| Summary: | Motivated, mainly, by the works of Fewster-Young and Tisdell [9,10]
and Orpel [30], as well as the papers by Karakostas [21,22,23],
we give sufficient conditions to guarantee the existence of (nontrivial)
solutions of the second-order Phi-Laplacian equation
$$
\frac{1}{p(t)}\frac{d}{dt}[p(t)\Phi(u'(t))]+(Fu)(t)=0,\quad\text{a.e. }
t\in[0,1]=:I,
$$
which satisfy the nonlocal boundary value conditions of the
limiting Sturm-Liouville form
$$
\lim_{t\to 0}[p(t)\Phi(u'(t))]=\int_0^1u(s)d\eta(s),\quad
\lim_{t\to 1}[p(t)\Phi(u'(t))]=-\int_0^1u(s)d\zeta(s).
$$
Here $\Phi$ is an increasing homeomorphism of the real line onto itself
and F is an operator acting on the function u and on its first
derivative with the characteristic property that $u\to p(Fu)$ is a
$C^0$-type, or $C^1$-type Caratheodory operator, a meaning introduced here.
Examples are given to illustrate both cases.
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