Shooting Type Methods

Abstract

We introduce one of the simplest topological methods, usually known as the shooting method, which basically consists in reducing a problem to a finite-dimensional equation for a certain parameter λ. Then, appropriate tools can be used, such as the Brouwer theorem or equivalent results. The chapter is intended to be self-contained and employs only concepts from basic calculus; for simplicity, the study of systems is restricted here to the two-dimensional case, for which we present a very elementary proof of the fixed point theorems we shall be using.

Appendices

Appendix

A few historical notes and further comments. The shooting method was introduced in 1905 by Severini [106], although, as mentioned in [47], the modern version of the method has its origins in a more sophisticated technique, known as the Ważewski method, which makes use of a topological lemma closely related to Brouwer’s theorem (e.g., [44]). Historical aspects of the two-point boundary value problems for ordinary equations are presented in [80].

As mentioned, the case n = 2 of Brouwer’s theorem is very special since its proof, far from being trivial as in the case n = 1, can still be performed using only basic tools. For instance, it is very easy to conclude that there are no retractions from B ₁(0) to S ¹ since otherwise there would be an epimorphism from the fundamental group of B ₁(0)—which is trivial—onto $\mathbb{Z}$, the fundamental group of the unit circumference. When n > 2, the fundamental group of S ⁿ⁻¹ is also trivial, so more information is needed. Elementary arguments are also possible, as seen, in the context of complex analysis and—amazingly—in game theory: indeed, the result can be deduced from the so-called Hex theorem, which, roughly speaking, establishes that any game of Hex has a winner [39]. It is also worth mentioning that Brouwer’s theorem allows one to prove another of the best known results in the topology of the plane, which can also be generalized for higher dimensions: the Jordan curve theorem.

Problems

One-Dimensional Shooting

• 1.1. Prove that the forced pendulum equation with friction

  $$\displaystyle{ u^{\prime\prime}(t) + au^{\prime}(t) + b\sin (u(t)) = p(t) }$$

  (1.23)

  with $p: [0,1] \rightarrow \mathbb{R}$ continuous admits at least one solution for the arbitrary Dirichlet conditions

  $$\displaystyle{u(0) = u_{0},\qquad u(1) = u_{1}.}$$

  Does an analogous result hold for the periodic conditions

  $$\displaystyle{u(0) = u(1),\qquad u^{\prime}(0) = u^{\prime}(1)?}$$

  Generalize for the problem $u^{\prime\prime}(t) + au^{\prime}(t) = f(t,u(t))$ with f bounded.

• 1.2. Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a C ¹ and bounded function such that

  $$\displaystyle{g(0) = 0\qquad \mbox{ and}\quad {[(2k - 1)\pi ]}^{2} <g^{\prime}(0) <{[2k\pi ]}^{2}}$$

  for some integer k. Prove that the Dirichlet problem

  $$\displaystyle{u^{\prime\prime}(t) + g(u(t)) = 0,\qquad u(0) = u(1) = 0}$$

  has at least two different nontrivial solutions.

  Hint: consider as in Sect. 1.1 the differentiable function $\phi: \mathbb{R} \rightarrow \mathbb{R}$ given by ϕ(λ) = u _λ (1). Then ϕ(0) = 0 and $\phi (-\lambda ) <0 <\phi (\lambda )$ for $\lambda>\| g\|_{\infty }$. Next, prove that ϕ′(0) = w ₀(1), where w ₀ is the unique solution of the linear problem

  $$\displaystyle{\left \{\begin{array}{l} w_{0}^{\prime\prime}(t) + g^{\prime}(0)w_{0}(t) = 0, \\ w_{0}(0) = 0,\quad w_{0}^{\prime}(0) = 1, \end{array} \right.}$$

  and verify that ϕ′(0) < 0. Now draw a graph of ϕ and deduce that it has at least two nontrivial zeros.

• 1.3. *(Strict upper and lower solutions) Let $f: [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ be continuous and locally Lipschitz on u, and assume there exist functions $\alpha,\beta: [0,1] \rightarrow \mathbb{R}$ such that

  $$\displaystyle{\mbox{ $\alpha $}(t) \leq \mbox{ $\beta $}(t),\qquad \alpha ^{\prime\prime}(t)> f(t,\alpha (t)),\qquad \beta ^{\prime\prime}(t) <f(t,\beta (t))}$$

  for all t and

  $$\displaystyle{\alpha (0),\alpha (1) \leq 0 \leq \beta (0),\beta (1).}$$

  Prove that the Dirichlet problem (1.1)–(1.2) admits at least one solution u with $\mbox{ $\alpha $}(t) \leq u(t) \leq \mbox{ $\beta $}(t)$ for all t. In particular, set $\mbox{ $\alpha $} \equiv -R$ and $\mbox{ $\beta $} \equiv R$ to obtain the Hartman condition (1.4).

