On periodic Ambrosetti-Prodi-type problems

1.   Introduction

In this paper, we consider the second-order periodic Ambrosetti-Prodi-type problem composed by the differential equation

where $f:[a, b]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ and $g:[a, b]\rightarrow \mathbb{R}^{+}$ are continuous functions, $\lambda \in \mathbb{R},$ and the periodic boundary conditions

Periodic problems are an important field of research, not only from a theoretical point of view but also by the huge sort of applications. They have been studied by many authors with different goals and applying a large panoply of methods. We mention only a few examples: results of nonexistence and multiplicity of solutions for strongly nonlinear differential equations ^[3]; sufficient conditions for the existence of periodic orbits as solutions of the $\phi$-Laplacian generalized Liénard equations ^[9], or as limit cycles ^[6]; solvability of higher-order fully differential equations ^[10], nonlinear oscillations of even-order differential equations ^[11]; cones theory applied to singular third order problems ^[12]; problems with anti-periodic boundary conditions ^[21].

Equations involving parameters as in (1.1) allow a so-called Ambrosetti-Prodi alternative, as it was introduced in ^[1]: There are values $\lambda _{0}$ and $\lambda _{1},$ of the real parameter $\lambda,$ such that the problem has no solution for $\lambda < \lambda _{0}$, at least one if $\lambda = \lambda _{0},$ or two solutions, for $\lambda _{0} < \lambda < \lambda _{1}.$ Since then, these problems have been studied in several cases and contexts, such as separated two-point or three-point boundary value problems ^[13,20]; Neuman's type conditions ^[19]; the periodic case ^[8,16,22]; parametric problems with the nonlinear Robin $(p, q)$-Laplace operator ^[18]; with adequate asymptotic properties ^[5]; applied to the fractional Laplacian ^[2], with asymptotic sign-changed nonlinearities ^[17], among others.

Recently, in ^[14,15], the authors suggested a method to apply lower and upper solutions to third-order periodic Ambrosetti-Prodi problems, based on a speed-growth condition on the nonlinearity, requiring that it grows with different velocities in the several variables. In this paper, we prove that such speed condition is no need for second-order periodic problems and the monotone properties on the nonlinearities are more general and not necessarily periodic.

This work is organized as follows: Section 2 contains the definitions of lower and upper functions, the Nagumo condition used, the corresponding a priori bound for the first derivative, and a classic localization theorem existent in the literature. In Section 3, a first discussion on the parameter is done, about the existence and non-existence of solution, and, in Section 4, this discussion is extended to the multiplicity of solutions. The last section contains a model to study the oscillation of a damped and forced pendulum, using a technique to estimate the critical values $\lambda _{0}$ and $\lambda _{1}$ of the parameter.

2.   Definitions and preliminary results

The functional frame work, followed in the paper, is the usual space of continuous functions $\mathcal{C}^{2}[a, b]$ with the correspondent norm

where

Lower and upper functions are considered as in the next definition:

Definition 2.1. The function $\alpha \in \mathcal{C}^{2}[a, b]$ is a lower solution of problems (1.1) and (1.2) if:

(i) $\alpha ^{\prime \prime }\left(x\right) +f\left(x, \alpha \left(x\right), \alpha ^{\prime }\left(x\right) \right) \geq \lambda$ $g(x),$

(ii) $\alpha \left(a\right) \leq \alpha \left(b\right),$ $\alpha ^{\prime }\left(a\right) \geq \alpha ^{\prime }\left(b\right).$

The function $\beta \in \mathcal{C}^{2}[a, b]$ is an upper solution of problems (1.1) and (1.2) if:

(iii) $\beta ^{\prime \prime }\left(x\right) +f\left(x, \beta \left(x\right), \beta ^{\prime }\left(x\right) \right) \leq \lambda g(x),$

(iv) $\beta \left(a\right) \geq \beta \left(b\right),$ $\beta ^{\prime }\left(a\right) \leq \beta ^{\prime }\left(b\right).$

The only growth condition to require to the nonlinearity in (1.1) is given by a Nagumo-type condition:

Definition 2.2. A continuos function $h:\left[0, 1\right] \times \mathbb{R} ^{2}\rightarrow \mathbb{R}$ verifies a Nagumo-type condition relatively to some continuos functions $\gamma, \Gamma,$such that

in the set

if there is a continuos function $\psi _{S}:\left[0, +\infty \right[ \rightarrow \left] 0, +\infty \right[$ such that

with

The a priori estimation provided by the Nagumo condition is given by next lemma, which proof is a particular case ($n = 2$) of 【^[7], Lemma 1.

