Singular \((k,n-k)\) boundary value problems between conjugate and right focal. (English) Zbl 0901.34026
Boundary value problems (1) \((-1)^{n- k}y^{(n)}= f(x, y)\), \(0< x< 1\), \(y^{(i)}(0)= 0\), \(0\leq i\leq k-1\), \(y^{(i)}(1)= 0\), \(q\leq j\leq n- k+ q- 1\), whith \(1\leq k\leq n-1\), \(0\leq q\leq k- 1\), and \(f(x, y)\) is singular at \(y= 0\), are solved by using a fixed point theorem for decreasing operators in a partially ordered Banach space. The problem (1) is reduced to a lower-order conjugate boundary value problem and this is approximated by a sequence of perturbed boundary value problems which contain no singularity. To each term of that sequence the fixed point theorem is applied and finally a positive solution \(y(x)\) to (1) is obtained such that \(y^{(i)}(x)> 0\) on \((0, 1)\), \(0\leq i\leq q\).
Reviewer: Walter Šeda (Bratislava)
MSC:
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 47J25 | Iterative procedures involving nonlinear operators |
| 34B27 | Green’s functions for ordinary differential equations |
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