Advanced
TRIPLE SOLUTIONS FOR THREE-ORDER PERIODIC BOUNDARY VALUE PROBLEMS WITH SIGN CHANGING NONLINEARITY†
TRIPLE SOLUTIONS FOR THREE-ORDER PERIODIC BOUNDARY VALUE PROBLEMS WITH SIGN CHANGING NONLINEARITY†
Journal of Applied Mathematics & Informatics. 2014. Jan, 32(1_2): 75-82
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : May 19, 2012
  • Accepted : October 18, 2013
  • Published : January 28, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
HUIXUAN TAN
HANYING FENG
XINGFANG FENG
YATAO DU

Abstract
In this paper, we consider the periodic boundary value problem with sign changing nonlinearity subject to the boundary value conditions: where is a positive constant and f ( t , u ) is a continuous function. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term f may change sign. AMS Mathematics Subject Classification : 34B18.
Keywords
1. Introduction
In this paper, we are concerned with the multiplicity of positive solutions of the nonlinear three-order periodic boundary value problem
PPT Slide
Lager Image
throughout this paper, we assume that
(H 1 )
PPT Slide
Lager Image
is continuous and there exists a constant L > 0 such that
PPT Slide
Lager Image
for all ( t , u ) ∈ [0, 2π]×[0, ∞];
(H 2 )There exist continuous, non-negative and non-decreasing functions g ( x ), h ( x ) on (0, ∞) such that g ( x ) ≤ f ( t , x ) + L h ( x ).
Nonlinear periodic boundary value problem have been extensively studied by many authors. The existence of solutions is one of the most important aspects of periodic boundary value problem(see [1] - [7] , [9] - [17] and references therein). In recent years, many authors take more interested in the two-order or four-order periodic boundary value problem ( [4] , [11] , [13] - [17] ). However, for three-order periodic boundary value problems , a few of authors have studied ( [9] , [10] , [12] ). For the periodic boundary value problem (1), different methods and techniques have been employed to discuss the existence of positive solutions. We recall the following three results. In [12] , Kong and Wang, by employing Schauder fixed point theorem together with priori estimates and perturbation technique, established the existence of at least one positive solution under suitable conditions of f . In [9] , by using Krasnoselskii fixed point theorem together with non-linear alternative of Leray-Schauder, the existence of positive periodic solutions have been discussed. Recently, In [10] , Yao obtained existence results for singular and multiple positive periodic solutions by applying Guo-Lakshmikantham fixed point index theory for cones.
Inspired and motivated by the work mentioned above, in this paper, we shall apply Leggett-Williams fixed point theorem to investigate the existence of at least three positive periodic solutions to (1). The interesting point is the nonlinear term f may change sign.
2. Background and definitions
The proof of our main result is based on the Leggett-Williams fixed-point theorem, which deals with fixed points of a cone-preserving operator defined on an ordered Banach space. For the convenience of readers, we present here the necessary definitions from cone theory in Banach spaces. These definitions can be found in the recent literature.
Definition 2.1. Let E be a real Banach space over R . A nonempty closed set P E is said to be a cone provided that
  • (i)αu+βv∈Pfor allu,v∈Pand allα≥ 0,β≥ 0, and
  • (ii)u,−u∈Pimpliesu= 0.
If P E is a cone, we denote the order induced by P on E by ≤. For u , v P , we write u v if and only if v u P .
Definition 2.2. The map ψ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that ψ : P → [0, ∞) is continuous and
PPT Slide
Lager Image
for all x , y P and 0 ≤ t ≤ 1.
Definition 2.3. Let 0 < a < b be given and let ψ be a nonnegative continuous concave functional on the cone P . Define the sets
PPT Slide
Lager Image
and P ( ψ , a , b ) by
PPT Slide
Lager Image
Next we state the Leggett-Williams fixed-point theorem. The proof can be found in Deimling’s text [8] .
Theorem 2.4 (Leggett-Williams Fixed-Point Theorem). Let E = ( E , ∥·∥) be a Banach space , P E is a cone in E . Let
PPT Slide
Lager Image
be a completely continuous operator and let ψ be a nonnegative continuous concave functional on P such that ψ ( u ) ≤ ∥ u ∥,
PPT Slide
Lager Image
Suppose that there exist 0 < r < a < b < c such that
  • (S1) {u∈P(ψ,a,b) |ψ(u) >a} ≠ ∅and ψ(Tu) >a for u∈P(ψ,a,b),
  • (S2)
  • (S3)ψ(Tu) >a for u∈P(ψ,a,c)with∥Tu∥ >b.