Lemma 2.1. Let $h:[a, b]\times \mathbb{R} ^{2}\rightarrow \mathbb{R}\ $be a continuous function verifying the Nagumo-type conditions (2.1) and (2.2) in $S.$Then there is $r > 0\ $such that every solution $y\left(x\right) \ \ $of (1.1) verifying

satisfies

Remark 2.1. This "a priori" bound is independently of $\lambda,$ since $\lambda$ belongs to a bounded set.

The existence and localization result will be based on Theorem 5.3 of ^[4] adapted to the Nagumo conditions (2.1) and (2.2), for the values of the parameter $\lambda$ such that there are upper and lower solutions of (1.1) and (1.2) according to Definition 2.1.

Theorem 2.1. ^[4] Let $f:[a, b]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ and $g:[a, b]\rightarrow \mathbb{R}^{+}$ be continuous functions. Assume that there are lower and upper solutions of problems (1.1) and (1.2), $\alpha (x)$ and $\beta (x),$ respectively, accordingly Definition 2.1, such that

and $f$ verifies Nagumo-type conditions (2.1) and (2.2) in

Then (1.1) and (1.2) have at least a solution $v(x)\in C^{2}(\left[0, 1\right])$ such that

The following example stresses how the localization provided by the lower and upper solutions technique could be helpful for the estimation of the parameter variation.

Example 2.1. Consider the problem composed by the differential equation

together with the periodic conditions

The functions $\alpha, \beta :\left[0, \pi \right] \rightarrow \mathbb{R}$ defined by

are, respectively, lower and upper solutions of problems (2.4) and (2.5) for

The above problem is a particular case of the initial one (1.1) and (1.2) with

As the assumptions of Theorem 2.1 are verified, then there is a solution $v(x)$ of (2.4) and (2.5), for $-66.164\leq \lambda \leq -52.537,$ lying in the strip

As it is illustrated graphically in Figure 1, this solution is not a trivial one, as no constant function is inside the strip.

3.   Existence and non-existence result

A first discussion on $\lambda$ about the existence and non-existence of a solution will be done in this section.

Theorem 3.1. Consider the assumptions of Theorem 2.1, $f:[a, b]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ and $g:[a, b]\rightarrow \mathbb{R}^{+}$ continuous functions, satisfying a Nagumo-type condition and assume that there are $\lambda _{1}\in \mathbb{R}$ and $m > 0$ such that

for every $x\in \left[a, b\right]$ and $y_{0}\leq -m.$

Then there is $\lambda _{0} < \lambda _{1}$ (eventually $\lambda _{0} = -\infty$) such that:

● for $\lambda < \lambda _{0},$ the problems (1.1) and (1.2) has no solution;

● for $\lambda _{0} < \lambda \leq \lambda _{1},$ the problems (1.1) and (1.2) has at least one solution.

Proof. Defining

by (3.1), there exists $x^{\ast }\in \left[a, b\right]$ such that

Thus $\beta \left(x\right) \equiv 0$ is an upper solution of (1.1) and (1.2). Moreover, the function $\alpha \left(x\right) \equiv -m$ is a lower solution of (1.1) and (1.2), as by (3.1),

Therefore, by Theorem 2.1, there is at least a solution of (1.1) and (1.2) with $\lambda = \lambda ^{\ast }$.

Next we show that the set of the parameters for which there is a solution, is a continuous set, that is assuming that the problem (1.1) and (1.2) have a solution for $\lambda = \xi < \lambda _{1},$ then it has at least one solution for $\lambda \in \left[\xi, \lambda _{1}\right]$.

