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Applied Mathematics Letters 17 (2004) 1391-1395
Applied
Mathematics
Letters
,
www.elsevier.com/locate/aml
Uniqueness Implies Existence for
Three-Point Boundary Value Problems
for Dynamic Equations
J. HENDERSON
D e p a r t m e n t of M a t h e m a t i c s
Baylor University
Waco, T X 76798-7328, U.S.A.
Johnny_Henderson©baylor. edu
C. C. TISDELL
School of Mathematics
The University of New South Wales
UNSW
Sydney 2052, Australia
cct©maths, unsw. edu. au
W. K. C. YIN
D e p a r t m e n t of M a t h e m a t i c s
L a G r a n g e College
LaGrange, G A 30240, U.S.A.
wyin©lagrange, edu
(Received and accepted August 2003)
Abstract--Shooting
methods are used to obtain solutions of the three-point boundary value
problem for the second-order dynamic equation, yAA = f(x, y, y~), y(xl) = Yl, y(x3) -- y(x2) = Y2,
where f : (a,b)v x ~2 ~ R is continuous, xl < x2 < xa in (a,b)v, Yl,Y2 E ~, and T is a time scale.
It is assumed such solutions are unique when they exist. (~) 2004 Elsevier Ltd. All rights reserved.
K e y w o r d s - - T i m e scale, Boundary value problem, Dynamic equation, Shooting method.
1. I N T R O D U C T I O N
This paper is devoted to boundary value problems for dynamic equations on time scales. It is
assumed that, by this time in the development of the theory, the reader is familiar with time
scale calculus and notation for delta differentiation, as well as concepts for dynamic equations on
time scales. Otherwise, the reader is referred to the introductory book on time scales by Bohner
and Peterson [1].
Let T be a nonempty closed subset of ]~ (i.e., ~F is a time scale), with endpoints a < b. For
notation, we use the convention that, for each subset S of ]~,
ST:,_gN'~.
0893-9659/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved.
doi: 10.1016/j .aml.2003.08.015
Typeset by AAz~-9-TEX
J. HENDERSONe$ al.
1392
/
In this paper, we address the question of the uniqueness of solutions implying the existence
solutions for multipoint boundary problems for the second-order dynamic equation,
yZX~ = f (x, y, y ~ ) ,
x E (a, b)v,
(1)
where cr denotes the forward jump operator. In particular, given Yl,Y2 E R and points xl,x2,
x3 E (a, b)v, with either Card (a, Xl)V _> 1 or Card (x3, b)v _> 1, and such that,
a < xl <
(xl) < x2 <
(x2) < x3 < b,
we consider uniqueness implies existence results for solutions of (1) satisfying the three-point
boundary conditions,
=
y ( x 3 ) - y ( x 2 ) = y2.
(2)
Two conditions that we assume throughout are as follows.
(A) f : (a, b)T × IR2 --+ R is continuous.
(B) Solutions of initial value problems for (1) are unique and exist on all of (a, b)v.
Multipoint boundary value problems for second-order differential equations have received considerable attention for several years with most results obtained for positive solutions. A sampling
of such work can found in the papers by Bai and Fang [2], Gupta and Trofimchuk [3], Ma [4],
and Yang [5]. In addition, Ma and Raffoul [6] used a Krasnosel'skii fixed-point theorem to
establish positive solutions of three-point boundary value problems for second-order difference
equations. And a couple of recent papers by Anderson [7,8] were devoted to m-point boundary
value problems for second-order delta-nabla dynamic equations on time scales.
The history of uniqueness implies existence results for boundary value problems also enjoys a
good history, first for differential equations, then finite difference equations, and recently, unifying results for dynamic equations. This spectrum of works is covered chronologically in the
order we list papers here by Lasota and Luezynski [9], Jackson and Schrader [10], Hartman [11],
Klaasen [12], Peterson [13], and Henderson [14] for ordinary differential equations, followed by
the list by Henderson [15,16] and Davis and Henderson [17] for finite difference equations, and
finally the results by Bohner and Peterson [1], Chyan [18], and Henderson and Yin [19,20] for
dynamic equations.
Our uniqueness conditions on solutions of (1),(2) will be stated in terms of generalized zeros
of functions.
