A third order nonlocal boundary value problem at resonance

E. Kaufmann, University of Arkansas at Little Rock, Little Rock, AR, U.S.A.

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 16., pp. 1-11.

Abstract: We consider the third-order nonlocal boundary value problem
\begin{eqnarray*}
&&u'''(t) = f(t, u(t)), \quad \mbox{a.e. in } (0, 1),\\
&&u(0) = 0, \, u'(\rho) = 0,\\
&&u''(1) = \lambda[u''],
\end{eqnarray*}
where $0 < \rho < 1,$ the nonlinear term $f$ satisfies Carath\'{e}odory conditions with respect to $L^1[0, T]$, $\lambda [v] = \int_0^1 \! v(t) \, \rm{d} \Lambda(t)$, and the functional $\lambda$ satisfies the resonance condition $\lambda[1]= 1$. The existence of a solution is established via Mawhin's coincidence degree theory.

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