Suppose that (1.1) and (1.2) have a solution $v_{\xi }\left(x\right)$, for $\xi$ $\leq \lambda _{1},$ that is

For $m > 0$ given by (3.1)$, \ $take $M > 0,$ large enough such that

As in (3.2), the function $\alpha \left(x\right) = -M$ is a lower solution of (1.1) and (1.2) for $\lambda \leq \lambda _{1},$ and $\beta \left(x\right) =$ $v_{\xi }\left(x\right)$ is an upper solution of (1.1) and (1.2), for every $\lambda \in \left[\xi, \lambda _{1} \right],$ because$,$ for $\xi \leq \lambda,$ we have

To apply Theorem 2.1, it remains to prove that

Suppose the inequality is not true. So, there is $x\in \left[a, b\right]$ such that

and define

By (3.3), $x_{0}\in \left] a, b\right[$, $v_{\xi }^{\prime }\left(x_{0}\right) = 0$ and $v_{\xi }^{\prime \prime }\left(x_{0}\right) \geq 0.$ So, by (3.1), we obtain the following contradiction:

Therefore, by Theorem 2.1, the problems (1.1) and (1.2) have at least a solution $v\left(x\right)$ for every $\lambda$ such that $\xi \leq \lambda \leq \lambda _{1}.$

Consider now the set

As $\lambda ^{\ast }\in S,$ then $S\neq \emptyset$ and define

Clearly, (1.1) and (1.2) have no solution for $\lambda < \lambda _{0}$ and, by the continuity of the parameters set, it has at least a solution for $\lambda \in \lbrack \lambda _{0}, \lambda _{1}]$.

Remark that, if $\lambda _{0} = -\infty$ then (1.1) and (1.2) have a solution for every $\lambda < \lambda _{1}.$

4.   Multiplicity theorem

The multiplicity of solutions is proven by topological degree theory applied to a homotopic, modified and perturbed problem.

Theorem 4.1. Let $f:[a, b]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ be a continuous function verifying the assumptions of Theorem 3.1, together with the monotone condition

If there are $B > 0$ such that every solution $v$ of (1.1) and (1.2) with $\lambda \leq \lambda _{1}$, verifies

and $\theta \in \mathbb{R}$ such that

for every $\left(x, y_{0}, y_{1}\right) \in \left[a, b\right] \times \left[-m, B\right] \times \mathbb{R}$, with $m$ given by (3.1), then $\lambda _{0}$, in Theorem 3.1, is finite and:

(1) If $\lambda < \lambda _{0}$, (1.1) and (1.2) have no solution;

(2) If $\lambda = \lambda _{0},$(1.1) and (1.2) have at least one solution;

(3) If $\lambda \in \left] \lambda _{0}, \lambda _{1}\right]$, (1.1) and (1.2) have at least two solutions.

Proof. For clarity, we divide the proof into several steps.

Step 1. Every solution $v$ of problems (1.1) and (1.2) for $\lambda \in \left[\lambda _{0}, \lambda _{1}\right]$ , satisfies

with $m$ given by (3.1) and $B$ by (4.1).

Assume, by contradiction, that there is a solution $v$ of (1.1) and (1.2) for $\lambda \in \left[\lambda _{0}, \lambda _{1}\right],$ and $\tau \in \left[a, b\right]$ such that

If $\tau \in \left] a, b\right[$, then $v^{\prime }\left(\tau \right) = 0,$ $v^{\prime \prime }\left(\tau \right) \geq 0,$ and, by (3.1) this contradiction holds

If $\tau = a$ or $\tau = b$,

then

and therefore,

Applying an analogous technique as in the previous theorem, it can be proved similar contradictions.

So, every solution $v$ of (1.1) and (1.2) with $\lambda \in \left[\lambda _{0}, \lambda _{1}\right],$ verifies

and, therefore, by (4.1),

Step 2.$\lambda _{0}$ is finite.

Assume that $\lambda _{0} = -\infty$. So, by Theorem 3.1, for every $\lambda \leq \lambda _{1},$ the problems (1.1) and (1.2) has at least a solution.

