# 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)

## 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 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 → ∞.

## Keywords

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.

## References

1. [1]
S. Cinquini, Sopra i problemi di valori al contorno per equazioni differenziali non lineari, Boll. Un. Mat. Ital. (1) 17 (1938), 99–105.
2. [2]
D.G. De Figueiredo and J.P. Gossez, Nonresonance below the first eigenvalue for a semilinear elliptic problem, Math. Ann. 281 (1988), 589–610.
3. [3]
M.L.C. Fernandes, P. Omani and F. Zanolin, On the solvability of a semilinear two-point BVP around the first eigenvalue,J. Differential and Integral Equations, to appear.Google Scholar
4. [4]
M.L.C. Fernandes and F. Zanolin, Periodic solutions of a second order di f ferential equation with one-sided growth restrictions on the restoring term, Archiv Math. (Basel) 51 (1988), 151–163.
5. [5]
A. Fonda and J.P. Gossez, Semicoercive variational problems at resonance,to appear.Google Scholar
6. [6]
A. Fonda and J. Mawhin, Quadratic forms, weighted eigenfunctions and boundary value problems for nonlinear second order ordinary differential equations, Proc. Royal Soc. Edinburgh 112A (1989), 145–153.
7. [7]
R.E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., 568, Springer, Berlin, 1977.Google Scholar
8. [8]
A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1930), 117–176.
9. [9]
L. Lichtenstein, Uber einige Existenzprobleme der Variationsrechnung. Methode der unendlichvielen Variablen, J. Reine Angew. Math. 145 (1915), 24–85.Google Scholar
10. [10]
J. Mawhin, Problèmes de Dirichlet variationnels non linéaires, Sémin. Math. Sup. 104, Presses Univ. Montréal, Montréal, 1987.Google Scholar
11. [11]
J. Mawhin, J. Ward and M. Willem, Variational methods and semilinear equations, Arch. Rat. Mech. Anal. 95 (1986), 269–277.
12. [12]
J. Mawhin and M. Willem, Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation,in “Nonlinear Analysis and Optimization,” Vinti ed., Lecture Notes in Math., 1107 Springer, Berlin, 1984, 181192.Google Scholar