Nonlinear Variational Two-Point Boundary Value Problems

  • J. Mawhin
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)


Let us consider the two-point boundary value problem
$$\begin{array}{*{20}{c}} {u''\left( x \right) = f\left( {x,u\left( x \right)} \right),x \in I,} \\ {u\left( 0 \right) = u\left( \pi \right) = 0} \end{array}$$
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
$$F(x,u) \geqslant A$$
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 0 1 (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 0 1 (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 → ∞.


Order Differential Equation Critical Point Theory Positive Density Coincidence Degree Nonresonance Condition 


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Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • J. Mawhin
    • 1
  1. 1.Institut MathématiqueUniversité de LouvainLouvain-la-NeuveBelgique

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