Journal of Mathematics

Volume 2013 (2013), Article ID 593285, 9 pages

http://dx.doi.org/10.1155/2013/593285

Research Article

Exact Multiplicity of Solutions for a Class of Singular Generalized One-Dimensional -Laplacian Problem

Youwei Zhang

Department of Mathematics, Hexi University, Gansu 734000, China

Received 11 November 2012; Accepted 29 May 2013

Academic Editor: Francisco B. Gallego

Copyright © 2013 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 describe the existence of positive solutions for a class of singular generalized one-dimensional -Laplacian problem. By applying the related fixed point theory in cone, some new and general results on the existence of positive solutions to the singular generalized -Laplacian problem are obtained. Note that the nonlinear term involves the first-order derivative explicitly.

1. Introduction

Recently, increasing attention is paid to question of positive solution for singular boundary value problems [1–11]. One notices that the singular boundary value problems for ordinary differential equation describe many phenomena in applied mathematics and physical science, which can be found in the theory of nonlinear diffusion generated by nonlinear sources and in the thermal ignition of gases [12–17]. Moreover, there are excellent results of nonsingular problem, see [18–21] and the references therein.

Moreover, the nonlocal -Laplacian problems for ordinary differential equation have been studied extensively. There are many papers dealing with the existence of positive solutions for the nonlocal -Laplacian boundary value problem, in which the nonlinear term is independent of the first-order derivative with different boundary conditions. For example, Ma et al. [22] established some existence results of positive solutions for the problem

The main tool is the monotone iterative technique. By applications of fixed point index theory, Cheung and Ren [23] investigated two classes of quasilinear nonlocal problems:

Sufficient conditions for the existence of twin positive solutions are established. In [16], Feng and Ge studied nonlocal problem for the one-dimensional -Laplacian

They obtained sufficient conditions for the existence of at least three solutions to the above problem by using the Avery and Peterson fixed point theorem.

However, there are not many concerning the boundary value problems, in which a generalized -Laplacian equation with a nonlocal boundary conditions. The motivation for the present work stems from many recent investigations in [16, 23–25] and references therein. Our purpose of this paper is to establish some sufficient conditions for the existence of triple positive solutions to a class of a singular generalized one-dimensional -Laplacian problem

where is -Laplacian operator; that is, , , , , and , , , , , , satisfy

), satisfy , ,(), is positive and nondecreasing on ,() may be singular at and/or , and , where ,() is an -Carathédory function; that is, for each , the mapping is Lebesgue measurable on ; for a.e. , the mapping is continuous on .

Under the above assumptions, some new and general results on the existence of multiple positive solutions to singular one-dimensional -Laplacian are obtained. Our results develop some results of Cheung and Ren [23] and include and improve the main results of Feng and Ge [16] and Zhang [25].

The rest of the paper is organized as follows. In Section 2, we provide some background material from the theory of cones in the Banach spaces, which are useful later. Some lemmas and criteria for the existence of three positive solutions for one-dimensional -Laplacian problems are established in Section 3. Finally, we give an example to illustrate our main results.

2. Preliminaries

Let and be nonnegative continuous convex functionals on a cone , a nonnegative continuous concave functional on , a nonnegative continuous functional on , and ,  ,  ,  and  positive numbers. We define the following convex sets and a closed set

Now we state the fixed point theorem due to Avery and Peterson [26].

Lemma 1. Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on , a nonnegative continuous concave functional on , and a nonnegative continuous functional on satisfying for , such that, for some positive numbers and ,

Suppose that is completely continuous and there are positive numbers , , and with such that

(), for ,() for with ,() and for with .

Then has at least three fixed points such that

Denote that .

Lemma 2. Assume that and hold. If , then the boundary value problem

has a unique solution

Proof. For a.e. , integrating (7) from to , in view of (8), we have

Integrating this from to yields

Again, it follows from (8) that

The proof is complete.

