A singular non-local problem at resonance. (English) Zbl 1260.34037

Kosmatov, Nickolai

J. Math. Anal. Appl. 394, No. 1, 425-431 (2012).

The author uses the coincidence degree theory to study the second-order non-local problem \[ \begin{aligned} x''(t)=f(t, x(t),x'(t))+e(t), \\ x'(0)=0,\;x(1)=\sum^{m-1}_{i=1}a_i x_i(\xi_i),\end{aligned} \] where where \(m \geq 3\), \(0 < \xi_1<\dotsb<\xi_{m-2} < 1\), and \(f\) satisfies the Carathéodory conditions and \(e\) is a locally Lebesgue-integrable function such that \((1-t)e\) is Lebesgue integrable. The author shows the existence of a solution whose derivative is singular at the right end-point of the interval. Under the non-local condition, the author gives a general way to ensure that the differential operator is a Fredholm operator of index zero.

Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B16	Singular nonlinear boundary value problems for ordinary differential equations
47H11	Degree theory for nonlinear operators
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
47E05	General theory of ordinary differential operators

Keywords:

coincidence degree; Carathéodory conditions; non-local problem resonance; singularity

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