International Journal of Analysis

Volume 2014 (2014), Article ID 638976, 18 pages

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

Research Article

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

Youwei Zhang

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

In view of the continuity of in , we can obtain that, for any . This means that the operator is continuous. It follows from the properties of the function that we can choose two sequences , . Obviously for any and as , respectively. Define

and an operator sequence by

Clearly, is a piecewise continuous function; the operator is well defined. Further, we have that is completely continuous.

Let and . We will show that approaches uniformly on as . From the absolute continuity of integral, we obtain

where . Green’s functions and (52) imply that, for any ; we have

For any , we get that and, for any , there is .

Similarly, for any , or , we can obtain that and , respectively.

The above argument implies that ; that is, the sequence uniformly approximates on any bounded subset of . Hence, is completely continuous.

4. Single Solution

Now, we are ready to present the sufficient conditions for the existence of at least one positive solution to the problem .

Theorem 8. Suppose hold; on any subinterval of and

()there exist a constant and a function which is integrable on , satisfying

()there exist constants , such that and satisfy

Then, the boundary value problem has at least one positive solution.

Proof. By the definition of operator and Lemma 7, it suffices to show that the conditions of Lemma 1 hold with respect to . Furthermore, from the fact that is Green’s function, it is not difficult to prove that a fixed point of is coincident with the solution of the boundary value problem , so we concentrate on the existence of the fixed point of the operator .

First of all, we show that

where

and . For convenience, we introduce notation as follows:

By the well-known Hölder’s inequality, for any and , by using hypothesis , we get

Consequently, .

Next, we verify that

where

In fact, hypothesis implies that, for any , , there exists such that and

Then by (62) and Lemma 4, we have

The argument guarantees that, for ,

Hence, (60) holds.

Finally, from the definitions of and , it is easy to check that

Note that (56) and (60) imply that there exist , such that and , , respectively. Thus, it follows that

By virtue of (66), the hypotheses of Lemma 1 are satisfied; we infer that there exists such that , and . In addition, condition guarantees that the fixed point is positive, so the problem has at least one positive solution. This completes the proof.

Remark 9. Since the definition of the operator , a slight modification of and

()there exist constants , such that and satisfy

which guarantee that the problem has at least one positive solution.

Furthermore, may be superlinear or sublinear; we can give the following existence results for the problem and the proof is omitted.

Theorem 10. Suppose hold, on any subinterval of and

() and , or() and .

Then, the problem has at least one positive solution.

5. Multiple Solutions

Let the nonnegative continuous convex functionals and , the nonnegative continuous concave functional , and the nonnegative continuous functional be defined on the cone by

To prove our main results, we recommend notation

Now, we are ready to apply Avery and Peterson fixed point theorem to the operator to give the sufficient conditions for the existence of at least three positive solutions to the problem .

Theorem 11. Let hold; on any subinterval of . Suppose that there exist numbers , , such that , and

() for ,() for ,() for .

Then, the has at least three positive solutions such that

Proof. By the definition of operator and its properties, it suffices to show that the conditions of Lemma 2 hold with respect to . Furthermore, from the fact that is the Green function, it is not difficult to prove that a fixed point of is coincident with the solution of the boundary value problem , so we concentrate on the existence of the fixed point of the operator .

The functionals are defined as (68); we also note that , , , and are continuous nonnegative functionals on satisfying for such that, for some positive numbers and , and , for , where . If , then . We get by (39). It follows from assumption that . On the other hand, for , we have proved that in Lemma 7 and thus

Therefore, .

Firstly, to verify the condition of Lemma 2 holds, we choose which is defined by (46); (47) gives the expression of . Set ; we have

Equation (72) implies that ; that is, . So ; it holds and . Hence, if , then , for . In view of assumption , we get

Thus, for each .

Next, we verify that the condition of Lemma 2 is satisfied. In fact, if with , then

So for any with .

Finally, we assert that the condition of Lemma 2 is fulfilled. It is clear that . Suppose that with ; by assumption , we have

Hence, for with .

Hence, the hypotheses of Lemma 2 are satisfied. Therefore, we can conclude that the operator has three positive fixed points . In addition, condition guarantees that those fixed points are positive, so the problem has at least three positive solutions , , satisfying , for , ,  and with .

6. Applications

The known Falkner-Skan equation

whose solution is the similarity solution of the two-dimensional incompressible laminar boundary-layer equations [4, 5]. When , which arises in the study of two-dimensional incompressible viscous flow past a thin semi-infinite flat plate , . The special case is Blasius’s equation, in which the wedge reduces to a flat plate. The special case is called Homann's equation, in which the wedge reduces to a flat plate. The special case is called Hiemenz’s equation. corresponds to flow toward the wedge, otherwise, to flow away from the wedge. When , taking , Blasius flow over a flat plate with a sharp edge as , flow over a wedge with half angle , as , Hiemenz flow toward a plane stagnation point as , flow into a corner with as , and no corresponding simple ideal flow as exist.

