Open Access

Positive solution for boundary value problems with p-Laplacian in Banach spaces

Boundary Value Problems20122012:51

DOI: 10.1186/1687-2770-2012-51

Received: 28 December 2011

Accepted: 30 April 2012

Published: 30 April 2012

Abstract

In this article, by using the fixed point theorem of strict-set-contractions operator, we discuss the existence of positive solution for boundary value problems with p-Laplacian

ϕ p u t + f u t = θ , 0 < t < 1 , u 0 = θ , u 1 = θ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equa_HTML.gif

in Banach spaces E, where: θ is the zero element of E. Although the fixed point theorem of strict-set-contractions operator is used extensively in yielding positive solutions for boundary value problems in Banach spaces, this method has not been used to study those boundary value problems with p-Laplacian in Banach spaces. So this article may be regarded as an illustration of fixed point theorem of strict-set-contractions operator in a new area.

MSC: 34B18.

Keywords

boundary value problems p-Laplacian positive solution strict-set-contractions

1 Introduction

In the last ten years, the theory of ordinary differential equations in Banach spaces has become an important new branch, so boundary value problems in Banach Space has been studied by some researchers, we refer the readers to [19] and the references therein.

For abstract space, it is here worth mentioning that Guo and Lakshmikantham [10] discussed the multiple solutions of the following two-point boundary value problems (BVP for short) of ordinary differential equations in Banach space
u t + f u t = θ , 0 < t < 1 , u 0 = θ , u 1 = θ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equb_HTML.gif
Very recently, by using the fixed-point principle in cone and the fixed-point index theory for strict-set-contraction operator, Zhang et al. [11] investigated the existence, nonexistence, and multiplicity of positive solutions for the following nonlinear three-point boundary value problems of n th-order differential equations in ordered Banach spaces
x ( n ) t + f ( t , x ( t ) , x ( t ) , , x ( n - 2 ) ( t ) ) = θ , t ( 0 , 1 ) , x ( i ) ( 0 ) = θ , i = 0 , 1 , 2 , , n - 2 , x ( n - 2 ) ( 1 ) = ρ x ( n - 2 ) ( η ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equc_HTML.gif

On the other hand, boundary value problems with p-Laplacian have received a lot of attention in recent years. They often occur in the study of the n-dimensional p-Laplacian equation, non-Newtonian fluid theory, and the turbulent flow of gas in porous medium [1219]. Many studies have been carried out to discuss the existence of solutions or positive solutions and multiple solutions for the local or nonlocal boundary value problems.

However, to the authors' knowledge, this is the first article can be found in the literature on the existence of positive solutions for boundary value problems with p-Laplacian in Banach spaces. As is well known, the main difficulty that appears when passing from p = 2 to p ≠ 2 is that, when p = 2, we can change the differential equation into a equivalent integral equation easily and therefore a Green's function exists, so we can easily prove the equivalent integral operator is a strict-set-contractions operator, which is a very important result for discussing positive solution for boundary value problems in Banach space. However, for p ≠ 2, it is impossible for us to find a Green's function in the equivalent integral operator since the differential operator (ϕ p (u'))' is nonlinear. To authors' knowledge, this is the first article to use the fixed point theorem of strict-set-contractions to deal with boundary value problems with p-Laplacian in Banach spaces. Such investigations will provide an important platform for gaining a deeper understanding of our environment.

Basic facts about an ordered Banach space E can be found in [1, 4]. Here we just recall a few of them. Let the real Banach spaces E with norm || ·|| be partially ordered by a cone P of E, i.e., x ≤ y if and only if y - x P , and P* denotes the dual cone of P. P is said to be normal if there exists a positive constant N such that θxy implies ||x|| ≤ N||y||, where θ denotes the zero element of E, and the smallest N is called the normal constant of P (it is clear, N ≥ 1). Set I = 0 [1], (C[I, E], ||·|| C ) is a Banach space with ||x|| C = maxtI||x(t)||. Clearly, Q = {x C[I, E]|x(t) ≥ θ for t I} is a cone of the Banach space C[I, E].

For a bounded set S in a Banach space, we denote by α(S) the Kuratowski measure of noncompactness. In this article, we denote by α(·) the Kuratowski measure of noncompactness of a bounded set in E and in C[I, E].

The operator T : DE(D E) is said to be a k-set contraction if T : DE is continuous and bounded and there is a constant k ≥ 0 such that α(T (S)) ≤ (S) for any bounded S D; a k-set contraction with k < 1 is called a strict set contraction.

