The Solutions of Mixed Monotone Fredholm-Type Integral Equations in Banach Spaces

By introducing new definitions of ϕ convex and -φ concave quasioperator and v0 quasilower and u0 quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new to ϕ convex and -φ concave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces