Existence of triple positive solutions for \((k,n-k)\) right focal boundary value problems (Q2763890)

The authors study the existence of multiple positive solutions to the \(n\)th-order nonlinear differential equation NEWLINE\[NEWLINE (-1)^{n-1}y^{(n)} = f(y(t)), \quad 0\leq t\leq 1, \tag{1}NEWLINE\]NEWLINE satisfying the \((1, n-1)\) right focal boundary conditions NEWLINE\[NEWLINE y(0) = y^{i}(1) = 0, \qquad 1\leq i\leq n-1, \tag{2}NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE y^{(i)}(0) = y^{(i)}(1) = 0, \qquad 0\leq i\leq k-1, \qquad k\leq j\leq n-1.NEWLINE\]