Fimkcialaj Ekvacioj, 38 (1995) 59-69
Existence and Uniqueness for Second Order
Boundary Value Problems
By
S. K, NTOUYAS and P. Ch. TSAMATOS
(University of Ioannina, Greece)
1. Introduction
The main purpose of the present paper is to get the solvability of the
for short)
following boundary value problem (
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
(E)
$(¥rho(t)¥mathrm{x}^{¥prime}(t))^{¥prime}+f(t,$
(BC)
$¥mathrm{x}(t)$
,
$¥mathrm{x}(¥sigma(t))$
,
$¥mathrm{x}^{¥prime}(t)$
,
$x^{¥prime}(g(t))=0$
$x(t)=¥phi_{1}(t)$
,
$t¥leq a$
$x(t)+¥gamma x^{¥prime}(b)=¥phi_{2}(t)$
,
$t$
$¥geq b$
,
,
$t¥in[a, b]$
$¥gamma¥geq 0$
is a continuous function, is a real valued
are continuous real
and
continuous and positive function defined on /,
valued functions defined and continuously differentiable on $E(a)$ , $E(b)$
respectively, where we assume that
Here,
$I$
$=[a, b]$
,
$f:I¥times(R^{n})^{4}¥rightarrow R^{n}$
$¥rho$
$g$
$¥sigma$
$-¥infty<r(a)=¥min_{t¥in I}$ $¥{¥sigma(t), g(t)¥}<a$
$b<r(b)=mt¥mathrm{a}¥mathrm{x}$
$¥{¥sigma(t), g(t)¥}<+¥infty$
and we set $E(a)=[r(a), a]$ , $E(b)=[b, r(b)]$ and $J=[r(a),
assume that the set { $t¥in I:g(t)=a$ or $g(t)=b$ } is finite.
equation (E) for
Also we
we mean a function $x¥in C(J, R^{n})$
which is piecewice twice differentiable on /, satisfies the
and the boundary conditions (BC) for $t¥in E(a)¥cup E(b)$ .
By a solution of the
$¥cap C^{1}(E(a)¥cup E(b), R^{n})$
r(b)]$ .
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
$(¥mathrm{E})-(¥mathrm{B}¥mathrm{C})$
$t¥in I$
has at least one solution, we
In order to show that the
use in this paper the “Leray-Schauder alternative” which follows immediately
from the Topological Transversality Theorem of Granas [1]. This method
to the establishment
reduces the problem of the existence of solutions of a
of suitable a priori bounds for solutions of these problems. For applications
of Topological Transversality method for ordinary differential equations we
refer the reader to [2, 6], whereas, for differential equations with delay or
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
$(¥mathrm{E})¥triangleleft ¥mathrm{B}¥mathrm{C})$
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
60
S. K. NTOUYAS and P. Ch. TSAMATOS
deviating arguments in [7, 8, 10, 11] and the references therein.
In Section 2 we prove the basic existence theorem by assuming a priori
bounds on solutions and their derivatives. In order to apply this basic
existence theorem, we establish a priori bounds for solutions and their
derivatives in Section 3. The required a priori bounds for the possible
estimates and a Nagumo type condition, by
solutions are obtained via
using some ideas from [5]. We prove also and uniqueness results. It is
noteworthy that our results for the choice $¥rho(t)=e^{kt}$ , $k¥neq 0$ ,
, $¥sigma(t)=g(t)=t$ ,
$¥gamma=0$ lead to the results of Mawhin [9], whereas for the choice $¥rho(t)=e^{kt}$ , $k¥neq 0$ ,
, $¥sigma(t)=-t$ , $¥gamma=0$ and independent of to the results of Gupta [3, 4].
