Existence Results for Differential Equations of Fourth Order with Non-Homogeneous Boundary Conditions

Abstract

The objective of this paper is to investigate the existence and uniqueness of solutions to fourth order differential equations
                                                           v⁽⁴⁾ (x) + f (x, v(x)) = 0, x∈[a,b],
satisfying the three-point non-homogeneous conditions
                                                 v(a) = 0, v′(a) = 0, v′′(a) = 0, v′(b) −α v′(ζ ) = μ,
where 0 ≤ a < ζ < b, the constants α, μ are real numbers and f : [a, b] × R → R is a continuous function. The framework for establishing the existence results is based on sharper estimates on the integral of the kernel to connect with fixed point theorems of Banach and Rus.