Then T has at least three fixed points
PPT Slide
Lager Image
such that u 1 ∥ < r , a < ψ ( u 2 ), ∥ u 3 ∥ > r , ψ ( u 3 ) < a .
3. Some preliminary results
Lemma 3.1 ( [12] ). If ρ ∈ (0, +∞), then the linear problem
PPT Slide
Lager Image
has a unique positive solution
PPT Slide
Lager Image
For every function u C [0, 2π], we define the operator
PPT Slide
Lager Image
where
PPT Slide
Lager Image
By a direct calculation, we can easily obtain
PPT Slide
Lager Image
Now, we consider the problem
PPT Slide
Lager Image
If u is a positive solution of problem (3), i.e. u ( t ) > 0 for t ∈ [0, 2π], it is easy to verify that y ( t ) = ( Ju )( t ) is a positive solution of problem (1). Therefore, we will concentrate the problem (3) for which we have the following result.
Lemma 3.2 ( [9 , 12] ). Let w ( t ) be a unique solution of (2), then problem (3) is equivalent to integral equation
PPT Slide
Lager Image
where
PPT Slide
Lager Image
Lemma 3.3 ( [9 , 12] ). Let
PPT Slide
Lager Image
then we have the estimates
PPT Slide
Lager Image
From [9] , we know that
PPT Slide
Lager Image
is a positive solution of (3) when y ( t ) is the solution of
PPT Slide
Lager Image
if the nonlinear term f satisfies the condition (H 2 ). So, we can transform the problem into discussing the existence of positive solutions for (5).
4. Existence of triple positive solutions
Let E = C [0, 2π] be endowed with the maximum norm,
PPT Slide
Lager Image
Define the cone P E by
PPT Slide
Lager Image
where σ = m/M .
Finally, let the nonnegative continuous concave functional ψ : P → [0, ∞) be defined by
PPT Slide
Lager Image
We notice that, for each u P , ψ ( u ) ≤ ∥ u ∥.
Theorem 4.1. Assume that (H 1 ), (H 2 ) hold. There exist constants
PPT Slide
Lager Image
such that
  • (H3)
  • (H4)
  • (H5)
  • (H6)
Then the boundary value problem (1) has at least three positive solutions u 1 , u 2 and u 3 satisfying ui = J ( yi ρω ), ∥ y 1 ∥ < r , a < ψ ( y 2 ) and y 3 ∥ > r with ψ ( y 3 ) < a where yi ( t ) is the solution of (5)(i=1, 2, 3).
Proof. Define the operator T : P P by
PPT Slide
Lager Image
where G ( t , s ) is the Green function given by (4). The boundary value problem (5) has a solution u = u ( t ) if and only if u solves the operator equation u = Tu . Thus we set out to verify that the operator T satisfies Theorem 2.1.
Firstly, we show that
PPT Slide
Lager Image
. In fact, if u P , then ( Ju )( t ) − ω ρω/ρ ω = 0. Thus from Lemma 3.3 and (H 1 ), it follows that ( Tu )( t ) ≥ 0, 0 ≤ t ≤ 2π. Let u P , then from Lemma 3.3, we have
PPT Slide
Lager Image
On the other hand, by Lemma 3.3 and (H 3 ) , for all u P ,
PPT Slide
Lager Image
So
PPT Slide
Lager Image
thus we have TP P .
If
PPT Slide
Lager Image
then ( Ju )( t ) − ω c/ρ ω . Therefore, by (H 2 ), (H 4 ) and Lemma 3.3, for 0 ≤ t ≤ 2π,
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
is well defined. It is easy to see that T is continuous and completely continuous since f : [0, 2π] × [0, ∞] → R is a continuous function.
In the same way, we can obtain that
PPT Slide
Lager Image
by (H 6 ) . So, condition (S 2 ) of Theorem 2.1 is satisfied.
Next we prove (S 1 ) of Theorem 2.1 holds. Choose
PPT Slide
Lager Image
It is easy to see that u 0 ( t ) ∈ P ( ψ , a , b ) and
PPT Slide
Lager Image
so { u P ( ψ , a , b ) | ψ ( u ) > a } ≠ ∅.
In fact, if u P ( ψ , a , b ), then
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
As a result, it follows from (H 2 ), (H 5 ) and Lemma 3.3 that, for 0 ≤ t ≤ 2π,
PPT Slide
Lager Image
Consequently condition (S 1 ) of Theorem 2.1 is satisfied.
We finally show that (S 3 ) of Theorem 2.1 also holds.
