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Solvability of a generalized third-order left focal problem at resonance in Banach spaces. (English) Zbl 1289.34061

The author deals with the third-order left focal BVP at resonance \[ \begin{aligned} & (p(t)u''(t))'-q(t)u(t)=f(t, u(t), u'(t), u''(t)), \quad t\in ]t_0, T[, \\ & m(u(t_0), u''(t_0))=0, \;n(u(T), u'(T))=0, \;l(u(\xi ), u'(\xi ), u''(\xi ))=0, \end{aligned} \] where \(m,n,l\) are linear maps, \(\xi \in (t_0,T)\), \(p\in C^1\bigl ((t_0,T), (0,\infty )\bigr )\), \(q\in L^1\bigl ([t_0,T],\mathbb {R}\bigr )\) and \(f\) is a Carathéodory function. Resonance here means that the above problem with \(q\equiv 0\), \(f\equiv 0\) has a nontrivial solution. The main results of the paper are proved via the Mawhin’s coincidence degree theory. More precisely, the above problem is converted into the operator equation \(L(u)=N(u)\) with \(L(u)=\bigl (pu'')\bigr )'\) and \(N(u)=f(t,u,u',u'')+qu\) to which the coincidence degree method is applied.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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