Differential and Integral Inequalities - Theory and Applications: Functional, Partial, Abstract, and Complex Differential Equations

311 Pages·1969·5.12 MB·English

by  V. Lakshmikantham and S. Leela (Eds.)

DIFFERENTIAL
AND
INTEGRAL
INEQUALITIES
Theory and Applications
Volume I1
FUNCTIONAL, PARTIAL, ABSTRACT, AND
COMPLEX DIFFERENTIAL EQUATIONS
V. LAKSHMIKANTHAM and S. LEELA
University of Rhode Island
Kingston, Rhode Island
A C A D E MI C P RES S New York and London 1969
0
COPYRIGHT 1969, BY ACADEMIPCR ESSI, NC.
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Preface
The first volume of Differential and Integral Inequalities: Theory
and Applications published in 1969 deals with ordinary differential
equations and Volterra integral equations. It consists of five chapters
and includes a systematic and fairly elaborate development of the
theory and application of differential and integral inequalities.
This second volume is a continuation of the trend and is devoted to
differential equations with delay or functional differential equations,
partial differential equations of first order, parabolic and hyperbolic
types respectively, differential equations in abstract spaces including
nonlinear evolution equations and complex differential equations. To
cut down the length of the volume many parallel results are omitted
as exercises.
We extend our appreciation to Mrs. Rosalind Shumate, Mrs. June
Chandronet, and Miss Sally Taylor for their excellent typing of the
entire manuscript.
V. LAKSHMIKANTHAM
S. LEELA
Kingston, Rhode Island
August, 1969
V
Chapter 6
6.0. Introduction
The future state of a physical system depends, in many circumstances,
not only on the present state but also on its past history. Functional
differential equations provide a mathematical model for such physical
systems in which the rate of change of the system may depend on the
influence of its hereditary effects. The simplest type of such a system
is a differential-difference equation
x’@) = .f(G x(t), x(t - T)),
where T > 0 is a constant. Obviously, for T = 0, this reduces to an
ordinary differential equation. More general systems may be described
by the following equation:
x’(4 = f(t,4 ,
where f is a suitable functional. The symbol xi may be defined in
several ways. For example, if x is a function defined on some interval
+ +
[to- T,t o a), a > 0, then, for each t E [to,t o a),
(i) xi is the graph of x on [t - T, t] shifted to the interval [-T, 01;
(ii) xi is the graph of x on [to - T, t].
In case (ii), f is a functional of Volterra type which is determined by t
< <
and the values of x(s), to - T s t. Systems of this form are called
delay-differential systems. In what follows, we shall, however, consider
the functional differential equations in which the symbol xL has the
meaning described by case (i) and study some qualitative problems by
means of the theory of differential inequalities.
In the present chapter, we consider existence, uniqueness, continua-
tion, and continuous dependence of solutions and obtain a priori bounds
3
4 CHAPTER 6
and error estimates. Asymptotic behavior and stability criteria are
included. An extension of topological principle to functional differential
equations, together with some applications, is discussed. Finally, we
dcvclop the theory of functional differential inequalities, introduce the
notion of maximal and minimal solutions, prove comparison theorems
in this setup, and give some interesting applications.
6.1. Existence
Given any T > 0, let VrL= C[[-T,O],R "] denote the space of
continuous functions with domain [-T, 01 and range in Rn. For any
element $ E P, define the norm
Suppose that x E C[[--7, a),R "]. For any t >, 0, we shall let xLd enote
a translation of the restriction of x to the interval [t - T, t]; more
specifically, xt is an element of Vnd efined by
In other words, the graph of xti s the graph of x on [t - T, t] shifted
to the interval [-T, 01. Let p > 0 be a given constant, and let
With this notation, we may write a functional differential system in
the form
x'(f) = f(t,X J. (6.1.1)
DEFINITIO6N.1 .1. A function x(t, , $0) is said to be a solution of (6. I .l)
with the given initial function do E C, at t = t, >, 0, if there exists a
number A > 0 such that
+
(i) x(t, ,$J is defined and continuous on [to- T, to A] and
x,(t, , $0) E c, for to < to < to -tA ;
(4 X&o , $0) = $0 ; +
(iii) the derivative of x(t, , $") at t, x'(t, ,# ,)(t) exists for t E [to,t o A)
+
and satisfies the system (6.1.1) for t E [to,t , A).
We now state the following well-known result.
