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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 An integro-differential equation is an equation which has both integrals and derivatives of an unknown function. The equation is of the form

where
, A priori a priori

In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience.
 bound on solutions, Essential maps, Topological to·pol·o·gy  
n. pl. to·pol·o·gies
1. Topographic study of a given place, especially the history of a region as indicated by its topography.

2.
 transversality Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology.  theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. .

1. Introduction

Differential inequalities have played a major role in the study of the existence, uniqueness, stability of solutions of ordinary differential equations ordinary differential equation

Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function).
. For more details, one can consult the monographs [?], [?], [?], [?] and the papers [?], [?] and references therein.

In this paper, we present a nonlinear A system in which the output is not a uniform relationship to the input.

nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input.
 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 infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ]) [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 A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .]

(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 lim
abbr.
Mathematics limit
.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 Noun 1. first derivative - the result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx
derivative, derived function, differential, differential coefficient
 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 de·note  
tr.v. de·not·ed, de·not·ing, de·notes
1. To mark; indicate: a frown that denoted increasing impatience.

2.
 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 summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) ].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 The Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain side conditions which are given on a hypersurface in the domain. It is an extension of the initial value 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 (mathematics) Banach space - A complete normed vector space. Metric is induced by the norm: d(x,y) = ||x-y||. Completeness means that every Cauchy sequence converges to an element of the space.  endowed en·dow  
tr.v. en·dowed, en·dow·ing, en·dows
1. To provide with property, income, or a source of income.

2.
a.
 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 in·vert  
v. in·vert·ed, in·vert·ing, in·verts

v.tr.
1. To turn inside out or upside down: invert an hourglass.

2.
 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 King Fahd University of Petroleum and Minerals (KFUPM or UPM) (Arabic:جامعة الملك فهد للبترول و  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 anal (a´n'l) relating to the anus.

a·nal
adj.
1. Of, relating to, or near the anus.

2.
. Appl., 197 (1996), 855-863.

[4] J. Dugundji and A. Granas, Fixed Point Theory , Monografie Mat. PWN In gaming, to trounce an opponent. To be "pwned" is to be defeated unmercifully. Pronounced "pone," "pwen," "pawn" or "pun," the derivation of the term is obscure. Some believe it came from a common typo of "own" because the o and p keys are next to each other. , Warsaw, 1982.

[5] M. Frigon, Application de la transversalite topologique a des problemes non lineaires pour des equations differentielles ordinaires, Dissert dis·ser·tate   also dis·sert
intr.v. dis·ser·tat·ed also dis·sert·ed, dis·ser·tat·ing also dis·sert·ing, dis·ser·tates also dis·serts
To discourse formally.
. 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 fink   Slang
n.
1. A contemptible person.

2. An informer.

3. A hired strikebreaker.

intr.v. finked, fink·ing, finks
1. To inform against another person.
, 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 differential equation

Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions.
, J. Math. Anal. Appl. 189 (1995), 128-144.

[9] W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
, 1970.

Abdelkader Boucherif ([dagger]) and Yawvi A. Fiagbedzi ([double dagger double dagger
n.
A reference mark () used in printing and writing. Also called diesis.

Noun 1.
])

Department of Mathematical Sciences

King Fahd University of Petroleum and Minerals

Dhahran 31261, Saudi Arabia Saudi Arabia (sä`dē ərā`bēə, sou`–, sô–), officially Kingdom of Saudi Arabia, kingdom (2005 est. pop.  

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

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

([double dagger]) E-mail:yawvi@kfupm.edu.sa
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Author:Fiagbedzi, Yawvi A.
Publication:Tamsui Oxford Journal of Mathematical Sciences
Geographic Code:7SAUD
Date:Nov 1, 2005
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