boundary value problem (BVP) for second order mixed-type functional differential equation ~(t). = f(t,x' ,Jr'), 0 ~ t ~ T. Existence principle and theorem for ...
Appl.
Math.
- JCU
12B(1997), 155-164
BOUNDARY VALUE PROBLEMS FOR SECOND ORDER MIXED-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS"
WENG PEIXUAN Abstract. We use the topological degree method to deal with the generalized Sturm-Liouville
boundary value problem (BVP) for second order mixed-type functional differential equation ~(t) = f ( t , x ' ,Jr'), 0 ~ t ~ T. Existence principle and theorem for solutions of the BVP are obtained.
1. Introduction
In recent years, boundary value problems associated with second order functional differential equations has attracted more attentions in the literature ( see [ 1 - - 6 ] ,
and [ 9 - - 1 0 ] ) . ~
Methods used are usually "fixed point methods" [ 1 , 4 , 9 ] , topological transversality method of Granas [ 2 , 3 , 5 , 6 ] and topological degree method [ 8 , 1 0 ] . In this paper, we apply the topological degree method to the existence of solutions of the boundary value problems for second order functional differential equation of the form 5?(t) = f ( t , S , x
ax(t) -- ilk(t) = l~(t),
f r
E
[[0,T3
•
--r~t~O;
(2)
Y x ( t ) ~- 3 x ( t )
where r , h , T , a , f l , Y , 3
(1)
e) , 0 ~ t ~ T ;
= v(t),
T~t~Tq-h;
are given real nonnegative constants, a z + flz :/: O, yz q_ 32 :/: 0 , f ( t , q g , C •
C,R],C
"-- C [ [ - -
r,h],R]
with the norm defined by I[~oll =
SUP_r~O~ h I~(O) [ and x t = x ( t q- O),Jc' = :k(t Jr- 0 ) , - - r
O
where U M = { x E C a [ - - r , T - f - h ] ;
is
a constant. P r o o f . Let ~l E [ 0 , 1 1 be fixed. For a n y x l , x 2 E UM C C1~ - r , T + h I we have IlF,x, - Fax~[I 0
2
2
($).~$($ -+- r2($)) -[- e ( $ ) x ( $ )
- ~ / ( ~ : ) x 4 ( ~ t) +
d(~)x2($)Jc~(~ ~- rz(~)) -j- e ( ~ ) x ( ~ )
Appl. Math.-JCU
164
as Ix(~)l >,~
[e(t)
.~/max~
min0g,gvl(t)
I
= M o, i . e .
Vol. 12, Ser.
( H e) is satisfied. O n the o t h e r h a n d , let
r / ( t ) - - max{ Ic(t) l , d ( t ) M o , a ( t ) M ~ q- b ( t ) ( M ~ -k M z ) 3 q- l e ( t ) 1}, 2
~(u)
= 1 q- u q- uX, M 1, M z defined as in ( 1 6 ) .
T h e n we have If(t,x',x')
f~
Since 0 r](t)dt < - t - oo and
f§
1
~-~du
[ ~ r / ( t ) ( 1 q-I1~'11 +
9t
2
IIx I1~) 9
= + co, (H3) is satisfied too and t h e n B V P ( 3 0 ) has
at least one solution.
References [-1-] Erbe, L . H . and Kong Qingkai, Boundary value problems for singular second order functional equations, J. Comp. Appl. Math. 53(1994),377-388. [-2]
Erbe, L.H. i Wang Zhicheng and Li Longtu, Boundary value problems for second order mixed type functional differential equations, Boundary Value Problems for Functional Equations, World Scientific, 143-151.
[3-1 Granas, A. , Guenther R.B. and Lee, J.W. , On a Theorem of S. Bernstein, Pacific J. Math. , 94 (1978) ,67-82. [41
Lalli, B.S. and Zhang,B. G. , Boundary value problems for second order functional differential equations, Ann. o f D i f f . Eqs. , 8(1992),261-268.
[-5-] Lee, J.W. and O ' Regan, D. , Existence results for differential delay equations-I, J. Differential E-
quations, 102(1993) ,342-359. [-6]
Lee, J.W. and O' Regan D. , Existence results for differential delay equations -If, Nonlinear Analysis, 17(1991),683-702.
[-71 Li Shenling and Wen Lizhi, Functional Differential Equations, Science and Technique Press of Hunan, Changsha, 1986. [-81
Li Zhengyuan and Ye Qixiao, Theory of Reaction-Diffusion Equations, Publishing House of Science, Beijing, 1990.
I-9-1 Ntouyas, S.K. , Sficas Y.G. and Tsamatos, P. Ch. , An existence principle for boundary value problems for second order functional differential equations, Nonlinear Analysis, 20(1993),215-222. [103
Qian Min and Li Zhengyuan, Rotating Vector-Field and Degree Theory, Publishing House of Beijing University, 1982.
[-11-1 Qian, X.Z. , Wang Z.C. and Li,L. T. , Periodic Boundary Value Problems for Second Order Functional Differential Equations, Lecture Notes in Pure and Applied Mathematics, Vol. 176, Dekker, New York, 231-236. Department of Mathematics, South China Normal University, Guangzhou 510631.