6.1. EXISTENCE 5
SCHAUDERF'SIX EDP OINT THEOREMA . continuous mapping of a
compact convex subset of a Banach space into itself has at least one
fixed point.
The following local existence theorem will now be proved.
THEORE6.M1.1 . LetfE C[J x C, , R"]. Then, given an initial function
+o E C, at t = to >, 0, there exists an a > 0 such that there is a solution
+
~ ( t, +,o ) of (6.1.1) existing on [to- T, to a).
Proof. Let a > 0 and define y E C[[to- 7, to + a],R n] as follows:
'1,
< <
r(t)= Idd"o(eO -), to), ttoo -e 7t < tto f tao
+
Then f(t,y t) is a continuous function of t on [to,t o a] and hence
<
IIf(t , yt)ll Ml . We shall show that there exists a constant
b E (0, P - II do(0)lI)
such that
Ilf(t 9 $) -f(t > rt) I/ < 1
whenever t E [to , to + a], $ E C, and 11 $ - y1/l o < b. Suppose that+ th is
is not true. Then for each k = 1, 2, ..., there would exist t, E [to , to a]
and +k E c, such that 11 $, - yt, 1l0 < 1/k and yet
llf(tk Y #k) -f(& ,YtJ 2 1.
We now choose a subsequence {tkJ such that limp+%t kD= t, exists,
and we have a contradiction to the continuity off at (tl , ytl).
< + +
It now follows that Ilf(t, $)/ M = Ml 1 whenever t E [to,t o a],
$ E C, and I/ $ - yt/ lo < b. Choose 01 = min(a, b/M). +
Let B denote the space of continuous functions from [to - 7, to a]
into R". For an element x E B, define the norm
Since the members of S are uniformly bounded and equicontinuous
6 CHAPTER 6
on [to- T, t, 4- 011, the compactness of S follows. A straightforward
computation shows that S is convex.
We now define a mapping on S as follows. For an element x E S, let
m,)
(4 $0 ; + < < +
(ii) T(x(t))= $do) &J(s, xs) ds, to t to a.
+
For every x E S and t E [to,t o a],
11 ~ ( t-&) ( o)lI < MI t - to 1 < Ma < b.
<
Thus xtE C, and /I x<l - yl 11" <b. He<n cef+(s , xs) is a continuous function
of s and IIf(s, x,)ll M for to s to 01. It therefore follows that
the mapping I' is well-defined on S. It is readily verified that T maps S
into itself and is continuous. An application of Schauder's fixed point
theorem now yields the existence of at least one function x E S such that
mu)
6)
= xt, ; < < +
(ii) T(x(t))= x(t), t, t to a,
which implies that
(9 xt, = $0 ; + < < +
(ii) x(t) = ~ ( 0 )J!J( S, xs>d s, to t to 01.
Since 3c" E S, the integrand in the foregoing equation is a continuous
function of s. Thus, for t, < t < to 4- 01, we can differentiate to obtain
x'(t) =f(t, x,), t" < t < to + a.
+
It follows that x(t, , $J is a solution of (6.1. l), defined on [to - T, to a),
and the proof is complete.
We consider next the existence of solutions of (6.1.1) for all t 3 to.
The following lemma is needed before we proceed in that direction.
LEMMA6.1 .1. Let rn E C[[t,- T, a),R ,], and satisfy the inequality
D-mft) < R(t, I m, lo), t -> t" ,
whereg t C[J x R, , R,]. Assume that r(t) = r(t, to , u,) is the maximal
solution of the scalar differential equation
u' = g(t, u), u(to) = U(, 2 0 (6.1.2)
existing for t >, to. Then,
m(f) .=, 4th t 2 t"
7
<
provided 1 m," lo u0
~
6.1. EXISTENCE 7
Proof. To prove the stated inequality, it is enough to prove that
m(t) < u(t, f, > u" , €1, t 2 to,
where u(t, to,u ,, , c) is any solution of
+ +
4
u' = g(t, E, u(t0) = uo €9
E > 0 being an arbitrarily small quantity, since
lim u(t, to , uo , 6) = r(t, to , uo).
e-30
The proof of this fact follows closely the proof of Theorem 1.2.1.
We assume that the set
Since g(t, u) 3 0, u(t, to,u ,, , c) is nondecreasing in t, and this implies,
from the preceding considerations, that
I mtl lo = 44 7 to , uo 9 4 = +l). (6.1.4)
Thus, we are led to the inequality
D-m(tl> G > I mtl I") = R(tl , U(t, 7 t, , uo , E)),
which is incompatible with (6.1.3). The set 2 is therefore empty, and
the lemma follows.
