Periodic solutions for some phi-Laplacian and reflection equations

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 first.


Introduction
The idea behind this paper appeared in another work of the authors [10] where the following lemmas were proved. Definition 1.1. If A ⊂ R, a function ϕ : A → A such that ϕ = Id and ϕ • ϕ = Id is called an involution.
Let us consider the problems x (t) = f (x(ϕ(t))), x(c) = x c (1) and Lemma 1.1 ( [10, Lemma 2.1]). Let (a, b) ⊂ R and let f : R → R be a diffeomorphism. Let ϕ ∈ 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.
Let us consider the equations and x (t) = f ( f −1 (x (t))) f (x(t))ϕ (t), x(a) = x(b) = f −1 (x (a)). . 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 [10], it is important to notice that 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 absolutely continuous and f locally Lipschitz.
Linear problems with involutions, similar to problem (1), have also been studied in [10][11][12]. Observe that, from problem (4) we have that So, clearly, problem (4) is equivalent to the problem Which involves the f −1 -Laplacian ( f −1 • x ) , although, contrary to most literature, the other term in the equation does not involve f −1 but f . As we will see, this is not more than a further generalization in the line of the p − q-Laplacian.
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 [5,[16][17][18]27] using variational methods. The existence of positive solutions is treated in [19], the existence of an exact number of solutions in [29] and topological existence results can be found in [28]. Anti-maximum principles and sign properties of the solutions are studied in [7,8].
In [14] they study a variant of the p-Laplacian equation with an approach based on variational methods, in [6] they study the eigenvalues of the Dirichlet problem and in [15] they find some oscillation criteria for equations with the p-Laplacian.
The φ -Laplacian is studied from different points of view in several papers, e. g. [1,9,13,21,23,25,26]. Actually, if we consider the problem with the f −1 -Laplacian and we assume there exist c 1 , c 2 ∈ R, c 1 < c 2 , such that a unique solution of problem (6) exists for every c ∈ [c 1 , < 0, then problem (4) must have at least a solution due to the continuity of x c on c and Bolzano's theorem. For this reason we will be interested in studying the properties of problem (6) and its solutions in this paper. In the sections to come we study this problem and more general versions of it.
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.

General solutions
First, we write in a general way the solutions of equations involving the g − f -Laplacian.
The following result holds immediately from the properties of continuous real functions. All the same, define G(t) := t g −1 ({0}) g −1 (s) d s and consider the problem for some fixed c 1 , c 2 ∈ R.
A solution x of problem (7) will be x ∈ C 1 (I) such that is absolutely continuous on I where I is an open interval with a ∈ I. The solution must further satisfy that the equation in problem (7) satisfied a. e and the initial conditions are satisfied as well.
Furthermore, if F(c 1 ) + G(g(c 2 )) < min{F(τ 1 ), F(τ 2 )}, then such solution is defined on the whole real line and is periodic of smallest period 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 + δ ), δ ∈ R + . The proof would be analogous for an interval of the kind (a − δ , a]. Let y(t) = g(x (t)). Then problem (7) is equivalent to .
so, integrating both sides from a to t, ). That is, undoing the change of variables, If c 1 = c 2 = 0 it is clear that the only possible solution is x ≡ 0. Assume, without loss of generality, that c 2 is non-negative and c 1 negative (the other cases are similar). If c 2 = 0 then, integrating (7), which implies x is positive in some interval [a, a + δ ). If c 2 is positive, then x has to be positive at least in some neighborhood of a, so, in a right neighborhood of a, we can solve for g • x in (9) as In order to solve for x in (10), we need F(c 1 ) + G(g(c 2 )) < G(σ 4 ). Then, Integrating between a and t, where H + is strictly increasing in its domain due to the positivity of the denominator in the integrand. Hence, for t sufficiently close to a, Therefore, a solution of problem (7) exists and is unique on an interval [a, a + δ ).
) . Now, we study the range of H + . g(x (t)) is positive as long as x (t) is negative. Hence, consider G is positive on non-zero values, so equation (9) implies that, F(x(t)) < G(g(c 2 )) + F(c 1 ) for all t ∈ (a,t 0 ).
On the other hand, since x is increasing in [a, +∞) and c 1 < 0, by equation (11) we have that x is increasing as long as x is positive. This means that, eventually (in finite time), x will be positive and therefore, x is decreasing in [ã, +∞) forã big enough, so there exists x (+∞) ≥ 0. If we assume Take M 3 in such a way. Then, integrating equation (7) between M 3 and M 3 + 1, a contradiction. Therefore, t 0 ∈ R.
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 Now, we have that H + is well defined at sup I (assuming it is defined continuous at that point).

