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Existence of triple positive solutions for $$(k,n-k)$$ right focal boundary value problems. (English) Zbl 0999.34019
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 $y^{(i)}(0) = y^{(i)}(1) = 0, \qquad 0\leq i\leq k-1, \qquad k\leq j\leq n-1.$

##### 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