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In this paper, we investigate the second-order problem with dependence on derivative in nonlinearity and Stieltjes integral boundary condition { −u′′(t) = f (t, u(t), u′(t)), t ∈ [0, 1], u(0) = α[u], u′(1) = 0, where f : [0, 1]×R+ ×R+ → R+ is continuous and α[u] is a linear functional. Some inequality conditions on nonlinearity f and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in C1[0, 1]. The conditions allow that f (t, x1, x2) has superlinear or sublinear growth in x1, x2. Some examples are given to illustrate the theorems respectively under multi-point and integral boundary conditions with sign-changing coefficients.

Positive solutions of second-order problem with dependence on derivative in nonlinearity under Stieltjes integral boundary condition