General Periodic Boundary Value Problem for Systems

(1)

General Periodic Boundary Value Problem for Systems

Mohammed Elnagi1,2

1_{Department of Mathematics, Faulty of Educations, Khartoum University, Omdurman, Sudan} 2_{Department of Mathematics, Northwest Normal University, Lanzhou, China}

Email: [email protected]

Received May 20,2012; revised July 2, 2012; accepted July 9, 2012

ABSTRACT

The paper deals with the existence of nonzero periodic solution of systems

 





 

2 _, _, _0, _, _0,

0 , 0

u k u f t u t T T

u u T u u T

 

,

  



 _ _

 





where k 0,π

T

 

 

 ,  , are n n real nonsingular matrices, u



u1, , un



,

 

,



1

 

, , ,

 

,





 

0, ,

n n

f t u  f t u  f t u C T  

 



  is periodic of period in the t variable are continuous and nonnegative functions. We determine the Green’s function and prove that the existence of nonzero periodic positive

solutions if one of

T

1 0

, lim i u

f t u

u 

 

  . In addition, if all

 

1 0

, lim i u

f t u

u 



  , i



1,,n



where 1 is the principle

eigenvalues of the corresponding linear systems. The proof based on the fixed point index theorem in cones. Applica-tion of our result is given to such systems with specific nonlinearities.

Keywords: Systems; Principle Eigenvalues; Positive Solutions; Green’s Function; Fixed Point Index Theorem in Cones

1. Introduction

In this paper, we study the existence of nonzero positive periodic solution of systems

 





 

2 _, _, _0, _, _0,

0 , 0

u k u f t u t T T

u u T u u

     

 

 

 

 T , (1.1)

where k 0,π

T

 

_ _ ,  ,



un



are real nonsingular

matrices, , and n n

1, ,

u u 

 

,



1

 

, , ,

 

,





 

0, ,

n n

f t u  f t u  f t u C T  _ _



are periodic of period in the t, are continuous and nonnegative functions and

T



0,



 

 .

Beginning with the paper of Erbe and Palamides [1], obtained the sufficient conditions for existence solution of the systems of nonlinear boundary value problem





 

0

 

1

 

, , , 0,1 ,

0 1 , 0

u f t u u t

u Q u u Q u

  

 

1 ,

 

 

 (1.2)

where _f _{: 0,1}

 

_E2n_En

0

Q

is continuous ( is n-di- mensional real Euclidean space) and 0 1 are

non-singular matrices, with orthogonal matrix, Erbe and

Palamides generalize earlier conditions of Bebernes and Schmitt [2] for periodic case. Erbe and Schmitt [3] ex-tend the results in [1] established the sufficient condi-tions for existence solution of the systems (1.2). The re-sults in [1,3] were obtained via a modifications of a de-gree-theoretic approach and Leray-Schuader degree and eliminates the modified function approach respectively. None of these earlier results use Green’s function and the first eigenvalues of the corresponding to the linear sys-tems of (1.1).

n E Q

,

Q

There has been progress in the study of the existence of positive solutions of system problem. If    I identity, then (1,1) reduces to the usual periodic bound-ary value problem for which the literature in both the scalar and systems versions is very extensive (We refer to [4-22] and references therein). For instance a recent paper, Wang [4] obtained the existence of periodic solu-tion of a class of non-autonomous second-order systems

 

, 0,



0,



,

uu f t u  t T T 0 where



1, , n



u u u ,

 

,



1

 

, , ,

 

,





 

0, ,



n n

n

f t u  f t u  f t u C T  

(2)

if

0

( , ) lim i 0

u

f t u u

  , i



1, , n



and f t u

 

, , is bounded

below or above for appropriate ranges of , via fixed point theorem in cones. Franco and Webb [5] established the existence of -periodic solutions for systems of (1.1) with

T I

  

k

identity, in the boundary conditions, where and f t u

 

,

t

is a continuous vector val-ued function, periodic in with period , and T f is allowed to have a singularity when . Non-singular systems, which are included in the same framework that we study here (i.e., they can be reduced to a Hammer-stein integral system with positive kernel), have been considered using some other approaches based on fixed point theorems in conical shells, but previously it has always been assumed that the nonlinearity

0

u

 

,

f t u has a constant sign behaviour: f t u_i

 

, _i 0 where

i for (see [6]) with



 

1, 1

 

 0 i n    I

identity, in the boundary conditions. For systems problem see also [7-12] and references therein.