$L^{2_{-}}$
$t¥in I$
$t¥in I$
$f$
$¥mathrm{x}^{¥prime}$
2. The basic existence theorem
Let B be the space
B
$=C(J, R^{n})¥cap C^{1}(E(a)¥cup E(b), R^{n})¥cap C^{1}(I, R^{n})$
with the norm
,
$x¥in B$
$||¥mathrm{x}||_{1}=¥max¥{¥max_{t¥in}|¥mathrm{x}(t)|,¥max_{t¥in E(a)¥cup E(b)}|¥chi^{¥prime}(t)|, mt¥mathrm{a}¥mathrm{x} |x^{¥prime}(t)|¥}$
.
The following Lemma is an immediate consequence of the Topological
Transversality Theorem of Granas [1, p. 61] known as “Leray-Schauder
alternative” [1, p. 61].
Lemma 2.1. Let $X$ be a convex subset of a normed linear space
and
assume
. Let $F:X¥rightarrow X$ be a completely continuous operator,
. it is
continuous and the image of any bounded set is included in a compact set, and let
$E$
$i.e$
$¥mathrm{O}¥in B$
{ $x¥in X:x=¥lambda Fx$ for some $0<¥lambda<1$ }.
has a fixed point.
Then either $E(F)$ is unbounded or
be a continuous function.
Theorem 2.2. Let
$E(F)=$
$F$
$f:I¥times(R^{n})^{4}¥rightarrow R^{n}$
there exists a constant
$K$
, such that
$||¥mathrm{x}||_{1}¥leq K$
for every
$(E_{¥lambda})$
(A)
solution
$x$
of
Assume that
the
,
$BVP$
$(¥rho(t)¥mathrm{x}^{¥prime}(t))^{¥prime}+¥lambda f(t,$
$x(t)$
,
$x(¥sigma(t))$
,
$x(t)+¥gamma x^{¥prime}(b)=¥phi_{2}(t)$ ,
$x(t)=¥phi_{1}(t)$
,
$x^{¥prime}(t)$
,
$x^{¥prime}(g(t))=0$
,
$t¥in E(a)$
$t¥in E(b)$
,
$¥gamma¥geq 0$
$t¥in I$
Second Order Boundary Value Problems
where
$¥lambda¥in(0,1)$
.
Then the
Proof. Define
$BVP$ $(E)-(BC)$
61
has at least one solution.
by
$T:B¥rightarrow B$
,
$Tx(t)=¥{¥omega¥phi_{2}¥phi_{1}(t)(t)’-(Tx)^{¥prime}(b)(t)+¥int_{a}^{b}G(t,s,)f(s, x(s), x(¥sigma(s)), ¥chi^{¥prime}(s), x^{¥prime}(g(s)))ds$
$ttt¥in E(b)¥in E(a)¥in I$
where
$¥omega(t)=¥phi_{1}(a)+¥frac{¥phi_{2}(b)-¥phi_{1}(a)}{¥rho(b)h(b)+¥gamma}¥rho(b)h(t)$
$t¥in I$
,
$h(t)=¥int_{a}^{t}¥frac{ds}{¥rho(s)}$
and
,
$t¥in I$
is the Green’s function which is given by the formula
$G$
$G(t, s)=-¥frac{1}{¥rho(b)h(b)+¥gamma}¥left¥{¥begin{array}{l}[¥rho(b)¥int_{a}^{b}¥frac{ds}{¥rho(s)}+¥gamma]h(t),¥¥[¥rho(b)¥int_{t}^{b}¥frac{ds}{¥rho(s)}+¥gamma]h(s),¥end{array}¥right.$
$sa$
$¥leq¥leq tt¥leq¥leq sb$
is clearly continuous. We shall prove that is completely continuous.
this purpose we consider a bounded sequence
in ,
.
