Solvability of a Third-Order Singular Generalized Left Focal Problem in Banach Spaces

We consider the existence of positive solution for a third-order singular generalized left focal boundary value problem with full derivatives in Banach spaces. Green’s function and its properties, explicit a priori, estimates will be presented. By means of the theories of the fixed point in cones, we establish some new and general results on the existence of single and multiple positive solutions to the third-order singular generalized left focal boundary value problem. Our results are generalizations and extensions of the results of the focal boundary value problem. An example is included to illustrate the results obtained. 1. Introduction Third-order differential equation describes many phenomena in applied mathematics, physical science, aeronautics, and applied mechanics such as the study of steady flows produced by free jets, wall jets, liquid jets, the flow past a stretching plate, and Blasius flow [1–6]. For two-dimensional flow of a fluid with small viscosity adhering to the flat plate, the simplified version, which has been derived by Blasius, only uses two unknowns and . Laminar flat plate flow across a flat plate can be expressed using a boundary-layer equation defined by Blasius. Simplified momentum equation is given by where is the velocity measured, is the transverse velocity component, is length of the plate, is the distance away from the plate, and is free-stream velocity. Assume that the leading edge of the plate is and the plate is infinity long; this equation can be simplified as where is the dimensionless stream function. For further simplification, we have the Blasius differential equation where the boundary conditions are , and or . For other related application results of the problem we refer to [7–10]. Recently, increasing attention is paid to the question of the solution for the third-order focal boundary value problem, especially for right focal problem. The main means are the Leray-Schauder continuation theorem, iteration of monotone mapping, upper and lower solutions method, fixed point theory, and so on. Many applications of the above tools of various nonlocal boundary conditions include recent works [11–13] and the reference therein. It is well known that a powerful tool for proving the existence of the solution to the focal boundary value problem is the fixed point theorem. In many cases, it is possible to find single or multiple solutions for the given problem. We would like to mention some results of [14, 15]. Anderson [14] has investigated the existence of a solution to a third-order generalized right focal problem for