Mathematical Computation March 2014, Volume 3, Issue 1, PP.1-4
Asymptotic Behavior for Solution of Nonlinear Differential Equation Zhimin Luo Department of Education, Luoding Polytechnic, Luoding 527200, Guangdong
Abstract This paper is concerned with asymptotic behavior of solutions of a class of second order nonlinear differential equation. Banach contraction principle plays a key role in our approach.By means of example, we show the usefulness of our result. Keyword：Equation; Asymptotic Behavior; Banach Contraction Principle
1 INTRODUCTION. The second order nonlinear differential equation
u f (t , u) ,
t 1
(E1)
and the rather general form
u f (t , u, u) ,
t 1
(E2)
are often used for mathematical modeling of some physical, chemical and biological systems. Asymptotic properties of solutions of (E1) and (E2) have been widely investigated by D.S.Cohen, M.Naito, F.M.Dnanan, O.Lipovan, S.P.Rogovchenko, O.G.Mustafa and Y.V.Rogovchenko. In [1, 2, 4, 5, 8], by Gronwall inequality and Bellman-Bihari inequality, these papers were concerned with sufficient conditions which ensure that some or all solutions will approach those of
u 0 ,
(a) (b)
lim u(t ) lim t
t
u(t ) a. t
These results showed the solutions of (E1) and (E2) satisfied
u(t ) at b as t . In recent years, having used fixed-point technique, O.G.Mustafa and Y.V.Rogovchenko [4, 7], M.Naito [6], and O.Lipovan [3] have investigated asymptotic behavior at infinity. They adopted the integration operators in Banach space, proved continuity and compactity of these integration operators, and then established the existence of solutions to the (E1) and (E2) that behave asymptotically using Schauder-Tickhonov theorem. In this paper, we also note the study of linear-like solutions of (E2). In our theorem1, under the conditions of
u f (t , u, u) h(t ) p1 ( ) p2 ( u ) t where pi (i 1, 2) satisfies Lipschitz condition pi (u) pi (v) L u v
(i 1, 2)
we also established the Banach Functional Space. To be different from [3, 6, 7], we only need to prove that the continuity operators are contract rather than compactity in the space. Using Banach Contraction Principle, we obtained asymptotic solutions of (E3).
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2 MAIN RESULTS THEOREM 1： Consider (E2) with f C 1, ) R R, R satisfying
u f (t , u, u) h(t ) p1 ( ) p2 ( u ) t
(i )
(1)
(ii ) p1 ， p2 C ( R , R ) and satisfying Lipschitz condition pi (u) pi (v) L u v , (i 1, 2) ,
where L 0 .
(2)
1 sh(s)ds
(iii) h C (1, ), R ) and
(3)
Then, for every a, b R ， (E2) has a solution u (t ) defined on an interval T , (T T (a, b)) such that
u(t ) (at b) 0
as t
(4)
Proof. Take a R and denote x(t ) u(t ) at , t 1 . Then ( E1 ) becomes
x(t ) f (t, x(t ) at, x(t ) a)
(5)
We notice that if for every b R ，equation (5) has a solution x(t ) such that
x(t ) b as t . Then theorem1 is proved. From the continuity of p1 ， p2 ,we denote
m
sup
1 a b 1
p1 ( ) ， m
sup
1 a 1
p2 ( )
Consider now space
X x(t ) C1 ([T , ), R) : x(t ) and x(t ) are bounded With the norm
x sup x(t ) x(t ) Clearly, space X ,
t 1
is Banach space.
Let K X be the closed and bounded set
K x X : x(t ) b 1; x(t ) 1, t 1 and define the map F : K X ,
Fx (t ) b (s t ) f (s, x(s) as, x(s) a)ds t
Before applying Banach Contraction Principle, we will proof the two propositions. Proposition2.1 F is well defined on K and maps K into itself. Let x K be arbitrary. By relation(1)、(3)、(6) we have that
Fx (t ) b s t
f (s, x(s) as, x(s) a) ds
t
x( s ) a ) p2 ( x( s) a ) ds ( s t )h(s) p1 ( s t -2www.ivypub.org/mc
(6)
(m m) ( s t )h(s)ds . t
( Fx)(t ) f (s, x(s) as, x(s) a) ds t
x( s ) h(s) p1 ( a ) p2 ( x(s) a ) ds s t
(m m) h( s)ds t
In view of relation (3) we can choose T 1 sufficiently large such that
(s t )h(s)ds t
This fact implies
Fx (t ) b
1 and m m
1 and ( Fx)(t ) 1
1
h(s)ds m m
as t T .
t
as
t 1.
Thus F is well defined on K and maps K into itself. Proposition1.2 F is contraction. Take x(t )、y(t ) K .We proof
0,1
Fx Fy x y
In fact，by relation(1)、(2)、(3) we have the following estimates
( Fx)(t ) ( Fy)(t )
s t f (s, x(s) as, x(s) a) f (s, y (s) as, y (s) a) ds t
( s t )h(s) [ p1 ( t
x( s ) y(s) a ) p1 ( a )] [ p2 ( x(s) a ) p2 ( y (s) a )] ds s s
( s t )h( s) L x( s) y( s) L x(s) y (s) ds t
( s t )h( s) L( x( s) y( s) x( s) y ( s) ) ds t
L sup x(t ) y(t ) x(t ) y (t ) (s t )h(s)ds t 1
t
L x y
(s t )h(s)ds t
By relation (3) we also choose T 1 sufficiently large such that
1
(s t )h(s)ds 1 L ， t
and therefore ( Fx)(t ) ( Fy)(t ) x y ,
where
L ,0 1 . 1 L
Thus, F is contraction. -3www.ivypub.org/mc
From the Banach Contraction Principle we conclude that K has exactly one fixed point in space X , , i.e.
x(t ) b (t s) f (s, x(s) as, x(s ) a)ds
as t 1 .
t
Hence x(t ) is a solution of equation (5) and lim x(t ) b .The proof is completed. t
Remark: In fact, the class of equations with nonlinear function f satisfying condition (i ) is in a sense the“biggest”[4]. Furthermore, this paper drops the requirement of Lipschitz condition on nonlinear functions p1 and p2 instead of requirement of non-decreasing condition on them in the above-mentioned literature.
3 EXAMPLE Example 1：Consider the nonlinear differential equation u f (t , u, u) 0 1 t 3 (u t 6 1 t 3 3 t 2 ) 2 (3 t ) 6 2 f (t , u, u ) 7 1 2 t 2 sin(u t 3t )(3 t ) 2 0
t 3
1 3 1 t 3, u t 6 t 3 t 2 6 2 1 3 1 t 3, u t 6 t 3 t 2 6 2
satisfying the assumptions of Theorem1. Now taking a 4, b 3 ，we obtain that
3 2 1 3 t t u (t ) 2 6 4t 3
1 t 3 3t
is a solution for t 1 and lim u(t ) 4t 3 . t
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