$T$
$T$
$B$
$¥{x_{v}¥}$
$||¥mathrm{x}_{v}||_{1}¥leq M$
where
$M$
,
for all
$v$
$¥mathrm{i}.¥mathrm{e}$
,
is a positive constant. Then we have
$||Tx_{v}||_{1}¥leq M_{0}$
,
where
$ M_{0}=¥max$
$K_{1}$
,
$¥{¥Theta K_{1}+A_{1}, ¥Theta K_{2}+A_{2}¥}$
,
constants with
$K_{2}$
$¥int_{a}^{b}|G(t, s)|ds$ $¥leq K_{1}$
,
$¥int_{a}^{b}|G_{t}(t, s)|ds$ $¥leq K_{2}$
,
$t¥in I$
,
$¥Theta=¥max$ $¥{|f(t, u, u_{1}, v, v_{1})| : t¥in I, |u|, |u_{1}|, |v|, |v_{1}|¥leq M¥}$
and
$A_{1}$
,
$A_{2}$
constants with
,
For
62
S. K. NTOUYAS and P. Ch. TSAMATOS
$¥sup¥{|¥phi_{1}(a)+¥frac{¥phi_{2}(b)-¥phi_{1}(a)}{¥rho(b)h(b)+¥gamma}¥rho(b)h(t)|$
:
$t¥in I¥}¥leq A_{1}$
$¥sup¥{|¥frac{¥phi_{2}(b)-¥phi_{1}(a)}{¥rho(b)h(b)+¥gamma}||¥frac{h(b)}{¥rho(t)}|:t¥in I¥}¥leq A_{2}$
Next we shall prove that the sequences
Indeed, for any ,
in
and arbitrary
and
we have
$¥{(Tx_{v})^{¥prime}¥}$
$¥{T¥mathrm{x}_{v}¥}$
$t_{1}$
$J$
$t_{2}$
$v$
.
are equicontinuous.
$|Tx_{v}(t_{1})-T¥mathrm{x}_{v}(t_{2})|=|¥int_{t_{1}}^{t_{2}}(Tx_{v})^{¥prime}(s)ds|¥leq¥hat{K}|t_{1}-t_{2}|$
where
$¥hat{K}=¥{K_{2}¥Theta+A_{2}, ¥max_{t¥in E(a)}|¥phi_{1}^{¥prime}(t)|, ¥max_{t¥in E(b)}|¥phi_{2}^{¥prime}(t)|¥}$
which proves that
in I and for arbitrary
is equicontinuous.
we have
$¥{TX_{¥mathcal{V}}¥}$
$v$
,
On the other hand for any
$|(Tx_{v})^{¥prime}(t_{1})-(Tx_{v})^{¥prime}(t_{2})|=|¥int_{t_{1}}^{t_{2}}(Tx_{¥mathcal{V}})^{¥prime¥prime}(s)ds|¥leq¥Theta|t_{1}-t_{2}|$
$t_{1}$
,
$t_{2}$
.
This relation and the fact that
are continuously differentiable functions
,
imply, obviously, that
is an equicontinuous sequence.
$¥phi_{1}$
$¥phi_{2}$
$¥{(Tx_{v})^{¥prime}¥}$
Thus the mapping
is completely continuous. Finally, we observe by
hypothesis that the set $E(T)=$ { $x¥in B:x=¥lambda Tx$ for some
} is bounded.
Hence, by Lemma 2.1 the operator
has a fixed point $x¥in B$ . This means
that the
has as least one solution. The proof of the theorem
is now complete.
$T$
$¥lambda¥in(0,1)$
$T$
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
$(¥mathrm{E})-(¥mathrm{B}¥mathrm{C})$
3. Applications
which
In order to apply Theorem 2.2 we must impose conditions on
imply the existence of the needed a priori bounds. In the next theorems we
assume that $¥phi_{1}(a)=¥phi_{2}(b)=0$ . With this restriction is no loss in generality,
since an appropriate change of variables reduces the problem with $¥phi_{1}(a)¥neq 0¥neq$
to this case.