Now, we give a general equation of (77), subject to the boundary conditions. But it is even more difficult to solve, because it is singular; the solution denotes the physically relevant solutions for and the dimensionless stream function for of (77), respectively.

Consider

where

It is easy to check that hypotheses , hold. Taking , , by some calculations, we have , , , , , and

then for and . By Theorem 8, problem (78) has at least one positive solution. These results accord with ; the solutions of (77) are known to exist [23].

On the other hand, let

It is easy to check that hypotheses hold and on . By some calculations, we have , , , , , , , , , , , , . If we choose , , , then satisfies

Then all hypotheses of Theorem 11 hold. Hence, problem (78) has at least three positive solutions , , and such that , , with , hold. These results coincide with ; the solutions of (77) are the dimensionless stream functions.

Conflict of Interests

The author declares that there is no conflict of interests.

Authors’ Contribution

The author plotted this paper carefully, gave a rigorous derivation process, read, and approved the final paper.

References

J. W. Bebernes and D. R. Kassoy, “A mathematical analysis of blowup for thermal reactions—the spatially nonhomogeneous case,” SIAM Journal on Applied Mathematics, vol. 40, no. 3, pp. 476–484, 1981.

A. Nachman and A. Callegari, “A nonlinear singular boundary value problem in the theory of pseudoplastic fluids,” SIAM Journal on Applied Mathematics, vol. 38, no. 2, pp. 275–281, 1980.

S. H. Lin, “Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics,” Journal of Theoretical Biology, vol. 60, no. 2, pp. 449–457, 1976.

M. Moghimi, H. Khoramishad, H. R. Massah, and S. M. Mortezaei, “Approximate analytical solution to flow over a flat plate by variational iteration method,” Mechanical and Aerospace Engineering Journal, vol. 2, pp. 63–69, 2006.

H. Nasr, I. A. Hassanien, and H. M. El-Hawary, “Chebyshev solution of laminar boundary layer flow,” International Journal of Computer Mathematics, vol. 33, pp. 127–132, 1990.

R. Naz, F. M. Mahomed, and D. P. Mason, “Symmetry solutions of a third-order ordinary differential equation which arises from Prandtl boundary layer equations,” Journal of Nonlinear Mathematical Physics, vol. 15, pp. 179–191, 2008.

R. P. Agarwal and D. O'Regan, Singular Differential and Integral Equations with Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.

A. Cabada and S. Lois, “Existence of solution for discontinuous third order boundary value problems,” Journal of Computational and Applied Mathematics, vol. 110, no. 1, pp. 105–114, 1999.

J. R. Graef and B. Yang, “Multiple positive solutions to a three point third order boundary value problem,” Discrete and Continuous Dynamical Systems, vol. 2005, pp. 337–344, 2005.

L. K. Jackson, “Existence and uniqueness of solutions of boundary value problems for third order differential equations,” Journal of Differential Equations, vol. 13, pp. 432–437, 1973.

D. R. Anderson and J. M. Davis, “Multiple solutions and eigenvalues for third-order right focal boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 135–157, 2002.

Y. Guo and W. Ge, “Positive solutions for three-point boundary value problems with dependence on the first order derivative,” Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 291–301, 2004.

Z. Liu, L. Debnath, and S. M. Kang, “Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem,” Computers & Mathematics with Applications, vol. 55, no. 3, pp. 356–367, 2008.

D. R. Anderson, “Green's function for a third-order generalized right focal problem,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 1–14, 2003.

F. M. Minhós, “On some third order nonlinear boundary value problems: existence, location and multiplicity results,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1342–1353, 2008.

R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, vol. 436 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

R. P. Agarwal, D. O'Regan, and V. Lakshmikantham, “Singular focal and higher order boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 2, pp. 215–228, 2000.

A. Cabada and S. Heikkilä, “Extremality and comparison results for discontinuous third order functional initial-boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 195–212, 2001.

P. J. Y. Wong and R. P. Agarwal, “Multiple positive solutions of two-point right focal boundary value problems,” Mathematical and Computer Modelling, vol. 28, no. 3, pp. 41–49, 1998.

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.

R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673–688, 1979.

R. I. Avery and A. C. Peterson, “Three positive fixed points of nonlinear operators on ordered Banach spaces,” Computers & Mathematics with Applications, vol. 42, no. 3–5, pp. 313–322, 2001.

R. L. Panton, Incompressible Flow, John Wiley & Sons, New York, NY, USA, 1996.