Brouwer’s Theorem and Related Results

• 1.4. Prove that all the theorems of Sect. 1.2.3 are equivalent, that is, that any of them can be used to prove any of the others. Moreover, prove that all of them are equivalent to the following statement: if $f: {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2}$ is continuous and there exists a constant C such that $\vert f(x) - x\vert \leq C$ for all x, then f has at least one zero.

• 1.5. Prove that the completeness axiom of the real numbers can be replaced by any of the statements mentioned in problem 1.4. Prove that, furthermore, it suffices to assume that all the mappings involved are of class C ².

• 1.6. Use the lemma in Sect. 1.4 to give a direct proof of each of the statements mentioned in problem 1.4.

• 1.7. (Hartman condition) Let $f: [0,1] \times {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2}$ be a C ¹ function, and assume there exists a positive constant R such that

  $$\displaystyle{f(t,u) \cdot u> 0\qquad \mbox{ for all }\,t \in [0,1],u \in \partial B_{R}(0).}$$

  Prove that the Dirichlet problem

  $$\displaystyle{ \left \{\begin{array}{l} u^{\prime\prime}(t) = f(t,u(t))\\ u(0) = u_{ 0},\quad u(1) = u_{1} \end{array} \right. }$$

  (1.24)

  has at least one solution for any $u_{0},u_{1} \in \overline{B_{R}(0)}$.

• 1.8. (Monotonicity condition) Let $f: [0,1] \times {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2}$ be a C ¹ function, and assume that

  $$\displaystyle{[f(t,u) - f(t,v)] \cdot (u - v)> 0}$$

  for all $(t,u),(t,v) \in [0,1] \times {\mathbb{R}}^{2}$. Prove that problem (1.24) has a unique solution for any $u_{0},u_{1} \in {\mathbb{R}}^{2}$.

• 1.9. *Extend problems 1.7 and 1.8 for f continuous and nonstrict inequalities.

Poincaré Mapping

• 1.10. Let $f: \mathbb{R} \times {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2}$ be a smooth function, T-periodic in t and sublinear in x, that is,

  $$\displaystyle{f(t + T,x) = f(t,x)\qquad \mbox{ for all $(t,x)$},}$$
  $$\displaystyle{\lim _{\vert x\vert \rightarrow \infty }\frac{f(t,x)} {\vert x\vert } = 0\qquad \mbox{ uniformly on $t$.}}$$

• 1. Prove that the problem

  $$\displaystyle{x^{\prime}(t) + x(t) = f(t,x(t))}$$

  admits at least one T-periodic solution.

• 2. Prove that the problem

  $$\displaystyle{x^{\prime}(t) - x(t) = f(t,x(t))}$$

  admits at least one T-periodic solution.

• 1.11. Let $f,g: \mathbb{R} \times {\mathbb{R}}^{2} \rightarrow \mathbb{R}$ be of class C ¹ and T-periodic in t. Furthermore, assume there exist R ₁, R ₂ > 0 such that

  $$\displaystyle{f(t,R_{1},y)f(t,-R_{1},y) <0}$$

  for each $t \in \mathbb{R}$ and $y \in \mathbb{R}$ such that $\vert y\vert \leq R_{2}$, and

  $$\displaystyle{g(t,x,R_{2})g(t,x,-R_{2}) <0}$$

  for each $t \in \mathbb{R}$ and $x \in \mathbb{R}$ such that | x | ≤ R ₁. Prove that the problem

  $$\displaystyle{\left \{\begin{array}{l} x^{\prime} = f(t,x(t),y(t))\\ y^{\prime} = g(t, x(t), y(t)) \\ \end{array} \right.}$$

  admits at least one T-periodic solution.

• 1.12. *(Adapted from [46]). Prove the following Lazer–Leach theorem [70]: let $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous and bounded, and assume that the limits

  $$\displaystyle{g(\pm \infty ):=\lim _{u\rightarrow \pm \infty }g(u)}$$

  exist. Furthermore, let $p \in C(\mathbb{R}, \mathbb{R})$ be $2\pi$-periodic, and define

  $$\displaystyle{A:=\int _{ 0}^{2\pi }p(t)\cos (t)\,dt,\qquad B:=\int _{ 0}^{2\pi }p(t)\sin (t)\,dt.}$$

  Then the problem

  $$\displaystyle{u^{\prime\prime}(t) + u(t) + g(u(t)) = p(t)}$$

  has at least one $2\pi$-periodic solution, provided that

  $$\displaystyle{ \sqrt{{A}^{2 } + {B}^{2}} <2\vert g(+\infty ) - g(-\infty )\vert. }$$

  (1.25)

  Hint: first, assume that g is smooth, and define as in Sect. 1.3.2 the mapping

  $$\displaystyle{F(x,y):= (y - u_{x,y}^{\prime}(2\pi ),u_{x,y}(2\pi ) - x).}$$

  Next, use the polar coordinates $(x,y):= (r\cos (\theta ),r\sin (\theta ))$ in order to prove that if r is large enough, then $F(x,y) \cdot (x,y)\neq 0$ for (x, y) ∈ ∂ B _r (0), so the result follows. The result for g continuous is deduced by an approximation argument.