With obvious changes, we can prove
COROLLAR6Y.1 .1. Let m E C[[t,- T, m), R,], and, for t > to ,
D-m(t) 3 -g(t, min m,(s)),
t&-TKS<t
8 CHAPTER 6
where g E C[J x R, , R,]. Let p(t) = p(t, to,u o) >, 0 be the minimal
solution of
u' = -g(t, u), u(t0) = uo > 0
existing for t 2 to . Then uo < min+GsGto mto(s)i mplies
P(t) < W), t 2 to -
The following variation of Lemma 6.1.1, whose assumptions are
weaker, is also useful for later applications.
LEMM6A.1 .2. Let m E C[[to- 7, co), R,], and let, for every t, > to
for which I mtl lo = m(tl), the inequality
<
D-Wd dtl , 4tl))
be verified, where g E C[J x R, , R,]. Then, the conclusion of Lemma
6.1.1 is valid.
The proof follows from Lemma 6.1.1 in view of the relation (6.1.4).
THEORE6.M1.2 . Let f E C[J x V", R"], and, for (t, 4) E J x Yn,
Ilf(4 4111 < g(t7 II d Ilo), (6.1.5)
where g E C[J x R4. , R,] and is nondecreasing in u for each t E J.
Assume that the solutions u(t) = u(t, to , uo) of (6.1.2) exist for all
t to . Then, the largest interval of existence of any solution ~(t,+,o)
of (6.1.1) is [to,m ).
Proof. Let x(to , $,,) be a solution of (6.1.1) existing on some interval
[t,,, p), where to ip < co. Assume that the value of /3 cannot
be increased. Define, for t E [to, p), m(t) = // ~ ( t, d,o )(t)I/,s o that
m, jj xl(t0,$o )ll. Using the assumption (6.1.5), it is easy to obtain,
:
for t > to,t he inequality
o-44 < sk 1 m, lo).
<
Choosing I m," lo = /I +,) Ijo uo , we obtain, by Lemma 6.1.1,
I/ "(to, +o)(t)ll < r(t, to , u,), t" < t < P. (6.1.6)
Since the function g(t, u) 3 0, r(t, to , uu) is nondecreasing, and, hence,
it follows from (6.1.6) that
11 %(to ,+")ll" < r(t, to , U"), to < t P. (6.1.7)
6.2. APPROXIMATE SOLUTIONS AND UNIQUENESS 9
For any t, , t, such that to < t, < t, < p, we have
which, in view of (6.1.7) and the monotonicity of g(t, u) in u, implies
-- r(t, > to , uo) - +I > to, uo).
Letting t, , t, -+ p-, the foregoing relation shows that limi+o- ~ ( t, ,+J (t)
exists, because of Cauchy's criterion for convergence. We now define
.(to , +o)(p) = limL+o-.( to ,+ O)(t)a nd consider lCIo = xs(to , do) as the
new initial function at t = /3. An application of Theorem 6.1.1 shows
+
that there exists a solution x(p, &) of (6.1.1) on [/3, p 011, 01 > 0. This
means that the solution afto,+ ,J can be continued to the right of p,
which is contrary to the assumption that the value of /3 cannot be
increased. Hence, the stated result follows.
COROLLAR6Y.1 .2. The conclusion of Theorem 6.1.2 remains true
even when the condition (6.1.5) is assumed to hold only for t E J and
4 satisfying I1 4 I10 = I1 +(O)ll.
6.2. Approximate solutions and uniqueness
DEFINITIO6N.2 .1. A function x(to , 4, , 6) is said to be an €-approximate
solution of (6.1.1) for t 2 to with the initial function +0 E C, at t = to if
(i) x(to ,+o , E) is defined and continuous on [to - T, a) and
c,
%(to 3 $0 9 ). E ;
( 4 .,,(to 9 90 7 ). = $0 ;
(iii) .(to , +,, , E) is differentiable on the interval [to,a ),e xcept for
an at most countable set S .a>nd satisfies
e
II XYtO I 4 0 9 (t)- f@, %(to , 40 I .))I1 E (6.2.1)
for t E [to,C O) - S.
In case E = 0, it is to be understood that S is empty and .(to , 40)
is a solution of (6.1.1).
We shall now give some comparison theorems on r-approximate
solutions of (6.1.1).