Existence of triple positive solutions for \((k,n-k)\) right focal boundary value problems. (English) Zbl 0999.34019

Chyan, Chuan Jen; Davis, John M.; Yin, William K. C.

Nonlinear Stud. 8, No. 1, 33-51 (2001).

The authors study the existence of multiple positive solutions to the \(n\)th-order nonlinear differential equation \[ (-1)^{n-1}y^{(n)} = f(y(t)), \quad 0\leq t\leq 1, \tag{1} \] satisfying the \((1, n-1)\) right focal boundary conditions \[ y(0) = y^{i}(1) = 0, \qquad 1\leq i\leq n-1, \tag{2} \] where \(f:\mathbb{R}\to [0, \infty)\) is continuous. They derive the best possible upper and lower bounds for the integral homogeneous problem. Using the Leggett-Williams fixed-point theorem, the existence of at least three positive solutions to (1)–(2) is proved. Finally, these results are extended to the more general \((k, n-k)\) right focal boundary conditions \[ y^{(i)}(0) = y^{(i)}(1) = 0, \qquad 0\leq i\leq k-1, \qquad k\leq j\leq n-1. \]

Reviewer: Nataliya Bantsur (Ternopil)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations

Keywords:

multiple solutions; positive solutions; right focal boundary value problem; Green’s function; fixed-points