Positive solutions for a system of p-laplacian boundary value problems

In this paper, we investigate the existence of positive solutions for a system of fourth order p-Laplacian boundary⎧ value problems −((−x ^′′′ ) ^{p −1} ) ^′ = f(t, x, x ^′ , y, y ^′ ), t ∈ [0, 1], ⎪⎨−((−y ^′′′ ) ^{p −1} ) ^′ = g(t, x, x ^′ , y, y ^′ ), t ∈ [0, 1], x(0) = x ^′ (1) = x ^′′ (0) = x ^′′′ (1) = 0, ⎪⎩y(0) = y ^′ (1) = y ^′′ (0) = y ^′′′ (1) = 0, where p > 1, f, g ∈ C([0, 1] × R ⁺ × R ⁺ × R ⁺ × R ⁺ , R ⁺ )(R ⁺ := [0, ∞)). Under some new general conditions on f and g, we use the fixed point index to establish two existence theorems for the above system. The interesting point lies in the fact that the nonlinear term f, g can be allowed to depend on the first derivative of the unknown functions, and this derivative dependence in systems is seldom considered in the literature.