Even in the scalar case the existence of periodic solu-tions for problems with nonsingular and singular case has commanded much attention in recent years (see [13-22] and references therein. In particular, in [13-15] fixed point theorems in conical shells are used to obtain exis-tence and multiplicity results, some of these are im-proved in this paper. In this notes, we prove result in the case where f has no singularity. In scalar case problem see [16-22] and references therein.

Motivated by these problems mentioned above, we study the existence of nonzero positive solution of (1.1) while we assume that if one components satisfy

 

0

1 0

, lim i u

f t u

u 

  , , and all components of

nonlinearity are



0 1, ,

i   n



 

1 0

, lim i u

f t u

u 

  , i



1, , n



, where

1

 , is the largest characteristic value of the linear system corresponding to (1.1). The approach is to use the theory of fixed point index for compact maps defined on cones [23]. To apply this theory one needs to find the Green’s function. Our purpose here is prove that (1.1) has non-trivial nonnegative solution, assuming the following conditions:

 

H1 0 k π T

  ,

 

H2  , 0, 0,  0,  0 and









2 1k    coskT 0,

   

3 ,



1

 

, , ,

 

,





 

0, ,



n n

H f t u  f t u  f t u C T  

are continuous and periodic of period in the vari-able and

T t

 

, 0

i

f t u  , i



1, , n



on any subinterval of

 

0,T .

 

H4 There exits i₀



1, , n



such that

 

0

1 0

, lim i u

f t u

u 

  ,

where



₁, ,



n n

u u  u _ and

1

sup

n

i i

u u





, and

1

 is the largest characteristic value of the linear system corresponding to (1.1),

 

H5 For all i



1, , n



,

 

1 0

, lim i u

f t u

u 

  ,

where



1, ,



n n

u u  u _ and

1

sup

n

i i

u





u , and

1

 is the largest characteristic value of the linear system corresponding to (1.1).

Remark 1.1. The assumptions

 

H4 and

 

H5 appeared in Lan [24].

Remark 1.2. The nonzero positive solution has been studied by Lan [24] and Hai and Wang [25].

Throughout this paper, we will use the notation



0,



  

 ,

1

n n

i

 





  , and denote by

1

n i i

u u





the usual norm of n



 for



1, ,



n n

u u u _ ,

 

,



1

 

, , , n

 

,



f t u  f t u  f t u and i



1, , n



.

2. Preliminaries

In this section, we shall introduce some basic lemmas which are used throughout this paper.

Lemma 2.1. Let

 

H1 and

 

H2 holds. Let









2 1k   osk

     c T then for viC

 

0,T ,

the periodic boundary value problems problem





 

2 _, _0, _,

0 , 0

i i i

i i i i

u k u v t T T

u u T u u

    



 _ _

 



0, ,

T

s

(2.1)

has a unique solution

 

0

, d

T

i i i

u 



G t s v ,

where

 

, 2 sin



_

 

_{ }

_



sin

_



 

_{ }

_



sin



_



_

,0 ,

2 sin sin sin , 0 .

k t s k T s t k T s t s t T

G t S

k s t k T s t k T t s t s T

    

           

  

           

 (2.1

*₎

(3)

where i



1, , n



.

Proof. Consider the scalar periodic boundary value problems of (2.1) and let u t₁

 

and be linearly distinct solutions of the scalar equation of (2.1) and con-sider the function

 

2

u t

 

   





1 2

1 2 1 2

,

2

F t s Au t Bu t

u t u s u s u t

u s u s u s u s

 

 _ 

_ 

  

 

where the positive sign is taken when , and the negative sign when

0 t s s t T



we can obtain this result by routine substitutions of scalar boundary conditions, we do not state it here.