$f$
$¥phi_{2}(b)$
$¥sigma$
Theorem 3.1. Let
$g:I$
are such that
,
$f:I$
$¥times(R^{n})^{4}¥rightarrow R^{n}$
continuous function,
be
$¥rightarrow R$
$|¥sigma^{¥prime}(t)|¥geq¥frac{1}{c_{1}}$
and
$|g^{¥prime}(t)|¥geq¥frac{1}{c_{2}}$
,
$t¥in I$
and
Second Order Boundary Value Problems
for some
$(H_{1})$
constants
$c_{1}>0$
and
63
$c_{2}>0$ .
Assume that:
There exist nonnegative constants
$A$
,
$B$
,
$C$
,
and
$D$
$G$
with
$4(A+B¥sqrt{c_{1}})¥frac{(b-a)^{2}}{¥pi^{2}}$
(3.1)
$+[4(C+D¥sqrt{c_{2}})+2B¥sqrt{2c_{1}(b-a)(r(b)-b)}]¥frac{b-a}{¥pi}<¥rho_{0}$
where
(3.2)
$(H_{2})$
such that
$¥rho_{0}=¥min$ $¥{¥rho(t):t¥in I¥}$
$¥langle u, f(t, u, u_{1}, v, v_{1})¥rangle¥leq A|u|^{2}+B|u||u_{1}|+C|u||v|+D|u||v_{1}|+G|u|$
There exist a continuous function
(3.3)
$h:R^{+}¥rightarrow R^{+}$
and a constant
$N$
such that
$¥int_{R^{2}M^{2}}^{N}b-aa¥frac{ds}{h(s)}¥geq 2R^{2}M^{2}$
where $ R=¥max$
$¥{¥rho(t):t¥in I¥}$
,
(3.4)
2
$[B¥sqrt{c_{1}}(||¥phi_{1}||+¥sqrt{2}||¥phi_{2}||)+D¥sqrt{c_{2}}(||¥phi_{1}^{¥prime}||+||¥phi_{2}^{¥prime}||)]¥frac{b-a}{¥pi}+¥frac{2G(b-a)¥sqrt{b-a}}{¥pi}$
$M=$
$¥rho_{0}-4(A+B¥sqrt{c_{1}})¥frac{(b-a)^{2}}{¥pi^{2}}-[4(C+D¥sqrt{c_{2}})+2B¥sqrt{2c_{1}(b-a)(r(b)-b)}]¥frac{b-a}{¥pi}$
and
(3.3)
for
$|¥langle v, f(t, u, u_{1}, v, v_{1})¥rangle|¥leq h(|¥rho(t)v|^{2})¥rho(t)|v|^{2}$
all
$t¥in I$
(Here
and
$|u|_{0}¥leq¥sqrt{b-a}M$ .
$||u||^{2}=¥int_{a}^{b}|u(t)|^{2}dt$
Then the
$BVP$ $(E)-(BC)$ ,
).
with
$¥phi_{1}(a)=¥phi_{2}(b)=0$
, has at least one solution.
Proof.
We need only to establish the a priori bounds for the
Let
be a solution of
. By taking the inner product
-(BC).