DEFINITION 1.1. For y : T --* R, c E • is a generalized zero (GZ) of y if either
(i) y(c) = O, or
(ii) y(p(c))y(c) < 0, where p(.) denotes the backward j u m p operator.
In view of this terminology, our uniqueness assumption on solutions of (1),(2) takes the following
form.
(C) Given points r l , r 2 , r 3 E (a,b)~- with a < rl < or(r1) < r2 < ~(r2) < r3 < b, if y(x)
and z(x) are solutions of (1), such that y(x) - z(x) has a GZ at rl, and [y(r3) - z(r3)][y(x) - z(x)] has a GZ at r2, then y(x) = z(x), for all x E (a, b)v.
REMARK. We notice, if (C) is satisfied, then for any points r~,r2,ra E (a,b)v with a < rl <
or(r1) < r2 <_ or(r2) < r3 < b, if y(x) and z(x) are solutions of (1), such that y(x) - z(x) has
overdetermined GZs at rl, r2, and r3, then y(x) -- z(x), for all x E (a, b)v.
Our final assumption involves a precompactness .condition on uniformly bounded sequences of
solutions of (1).
(D) If {yk(x)} is a sequence of solutions of (1) for which there exists an interval [c,d]v, with
Card(c,d)v > 2, and there exists an M > 0, such that lyk(x)l <_ M , for all x E [c,d]v
and for all k E N, then there exists a subsequence {Ykj (x)}, such that {Yk~
zx~(x)} converges
uniformly on [c, d]v, i = 0, 1.
1393
Uniqueness Implies Existence
/
In Section 2, we prove that solutions of (1),(2) depend continuously on boundary conditions.
We state a similar resuk for solutions of (1) satisfying overdetermined three-point boundary
conditions due to their uniqueness as well. In Section 3, we invoke our assumptions, along with
the continuous dependence results, to obtain the existence of solutions of (1),(2).
2. C O N T I N U O U S
DEPENDENCE
In this section, we show that solutions of (1),(2) depend continuously on boundary conditions.
THEOREM 2.1. Assume that with respect to (1), Conditions ( A ) - ( C ) are satisfied. Given a
solution y(x) of (1) on (a, b)v, an interval [c, d]~ c (a, b)v, points xl < ~(xl) < x2 <_ or(x2) < x3
belonging to (c,d)~, and an e > O, there exists a 5(e,[c,d]v) > O, such that if ]x~ -t~] < 5,
i = 1,2,3, and tl <<_(7(tl) < t2 < ~(t2) < t3 belong to (c,d)v, and if ly(xl) - zl[ < 5 and
[y(x3)--y(x2)--Z21 < 5, then there exists a solution Z(X) o f ( l ) s a t i s ~ i n g z ( t l ) = Zl, z(t3)--z(t2) =
z2, and lyeS(x) - zA~(x)[ < e on [c,d]v, i = 0,1.
PROOF. Fix a point Po E (a, b)v. Next, define the set
G = {(xl, x2, x3, cl, c2) ] xl < a(xl) < x2 <_ a(x2) < x3 belong to (a, b)v, and cl, c2 E ]~}.
G is an open subset of N 5. Next, define a mapping ¢ : G --~ ]~5 by
¢(Xl,X2, X3, Cl,C2) = (Xl,X2,X3,U(Xl),U(X3)--U(X2)),
where u(x) is the solution of (1) satisfying the initial conditions u(po) = el, uA(po) = e2.
Condition (B) implies the continuity of solutions of initial value problems for (1) with respect to
initial conditions, from which follows the continuity of ¢. Moreover, Condition (C) implies that ¢
is one-one. By the Brouwer theorem on invariance of domain [21], it follows that ¢(G) is an open
subset of ]~5 that ¢ is a homeomorphism from G to ¢(G), and that ¢ - 1 is continuous on ¢(G).
The theorem then follows as a direct consequence of the continuity of ¢-1.
|
In conjunction with the remark after the statement of Hypothesis (C) in the Introduction, an
ana][ogous argument could be used for the continuous dependence of solutions of (1) satisfying
the overdetermined boundary conditions,
y(
l)=yl,
y(x2)=y ,
(3)
where Xl _< cr(xl) < x2 < ~(x2) < x3 belong to (a, b)T, and Yl, Y2, Y3 E N.
