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We establish existence
results for two point boundary value problems for second order ordinary
differential equations of the form y′′ = f(x,y,y′) x ∈ [0,1], where f satisfies the
Carathéodory measurability conditions and there exist lower and upper solutions.
We consider boundary conditions of the form G((y(0),y(1));(y′(0),y′(1))) = 0 for
fully nonlinear, continuous G and of the form (y(i),y′(i)) ∈𝒥 (i), i = 0,1
for closed connected subsets 𝒥 (i) of the plane. We obtain analogues of our
results for continuous f . In particular we introduce compatibility conditions
between the lower and upper solutions and : (i) G; (ii) the 𝒥 (i), i = 0,1.
Assuming these compatibility conditions hold and, in addition, f satisfies
assumptions guarenteeing a’priori bounds on the derivatives of solutions we
show that solutions exist. As an application we generalise some results of
Palamides.
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