## On multi-point boundary value problems for linear ordinary differential equations with singularities.(English)Zbl 1058.34012

The authors investigate the singular linear differential equation $u^{(n)}= \sum_{i=1}^n p_i(t)u^{(i-1)}+q(t) \tag{1}$ on $$[a,b]\subset \mathbb{R}$$, where the functions $$p_i$$ and $$q$$ can have singularities at $$t=a, t=b$$ and $$t=t_0\in (a,b)$$. This means that $$p_i$$ and $$q$$ are not integrable on $$[a,b]$$. Equation (1) is studied with the boundary conditions $u^{(i-1)}(t_0)=0 \text{ for } 1\leq i\leq n-1,\;\sum_{j=1}^{n-n_1}\alpha_{1j}u^{(j-1)}(t_{1j}) + \sum_{j=1}^{n-n_2}\alpha_{2j}u^{(j-1)}(t_{2j})=0, \tag{2}$ or $u^{(i-1)}(a)=0\text{ for }1\leq i\leq n-1,\;\sum_{j=1}^{n-n_0}\alpha_{j}u^{(j-1)}(t_{j})=0, \tag{3}$ where $$t_{1j}, t_{2j}, t_j$$ are certain interior points in $$(a,b)$$. The authors introduce the Fredholm property for these problems which means that the unique solvability of the corresponding homogeneous problem implies the unique solvability of the nonhomogeneous problem for every $$q$$ which is weith-integrable on $$[a,b]$$. Then, for the solvability of a problem having the Fredholm property, it sufficies to show that the corresponding homogeneous problem has only the trivial solution. In this way, the authors prove main theorems on the existence of a unique solution of (1),(2) and of (1),(3). Examples verifying the optimality of the conditions in various corollaries are shown as well.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations
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