Lemma 2.2. Let conditions ( 1)H hold, then G t s_i

 

, , , is continuous and positive on



1, ,

i  n



   

0,T  0,T ,

and we can find it’s positive minimum value Ai and

maximum value Bi of G ti

 

,s , i



1,,n



by

 

, [0, ] 0 i min i

t s T ,

A G t s



  ,

 

,

, [0, ]max ,

i i

t s T

B G



 t s

and

0 i

i i

A B



  , i



1, , n



.

Proof. It is easy to check that , is continuous and positive on



,

i G t s



   

0,T  0,T , i



1, , n



. 

It is clear that the problem (1.1) has a solution if and only if solves the operator equation

u u

 



 



0

: , ,

T

i i d

Au t 



G t s f s u s s (I)

It is easy to verify that the operator A is completely continuous.

We define :



 

0, ; n





 

0, ; n



L C T  C T  corre-sponding to linear equation of (1.1) by

 

   

 

1 1

0 0

, d , , , d

0,

T T

n n

Lu t G t s u s s G t s u s s

t T











,

(2.2)

where

 



₁

 

, ,

 



n and

n

u t  u t  u t  G t si

 

, is

Green’s function define in (2.1*_{), and define}

  

1 1

   

1

0

, d

T

u t G t s u s s

 

_

and



 

   

0

, d

T

n n n

u t G t s u s s

 

_

,

where :C



 

0,T



C



 

0,T



is completely con-tinuous.

Remark 2.1. Equations (2.2) appeared in [26].

It is known that :C



 

0,T



C



 

0,T



, is a

bounded and surjective linear operator and has a unique extension, denoted by , to C



 

0,T



. We write

 

C



 

0,T



0 0 ,

e v v t, 1v . (2.3)

It is known that is an interior point of the positive cone in

e

1

P



 



 

0,

C T , where





 

 





1 0, .

P uC T :u t 0,t 0,T (2.4)

Lemma 2.3. [24] :C



 

0,T



C



 

0,T



is a compact linear operator such that 

 

P1 P1 and for each

 

1\ 0

vP there exists  _v 0 such that  v _ve. By Lemma 2.3 and the well-known Krein-Rutman theorem (see [23, Theorem 3.1] or [27], it is easy to see that 1



0,



and there exists 1P1\ 0

 

1

such that

1 1

    (2.5) where ₁1r

 

 and r

 

 is the spectral radius of

.

We use the following maximum norm in _n:







i 1, ,



max i :

u  u n (2.6)

where u t

 





u t1

 

, , un

 

t



 

. We denote by







; n



C 0,T  the Banach space of continuous functions from

 

0,T into _n with norm







,



max _i :i 1,

u  u  n ,

where u t

 





u t₁

 

,...,u_n

 

t



for t



0,T



. We use the standard positive cone in

defined by



 



0, ; n

C T 

 





; n

T  0,

PC (2.7)

We can write :



 

0 n





 

0, ; n



L C ,T ; C T  de-fined in (2.2) as operator equations

    

Lu t : 



u1 t



,,



un

 

t



(2.8)

where

  

u1 t and



un



 

t are define above and

define a Nemytskii operator

  

Fu t :



f t u t1



,

 



, , fn



t,u t

 



. (2.9)

It is easy to verify that (1.1) is equivalent to the fol-lowing fixed point equation:

    

: Au t

 

,t

 

0,

u t  LF t   T (2.10) Note that (2.10) same as

 

I .

Recall that a solution of (1.1) is said to be a nonzero positive solution if



 



0, ; u C T n



 

\ 0

uP ;that is,



 

0,



; n



u C T  and u t

 





u t1

 

, , un

 

t



satisfies u ti

 

0 for t

 

0,T and i



1, , n



and there exists such that k



1,,n



uk 0 on

 

0,T .

(4)



:



P_  uP u  , P_ 



uP u: 



and



:



P_ u P u 

    .

We need some results from the theory of the fixed point index for compact maps defined on cones in a Ba-nach space X (see [23]).

Lemma 2.4. Assume that A P: _P is a compact map. Then the following results hold:

1) If there exists x0P\ 0

 

such that uAuvx0

for uP_ and v0, then . Au



,



0 i_ A P_ 

2) If u for uP_ and 



0,1



, then



i_ A P, _



1.



, i A P

3) If _ _



1 and i_



A P, _₀



0 for some





0 0,

   , then A has a fixed point in

0

\

P_ P_ . Now, we are in a position to give our main result and proof analogous results were established in [24].

