039-2043 (1962) MathSciNetView ArticleMATHGoogle Scholar
- Veech, WA: Almost automorphic functions on groups. Am. J. Math. 87, 719-751 (1965) MathSciNetView ArticleMATHGoogle Scholar
- Zou, QF, Ding, HS: Almost periodic solutions for a nonlinear integro-differential equation with neutral delay. J. Nonlinear Sci. Appl. 9(6), 4500-4508 (2016) MathSciNetMATHGoogle Scholar
- Cooke, KL, Kaplan, JL: A periodicity threshold theorem for epidemics and population growth. Math. Biosci. 31(1-2), 87-104 (1976) MathSciNetView ArticleMATHGoogle Scholar
- Ait Dads, EH, Ezzinbi, K: Existence of positive pseudo almost periodic solution for a class of functional equations arising in epidemic problems. Cybern. Syst. Anal. 30(6), 900-910 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Ait Dads, EH, Ezzinbi, K: Almost periodic solution for some neutral nonlinear integral equation. Nonlinear Anal. 28(9), 1479-1489 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Ait Dads, EH, Ezzinbi, K: Boundedness and almost periodicity for some state-dependent delay differential equations. Electron. J. Differ. Equ. 2002(67), 1 (2002) MathSciNetMATHGoogle Scholar
- Fink, A: Almost Periodic Differential Equations. Lecture Notes in Mathematics, vol. 377. Springer, Berlin (1974) MATHGoogle Scholar
- Bochner, S: Continuous mappings of almost automorphic and almost periodic functions. Proc. Natl. Acad. Sci. USA 52(4), 907-910 (1964) MathSciNetView ArticleMATHGoogle Scholar
- N’Guérékata, GM: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Springer, Berlin (2001) View ArticleMATHGoogle Scholar
- Zaidman, S: Almost-Periodic Functions in Abstract Spaces. Pitman Advanced Publishing Program, vol. 126 (1985) MATHGoogle Scholar
- Es-sebbar, B: Almost automorphic evolution equations with compact almost automorphic solutions. C. R. Math. 354(11), 1071-1077 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Basit, B, Günzler, H: Spectral criteria for solutions of evolution equations and comments on reduced spectra. (2010). arXiv:1006.2169
- Diagana, T: Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces. Springer, Berlin (2013) View ArticleMATHGoogle Scholar
- Nicholson, AJ: Compensatory reactions of populations to stresses, and their evolutionary significance. Aust. J. Zool. 2(1), 1-8 (1954) View ArticleGoogle Scholar
- Nicholson, AJ: An outline of the dynamics of animal populations. Aust. J. Zool. 2(1), 9-65 (1954) View ArticleGoogle Scholar
- Gurney, WSC, Blythe, SP, Nisbet, RM: Nicholson’s blowflies revisited. Nature 287, 17-21 (1980) View ArticleGoogle Scholar
- Chen, Y: Periodic solutions of delayed periodic Nicholson’s blowflies models. Can. Appl. Math. Q. 11(1), 23-28 (2003) MathSciNetMATHGoogle Scholar
- Kolmanovskii, VB, Nosov, VR: Stability of Functional Differential Equations, vol. 180. Academic Press, San Diego (1986) Google Scholar