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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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