Theorem 2.1. Assume that

 

H1 -



H5



holds. be the same as in (2.5). Assume that the fol-lowing conditions hold: 1



0,

  



 

0

fi

 . There exist i0



1, , n



,  0 and

0 0

  such that fi₀

  

t u,   1



ui0 for t

 

0,T

and all n with



u u



0,₀



.

 

fi _₁

 _{. There exist}_{ }₀_and 1 0

  such that

for i



1,,n



, f t ui

  

,   1



u for t

 

0,T

and all n with





u u 1. Then (1.1) has a nonzero

positive solution in P.

Proof. By Lemma 2.1, Lemma 2.2 and Lemma 2.3,

 







 

: 0, ; n 0, ; L C T  C T

 

L P P



n

 is compact and satis-fies .

This, together with the continuity of fi in ( 3)H ,

implies that A P: P is compact. Without loss of generalization, we assume that uAu for uP_₀ . Let 



1, ,1



, where 1 is the same as in (2.5).

We prove that

0

,

uAuv  u P v_ 0. (2.11) In fact, if not, there exist

0

uP and such

that v 0



uAuv. Then

 



 





0 0 , 1,

i i

u t   f t u t v  t 0,T



(2.12) It follows that ui₀

 

t v1 for  t

 

0,T . Let

 

 



0 0 0 1



sup 0 :ui t , t 0,T

        .

Then 0   v  and ui₀

 

t 1 ,  t

 

0,T .

This, together with (2.12),

 

₀

0

fi

 and (2.5), implies

that for all t

 

0,T

 



  











 







0 1 0 1

1 1 1 .

i i

u t u t t

t

     

    

     

 

1

Hence, we have 



   1



1



, a contradiction. It

follows from (2.11) and Lemma 2.4 (1) i_



A P, _



0.

For each i



1, , n



, by the continuity of fi, there

exists bi0 such that f t ui

 

, bi for t

 

0,T , n

u_ withu ₁.

This, together with

 

1

i

f 

 _{implies that, for each}



1, ,



i  n fi

 

t u,  bi



 ₁



u for t

 

0,T and

all

n

u (2.13)

Since







r  1  





 1

  

r  1,







1

I   













1

exists and is bounded and satisfies

1 1

I     P P₁. Let bib_i for t

 

0,T ,









1 max bi :i 1, ,n

    

and









1

1 ˆ1

I

_ _  _{ }   _,

where ˆ1

 

t 1 for t

 

0,T . Let   . We prove





, , 0,

uAu uP_  1 .

Indeed, if not, there exist uP_ and  



0,1



such that uAu . By (2.13), we have for each



1, ,



i  n ,

 

1



1



 

i

u t     u t for t

 

0,T ,

where u t

 

max



u ti

 

:i



1, , n





. Taking the

maximum in the above inequality implies that

 

1



1



 

u t     u t

for t

 

0,T , and







I  1 



u t

 

1

for t

 

0,T . Since









1

1 1 1

I     PP,

 





1

 

1 ˆ1

u t  I      t for t

 

0,T .

Hence, we have

 

 





max : 0,

u u t t T

_ _ _ ___.

(5)



,



i_ A P_ 

0

\

P P

1. By Lemma 2.4 (3), (1.1) has a solution in

  .

3. Application

Let the systems







 







 

2 1/3 1/3 _ˆ _, _0, _,

0 , 0

i i i i i i

u k u a u b u h u t t T T

u u T u u

      

 

 

 

0, ,

T



 (3.1)

where k 0,π

T

 

 

 , uˆi



u1, , ui1,ui1, , un



and



n

0



 



n



1, ,

i 

1

: n i

h _ 

. Assume that the following conditions hold: 1) For each , and

is continuous and let





1, ,

i  n

n





,

i i

a b 

 





 







_ˆ _ˆ 1

: sup : n , 1, ,

i i i

h u t u t i n

 



    .

2) There exists i0



1, , such that ai0bi0 0



,n 0

 

and 





 

t



:u tˆi

 





.

1 ˆ

: min n , 1,

i i

h u i

 











n

Then equation (3.1) have a nonzero positive solution in P.