If we define
H − is strictly decreasing in its domain and x(t) = H −1 − (t) for t ∈ [t 0 ,t 1 ]. We can again deduce that Using the positivity and growth conditions of the functions involved, it is easy to check that .
Defining T := b − a and extending x periodically in the following way (we have x already defined in [a, a + T ]), where t := sup{k ∈ Z : k ≤ t}, it is easy to check that x, extended in such a way, is a global periodic solution of problem (7). Take z(t) := x(t − T ), t ∈ R, we show that z is a solution of the problem in [a + T, a + 2T ].
This is equivalent to (g • z ) (t) + f (z(t)) = 0 for a. e t ∈ R. Also, Remark 2.1. A similar argument can be done 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).  (7) is equivalent to with v(t) = x(t) − s 0 . Hence, we can apply Theorem 2.2 to this case.
Remark 2.3. Using the notation of the Theorem, the explicit form of the solution of problem (7) is given by Remark 2.4. Consider the following particular case of problem (7) with f (0) = 0, g(0) = 0, f and g increasing and the hypothesis for a unique global solution of the following problem are satisfied in Theorem 2.2.
It is clear that, in the case g(x) = f (x) = x, the unique solution of problem (12) is sin(t), which suggests the definition of the sin g, f function as the unique solution of problem (12) for general g and f . Correspondingly, arcsin + g, f (r) := H + (r) This function, defined as such, coincides with the arcsin p function defined in [4,24] for the p-Laplacian f (x) = g(x) = |x| p−2 x, the function arcsin p,q defined in [3,20,22] for the p − q-Laplacian f (x) = |x| q−2 x, g(x) = |x| p−2 x, which first appeared with a slightly different definition in [17], and the hyperbolic version of this function, also in [3,22], which corresponds to the case f (x) = |x| q−2 x, g(x) = −|x| p−2 x. [30] derives generalized Jacobian functions in a similar way, defining of which the inverse (see [30,Proposition 3.2]) is precisely a solution of where f r is the r-Laplacian for r = p, q and p * p = p * + p. Observe this case is also covered by our definition.
In all of the aforementioned works they are interested on the inverse of the arcsin g, f function, the sin g, f function, which they extend to the whole real line by symmetry and periodicity. Observe that in our case f and g need not to be odd functions, contrary to the above examples, but we can still give the definition of the sin g, f function in the whole real line. Also, this lack of symmetry gives rise to a richer set of right inverses of sin g, f , for instance, arcsin − g, f (r) := H − (r), In general, if we have a problem of the kind Φ((g • x ) , x(t)) = 0; x(0) = 0, x (0) = 1, and we know it has a unique solution in a neighborhood of 0, then we can define sin g,Φ as such unique solution and its inverse, in a neighborhood of 0, arcsin g,Φ .
We now study the periodicity of the solutions of problem (7) with the functions and constants defined in the previous section.

A particular case
Having in mind problem (6), we now consider a particular case of problem (7) for the rest of this section. Assume f is invertible and both f and f −1 are continuous. For convenience, assume also that f is increasing and f (0) = 0. Consider the following problem.
The following corollary is just the restatement of Theorem 2.2 for this particular case. Furthermore, if (1 + λ −1 )F(c) < min{F(τ 1 ), F(τ 2 )}, then such solution is defined on R and is periodic of first period T := There are some particular cases where the formula (14) can be simplified.
If f is odd then F is even and, with the change of variables r = c s, we have that expression (14) becomes .
We can also consider the dependence of T on λ . We do this study for this particular example and in the following section we develop a general theory.

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.
We continue to assume the hypotheses for (13) and further assume that f is a differentiable function. Let us divide the interval of integration in equation (8) Observe that F is injective restricted to any of the two intervals. For the nonnegative interval, taking the change of variables we have that All the same, with the change of variables . Then Therefore, Observe that α, f | [0,1] , f , F, F −1 + , β + , ∂ β − ∂ λ , γ + , ∂ γ + ∂ λ are non-negative, while ∂ α ∂ λ , F −1 − , β − , ∂ β + ∂ λ , γ − , ∂ γ − ∂ λ are non-positive. In general we cannot tell the sign of T (λ , c) from this expression, but making certain assumptions we can simplify it to derive information.

Problems with reflection
Let us consider again the problem that motivated this paper in the Introduction, the obtaining of solutions of problem (3) in the case ϕ(t) = −t. Hence, consider all of the problems (1)-(5) in the case ϕ(t) = −t.
Observe that Lemma 1.1 (following Remark 1.1) can be trivially extended to the following lemma.
Lemma 3.1. Let f : (τ 1 , τ 2 ) → (σ 1 , σ 2 ) an locally Lipchitz 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.
3.2 because the function determining the period has a simple inverse. Consider now the problem There is a unique solution for problem (17) for p ∈ (0, 2) ∪ (2, +∞). Just take the unique solution of problem (16) with