A note on the second order boundary value problem on a half-line.

McArthur, Summer; Kosmatov, Nickolai

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This paper deals with the generalized nonlinear third-order left focal problem at resonance $\{\begin{matrix} {(p (t) u^{''} (t))}^{'} - q (t) u (t) = f (t, u (t), u^{'} (t), u^{''} (t)), t \in] t_{0}, T [, m (u (t_{0}), u^{''} (t_{0})) = 0, n (u (T), u^{'} (T)) = 0, l (u (ξ), u^{'} (ξ), u^{''} (ξ)) = 0, \end{matrix}$ where the nonlinear term is a Carathéodory function and contains explicitly the first and second-order derivatives of the unknown function. The boundary conditions that we study are quite general, involve a linearity and include, as particular cases, Sturm-Liouville boundary conditions. Under certain growth conditions on the nonlinearity, we establish the existence of the nontrivial solutions...

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