of the equation
with $x(t)$ , integrating by parts over I and use of the
boundary conditions and (3.2), we get
$(E_{¥lambda} )$
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
$(E_{¥lambda})-(BC)$
$¥mathrm{x}$
$(E_{¥lambda})$
$¥int_{a}^{b}¥rho(t)|x^{¥prime}(t)|^{2}dt¥leq¥rho(b)¥langle¥chi^{¥prime}(b), ¥mathrm{x}(b)¥rangle+A¥int_{a}^{b}|x(t)|^{2}dt+B¥int_{a}^{b}.|x(t)||x(¥sigma(t))|dt$
$+C¥int_{a}^{b}|¥mathrm{x}(t)||¥chi^{¥prime}(t)|dt+D¥int_{a}^{b}|¥mathrm{x}(t)||¥chi^{¥prime}(g(t))|dt+G¥int_{a}^{b}|¥mathrm{x}(t)|dt$
64
S. K. NTOUYAS and P. Ch. TSAMATOS
which implies, by Cauchy-Schwarz inequality and
$¥gamma>0$
(if
$¥gamma=0$
then $x(b)=0$)
$¥rho_{0}||¥mathrm{x}||^{2}¥leq-¥frac{1}{¥gamma}¥rho(b)|¥mathrm{x}(b)|^{2}+A||¥mathrm{x}||^{2}+B||¥mathrm{x}||(¥int_{a}^{b}|¥mathrm{x}(¥sigma(t))|^{2}dt)^{¥frac{1}{2}}$
(3.6)
$+C||¥mathrm{x}||||¥chi^{¥prime}||+D||¥mathrm{x}||(¥int_{a}^{b}|¥chi^{¥prime}(g(t))|^{2}dt)^{¥frac{1}{2}}+G||x||¥sqrt{b-a}$
But
$¥int_{a}^{b}|x(¥sigma(t))|^{2}dt¥leq|¥int_{a}^{b}¥frac{1}{¥sigma(t)}’|x(¥sigma(t))|^{2}d(¥sigma(t))|$
$¥leq¥int_{¥sigma(I)}|¥mathrm{x}(t)|^{2}dt$
$=c_{1}(¥int_{a}^{b}|x(t)|^{2}dt+¥int_{E(a)}|x(t)|^{2}dt+¥int_{E(b)}|x(t)|^{2}dt)$
$=c_{1}[||¥mathrm{x}||^{2}+||¥phi_{1}||^{2}+¥int_{E(b)}|¥phi_{2}(t)-¥gamma x^{¥prime}(b)|^{2}dt]$
(3. 7)
$¥leq c_{1}$
$[||x||^{2}+||¥phi_{1}||^{2}+2||¥phi_{2}||^{2}+2¥gamma^{2}|¥chi^{¥prime}(b)|^{2}(r(b)-b)]$
$=c_{1}$
$[||x||^{2}+||¥phi_{1}||^{2}+2||¥phi_{2}||^{2}+2|x(b)^{2}(r(b)-b)]$
(since
$¥leq c_{1}$
$x(b)=-¥gamma x^{¥prime}(b)$
$[||¥mathrm{x}||^{2}+||¥phi_{1}||^{2}+2||¥phi_{2}||^{2}+2||¥chi^{¥prime}||^{2}(r(b)-b)(b-a)]$
(by the Wirtinger’s inequality
and likewise
(3.8)
)
$|x|_{0}=¥sup$
{
$|x(t)|$
:
$t¥in I¥}¥leq¥sqrt{b-a}||x^{¥prime}||$
)
$¥int_{a}^{b}|x^{¥prime}(g(t))|^{2}dt¥leq c_{2}$ $[||¥chi^{¥prime}||+||¥phi_{1}^{¥prime}||^{2}+||¥phi_{2}^{¥prime}||^{2}]$
Subtitute (3.7) and (3.8) into (3.6) to obtain
$¥rho_{0}||x^{¥prime}||^{2}¥leq A||¥mathrm{x}||^{2}$
$+B¥sqrt{c_{1}}||x||¥{||¥mathrm{x}||+||¥phi_{1}||+¥sqrt{2}||¥phi_{2}||+¥sqrt{2(r(b)-b)(b-a)}||¥chi^{¥prime}||¥}$
$+C||x||||x^{¥prime}||+D¥sqrt{c_{2}}||x||¥{||¥chi^{¥prime}||+||¥phi_{1}^{¥prime}||+||¥phi_{2}^{¥prime}||¥}+G||¥mathrm{x}||¥sqrt{b-a}$
Next by applying Wirtinger’s inequality
$||x||^{2}¥leq¥frac{4(b-a)^{2}}{¥pi^{2}}||x^{¥prime}||^{2}$
we get,
Second Order Boundary Value Problems
65
$¥rho_{0}||x^{¥prime}||^{2}¥leq 4(A+B¥sqrt{c_{1}})¥frac{(b-a)^{2}}{¥pi^{2}}||x^{¥prime}||^{2}+2[B¥sqrt{c_{1}}(||¥phi_{1}||+¥sqrt{2}||¥phi_{2}||)$
$+D¥sqrt{c_{2}}(||¥phi_{1}^{¥prime}||+||¥phi_{2}^{¥prime}||)¥frac{b-a}{¥pi}||¥chi^{¥prime}||+4(C+D¥sqrt{c_{2}})¥frac{b-a}{¥pi}||¥chi^{¥prime}||^{2}$
$+B¥sqrt{2c_{1}(r(b)-b)(b-a)}¥frac{2(b-a)}{¥pi}||x^{¥prime}||^{2}+¥frac{2G(b-a)¥sqrt{b-a}}{¥pi}||¥chi^{¥prime}||$
Therefore we deduce
$¥{¥rho_{0}-4(A+B¥sqrt{c_{1}})¥frac{(b-a)^{2}}{¥pi^{2}}$
?
