The General Case: Nontriviality.

We assume that H is #math37##tex2html_wrap_inline798#A_∞, B_∞#tex2html_wrap_inline799#-subquadratic at infinity, for some constant symmetric matrices #math38#A_∞ and #math39#B_∞, with #math40#B_∞ - A_∞ positive definite. Set:

#math41#

γ :	=	smallest eigenvalue of B_∞ - A_∞	(1)
λ :	=	largest negative eigenvalue of I>J#tex2html_wrap_indisplay810# + A_∞ .	(2)

Theorem 21 tells us that if #math42#λ + γ ;SPMlt; 0, the boundary-value problem:

#math43#

#tex2html_wrap_indisplay814#	=	JH'(x)
x(0)	=	x(T)

(3)

has at least one solution #math44##tex2html_wrap_inline821#, which is found by minimizing the dual action functional:

#math45#

ψ(u) = #tex2html_wrap_indisplay823##tex2html_wrap_indisplay824##tex2html_wrap_indisplay825##tex2html_wrap_indisplay826#Λ_o^-1u, u#tex2html_wrap_indisplay827# + N^*(- u)#tex2html_wrap_indisplay828#dt (4)

on the range of Λ, which is a subspace #math46#R(Λ)_L² with finite codimension. Here

#math47#

N(x) : = H(x) - #tex2html_wrap_indisplay832##tex2html_wrap_indisplay833#A_∞x, x#tex2html_wrap_indisplay834#

(5)

is a convex function, and

#math48#

N(x)≤#tex2html_wrap_indisplay836##tex2html_wrap_indisplay837##tex2html_wrap_indisplay838#B_∞ - A_∞#tex2html_wrap_indisplay839#x, x#tex2html_wrap_indisplay840# + c ∀x .

(6)

#proposition65#

#proof76#

#corollary127#

We recall once more that by the integer part #math49#E[α] of #math50#α∈#tex2html_wrap_inline843#, we mean the #math51#a∈#tex2html_wrap_inline845# such that #math52#a ;SPMlt; α≤a + 1. For instance, if we take #math53#a_∞ = 0, Corollary 2 tells us that #math54##tex2html_wrap_inline849# exists and is non-constant provided that:

#math55#

#tex2html_wrap_indisplay851#b_∞ ;SPMlt; 1 ;SPMlt; #tex2html_wrap_indisplay852#

(7)

#math56#

T∈#tex2html_wrap_indisplay854##tex2html_wrap_indisplay855#,#tex2html_wrap_indisplay856##tex2html_wrap_indisplay857# .

(8)

#proof169#

#lemma204#

#proof209#