International Journal of Analysis

Volume 2014, Article ID 638976, 18 pages

http://dx.doi.org/10.1155/2014/638976

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

Department of Mathematics, Hexi University, Gansu 734000, China

Received 9 February 2014; Accepted 5 May 2014; Published 26 May 2014

Academic Editor: Chuanxi Qian

Copyright © 2014 Youwei Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 certain constraints placed on nonnegative ; the main approach is the Leggett and Williams fixed point theorem. Minhós [15] has proved an Ambrosetti-Prodi type result for the third-order fully nonlinear right focal problem where and are continuous functions and . The author has studied the existence, nonexistence, and multiplicity of solutions as variable . Further, we may refer to recent contributions of Agarwal [16] and Agarwal et al. [17], Cabada and Heikkilä [18], and Wong and Agarwal [19] for the right focal problem. Accordingly, a third-order left focal problem for differential equation plays a role in many related fields too, even less attention has been focused, however, on third-order left focal problem. This paper will consider the existence of a positive solution for the following third-order singular generalized left focal boundary value problem with full derivatives in Banach spaces where , and satisfy ) and ,() with ,() may be singular at and/or and , ().

The novelty of our problem is to show the existence of solution of the left focal boundary value problem; we will concentrate on the solvability of the third-order singular generalized left focal problem for differential equation. To the best of our knowledge, it is the first time that the third-order left focal problem has been discussed successfully. As one might expect, the technique of finding Green’s function for the corresponding homogeneous left focal problem, as the kernel of an integral equation of Hammerstein type, has proven to be an effective method in most of third-order left focal problems. For this, we will employ known sign properties of Green’s function and its partial derivative cleverly. Note that the solution of the problem lacks concavity, which is slightly different from right focal problem. In order to overcome this difficulty, we will establish the appropriate a priori estimates of the solution. On the other hand, the condition on nonlinearity required in this paper is different from those of existing papers, which motivated us to consider the above left focal problem, since may be superlinear (sublinear). As usual, the function in the problem is called superlinear if and ; it is called sublinear when and , where for . Based on the known results of the singular multipoint boundary value problem and high-order right focal problem for differential equation, this study is devoted to proving the existence of solution for the third-order singular generalized left focal problem , and the full derivatives are involved in the nonlinear term explicitly. Under the suitable limit and growth conditions on the nonlinearity in cones, we will establish some new and general results on the existence of single and multiple positive solutions to the problem ; our results develop some results of the third-order focal problem and improve the results of third-order singular problem. The author believes that those results are useful in many applications for temporary new fields. Now we state the fixed point theorems.

Lemma 1 (see [20, 21]). *Let be a cone in a Banach space . Assume that are open subsets of with . If is a completely continuous operator such that either *()* for all and for all or*()* for all and for all .**
Then, has a fixed point in .*

Let and be nonnegative continuous convex functionals on a cone , let be a nonnegative continuous concave functional on , let be a nonnegative continuous functional on , and let , and be positive numbers. Define the convex sets as follows: and a closed set

Lemma 2 (see [22]). *Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on satisfying for such that for some positive numbers and **
for all . Suppose that is completely continuous and there exist positive numbers , , and with such that *()* and for ,*()* for with ,*()* and for with .**
Then, has at least three fixed points such that
*

#### 2. Preliminaries

We will employ several lemmas; these lemmas are based on the corresponding linear third-order generalized left focal boundary value problem.

Lemma 3. *Let , hold. Green’s function for the boundary value problem
**
is given as
**
where
*

*Proof. *First, we check that is defined for each . Obviously, , . At ,
Next, we check that satisfies the boundary conditions of problem (11). For convenience, we note that

For , by (12) and (16), we have
for , we get

For , by (12) and (15), we have ; for , we get .

For , by (12) and (15), we have
for , we get
Hence, satisfies the boundary conditions of problem (11).

Lemma 4. *Let , hold. If , then Green’s function as given in (12) satisfies
**
where
*

*Proof. *Condition implies that and for . To prove the nonnegative property of Green’s function, combining with (15) and (16), we divide it into two cases. *Case 1*. For ,
imply that is nondecreasing and concave in on ; it follows that for each :
Likewise,
guarantee that is nonincreasing and concave in on ; we have for each :
From the above argument, we obtain that
we have (29), since .*Case 2*. For ,
imply that is nonincreasing and convex in on ; it follows that for each :
Likewise,
guarantee that is nondecreasing and convex in on ; it follows that for each :
So we get
we have (35), since .

It is easy to show that . Therefore, we obtain that for each ; (27)–(36) imply the desired conclusion.

#### 3. A Priori Estimates

Lemma 5. *Let , hold. If with , then a solution of the problem
**
satisfies
**
where , .*

*Proof. *A solution of problem (37) is , where is defined by (12); it is not difficult to show that the results hold by Lemma 4 and we omit the proof process.

Lemma 6. *Let , hold. If with , then a solution of problem (37) satisfies
**
where
*

*Proof. *According to the nonnegative solution of problem (37), we establish a priori estimates as follows. For , we can check that ; note that , so

For , we can check that ; notice that and thus

For , we can check that ; note that and so

For , we can check that ; then
From the above argument, we can obtain the desired result (39).

Let be the Banach space with the norm , where . Set a cone in In view of the assumptions and , take where , , , and is an arbitrary positive constant. By some simplifying, we obtain that . Set a constant where . It is not difficult to check that is nonnegative on and satisfies , , . Under the assumption , we have ; that is, , so . By assumption , both and hold. Thus, we can define an operator by

Lemma 7. *Let hold. Then, is completely continuous.*

*Proof. *Since conditions hold, by Lemma 5, we observe that for ; , , are satisfied, so . Since may be singular at and/or , we will take the arguments to show that the operator is completely continuous. Assume that satisfy ; then there exists such that for arbitrary positive integer . One has that