$[4(C+D¥sqrt{c_{2}})+2B¥sqrt{2c_{1}(r(b)-b)(b-a)}]¥frac{(b-a)}{¥pi}¥}||¥chi^{¥prime}||$
$¥leq 2[B¥sqrt{c_{1}}(||¥phi_{1}||+¥sqrt{2}||¥phi_{2}||)$
$+D¥sqrt{c_{2}}(||¥phi_{1}^{¥prime}||+||¥phi_{2}^{¥prime}||)]¥frac{b-a}{¥pi}+¥frac{2G(b-a)¥sqrt{b-a}}{¥pi}$
which implies, by (3.1) and (3.4)
(3.8)
$||x^{¥prime}||¥leq M$
By the Wirtinger’s inequality
$|x|_{0}=¥sup$
.
$¥{|x(t)|:t¥in I¥}¥leq¥sqrt{b-a}||¥chi^{¥prime}||$
$|¥mathrm{x}|_{0}¥leq¥sqrt{b-a}M=M_{1}$
.
Also, (3.8) implies, by the mean value theorem, that there exists
$t_{0}¥in I$
such that
$(b -a)|¥chi^{¥prime}(t_{0})|^{2}¥leq M^{2}$
or
(3.9)
$¥rho^{2}(t_{0})|x^{¥prime}(t_{0})|^{2}¥leq¥frac{R^{2}M^{2}}{b-a}$
Now, taking the inner product of
$(E_{¥lambda})$
with
$¥mathrm{x}^{¥prime}(t)$
we have, by (3.5)
$|¥frac{d}{dt}|¥rho(t)x(t)|^{2}|¥leq 2h(|¥rho(t)x^{¥prime}(t)|^{2})¥rho^{2}(t)|x^{¥prime}(t)|^{2}$
or
(3. 10)
we have
$|¥frac{d}{dt}¥int_{a}^{|¥rho(t)x^{¥prime}(t)|^{2}}¥frac{ds}{h(s)}|¥leq 2|¥rho(t)¥chi^{¥prime}(t)|^{2}$
S. K. NTOUYAS and P. Ch. TSAMATOS
66
Integrating (3.10) and using (3.9) we get
$¥int_{a}^{|¥rho(t)¥mathrm{x}^{¥prime}(t)|^{2}}¥frac{ds}{h(s)}¥leq¥int_{a}^{|¥rho(t_{0})x^{¥prime}(t_{¥mathrm{o}})|^{2}}¥frac{ds}{h(s)}+2¥int_{a}^{b}|¥rho^{2}(t)¥mathrm{x}^{¥prime}(t)|^{2}dt$
$¥leq¥int_{a}^{|¥rho(t_{¥mathrm{o}})x^{¥prime}(t_{¥mathrm{O}})|^{2}}¥frac{ds}{h(s)}+2R^{2}M^{2}$
$¥leq¥int_{a}^{¥frac{R^{2}M^{2}}{b- a}}¥frac{ds}{h(s)}+¥int_{R^{2}M^{2}}^{N}b-aa¥frac{ds}{h(s)}=¥int_{a}^{N}¥frac{ds}{h(s)}$
Hence
$|¥rho(t)x^{¥prime}(t)|^{2}¥leq N$
or
$¥rho_{0}^{2}|¥chi^{¥prime}(t)|^{2}¥leq|¥rho(t)x^{¥prime}(t)|^{2}¥leq N$
which implies
$|¥chi^{¥prime}|_{0}¥leq¥frac{1}{¥rho_{0}}¥sqrt{N}=M_{2}$
,
$t¥in I$
.