In this article, we will consider the boundary value problems with p-Laplacian
( ϕ p ( u ( t ) ) ) + f ( u ( t ) ) = θ , 0 < t < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ1_HTML.gif
(1)
u ( 0 ) = θ , u ( 1 ) = θ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ2_HTML.gif
(2)

in Banach spaces E, where ϕ p (s) = sp-1, p > 1, (ϕ p )-1 = ϕ q , 1 p  +  1 q = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq1_HTML.gif, θ is the zero element of E, f C(P, P).

A function u is called a positive solution of BVP (1) and (2) if it satisfies (1) and (2) and u Q, u(t) Q.

The main tool of this article is the following fixed point Theorems.

Theorem 1. [5] Let K be a cone in a Banach space E and K r, R = {x K, r ≤ ||x|| ≤ R}, R > r > 0. Suppose that A : K r, R K is a strict-set contraction such that one of the following two conditions is satisfied:
( a ) Ax x , x K , x = r ; A x x , x K , x = R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equd_HTML.gif
( b ) A x x , x K , x = r ; A x x , x K , x = R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Eque_HTML.gif

Then, A has a fixed point x K r, R such that r ≤ ||x|| ≤ R.

2 Preliminaries

Lemma 2.1. If y C[I, E], then the unique solution of
( ϕ p ( u ( t ) ) ) + y ( t ) = θ , 0 < t < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ3_HTML.gif
(3)
u ( 0 ) = θ , u ( 1 ) = θ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ4_HTML.gif
(4)
is
u ( t ) = t 1 ϕ q 0 s y ( τ ) d τ d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equf_HTML.gif

Lemma 2.2. If y Q, then the unique solution u of the problem (3) and (4) satisfies u(t) ≥ θ, t I, that is u Q.

Lemma 2.3. Let δ ( 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq2_HTML.gif, J δ = [δ, 1-δ], then for any y Q, the unique solution u of the problem (3) and (4) satisfies u(t) ≥ δu(s), t J δ , s I.

Lemma 2.4. We define an operator T by
( T u ) ( t ) = t 1 ϕ q 0 s f ( u ( τ ) ) d τ d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ5_HTML.gif
(5)

Then u is a solution of problem (1) and (2) if and only if u is a fixed point of T.

In the following, the closed balls in spaces E and C[I, E] are denoted by T r = {x E|||x|| ≤ r} (r > 0) and B r = {x C[I, E]|||x|| c r}, M = sup {||f(u)||: u Q B r }.

Lemma 2.5. Suppose that, for any r > 0, f is uniformly continuous and bounded on P T r and there exists a constant L r with
( q - 1 ) M q - 2 L r < 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ6_HTML.gif
(6)
such that
α ( f ( D ) ) L r α ( D ) , D P T r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ7_HTML.gif
(7)

Then, for any r > 0, operator T is a strict-set-contraction on D P T r .

Proof. Since f is uniformly continuous and bounded on P T r , we see from Lemma 2.4 that T is continuous and bounded on Q B r . Now, let S Q B r be given arbitrary, there exists a partition S = i = 1 m S i . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq3_HTML.gif We set α{y : y S} = α(S

By virtue of Lemma 2.4, it is easy to show that the functions {Ty|y S} are uniformly bounded and equicontinuous, and so by [11],
α ( T ( S ) ) = sup t I α ( T ( S ( t ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ8_HTML.gif
(8)

where T (S(t)) = {Tu(t)|u S, t is fixed} P T r for any t I.

Let u1,u2 S i ,
| ( T u 1 - T u 2 ) ( t ) | = t 1 ϕ q 0 s f ( u 1 ( τ ) ) d τ - ϕ q 0 s f ( u 2 ( τ ) ) d τ d s t 1 ϕ q 0 s f ( u 1 ( τ ) ) d τ - ϕ q 0 s f ( u 2 ( τ ) ) d τ d s ( q - 1 ) M q - 2 t 1 0 s f ( u 1 ( τ ) ) - f ( u 2 ( τ ) ) d τ d s ( q - 1 ) M q - 2 t 1 0 s d τ d s max 0 t 1 f ( u 1 ( t ) ) - f ( u 2 ( t ) ) 1 2 ( q - 1 ) M q - 2 max 0 t 1 f ( u 1 ( t ) ) - f ( u 2 ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equg_HTML.gif
So, we have
α ( T u ) 1 2 ( q - 1 ) M q - 2 α ( f ( S ) ) 1 2 ( q - 1 ) M q - 2 L r α ( B ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equh_HTML.gif
where B = {y(s)| s I, y S} P T r . Similarly, to the proof of [10], we have α(B) ≤ 2α(S)·It follows from (6), (7), and (8), that
α ( T ( S ) ) < ( q - 1 ) M q - 2 L r α ( S ) < α ( S ) , S Q B r , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equi_HTML.gif

and consequently T is a strict-set-contraction on S Q B r because of (q-1)Mq-2L r < 1.    □

