Periodic solutions for some phi-Laplacian and reflection equations
- Alberto Cabada^{1}Email author and
- Fernando Adrián Fernández Tojo^{1}
https://doi.org/10.1186/s13661-016-0565-z
© Cabada and Fernández Tojo 2016
Received: 11 November 2015
Accepted: 16 February 2016
Published: 29 February 2016
Abstract
This work is devoted to the study of the existence and periodicity of solutions of initial differential problems, paying special attention to the explicit computation of the period. These problems are also connected with some particular initial and boundary value problems with reflection, which allows us to prove existence of solutions of the latter using the existence of the former.
Keywords
1 Introduction
The idea behind this paper appeared in another work of the authors [1] where the following lemmas were proved.
Definition 1.1
If \(A\subset {\mathbb {R}}\), a function \(\varphi:A\to A\) such that \(\varphi\ne \operatorname {Id}\) and \(\varphi\circ\varphi=\operatorname {Id}\) is called an involution.
Lemma 1.1
([1], Lemma 2.1)
Let \((a,b)\subset {\mathbb {R}}\) and let \(f:{\mathbb {R}}\to {\mathbb {R}}\) be a diffeomorphism. Let \(\varphi\in {\mathcal {C}}^{1}((a,b))\) be an involution. Let c be a fixed point of φ. Then x is a solution of the first order differential equation with involution (1) if and only if x is a solution of the second order ordinary differential equation (2).
Furthermore, a version of Lemma 1.1 can be proved for the case with periodic boundary value conditions.
Lemma 1.2
([1], Lemma 2.2)
Let \([a,b]\subset {\mathbb {R}}\) and let \(f:{\mathbb {R}}\to {\mathbb {R}}\) be a diffeomorphism. Let \(\varphi\in {\mathcal {C}}^{1}([a,b])\) be an involution such that \(\varphi ([a,b])=[a,b]\). Then x is a solution of the first order differential equation with involution (3) if and only if x is a solution of the second order ordinary differential equation (4).
Remark 1.1
Although not stated in [1], it is important to notice that, taking into account Theorem 9.3 in [2] and Theorem 6.93 in [3], the proofs of Lemmas 1.1 and 1.2 are still valid if we weaken the regularity hypothesis on f and \(f^{-1}\) to f and \(f^{-1}\) being absolutely continuous and f locally Lipschitz.
Problems concerning the ϕ-Laplacian (or, particularly, the p-Laplacian) have been studied extensively in recent literature. Drábek, Manásevich and others study the eigenvalues of problems with the p-Laplacian in [6–10] using variational methods. The existence of positive solutions is treated in [11], the existence of an exact number of solutions in [12] and topological existence results can be found in [13]. Anti-maximum principles and sign properties of the solutions are studied in [14, 15]. In [16] one studies a variant of the p-Laplacian equation with an approach based on variational methods, in [17] one studies the eigenvalues of the Dirichlet problem, and in [18] one finds some oscillation criteria for equations with the p-Laplacian.
In the following section we will study the existence, uniqueness and periodicity of solutions of problem (7) and in Section 3 we will apply these results to the case of problems with reflection.
2 General solutions
First, we write in a general way the solutions of equations involving the g-f-Laplacian.
Let \(\tau_{i},\sigma_{i}\in[-\infty,\infty]\), \(i=1,\dots,4\), \(\tau _{1}<\tau_{2}\), \(\sigma_{1}<\sigma_{2}\), \(\tau_{3}<\tau_{4}\), \(\sigma _{3}<\sigma _{4}\). Let \(f:(\tau_{1},\tau_{2})\to(\sigma_{1},\sigma_{2})\) and \(g:(\tau _{3},\tau_{4})\to(\sigma_{3},\sigma_{4})\) be invertible functions such that f and \(g^{-1}\) are continuous. Assume there is \(s_{0}\in(\tau_{1},\tau _{2})\) such that \(f(s_{0})=0\) and define \(F(t):=\int_{s_{0}}^{t}f(s)\,\mathrm {d}s\). Observe that F is 0 at \(s_{0}\) and of constant sign everywhere else. The following lemma is a straightforward application of the properties of the integral.