Define

and take $\lambda$ sufficiently small such that

For every solution $v$ of (1.1) and (1.2), by (4.2),

and, by (1.2), there exists $\xi \in \left] a, b\right[$ such that $v^{\prime }\left(\xi \right) = 0.$

For $x < \xi$

For $x\geq \xi$

Choose $J = \left[a, \frac{3a+b}{4}\right],$ or $J = \left[\frac{a+3b}{4}, b \right]$, such that $\left\vert \xi -x\right\vert \geq \frac{b-a}{4}$, for every $x\in J.$ If $J = \left[a, \frac{3a+b}{4}\right],$ then

for $J = \left[\frac{a+3b}{4}, b\right],$

In the first case, by (4.1) and (4.3),

which contradicts (4.1).

For $J = \left[\frac{a+3b}{4}, b\right]$ a similar contradiction is achieved, and, therefore, $\lambda _{0}$ is finite.

Step 3. For $\lambda \in \left] \lambda _{0}, \lambda _{1} \right]$, (1.1) and (1.2) has at least two solutions.

As $\lambda _{0}$ is finite, by Theorem 3.1, for $\lambda _{-1} < \lambda _{0}$, the problems (1.1) and (1.2), have no solution for $\lambda\; = \; \lambda_{-1}$.

By Lemma 2.1 and Step 1, we can take $m_{1} > 0$ large enough such that $\left\Vert v^{\prime }\right\Vert < m_{1},$ for every solution $v$ of (1.1) and (1.2), with $\lambda \in \left] \lambda _{-1}, \lambda _{1}\right].$

Consider the operators

and, for $\mu \in \left[0, 1\right]$,

where

and

As $\mathcal{L}$ has a compact inverse it can be considered the completely continuous operator

defined by

Let $B_{1}: = \max \left\{ m, \left\vert B\right\vert \right\}$ and define the set

By Step 1, if $v$ is a solution of (1.1) and (1.2) with $\lambda \in \left] \lambda _{-1}, \lambda _{1}\right],$ then $v\notin \partial \Omega$, the topological degree is well defined, and

Considering the convex combination of $\lambda _{-1}$ and $\lambda _{1}$,

and the corresponding homotopic problems (1.1) and (1.2) for $\lambda = \Pi \left(\gamma \right)$, the topological degree $d\left(\Psi _{\Pi \left(\eta \right) }, \Omega _{2}\right)$ is well defined for $\eta \in \left[0, 1\right]$ and $\lambda \in$ $\left] \lambda _{-1}, \lambda _{1} \right].$

By (4.4) and the invariance of the degree under homotopy

with $\lambda \in \left] \lambda _{-1}, \lambda _{1}\right].$

Take $s\in \left] \lambda _{0}, \lambda _{1}\right] \subset \left] \lambda _{-1}, \lambda _{1}\right]$ and, by Theorem 3.1, let $v_{s}$ be the correspondent solution of (1.1) and (1.2) for $\lambda = s.$

Consider $\delta > 0$, small enough, such that

Then $v_{\ast }\left(x\right) : = v_{s}\left(x\right) +\delta, \forall x\in \left[a, b\right],$ is an upper solution of (1.1) and (1.2), with $s < \lambda < \lambda _{1},$ as

and, for the boundary conditions,

The constant function equal to $-m,$ is a lower solution of (1.1), (1.2), with $\lambda \in \left[s, \lambda _{1}\right],$ as, by (3.1),

and the boundary conditions trivially hold.

So, by Theorem 2.1,

and as $\lambda$ belongs to a bounded set, by Lemma 2.1 and Remark 2.1, there are $m_{\delta } > 0$ and a set

such that

Considering $m_{\delta }$ sufficiently large such that $\Omega _{\delta }\subset \Omega _{2},$ then, by (4.5) and the additivity of the topological degree,

So, the problems (1.1) and (1.2) has at least two solutions $u$ and $v$ such that $u\in \Omega _{\delta }$ and $v\in \Omega -\overline{\Omega _{\delta }}$ for $\lambda \in \left[s, \lambda _{1}\right],$ for $s$ is arbitrary in $]\lambda _{0}, \lambda _{1}].$

Step 4. For $\lambda = \lambda _{0},$ (1.1) and (1.2) has at least one solution.