3 Existence of positive solution to BVP (1) and (2)

In the following, for convenience, we set
f β = lim sup u β f ( u ) ϕ p u , f β lim inf u β f ( u ) ϕ p u , ( ψ f ) β = lim  inf u β ψ f u ϕ p u , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equj_HTML.gif

where β = 0 or ∞, ψ P* and ||ψ|| = 1.

Furthermore, we list some condition:

(H1): For any r > 0, f is uniformly continuous and bounded on P T r and there exists a constant L r with (q - 1)Mq-2L r < 1 such that
α ( f ( D ) ) L r α ( D ) , D P T r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equk_HTML.gif

Theorem 3.1. Let (H1) hold, cone P be normal. If ϕ q ( f 0 ) < 1 < 1 2 δ ϕ q 1 2 - δ ψ f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq4_HTML.gif, then BVP (1) and (2) has at least one positive solution.

Proof. Set
K = { u Q | u ( t ) δ u ( s ) , t J δ , s I } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equl_HTML.gif
It is clear that K is a cone of the Banach space C[I, E] and K Q. By Lemma 2.4, we know T (Q) K, and so
T K K . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equm_HTML.gif
We first assume that ϕ q (f0) < 1 Then, there exists a constant r ̄ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq5_HTML.gif such that, for any u K, u r ̄ 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq6_HTML.gif we have ||f(u)|| ≤ (f0+ε1)ϕ p (||u||), where ε1 > 0 satisfies ϕ q (f0 + ε1) ≤ 1. Let r 1 ( 0 , r ̄ 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq7_HTML.gif then for any t I, u K, ||u|| C = r1, we have
( T u ) ( t ) 0 1 ϕ q 0 s f u τ d τ d s ϕ q 0 1 f 0 + ε 1 ϕ p u d s ϕ q f 0 + ε 1 u C u C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ9_HTML.gif
(9)

i.e., u K, ||u|| C = r1 implies ||Tu|| C ≤ ||u|| C ·

On the other hand, since 1 < 1 2 δ ϕ q 1 2 - δ ψ f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq8_HTML.gif, there exists r ̄ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq9_HTML.gifsuch that
ψ f u ( t ) ( ψ f ) - ε 2 ϕ p u , t I , x K , u r ̄ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equn_HTML.gif

where ε2 > 0 satisfies 1 2 δ ϕ q 1 2 - δ ( ψ f ) - ε 2 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq10_HTML.gif.

Choose r 2 = max { 2 r 1 , r ̄ 2 δ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq11_HTML.gif, then, for any t J δ , u K, ||u|| C = r2, we have
u ( t ) δ u C δ r 2 r ̄ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equo_HTML.gif
then,
( T u ) 1 2 ψ T u 1 2 = 1 2 1 ϕ q 0 s ψ ( f ( u ( τ ) ) ) d τ d s 1 2 1 ϕ q δ 1 2 ψ ( f ( u ( τ ) ) ) d τ d s 1 2 1 ϕ q δ 1 2 ( ( ψ f ) - ε 2 ) ϕ p u d τ d s 1 2 1 ϕ q δ 1 2 ( ( ψ f ) - ε 2 ) ϕ p δ u C d τ d s = 1 2 δ ϕ q 1 2 - δ ( ψ f ) - ε 2 ) u C u C , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equ10_HTML.gif
(10)
i.e., for any u K, ||u|| C = r2, we have
T u C u C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_Equp_HTML.gif

On the other hand, by Lemma 2.5, T is a strict set contraction from B r ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq12_HTML.gif into B r ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq12_HTML.gif. Consequently, Theorem 1 implies that T has a fixed point in B r ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-51/MediaObjects/13661_2011_Article_135_IEq12_HTML.gif, and the proof is complete. □

Declarations

Acknowledgements

This study was sponsored by the National Natural Science Foundation of China (No. (11071014))and the Tianjin City High School Science and Technology Fund Planning Project (No. (20091008)) and Tianyuan Fund of Mathematics in China (No. (11026176)) and Natural Science Foundation of Shandong Province of China (No. (ZR2010AM035)). The authors thank the referee for his/her careful reading of the article and useful suggestions.