Lemma 3. Assume that () and () hold. If , then the unique solution of the problems (7) and (8) satisfies

where .

Proof.  The proof is similar to Lemma  3.6 in [25], we omit it.

Let be the Banach space with the norm , where . Define a cone by

Let the assumptions () and () hold. Denote that

Evidently, . We take

where satisfies , is constant. It is not difficult to check that for a.e. , , for a.e. , and satisfies , . That is to say . Hence, .

To obtain the existence of solutions for the problem , the following priori estimate is needful.

Lemma 4. Assume that () holds. If , then

where .

Proof. The proof is similar to Lemma  3.1 in [21], we omit it.

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

In the view of assumption (), that is, and (). Thus, we can define an operator by

Lemma 5. Assume that hold. Then is completely continuous.

Proof. For any , by the definition of , we see that , . From and Lemma 3, we have , implies that is concave on , , , and . So we can conclude that . In what follows, we prove that is compact. Let , there exists a set such that . For any , the assumptions imply that

We take the arguments to show that the operator is completely continuous. In view of Lemma 4 and the continuity of in and , assume that satisfy . Moreover, is an increasing and continuously function, and we obtain that

Therefore,

This means that the operator is continuous.

Since may be singular at and/or , choose two sequences , satisfying for any , such that and as , respectively. Define

and an operator sequence by

Clearly, is a piecewise continuous function, and the operator is well defined. Further, we can see that is completely continuous.

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

where . For any and a.e. , one has that

For and a.e. , we have the same result. It is easy to see that, for any and a.e. , there is . Similarly, we can obtain that, for any and a.e. , , , respectively,

In view of the continuity of function , from the above arguments, we have

That is to say, the sequence is uniformly an approximate on any bounded subset of . Applying the Arzela-Ascoli lemma to the operator , we can conclude that is relatively compact; that is, is completely continuous.

3. Main Results

We are ready to apply the Avery and Peterson fixed point theorem to the operator to give sufficient conditions for the existence of at least three positive solutions to the problem . For convenience, we introduce the following notation:

Theorem 6. Suppose hold and for a.e. . If there exist positive numbers , , and with such that the following conditions are satisfied:

(), ,(), ,(), ,

then the problem has at least three positive solutions , , and satisfying

Proof. It is known that boundary value problem has a solution if and only if solves the operator equation . By the definition of operator , from Lemma 4 and the concavity of , the functionals defined above satisfy , for any . We also note that for , which suffices to show that the conditions of Lemma 1 hold with respect to . First of all, we show that if the assumption () is satisfied, then

In fact, for , there is . With Lemma 4, there is , and the assumption () implies that for a.e. . On the other hand, for , we obtain that , and so is concave on , and there is . By using of is positive and monotone increasing on a.e. , and so

Therefore, (31) is satisfied.

To check if the condition in Lemma 1 is fulfilled, we choose , where is defined by (16). From the hypothesis conditions of the theorem, we have

Equation (33) implies that . That is, . So for , and hold. Hence, for , we see that and for a.e. . Hence by the assumption (), one has that for a.e. . By the definition of the functional , we obtain

Therefore, we get for .

Next, we show that the condition in Lemma 1 holds. In fact, if with , then

Finally, we assert that the condition in Lemma 1 also holds. Since , there holds that . Assume that with . Then, by the assumption we get

Thus, the conditions in Lemma 1 hold, and so the problem has at least three positive solutions , , and such that (30) holds.

Similar to the above arguments, we will discuss the problem

Now, we replace the assumption with the following:

() , satisfy , .

We only give the main results of the problem , and the proofs are omitted.

Lemma 7. Assume that , , , and hold. Then the problem has a unique solution

The cone is defined by

Lemma 8. Assume that holds. If , then

where and .