Suppose that u P ( ψ , a , c ) with ∥ Tu ∥ > b . Then by Lemma 3.3, we have
PPT Slide
Lager Image
Thus condition (S 3 ) of Theorem 2.1 is also satisfied. Therefore an application of Theorem 2.1 leads to the conclusion that the boundary value problem (5) has at least three positive solutions y 1 , y 2 and y 3 satisfying ∥ y 1 ∥ < r , a < ψ ( y 2 ) and ∥ y 3 ∥ > r with ψ ( y 3 ) < a . Thus the boundary value problem (1) has at least three positive solutions u 1 , u 2 and u 3 satisfying ui = J ( yi ρω ).
BIO
Huixuan Tan received M.SC. from Dalian University of Thechnology. She is currently a instructor in the Department of Mathematics at Shijiazhuang Mechanical Engineering College. Her research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
e-mail: tanhuixuan 2004@163.com
Hanying Feng received M.SC. from Anhui University and Ph.D. from Beijing Institute of Thechnology.He is currently a professor in the Department of Mathematics at Shijiazhuang Mechanical Engineering College. His research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
e-mail: fhanying@yahoo.com.cn
Xingfang Feng received M.SC. from Hebei Normal University. She is currently a instructor in Department of Mathematics at Shijiazhuang Mechanical Engineering College. Her research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
e-mail: fxfg651@163.com
Yatao Du received M.SC. and Ph.D. from Hebei Normal University. She is currently a instructor in Department of Mathematics at Shijiazhuang Mechanical Engineering College. Her research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
e-mail: dyt77@sina.com
References
Nieto J.J. (2002) Differential inequalities for functional perturbations of first-order ordinary differential equation Appl. Math. Lett. 15 173 - 179    DOI : 10.1016/S0893-9659(01)00114-8
Amster P. , De Npoli P. , Mariani M.C. (2005) Periodic solutions of a resonant third-order equation Nonlinear Anal. 60 399 - 410    DOI : 10.1016/j.na.2003.03.001
Cabada A. (1994) The method of lower and upper solutions for second third, fourth and higher order boundary value problem J. Math. Anal. Appl. 185 302 - 320    DOI : 10.1006/jmaa.1994.1250
Jiang D. (1998) On the existence of positive solutions to second order periodic BVPs Acta Math. Sinica(New Ser). 18 31 - 35
Jiang D. , Chu J. , O’Reganb D. , Agarwal R.P. (2003) Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces J. Math. Anal. Appl. 286 563 - 576    DOI : 10.1016/S0022-247X(03)00493-1
Li Y.X. (2004) Positive solutions of higher-order periodic boundary value problems Comput. Math. Appl. 48 153 - 161    DOI : 10.1016/j.camwa.2003.09.027
Nieto J.J. , Rodriguez-Lopez R. (2003) Remarks on periodic boundary value problems for functional differential equations J. Comput. Appl. Math. 158 339 - 353    DOI : 10.1016/S0377-0427(03)00452-7
Deimling K. 1985 Nonlinear Functional Analysis Springer-Verlag New York
Chu J.F. , Zhou Z.H. (2006) Positive solutions for singular non-linear third-order periodic boundary problems Nonlinear Anal. 64 1528 - 1542    DOI : 10.1016/j.na.2005.07.005
Yao Q.L. (2010) Positive solutions of nonlinear three-order periodic boundary value problem Acta Math. Sci. Ser. A Chin. Ed. 30 1495 - 1502
Cao J.Y. , Wang Q.Y. (2010) Existence of positive solutions to a class of two-order differential equation with two-point boundary value problems J. Huaqiao Univ. Nat. Sci. Ed. 31 113 - 117
Kong L.B. , Wang S.T. , Wang J.Y. (2001) Positive solution of a singular nonlinear third-order periodic boundary value problem J. Comput. Appl. Math. 132 247 - 253    DOI : 10.1016/S0377-0427(00)00325-3
Rudolf B. , Kubacek Z. (1990) Nieto’s paper: Nonlinear second order periodic boundary value problem J. Math. Anal. Appl. 146 203 - 206    DOI : 10.1016/0022-247X(90)90341-C
Cabada A. , Nieto J.J. (1990) A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems J. Math. Anal. 151 181 - 189    DOI : 10.1016/0022-247X(90)90249-F
Wang M.X. , Cabada A. , Nieto J.J. (1993) Monotone methed for nonlinear second order periodic boundary value problems with Caratheodory functions Ann. Polon. Math. 58 221 - 235
Kong L.B. , Jiang D.Q. (1998) Multiple positive solutions of a nonlinear fourth order periodic boundary value problem Ann. Polon. Math. 69 265 - 270
Wang J.Y. , Jiang D.Q. (1998) A singular nonliear second-order periodic boundary value problem J. Tohoku Math. 50 203 - 210    DOI : 10.2748/tmj/1178224974