Consequently the required a priori bounds are established and the proof of
the theorem is complete.
The next theorem concerns uniqueness results for the
in
$
¥
gamma=0$
in
.
We
remark
that
this
case
the
corresponding
the case when
Wirtinger’s inequality becomes
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
$(¥mathrm{E})-(¥mathrm{B}¥mathrm{C})$
$||x||^{2}¥leq¥frac{(b-a)^{2}}{¥pi^{2}}||¥chi^{¥prime}||^{2}$
and relation (3.1)
(3.1)’
$¥sigma$
$(A+B¥sqrt{c_{1}})¥frac{(b-a)^{2}}{¥pi^{2}}+(C+D¥sqrt{c_{2}})¥frac{b-a}{¥pi}<¥rho_{0}$
Theorem 3.2. Let
, $g:I$
are such that
$f:I¥times(R^{n})^{4}¥rightarrow R^{n}$
be
continuous function,
$¥rightarrow R$
$|¥sigma^{¥prime}(t)|¥geq¥frac{1}{c_{1}}$
for some
constants
Assume that:
$c_{1}>0$
and
and
$c_{2}>0$ .
$|g^{¥prime}(t)|¥geq¥frac{1}{c_{2}}$
,
$t¥in I$
and
67
Second Order Boundary Value Problems
There exist nonnegative constants
$(H_{3})$
$A$
,
$B$
,
and
$C$
$D$
satisfying (3.1)’ and such
that
(3.11)
$¥langle u -x, f(t, u, u_{1}, v, v_{1})-f(t, x, x_{1}, y, y_{1})¥rangle$
$¥leq A|u-x|^{2}+B|u-x||u_{1}-¥mathrm{x}_{1}|+C|u-x||v-y|+D|u-x||v_{1}-y_{1}|$
Then the
solution.
$BVP$ $(E)-(BC)$
Proof.
Let
$u$
,
$x$
with
and
$¥phi_{1}(a)=¥phi_{2}(b)=0$
be two solutions of the
$¥mathrm{B}¥mathrm{V}¥mathrm{P}$
has at most one
$¥gamma=0$
$(¥mathrm{E})-(¥mathrm{B}¥mathrm{C})$
.
Then we get
$0=-¥int_{a}^{b}¥langle(¥rho(t)u^{¥prime}(t))^{¥prime}- (¥rho(t)x^{¥prime}(t))^{¥prime}, u(t)-¥mathrm{x}(t)¥rangle dt$
$-¥int_{a}^{b}¥langle f(t, u(t), u(¥sigma(t)), y^{¥prime}(t), u^{¥prime}(g(t)))-f(t, x(t), ¥mathrm{x}(¥sigma(t)), x^{¥prime}(t), ¥mathrm{x}^{¥prime}(g(t)))¥rangle dt$
$¥geq¥rho_{0}¥int_{a}^{b}|u^{¥prime}(t)-¥mathrm{x}^{¥prime}(t)|^{2}dt-A¥int_{a}^{b}|u(t)-x(t)|^{2}dt$
?
?
