An integro-differential inequality with application *.

Abstract

An integro-differential inequality is proposed together with an application to initial value problems.

Keywords and Phrases: Integro-differential equations, A priori bound on solutions, Essential maps, Topological transversality theorem.

1. Introduction

Differential inequalities have played a major role in the study of the existence, uniqueness, stability of solutions of ordinary differential equations. For more details, one can consult the monographs [?], [?], [?], [?] and the papers [?], [?] and references therein.

In this paper, we present a nonlinear integro-differential inequality and obtain some existence results for initial value problems for systems of first order integro-differential equations. Several papers have been devoted to linear as well as nonlinear integro-differential inequalities (see [?] for references). However, our assumptions are less restrictive and quite general. Also, our work is motivated by the recent results in [?].

2. A Nonlinear Integro-differential Inequality

We say that [omega] belongs to the class [OMEGA] if

(i) [omega]: [0, [infinity]) [right arrow] (0, [infinity]) is continuous and non-decreasing

(ii) There exists a positive continuous function R such that for each [delta] > 0,

[e.sup.-[delta]t][omega]([z]) [less than or equal to] R(t)[omega]([e.sub.-[delta]t][z])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(iii) [[integral].sub.[infinity].sub.0] [d[sigma]/[omega]]([sigma]) = [infinity]

We refer the reader to [?] for examples of such functions [omega]. Notice that E: [[??].sub.+] [right arrow] [[??].sub.+] defined by

E(l) = [[integral].sup.l.sub.0] d[sigma]/[omega]([sigma])

is strictly increasing and [lim.sub.l[right arrow][infinity]] E(l) = +[infinity]. Thus, [E.sub.-1] is well-defined and strictly increasing on (0, [infinity]).

Theorem 2.1. Let u be a non-negative function defined on [0, T] with a continuous first derivative such that u(0) = [u.sub.0] and

u'(t) [less than or equal to] au(t) + [[integral].sup.t.sub.0] k(s)[beta](s)[omega](u(s))ds (2.1)

where a > 0,k [member of] C([0, T]; [[??].sub.+]), [beta] [member of] [L.sup.1]([0,T]; [[??].sub.+]) and [omega] [member of] [OMEGA]. Then there exists a continuous function [??]: [0, T] [right arrow] [[??].sub.+] depending only on a, k, [beta] and R such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

for all t [member of] [0, T].

Proof. Let v(t) denote the right hand side of (??), namely,

v(t) = au(t) + [[integral].sup.t.sub.0] k(s)[beta](s)[omega](u(s))ds (2.3)

Then

u(t) [less than or equal to] v(t)/a, u'(t) [less than or equal to] v(t), v(0) = a[u.sub.0]

and

v'(t) = au'(t) + k(t)[beta](t)[omega](u(t)).

Hence

v'(t) [less than or equal to] av(t) + k(t)[beta](t)[omega](u(t)). (2.4)

Let [z](t) = v(t)[e.sup.-at]. Then (??) yields

z'(t) [less than or equal to] k(t)[beta](t)[e.sup.at][omega](v(t)/a)

Since [omega] [member of] [OMEGA]

z'(t) [less than or equal to] k(t)[beta](t)R(t)[omega]([e.sup.-at]v(t)/a) [less than or equal to] k(t)[beta](t)[omega](z(t)/a)

Letting Z(t) = z(t)/a, we obtain

Z'(t) [less than or equal to] [k(t)/a][beta](t)R(t)[omega](Z(t)), 0 [less than or equal to] t [less than or equal to] T (2.5)

It follows from (??) that

Z'(t)/[omega](Z(t)) [less than or equal to] [k(t)/a][beta](t)R(t), 0 [less than or equal to] t [less than or equal to] T (2.6)

