Abstract:
We establish the global structure of positive solutions
of fourth-order periodic boundary-value problems
$u''''(t)+Mu(t)=\lambda f(t,u(t))$,
$t\in[0,T]$,
$u^{k}(0)=u^{(k)}(T)$,
$k=0,1,2,3,$
with
$M\in\big(0,4({2\pi M_4}/{T})^4\big)$
and
$u^{(4)}(t)-Mu(t)+\lambda g(t,u(t))=0$,
$t\in[0,T]$,
$u^{k}(0)=u^{(k)}(T)$,
$k=0,1,2,3,$
with
$M\in \big(0,({2\pi M_4}/{T})^4\big)$;
here
$g, f\in
C([0,T]\times[0,\infty),[0,\infty))$,
$M$
is constant,
and
$\lambda>0$
is a real parameter.
The main results are based on a global bifurcation
theorem.