Proof. For each i



1, , , we define a function

 

, : 0,

 

n i

f t u T __

1/3 1/

by

 

_,



3





_ˆ

 



i i i i i i i .

f t u  a u b u h u t

Let 0 and





3\2

0 0 0

1 0  min 1, ai bi 

 

   



    _  

 

 

 

_.

Then for t

 

0,T and un_ with u



0,₀



and ui0,

 





0 0 0 0

0 0

0 0 0

3\2 3\2 3\2 3\2

0 0

1

,

.

i i i i

i i

i

a b a b

0

i

f t u u u

u u

u

 

 

 

   

_  _   

 

 

Hence,



 

₀



0

i f

 holds. Let 10,





3\2 0 0 1

( )

max ai bi , 1, ,

i n

 



_ _ _ 

 

    

 

 

 

 .

Then for t

 

0,T and un with u



1,



,

 

_, 1/3 1/3 1/3 1/3

i i i i i i i i i i

f t u _a u b u _u a u b u 

for i



1, , n



it follows that

 

1

i

f

 



holds. The re-sult follows from Theorem 2.1.

REFERENCES

[1] L. H. Erbe and P. K. Palamides, “Boundary Value Prob-lems for Second-Order Differential Systems,” Journal of Mathematical Analysis and Applications, Vol. 127, No. 1,

1987, pp. 80-92.

doi:10.1016/0022-247X(87)90141-7

[2] J. W. Bebernes and K. Schmitt, “Periodic Boundary Value Problems for Systems of Second Order Differential Equa-tions,” Journal of Differential Equations, Vol. 13, No. 1,

1973, pp. 32-47. doi:10.1016/0022-0396(73)90030-2

[3] L. H. Erbe and K. Schmitt, “Boundary Value Problems for Second-Order Differential Systems,” In: V. Lakshmi-kantham, Ed., Nonlinear Analysis and Applications, New

York and Basel, 1987, pp. 179-183.

[4] H. Wang, “Periodic Solutions to Non-Autonomous Sec-ond-Order Systems,” Nonlinear Analysis, Vol. 71, No. 3-4, 2009, pp. 1271-1275. doi:10.1016/j.na.2008.11.079

[5] D. Franco and J. R. L. Webb, “Collisionless Orbits of Singular and Nonsingular Dynamical Systems,” Discrete and Continuous Dynamical Systems, Vol. 15, No. 3, 2006, pp. 747-757. doi:10.3934/dcds.2006.15.747

[6] R. P. Agarwal, D. O’Regan and P. J. Y. Wong, “Con-stant-Sign Solutions of a System of Fredholm Integral Equations,” Acta Applicandae Mathematicae, Vol. 80, No. 1, 2004, pp. 57-94.

doi:10.1023/B:ACAP.0000013257.42126.ca

[7] D. ORegan and H. Wang, “Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equa-tions,” Positivity, Vol. 10, No. 2, 2006, pp. 285-298. doi:10.1007/s11117-005-0021-2

[8] X. Lin, D. Jiang, D. ORegan and R. Agarwal, “Twin Positive Periodic Solutions of Second Order Singular Differential Systems,” Topological Methods in Nonlinear Analysis, Vol. 25, 2005, pp. 263-273.

[9] L. H. Erbe and K. Schmitt, “On Solvability of Boundary Value Problems for Systems of Differential Equations,”

Journal of Applied Mathematics and Physics, Vol. 38,

1987.

[10] H. Wang, “Positive Periodic Solutions of Singular Sys-tems with a Parameter,” Journal of Differential Equations,

Vol. 249, No. 12, 2010, pp. 2986-3002.

doi:10.1016/j.jde.2010.08.027

[11] X. Li and Z. Zhang, “On the Existence of Positive Peri-odic Solutions of Systems of Second Order Differential Equations,” Mathematische Nachrichten, Vol. 284, No.