T H E O R E M 2.2. Assume the hypothesis of Theorem 2.1. Given a solution y(x) of (i) on (a, b)T, an
interval [c, d]~ C (a, b)v, points xl ~_ o-(xl) < x2 _~ or(x2) < x3 belonging to (c, d)v, and an e > O,
there exists a 5(e, [c, d]~r) > 0, such that if Ix~-til < 5, i = 1, 2, 3, and tl ~_ (r(tl) < t2 <_ ~(t2) < t3
belong to (c, d)y, and if
- zd < 5, i = 1, 2, 3, then there exists a solution z(x) of (1)
sati,~fying z(ti) = zi, i = 1, 2, 3, and
y z ~ ( x ) - z z~(x) <E,
3. E X I S T E N C E
on[c,a~v,
OF
i=0,1.
SOLUTIONS
In this section, we invoke a shooting method to establish solutions of (1),(2) exist under our
uniqueness assumptions.
J. HENDERSON et al.
1394
!
THEOREM 3.1. Assume that with respect to (1), Conditions (A)-(D) are satis/~ed. Given points
xl < a(xl) < x2 < a(x2) < x3 beIonging to (a, b)v with Card (a, xl)v _> 1 or Card (x3, b)T > 1,
and yl, Y2 E ]~, there exits a unique solution of (1), (2) on (a, b)v.
PROOF. Let xl < x2 < x3 in (a, b)T and Yl,Y2 C ]~ be selected as in the statement of the theorem.
Let z(x) denote the solution of the initial value problem for (1) satisfying initial conditions at x~,
Z(Xl) =
=
o.
Next, define the set
S = {y(x3) I y(x) is a solution (X), and y(x~) = z(xi), i = 1, 2}.
We observe that S is nonempty, since z(x3) e S. In addition, by applying Theorem 2.2, we
conclude that S is an open subset of ]~. The remainder of the proof is devoted to showing that S
is also a closed subset of ]~. To that end, we assume for contradiction purposes that S is not
closed. Then, there exists an r0 E :~ \ S and a strictly monotone sequence {rk} C S, such
that limk_.~ rk = r0.
We may assume without loss of generality that r k T to. By the definition of S, we denote, for
each k E N, by uk(x) the solution of (1) satisfying
uk(x{) : z(xi),
i : 1, 2,
Uk(X3) : rk.
and
By (B), we know that u~(x{) # uk~+~(x{), i = 1, 2, and since rk+~ > rk, we have from the remark
in the Introduction that
and
uk(x) > uk+~(x),
on (x~,x2)~.
Consequently, from (D) and the fact that r0 ~ S, we may conclude that, either
(a) if Card (a, x~)v _> 1, then there exist a < 7-~ < x~ < 7-2 < x2 < 7-3 < x3, such that
T
i = 1, 3,
or
(b) if Card (x3, b)v > 1, then there exist xl < T2 < x2 < 7-3 < x~ < T4 < b, such that
uk( 2) $
uk(Ti) T +co,
i = 3, 4.
Now, let w(x) be the solution of the initial value problem for (1) satisfying the initial conditions
at x3,
=
ro,
=
o.
It follows from either (a) or (b) that, for some K large, there exist points in the respective cases
(a.1) ~-1 < q l - < ~ - 2 < q 2 < - ~ - 3 < q 3 - < x 3 , s o t h a t u g ( x ) - w ( x ) h a s G Z s a t q i ,
i=l,2,3,
or
(b.1) ~-2 < Pl _< ~-3 < P2 _< x3 < p3 <_ T4, SO that Ug(X) -- w(x) has GZs at pi, i = 1, 2, 3,
which, in either case contradicts the remark in the Introduction. Thus, S is also a closed subset
of ]~.
In summary,
S is a nonempty
subset of ~ that i~ both open
choosing r -- z(x2) + Y2 E S, there is a corresponding solution
y(x
The proof is complete.
) -
(x2) =
y2.
and closed. We have S _-- ~. By
of (I), such that
y(x)
Uniqueness Implies Existence
1395
I
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