On Solutions for a Generalized Differential Equation Arising in Boundary Layer Problem

+(1/(n+1))xx 󸀠󸀠 = 0, x(0) = x󸀠(0) = 0, x󸀠(∞) = 1, arising in the study of a laminar boundary layer for a class of non-Newtonian fluids. In [4], the author considers the equation [a(t)x]󸀠+xx = 0, which describes boundary layer flows with temperature dependent viscosity. It is our goal to study the existence of solutions to the generalized boundary value problem consisting of the nonlinear third order differential equation


Introduction
The steady motion in the boundary layer along a thin flat plate which is immersed at zero incidence in a uniform stream with constant velocity can be described [1] in terms of the solution of the differential equation: which satisfies the boundary conditions This problem was first solved numerically by Blasius [2] and is the subject of a vast literature.Some generalizations of the Blasius equation can be found in [3][4][5].In [3], the authors investigate the model +(1/(+1))  = 0, (0) =   (0) = 0,   (∞) = 1, arising in the study of a laminar boundary layer for a class of non-Newtonian fluids.In [4], the author considers the equation [()  ] +  = 0, which describes boundary layer flows with temperature dependent viscosity.
For the related Falkner-Skan equation [6]   +  +(1− (  ) 2 ) = 0 the similar generalization was given in [7].Falkner-Skan equation describes the steady two-dimensional flow of a slightly viscous incompressible fluid past a wedge of angle  (0 ≤  < 2).The shooting method [8] is used for treating the existence and the number of solutions to boundary value problem.The shooting method reduces solving a boundary value problem to solving of an initial value problem.So we consider solution (, ) of the auxiliary initial value problem for (3) with initial data and we are looking for  1 and  2 such that   (∞,  1 ) > 1 and   (∞,  2 ) < 1. Applying the intermediate value theorem, continuity of (, ) with respect to  leads to the existence of at least one  * such that   (∞,  * ) = 1.
The paper is organized as follows.Section 2 contains some auxiliary results.Section 3 is devoted to the properties of solutions of initial value problem (3), (4).In Section 4 we consider dependence of solutions on initial data.In Section 5 we deal with solutions to boundary value problem (3), (2).Also one example is given to illustrate the results.The ideas of the proofs of some results are taken from [6].

Proposition 2. If a function 𝑓(𝑥) satisfies assumptions (H2) and (H3), then 𝑓(𝑥) is strictly increasing.
Proof.We can obtain that the function () is strictly increasing repeating the arguments used in the proof of the previous proposition.
For  < 0 the proof is analogous.
Proposition 5. Let () be a solution of (3).If one of the functions (),   (), or   () tends to infinity as  →  * then the others also tend to infinity.
Thus,   < −  for  >  0 .Dividing the last inequality by   and integrating from  0 to , we obtain Since Therefore, there exists a positive constant  such that lim  → +∞   (, ) = .

Scaling Formula
Proposition 8. Suppose that conditions (H1) and (H3) are fulfilled.If () is a solution of (3), then the function where  > 0 is an arbitrary constant, is also a solution of (3).
Proposition 9. Suppose that conditions (H1) and (H3) are fulfilled.If (,  0 ) is a solution of (3) such that then every solution of (3) which has a double zero at  = 0 and the second derivative  at  = 0 of the same sign as  0 ( 0 > 0) can be expressed via solution (,  0 ) as Proof.The proof follows from Proposition 8 and direct substitution.So, .
The proof is complete.