11-12, 2011, pp. 1472-1482.

doi:10.1002/mana.200710145

[12] Z. Cao and D. Jiang, “Periodic Solutions of Second Order Singular Coupled Systems,” Nonlinear Analysis, Vol. 71,

No. 9, 2009, pp. 3661-3667.

doi:10.1016/j.na.2009.02.053

[13] J. R. Graef, L. Kong and H. Wang, “A Periodic Boundary Value Problem with Vanishing Greens Function,” Ap-plied Mathematics Letters, Vol. 21, No. 2, 2008, pp. 176-

180. doi:10.1016/j.aml.2007.02.019

[14] I. Rachunkova, M. Tvrdy and I. Vrkoc, “Existence of Nonnegative and Nonpositive Solutions for Second Order Periodic Boundary Value Problems,” Journal of Differen-tial Equations, Vol. 176, No. 2, 2001, pp. 445-469. doi:10.1006/jdeq.2000.3995

(6)

Equations, Vol. 190, No. 2, 2003, pp. 643-662. doi:10.1016/S0022-0396(02)00152-3

[16] F. Li and Z. Liang, “Existence of Positive Periodic Solu-tions to Nonlinear Second Order Differential EquaSolu-tions,”

Applied Mathematics Letters, Vol. 18, No. 11, 2005, pp.

1256-1264. doi:10.1016/j.aml.2005.02.014

[17] D. Jiang, J. Chu, D. ORegan, R. Agarwal, “Multiple Posi-tive Solutions to Superlinear Periodic Boundary Value Problem, with Repulsive Singular Forces,” Journal of Mathematical Analysis and Applications, Vol. 286, No. 2,

2003, pp. 563-576. doi:10.1016/S0022-247X(03)00493-1

[18] X. Li and Z. Zhang, “Periodic Solutions for Second-Order Differential Equations with a Singular Nonlinearity,”

Nonlinear Analysis, Vol. 69, No. 11, 2008, pp. 3866-3876. doi:10.1016/j.na.2007.10.023

[19] D. Jiang, J. Chu and M. Zhang, “Multiplicity of Positive Periodic Solutions to Superlinear Repulsive Singular Equations,” Journal of Differential Equations, Vol. 211,

No. 2, 2005, pp. 282-302. doi:10.1016/j.jde.2004.10.031

[20] P. J. Torres and M. Zhang, “A Monotone Iterative Sch- eme for a Nonlinear Second Order Equation Based on a Geeralized Anti-Maximum Principle,” Mathematische Nachrichten, Vol. 251, No. 1, 2003, pp. 101-107. doi:10.1002/mana.200310033

[21] R. Ma, “Nonlinear Periodic Boundary Value Problems with Sign-Changing Green’s Function,” Nonlinear Analy-sis, Vol. 74, No. 5, 2011, pp. 1714-1720.

doi:10.1016/j.na.2010.10.043

[22] B. Liu, L. Liu and Y. Wu, “Existence of Nontrivial Peri-odic Solutions for a Nonlinear Second Order PeriPeri-odic Boundary Value Problem,” Nonlinear Analysis, Vol. 72,

No. 7-8, 2010, pp. 3337-3345.

doi:10.1016/j.na.2009.12.014

[23] H. Amann, “Fixed Point Equations and Nonlinear Eigen-value Problems in Ordered Banach Spaces,” SIAM Re-view, Vol. 18, No. 4, 1976, pp. 620-709.

[24] K. Q. Lan, “Nonzero Positive Solutions of Systems of Elliptic Boundary Value Problems,” AMS, Vol. 139, No.

12, 2011, pp. 4343-4349.

doi:10.1090/S0002-9939-2011-10840-2

[25] D. D. Hai and H. Wang, “Nontrivial Solutions for p-Lap- lacian Systems,” Journal of Mathematical Analysis and Applications, Vol. 330, No. 1, 2007, pp. 186-194. doi:10.1016/j.jmaa.2006.07.072

[26] K. Q. Lan and W. Lin, “Multiple Positive Solutions of Systems of Hammerstein Integral Equations with Appli-cations to Fractional Differential Equations,” Journal London Mathematical Society, Vol. 83 No. 2, 2011, pp.

449-469. doi:10.1112/jlms/jdq090

[27] R. D. Nussbaum, “Eigenvectors of Nonlinear Positive Operators and the Linear Krein-Rutman Theorem, in Fixed Point Theory,” In: E. Fadell and G. Fournier, Eds.,