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Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives. (English) Zbl 1154.34016

The authors consider the higher order \(m\)-point boundary value problem
\[ (x^{(n)})(t)+\lambda f(t, x(t), x'(t), \cdots, x^{(n-2)}(t))=0, \quad t\in (0,1), \]
\[ x^{(i)}(0)=0, \quad 0\leq i\leq n-3,\quad x^{(n-1)}(0)=\sum^{m-2}_{i=1}a_i x^{(n-1)}(\xi_i), \]
\(x^{(n-2)}(1)=\sum^{m-2}_{i=1}b_i x^{(n-2)}(\xi_i)\), where \(0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1\) with \(0<\sum^{m-2}_{i=1}a_i<1\) and \(0<\sum^{m-2}_{i=1}b_i<1\), \(f\in C[(0,1)\times \mathbb{R}^{n-1}, \mathbb{R}]\) satisfies \[ f(t,u_1, \dots, u_{n-1})\geq p(t) \text{ and } p\in L^1[(0,1), (0,+\infty)], \] and \(f\) may be singular at \(t=0\) and/or \(t=1\). He proves the existence of positive solutions by applying a fixed point theorem in cones. For earlier results, see R. Ma and N. Castaneda [J. Math. Appl. Anal. 256, 556–567 (2001; Zbl 0988.34009)].
Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators

Citations:

Zbl 0988.34009
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References:

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