Nonlinear Variational Two-Point Boundary Value Problems

Abstract

Let us consider the two-point boundary value problem

$$\begin{array}{*{20}{c}} {u''\left( x \right) = f\left( {x,u\left( x \right)} \right),x \in I,} \\ {u\left( 0 \right) = u\left( \pi \right) = 0} \end{array}$$

(1)

where I = [0, π], f: I × ℝ → ℝ is a Caratheorody function. As early as in 1915, Lichtenstein [9] observed that if the function F: I × ℝ → ℝ defined by

$$F(x,u) = \int_0^u {f(x,u)du} $$

for some real number A and all (x, u) ∈ I × ℝ, then the corresponding action integral

$$\varphi :u\mapsto \int_{I}{[(1/2){{(u'(x))}^{2}}}+F(x,u(x))]dx$$

is such that

$$F(x,u) \geqslant A$$

(2)

for some real number A and all (x, u) ∈ I × ℝ, then the corresponding action integral

$$\varphi :u \mapsto \int_I {[(1/2){{(u'(x))}^2}} + F(x,u(x))]dx$$

is bounded below on the set C ¹₀ (I) of functions u of class C ¹ on I which vanish at 0 and π,, and hence φ has an infimum d on C ¹₀ (I). Lichtenstein introduced then a minimizing sequence (u _n ) such that φ(u _n ) → d as n →∞, namely the one coming from the associated Ritz method. Writing

$${u_n}(x) = \sum\limits_{j = 1}^\infty {({u_{n,j}}/j)\sin jx,} $$

so that

$$\sum\limits_{j = 1}^\infty {{u_{n,j}}\sin jx} $$

is the Fourier series of u _n ′, Lichtenstein proved the existence of a subsequence (${u_{{n_k}}}$ ) such that, for each j ∈ ℕ*, ( ${u_{{n_k},}}_j$) converges to some u _j as k → ∞.