An integration from 0 to t gives

[[integral].sup.t.sub.0] Z'(s)/[omega](Z(s))ds [less than or equal to] [1/a] [[integral].sup.t.sub.0] k(s)[beta](s)R(s)ds, 0 [less than or equal to] t [less than or equal to] T (2.7)

Consequently, making a change of variable in the first integral,

[[integral].sup.Z(t).sub.u0] [d[sigma]/[omega]([sigma])] [less than or equal to] [1/a] [[integral].sup.t.sub.0] k(s)[beta](s)R(s)ds (2.8)

Therefore, from the monotonicity of E,

Z(t) [less than or equal to] [E.sup.-1] (E([u.sub.0]) + [1/a] [[integral].sup.t.sub.0] k(s)[beta](s)R(s)ds), 0 [less than or equal to] t [less than or equal to] T (2.9)

Now, we have

u(t) [less than or equal to] [v(t)/a] = z(t)[e.sup.at]/a = Z(t)[e.sup.at] 0 [less than or equal to] t [less than or equal to] T

This shows that [??]: [0, T] [right arrow] [[??].sub.+] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10)

is continuous and is such that u(t) [less than or equal to] [??](t) [for all]t [member of] [0, T]. This completes the proof of the Theorem.

3. Application

In this section, we consider an initial value problem for first order integro-differential equations in [[??].sup.n]. For x [member of] [[??.sup.n], we define .??] = [([summation].sup.n.sub.i=1] [[absolute value of [x.sub.i]].sup.2]).sup.1/2]. Consider the following

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.11)

where A(*) is a continuous n x n matrix, K : [0, T] x [0, T] [[??].sup.nxn] is continuous and f is an [L.sup.1]--Caratheodory function; that is,

(a) f(*, x) : [0, T] [right arrow] [[??.sup.n] is measurable for every x [member of] [[??.sup.n].

(b) f(t, x) : [[??.sup.n] [right arrow] [[??.sup.n] is continuous for almost all t [member of] [0, T].

(c) For each [rho] > 0, there exists [h.sub.[rho]] [member of] [L.sup.1] ([0, T];[[??.sub.+]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for almost all t [member of] [0, T].

We shall study the Cauchy problem (??) under the following assumptions:

(H1) The matrix A(*) is continuous on [0, T].

H2) K : [0, T] x [0, T] [right arrow] [[??].sup.nxn] is continuous.

(H3) f : [0, T] x [[??].sup.n] [right arrow] [[??].sup.n] is an [L.sup.1]--Caratheodory function satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all (t, x) [member of] [0, T] x [[??].sup.n] where q [member of] [L.sup.1]([0, T];[[??].sub.+]) and [omega] [member of] [OMEGA].

Theorem 3.1. Assume (H1), (H2) and (H3)are satisfied. Then problem (??) has at least one solution.

Proof. Consider the one parameter family of problems

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.12)

for 0 [less than or equal to] [lambda] [less than or equal to] 1. It is clear that (??) reduces to (??) when [lambda] = 1. Claim 1. Solutions of (??) are a priori bounded independently of [lambda]. For [lambda] = 0, the only solution of (??) is x(t) = 0 which is obviously bounded independently of [lambda]. So we consider 0 < [lambda] [less than or equal to] 1. Let [parallel][??][parallel] = [<x(t),x(t)>.sup.1/2] where <, > denotes the inner product in [[??].sup.n]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all t [member of] J := {t [member of] [0, T] : [parallel]x(t)[paralle]] > 0}. Hence, by Cauchy inequality, [parallel][??](t)[parallel]]' [less than or equal to] [parallel][??](t)[parallel] for all t [member of] J := {t [member of] [0,T] : [parallel][??](t)[parallel] > 0}. Let

[A.sub.0] = max {[parallel]A(t)[parallel] [member of] [0, T]}, [K.sub.0](s) := max {[parallel]K(t, s)[parallel]; t [member of] [0, T]}.

