has at least one solution x**, which is found by minimizing the dual action functional:**

$$\psi(u)=\int_{o}^{T}\left[{\frac{1}{2}}\left(\Lambda_{o}^{-1}u,u\right)+N^{*}(-u)\right]d t$$
$$\left(4\right)$$
space $R(A)^2_L$ with finite. 
$$\left({\bar{5}}\right)$$

```
on the range of Λ, which is a subspace R(Λ)
                                           2
                                            
                                           L with finite codimension. Here

```

$$N(x):=H(x)-{\frac{1}{2}}\left(A_{\infty}x,x\right)$$

is a convex function, and

$$N(x)\leq\frac{1}{2}\left(\left(B_{\infty}-A_{\infty}\right)x,x\right)+c\quad\forall x\ .$$
((B∞ − A∞) x, x) + c ∀x . (6)
Proposition 1. Assume H′(0) = 0 and H(0) = 0**. Set:**

$$d\;H(0)=0.\;S e t:$$
$$\delta:=\operatorname*{lim}_{x\to0}\operatorname*{inf}2N(x)\left\|x\right\|^{-2}\ .$$
−2. (7)
If γ < −λ < δ, the solution u **is non-zero:**

$$(6)$$
$$\left(7\right)$$
$$\overline{{{x}}}(t)\neq0\quad\forall t\ .$$
$\left(8\right)$. 
$$\left({\mathfrak{g}}\right)$$

$$(10)$$
x(t) 6= 0 ∀t . (8)
Proof. **Condition (7) means that, for every** δ
′ > δ, there is some ε > **0 such that**

$$\|x\|\leq\varepsilon\Rightarrow N(x)\leq{\frac{\delta^{\prime}}{2}}\left\|x\right\|^{2}\ .$$
. (9)
It is an exercise in convex analysis, into which we shall not go, to show **that**
this implies that there is an η > **0 such that**

$$f\left\|x\right\|\leq\eta\Rightarrow N^{*}(y)\leq{\frac{1}{2\delta^{\prime}}}\left\|y\right\|^{2}\ .$$

Fig. 1. This is the caption of the figure displaying a white eagle and a **white horse on**
a snow field Since u1 is a smooth function, we will have khu1k∞ ≤ η for h **small enough,**
and inequality (10) will hold, yielding thereby:

$$\psi(hu_{1})\leq\frac{h^{2}}{2}\frac{1}{\lambda}\left\|u_{1}\right\|_{2}^{2}+\frac{h^{2}}{2}\frac{1}{\delta^{\prime}}\left\|u_{1}\right\|^{2}.\tag{11}$$

8