$B¥int_{a}^{b}|u(t)-¥mathrm{x}(t)||u(¥sigma(t))-x(¥sigma(t))|dt$
$C¥int_{a}^{b}|u(t)-x(t)||u^{¥prime}(t)-x^{¥prime}(t)|dt-D¥int_{a}^{b}|u(t)-x(t)||u^{¥prime}(g(t))-x^{¥prime}(g(t))|dt$
$¥geq[¥rho_{0}-(A+B¥sqrt{c_{1}})¥frac{(b-a)^{2}}{¥pi^{2}}-(C+D¥sqrt{c_{2}})¥frac{b-a}{¥pi}]||u^{¥prime}-x^{¥prime}||^{2}$
$¥geq[¥rho_{0}-(A+B¥sqrt{c_{1}})¥frac{(b-a)^{2}}{¥pi^{2}}-(C+D¥sqrt{c_{2}})¥frac{b-a}{¥pi}]¥frac{¥pi}{(b-a)}||u-X||^{2}$
Therefore by
the theorem.
$¥sigma$
$(3.1)^{¥prime}$
we conclude that
Corollary 3.3. Let
, $g:I$
are such that
$u(t)=x(t)$
$f:I¥times(R^{n})^{4}¥rightarrow R^{n}$
for every
be
$t¥in I$
, which proves
continuous function,
and
$¥rightarrow R$
$|¥sigma^{¥prime}(t)|¥geq¥frac{1}{c_{1}}$
for some
and
$|g^{¥prime}(t)|¥geq¥frac{1}{c_{2}}$
,
$t¥in I$
and $c_{2}>0$ .
Assume that conditions
and
hold. Then the
$¥phi_{1}(a)=¥phi_{2}(b)=0$ and $¥gamma=0$ has a unique solution.
constants
$c_{1}>0$
$(H_{1})$
$(H_{3})$
$BVP$ $(E)-(BC)$
with
S. K. NTOUYAS and P. Ch. TSAMATOS
68
Proof.
By (3.11) with $x=x_{1}=y=y_{1}=0$ , we obtain
$¥langle u, f(t, u, u_{1}, v, v_{1})¥rangle$
$¥leq A|u|^{2}+B|u||u_{1}|+C|u||v|+D|u||v_{1}|+|u||f(t, 0, 0, 0, 0)|$ .
That is, condition (3.2) with
complete the proof.
Remark 3.4.
If
$¥rho(t)=e^{kt}$
,
$ G=¥max$ $¥{|f(t,
$k¥neq 0$
,
0,0,0, 0)|, t¥in I¥}$
$t¥in I_{1}=[0, ¥pi]$
holds.
This
equation (E) gives
$x^{¥prime¥prime}(t)+kx^{¥prime}(t)+h(t, x(t), ¥mathrm{x}(¥sigma(t)), x^{¥prime}(t), x^{¥prime}(g(t)))=0$
$(E_{1})$
where
.
$h(t, x(t), x(¥sigma(t)), x^{¥prime}(t), ¥mathrm{x}^{¥prime}(g(t)))=e^{kt}f(t, ¥mathrm{x}(t), x(¥sigma(t)), x^{¥prime}(t), x(g(t)))$
our Theorems 3.1 and 3.2 immediately imply
and 2 respectively of Mawhin [9]. Also if $¥sigma(t)=-t$ ,
$t¥in I_{2}=[-1,1]$ ,
. if we have boundary value problems involving reflection
of the arguments, and is independent of , our Theorems 3.1 and 3.2 imply
immediately the results of Gupta [3] and [4] for constant matrix $A$ .
Moreover if
Theorems 1
$¥sigma(t)=g(t)=t$
$¥mathrm{i}.¥mathrm{e}$
$f$
$x^{¥prime}$
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
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Second Order Boundary Value Problems
nuna adreso:
Department of Mathematics
University of Ioannina
451 10 Ioannina
Greece
(Ricevita la 8-an de februaro, 1993)
(Reviziita la 6-an de decembro, 1993)