Take a sequence $(\lambda _{n})$ with $\lambda _{n}\in]\lambda _{0}, \lambda _{1}]$ and $\lim$ $\lambda _{n} = \lambda _{0}.$ By Theorem 3.1, for each $\lambda = \lambda _{n},$ the problems (1.1) and (1.2) has a solution $v_{n}.$ From the bounds given in Step 1, $\left\Vert v_{n}\right\Vert < B_{1},$ $\left\Vert v_{n}^{\prime }\right\Vert < B_{1}$ independently of $n,$ and, by Remark 1, there is $k > 0$ sufficiently large such that $\left\Vert v_{n}^{\prime \prime }\right\Vert < k,$ independently of $n.$

Then sequences $(v_{n})$ and $(v_{n}^{\prime }),$ $n\in \mathbb{N}$, are bounded in $C([a, b]).$ By the Arzèla-Ascoli theorem, we can take a subsequence of $(v_{n})$ that converges in $C^{2}([a, b])$ to a solution $v_{0}(t)$ of (1.1) and (1.2) for $\lambda = \lambda _{0}$.

Therefore, there exists at least one solution for $\lambda = \lambda _{0}.$

5.   Oscillation of a damped and forced pendulum

Consider the oscillation of a damped and forced pendulum given by the differential equation

with $k < 0,$ where $u(x)$ represents the angle between the string and the vertical, $m,$ the mass of the pendulum, $g,$ the gravity acceleration, $r$ the string length, $\lambda$ a real parameter, and $p(x)$ a weight positive and continuous function, subject to the periodic conditions

Remark that this is a particular case of the problems (1.1) and (1.2) with

$a = 0$ and $b = 1.$

As a numerical example consider $k = -1, m = 1, r = 5, g = 9.8$ and $p(x) = x^{2}+3,$ that is, the equation

Therefore the functions

are, respectively, lower and upper solutions of (5.1) and (5.2), for $-0.336\, 56\leq \lambda \leq 0.301\, 99.$

The function $f$ verifies a Nagumo-type growth condition in the set

as

and

By Theorem 2.1, there is a solution $u(x)$ such that

Remark that, from this localization property, it is clear that this solution $u(x)$ is not a constant function, that is, a trivial periodic solution, as it can be seen in Figure 2.

Moreover the assumptions of Theorem 3.1 are satisfied for

and $\lambda _{0}\leq -0.336\, 56.$

Restricting the set of solutions $u$ of (5.1) and (5.2) to $\overline{S},$ then $\left\vert u(x)\right\vert \leq 1.078\, 9.$ Condition (4.2) holds with $r > 0$ given by Lemma 2.1, and $\theta \leq -\frac{r+2}{3} < 0,$ we have

for $(x, y_{0}, y_{1})\in \left[0, 1\right] \times \left[0, 1.078\, 9\right] \times \left[-r, r\right].$

So, Theorem 4.1 holds, $\lambda _{0}$ is finite and for $\lambda \in]\lambda _{0}, \lambda _{1}],$ the problems (5.1) and (5.2) have at least two solutions.

6.   Discussion

The lower and upper solutions method is an adequate tool to study periodic problems, or other types of boundary value problems, mainly because it provides, not only the existence but also, the solutions localization. This technique plays a key role to prove the Ambrosetti-Prodi alternative, not only for the existence and non-existence cases but also for the multiplicity.

Moreover, from the localization part, it can be easily seen if the periodic solution is, eventually, trivial or if it is nontrivial. This happens, in the periodic case, since there are no horizontal lines inside the admissible region.

7.   Conclusions

In recent papers on third-order periodic problems ^[14,15], it was shown that the nonlinearity must have different growth speeds on some variables, to have more than one solution.

This work proves that, for second-order periodic problems, such property is not needed to discuss the solution multiplicity, based on the parameter variation.

The example and the application suggest a method to estimate the critical values of the parameter, exploiting the localization region, given by the lower and upper solutions method.

Acknowledgment

This research was supported by national funds through the Fundação para a Ciência e Tecnologia, FCT, under the project UIDB/04674/2020.

Conflict of interest

There is no conflict of interest.