For $p>1, $ and $\phi_p (s) = |s| ^{p-2} s,$ we consider
the
equation
$(\phi_p (x'))' + \alpha \phi_p (x^+ ) - \beta \phi_p (x^- )
= f(t,x),$
where $ x^{+}=\max\{x,0\}$; $x^{-} =\max\{-x,0\},$ in a
situation of resonance or near resonance for the period $T,$ i.e. when
$\alpha,\beta$ satisfy exactly or approximately the equation
$\frac{\pi_p }{\alpha^{1/p}} + \frac{\pi_p}{\beta^{1/p}} =
\frac{T}{n},$
for some integer $n.$
We assume that $f$ is continuous, locally Lipschitzian
in $x,$ $T$-periodic in
$t,$ bounded on
$\mathbf R^2,$ and having limits $f_{\pm}(t)$ for $x \to \pm \infty,$ the
limits being uniform in $t.$ Denoting by $v $ a solution of the
homogeneous equation
$(\phi_p (x'))' + \alpha \phi_p (x^+ ) - \beta \phi_p (x^- ) = 0,$
we study the existence of $T$-periodic solutions by means of the function
$ Z (\theta) = \int_{\{t\in I | v_{\theta }(t)>0\}}
f_{+}(t)v(t + \theta) dt +
\int_{\{t\in I | v_{\theta }(t)<0\}} f_-(t) v (t + \theta) dt,$
where $ I \stackrel{def}{=} [0,T].$
In particular, we prove the existence of $T$-periodic solutions at
resonance
when $Z$ has $2z$ zeros in the interval $[0,T/n),$ all zeros being
simple, and $z$
being different from $1.$
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