Existence and iteration of positive solutions for a mixed-order four-point boundary value problem with p-Laplacian

In this paper, we firstly obtain the existence of the monotone positive solutions and establish a corresponding iterative scheme for the following mixed-order four-point boundary value problem with p-Laplacian (φp(D α 0+u(t))) ′ + a(t)f(t, u(t), u′(t)) = 0, 0 < t < 1, u′(0)− βu(ξ) = 0, u′′(0) = 0, u′(1) + γu(η) = 0. Unlike many other fractional boundary value problem with p-Laplacian, the nonlinear term involves the first-order derivative explicitly, so it is hard to get positive solutions for the problem. The main tool used here is the monotone iterative technique. By the fixed point theorem due to Avery and Peterson, we obtained some sufficient conditions that guarantee the existence of at least three positive solutions to the above boundary value problem. Meanwhile, we give an example to demonstrate the use of the main results of this paper. Key–Words: Iteration; Mixed-order four-point boundary value problem; Positive solutions; p-Laplacian.