Then using (??) and assumption (H3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.13)

Application of Theorem ?? to (??) yields [parallel]x(t)[parallel] [less than or equal to] [??](t) where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.14)

Let [M.sub.0] = [sup.sub.0[less than or equal to]t[less than or equal to]T] [??](t). Then [parallel]x(t)[parallel] [less than or equal to] [M.sub.0] independently of [lambda]. Furthermore,

[parallel]x[[parallel].sub.0] [less than or equal to] [M.sub.0] (3.15)

where [parallel]v[[parallel].sub.0] = sup {[parallel]v(t)[parallel]; t [member of] [0, T]} for any v [member of] C([0, T];[[??].sup.n]). Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.16)

Claim 2. Let X := {x [member of] [C.sup.1]([0, T];[[??].sup.n]); x(0) = 0} be the Banach space endowed with the norm [[parallel]c[parallel].sub.1] = [parallel]x[[parallel].sub.0] + [paralle]x'[[parallel].sub.0]. It is clear that the operator L : X [right arrow] C([0, T];[[??].sup.n]) defined by Lx(t) = x'(t) is invertible and [L.sup.-1] is bounded. Consider the operator H : [0, 1] x X [right arrow] C([0, T];[[??].sup.n]) given by

H([lambda], x)(t) = [lambda][L.sup.-1]F(x)(t), 0 [less than or equal to] t [less than or equal to] T (3.17)

where

F(x)(t):= A(t)x(t) + [[integral].sup.t.sub.0] K(t,s)f(s,x(s))ds

One can easily show that H([lambda], *) is a compact homotopoy without fixed points on the boundary of the set U := {x [member of] X : [[parallel]x[[parallel].sub.1] < 1 + [M.sub.0] + [M.sub.1]} where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In fact, we can take

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(see property (c) of a Caratheodory function). We now use the topological transversality theorem (see [?], [?] for definitions and details) to prove that H(1, *) is essential because H(0, *) [equivalent to] 0 is essential. Therefore, H(1, *) has a fixed point in U which is a solution of problem (??) with [lambda] = 1, i.e., a solution of (??).

Remark 3.1 We can follow the arguments in [?] with minor modifications to obtain a solution of (??) for all t > 0. Also, we can consider the case of a non-zero initial condition.

Acknowledgements

The authors are grateful to King Fahd University of Petroleum and Minerals for its constant support.

References

[1] R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, Kluwer Academic Publishers, Dordrecht, 1995.

[2] D. Bainov and P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992.

[3] A. Constantin, Topological transversality: Application to integrodifferential equation, J. Math. Anal. Appl., 197 (1996), 855-863.

[4] J. Dugundji and A. Granas, Fixed Point Theory , Monografie Mat. PWN, Warsaw, 1982.

[5] M. Frigon, Application de la transversalite topologique a des problemes non lineaires pour des equations differentielles ordinaires, Dissert. Math. 296, P.W.N., Warsaw, 1990.

[6] M. Medved, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. Appl. 214 (1997), 349-366.

[7] D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities Involving Functions and Their Integrals and Their Derivatives, Kluwer Academic Publishers, Dordrecht, 1991.

[8] B. G. Pachpatte, On some new inequalities in the theory of differential equations, J. Math. Anal. Appl. 189 (1995), 128-144.

[9] W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York, 1970.

Abdelkader Boucherif ([dagger]) and Yawvi A. Fiagbedzi ([double dagger])

Department of Mathematical Sciences

King Fahd University of Petroleum and Minerals

Dhahran 31261, Saudi Arabia

* Mathematics Subject Classification. Primary 34B10, 34B15; Secondary35B18.

([dagger]) Corresponding author Email:aboucher@kfupm.edu.sa

([double dagger]) E-mail:yawvi@kfupm.edu.sa
No portion of this article can be reproduced without the express written permission from the copyright holder.
srsr123 (Member): request 1/25/2008 2:25 AM
salm
I can not understand what did the question marks indicate
to.