Lemma 2.1
If f is continuous, invertible and increasing (decreasing) then \(F_{-}\equiv F| _{(-\infty,s_{0}]}\) is strictly decreasing (increasing) and \(F_{+}\equiv F| _{[s_{0},+\infty)}\) is strictly increasing (decreasing). Furthermore, if \(\tau_{1}=-\infty\), \(F(-\infty)=+ \infty\) (−∞) and if \(\tau_{2}=+\infty\), \(F(+\infty)=+ \infty\) (−∞).
Definition 2.1
A solution x of problem (7) will be \(x\in {\mathcal {C}}^{1}(I)\), such that \(g\circ x'\) is absolutely continuous on I, where I is an open interval with \(a\in I\). The solution must further satisfy the requirement that the equation in problem (7) holds a.e. and the initial conditions are satisfied as well.
Theorem 2.2
Let \(f:(\tau_{1},\tau_{2})\to(\sigma _{1},\sigma_{2})\) and \(g:(\tau_{3},\tau_{4})\to(\sigma_{3},\sigma_{4})\) be invertible functions such that f and \(g^{-1}\) are continuous and assume \(0\in(\tau_{1},\tau_{2})\cap(\tau_{3},\tau_{4})\), \(f(0)=0\), \(g(0)=0\), f and g increasing, \(F(c_{1})+G(g(c_{2}))<\min\{G(\sigma _{3}),G(\sigma_{4})\}\). Then there exists a unique local solution of problem (7).
Proof
For the first part of the theorem and without loss of generality, we will prove the existence of solution in an interval of the kind \([a,a+\delta)\), \(\delta\in {\mathbb {R}}^{+}\). The proof would be analogous for an interval of the kind \((a-\delta, a]\).
On the other hand, since x is increasing in \([a,+\infty)\) and \(c_{1}<0\), by equation (11) we see that \(x'\) is increasing as long as x is negative. This means that, eventually (in finite time), x will be positive and therefore, \(x'\) is decreasing in \([\tilde{a},+\infty)\) for ã big enough, so there exists \(x'(+\infty )\ge 0\). If we assume \(x'(+\infty)=\epsilon >0\), this implies that \(x(+\infty )=+\infty\), for there would exist \(M\in {\mathbb {R}}\) such that \(x'(t)>\epsilon /2\) for every \(t\ge M\), so \(x'(+\infty)=0\). Taking the limit \(t\to+\infty \) in equation (9), \(x(+\infty )=F^{-1}_{+}(G(g(c_{2}))+F(c_{1}))\).
Observe that \(x'(t_{0})=0\), so x attains its maximum at \(t_{0}\) and \(x(t_{0})=F^{-1}_{+}(G(g(c_{2}))+F(c_{1}))\) by equation (9), that is, \(x(t_{0})=\sup I\). In order for this value to be well defined it is necessary that \(G(g(c_{2}))+F(c_{1})\le F(\tau_{2})\).
Remark 2.1
A similar argument can be given for the case f and g have different growth type (e.g. f increasing and g decreasing), but taking the negative branch of the inverse function \(G^{-1}\) in (11).
Remark 2.2
Remark 2.3
Remark 2.4
2.1 A particular case
The following corollary is just the restatement of Theorem 2.2 for this particular case.
Corollary 2.3
Let \(f:(\tau_{1},\tau_{2})\to(\sigma _{1},\sigma_{2})\) be an invertible function such that f is continuous and assume \(0\in(\tau_{1},\tau_{2})\), \(f(0)=0\), and f increasing, \(\lambda >0\), \((1+\lambda )F(c)<\min\{F(\tau_{1}),F(\tau_{2})\}\). Then there exists a unique local solution of problem (13).
There are some particular cases where the formula (14) can be simplified.