In what follows, we define integral operators and , , and :

Theorem 9. Suppose that , , , and hold and for a.e. . If there exist positive numbers , , , and with such that the following conditions are satisfied:

,,,

then the problem has at least three positive solutions , , and satisfying

From Theorem 6 or Theorem 9, we can see that, when assumptions like , , and are imposed appropriately on the nonlinear term , we also establish the existence results of an arbitrary odd number of positive solutions of the problem or . Here, we only present the form of Theorem 6.

Theorem 10. Suppose that hold and for a.e. . If there exist positive numbers , , , and , with

such that the following conditions are satisfied:

(), ,(), ,(), ,

then the problem has at least positive solutions.

4. Example

Let , , and . Consider the boundary value problem

where

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

Thus all assumptions of Theorem 6 hold, and so the problem (44) has at least three positive solutions , , and such that for , ,   with , and .

References

R. P. Agarwal and D. O'Regan, “Some new results for singular problems with sign changing nonlinearities,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 1–15, 2000.

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

R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, 1999.

H. Asakawa, “Nonresonant singular two-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 44, no. 6, pp. 791–809, 2001.

R. Ma and D. O'Regan, “Solvability of singular second order -point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 124–134, 2005.

D. O'Regan, Theory of Singular Boundary Value Problem, World Scientific, Hackensack, NJ, USA, 1994.

S. D. Taliaferro, “A nonlinear singular boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 3, no. 6, pp. 897–904, 1979.

Y. Wang and W. Ge, “Existence of triple positive solutions for multi-point boundary value problems with a one dimensional -Laplacian,” Computers & Mathematics with Applications, vol. 54, no. 6, pp. 793–807, 2007.

Q. Yao, “Existence and iteration of symmetric positive solutions for a singular two-point boundary value problem,” Computers & Mathematics with Applications, vol. 47, no. 8-9, pp. 1195–1200, 2004.

Y. Zhang, “Positive solutions of singular sublinear Emden-Fowler boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 1, pp. 215–222, 1994.

Y. Zhang, “A multiplicity result for a singular generalized Sturm-Liouville boundary value problem,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 132–140, 2009.

N. Anderson and A. M. Arthurs, “Analytical bounding functions for diffusion problems with Michaelis-Menten kinetics,” Bulletin of Mathematical Biology, vol. 47, no. 1, pp. 145–153, 1985.

D. Aronson, M. G. Crandall, and L. A. Peletier, “Stabilization of solutions of a degenerate nonlinear diffusion problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 6, no. 10, pp. 1001–1022, 1982.

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.

H. Feng and W. Ge, “Existence of three positive solutions for -point boundary-value problems with one-dimensional -Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 7, pp. 2017–2026, 2008.

S. Lin, “Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics,” Journal of Theoretical Biology, vol. 60, pp. 449–457, 1976.

C. P. Gupta, “A generalized multi-point boundary value problem for second order ordinary differential equations,” Applied Mathematics and Computation, vol. 89, no. 1–3, pp. 133–146, 1998.

L. Kong and Q. Kong, “Multi-point boundary value problem of second order differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, pp. 909–931, 2004.

N. Kosmatov, “Symmetric solutions of a multi-point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 25–36, 2005.

Y.-W. Zhang and H.-R. Sun, “Three positive solutions for a generalized Sturm-Liouville multipoint BVP with dependence on the first order derivative,” Dynamic Systems and Applications, vol. 17, no. 2, pp. 313–324, 2008.

D.-X. Ma, Z.-J. Du, and W.-G. Ge, “Existence and iteration of monotone positive solutions for multipoint boundary value problem with -Laplacian operator,” Computers & Mathematics with Applications, vol. 50, no. 5-6, pp. 729–739, 2005.

W.-S. Cheung and J. Ren, “Twin positive solutions for quasi-linear multi-point boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 1, pp. 167–177, 2005.

R. Avery and J. Henderson, “Existence of three positive pseudo-symmetric solutions for a one-dimensional -Laplacian,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 395–404, 2003.

Y. Zhang, “Existence and multiplicity results for a class of generalized one-dimensional -Laplacian problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 748–756, 2010.

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.