Authors’ Affiliations

(1)
College of Science, Tianjin University of Technology
(2)
School of Science, Beijing Institute of Technology

References

  1. Lakshmikanthan V, Leela S: Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford; 1981.
  2. Gupta CP: A note on a second order three-point boundary value problem. J Math Anal Appl 1994, 186(1):277-281. 10.1006/jmaa.1994.1299MathSciNetView Article
  3. Ma RY, Castaneda N: Existence of solutions of nonlinear m-point boundary-value problems. J Math Anal Appl 2001, 256(2):556-567. 10.1006/jmaa.2000.7320MathSciNetView Article
  4. Guo DJ, Lakshmikantham V, Liu XZ: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers, Dordrecht; 1996.View Article
  5. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Inc., NewYork; 1988.
  6. Zhao YL, Chen HB: Existence of multiple positive solutions for m-point boundary value problems in Banach spaces. J Comput Appl Math 2008, 215(1):79-90. 10.1016/j.cam.2007.03.025MathSciNetView Article
  7. Liu B: Positive solutions of a nonlinear four-point boundary value problems in Banach spaces. J Math Anal Appl 2005, 305(1):253-276. 10.1016/j.jmaa.2004.11.037MathSciNetView Article
  8. Liu Y: Multiple positive solutions to fourth-order singular boundary value problems in abstract space. Electron J Diff Equ 2004, 2004(120):1-13.
  9. Feng M, Zhang X: Multiple solutions of two-point boundary value problem of fourth-order ordinary differential equations in Banach space. Acta Anal Funct Appl 2004, 6: 56-64. (in Chinese)MathSciNet
  10. Guo D, Lakshmikantham V: Multiple solutions of two-point boundary value problem of ordinary differential equations in Banach space. J Math Anal Appl 1988, 129(1):211-222. 10.1016/0022-247X(88)90243-0MathSciNetView Article
  11. Zhang X, Feng M, Ge W: Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces. Nonlinear Anal Theory, Methods & Appl 2009, 70(2):584-597. 10.1016/j.na.2007.12.028MathSciNetView Article
  12. O'Regan D: Some general existence principles and results for ( ϕ ( y' ))' = qf ( t, y, y' ), 0 < t < 1. SIAM J Math Anal 1993, 24(3):648-668. 10.1137/0524040MathSciNetView Article
  13. del Pino M, Drábek P, Manásevich R: The Fredholm alternative at the first eigenvalue for the one-dimensional p-Laplacian. J Diff Equ 1999, 151(2):386-419. 10.1006/jdeq.1998.3506View Article
  14. Cabada A, Pouso RL: Existence results for the problem ( ϕ ( y '))' = f ( t, y, y' ) with nonlinear boundary conditions. Nonlinear Anal Theory Methods & Appl 1999, 35(2):221-231. 10.1016/S0362-546X(98)00009-1MathSciNetView Article
  15. Lü H, Zhong C: A note on singular nonlinear boundary value problems for the one-dimensional p-Laplacian. Appl Math Lett 2001, 14(2):189-194. 10.1016/S0893-9659(00)00134-8MathSciNetView Article
  16. Li Y, Zhang T: Multiple positive solutions for second-order p-Laplacian dynamic equations with integral boundary conditions. Boundary Value Problems 2011., 2011: Article ID 867615, 17
  17. Tian Y, Ge W: Periodic solutions of non-autonomous second-order systems with a p-Laplacian. Nonlinear Anal Theory Methods & Appl 2007, 66(1):192-203. 10.1016/j.na.2005.11.020MathSciNetView Article
  18. Ji D, Ge W: Existence of multiple positive solutions for Sturm-Liouville-like four-point boundary value problem with p-Laplacian. Nonlinear Anal Theory Methods & Appl 2008, 68(9):2638-2646. 10.1016/j.na.2007.02.010MathSciNetView Article
  19. Ji D, Yang Y, Ge W: Triple positive pseudo-symmetric solutions to a four-point boundary value problem with p-Laplacian. Appl Math Lett 2008, 21(3):268-274. 10.1016/j.aml.2007.03.019MathSciNetView Article

Copyright

© Ji and Ge; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.