Example 2.1
Remark 2.5
If a continuous function f satisfies \(f(rt)=h(r)f(t)\), we can obtain the explicit expression of f. Let \(c=f(1)\), \(g(t):=f(t)/f(1)\), and \(\alpha =\ln g(e)\). Then \(g(t s)=g(t)g(s)\). Also, for \(t\ne0\), \(1=g(1)=g(t/t)=g(t)g(1/t)\), and therefore \(g(t^{-1})=g(t)^{-1}\). If \(n\in {\mathbb {N}}\), \(g(t^{n})=g(t)^{n}\), so, for \(t\ge0\), \(g(t)=g(t^{\frac {n}{n}})=g(t^{\frac{1}{n}})^{n}\), and \(g(t^{\frac{1}{n}})=g(t)^{\frac {1}{n}}\). Hence, \(g(t^{\frac{p}{q}})=g(t)^{\frac{p}{q}}\) for every \(p,q\in {\mathbb {N}}\), \(q\ne0\), and, by the density of \({\mathbb {Q}}\) in \({\mathbb {R}}\) and the continuity of f, \(g(t^{r})=g(t)^{r}\) for all \(t\ge0\), \(r\in {\mathbb {R}}^{+}\).
2.2 Dependence of T on λ and c
Based on the approach used in Example 2.1, we study now the dependence of T on λ and c in a general way. For simplicity, we will assume \(c>0\). For the case \(c<0\), just do the change of variable \(y(t)=-x(t)\).
Example 2.2
Example 2.3
3 Problems with reflection
Let us consider again the problem that motivated this paper in the Introduction, obtaining solutions of problem (3) in the case \(\varphi(t)=-t\). Hence, consider all of the problems (1)-(5) in the case \(\varphi(t)=-t\).
Observe that Lemma 1.1 (following Remark 1.1) can be trivially extended to the following lemma.
Lemma 3.1
Let \(f:(\tau_{1},\tau_{2})\to(\sigma_{1},\sigma_{2})\) a locally Lipschitz a.c. function with a.c. inverse. Then x is a solution of the first order differential equation with involution (3) if and only if x is a solution of the second order ordinary differential equation (4).
As was shown in Section 1, problem (4) is equivalent to problem (5). We can now state the following corollary of Theorem 2.2 regarding the periodicity of problem (3) as foreseen in Section 1.
Corollary 3.2
Let \(f:(\tau_{1},\tau_{2})\to(\sigma _{1},\sigma_{2})\) an increasing locally Lipschitz a.c. function with a.c. inverse such that \(0\in(\tau_{1},\tau_{2})\), \(f(0)=0\), and \({c}>0\). Assume \(2F({c})<\min\{F(\tau_{1}),F(\tau_{2})\}\). Then, if \(x_{c}(t)\) is a solution of problem (6) and we assume there exist \(\overline {c}_{1},\overline{c}_{2}\in {\mathbb {R}}\), \(\overline{c}_{1}<\overline{c}_{2}\), such that \(2\max\{ F(\overline{c}_{1}),F(\overline{c}_{2})\}<\min\{F(\tau_{1}),F(\tau_{2})\}\) and \((x_{\overline {c}_{1}}(b)-\overline{c}_{1})(x_{\overline{c}_{2}}(b)-\overline{c}_{2})<0\), then problem (3) must have at least a solution.
We now give an example in which there is no need to find \(\overline {c}_{1},\overline {c}_{2}\in {\mathbb {R}}\) under the conditions of Corollary 3.2 because the function determining the period has a simple inverse.
Example 3.1
The diffeomorphisms f in this example has been widely studied by Bereanu and Mawhin (see, for instance, [32, 33]) and is a type of singular ϕ-Laplacian known as the mean curvature operator of the Minkowski space. Its inverse, the mean curvature operator of the Euclidean space, also studied in [32], appears in Example 2.3.
Declarations
Acknowledgements
The work was partially supported by FEDER and Ministerio de Economía y Competitividad, Spain, project MTM2013-43014-P. The second author was supported by FPU scholarship, Ministerio de Educación, Cultura y Deporte, Spain and Xunta de Galicia (Spain), project EM2014/032.
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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