About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their app

About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their app

About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their app

About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their app

Nonlinear Analysis66(2007)2140–2165

http://m.wendangku.net/doc/85e9ed270722192e4536f6fa.html/locate/na

About a generalization of Bellman–Bihari type

inequalities for discontinuous functions and their

applications

Yu.A.Mitropolskiy a,G.Iovane b,?,S.D.Borysenko c

a International Mathematical Center of the National Academy of Sciences of Ukraine,Tereshcenkivska,3,Kyiv,Ukraine

b Department of Engineering of Informatics and Applied Mathematics,University of Salerno,Ponte don Melillo str.,

84084Fisciano,Salerno,Italy

c Department of Differential Equations,National Technical University of Ukraine,“KPI”37Peremohy Prosp.

03056Kyiv,Ukraine

Received3March2006;accepted3March2006

Abstract

In the present paper we introduce the conditions of solvability for Chaplygin’s problem with discontinuous functions in two independent variables,satisfying integro-sum inequalities.The new type of nonlinear integral and Wendroff’s inequality for discontinuous functions are investigated.As applications, the conditions of boundedness solutions of partial differential equations of hyperbolic type with impulse in?uence on some hypersurfaces{Γj}?R2+are obtained.Some historical aspects of the theory of integro-sum inequalities are presented.

c 2006Elsevier Ltd.All rights reserved.

MSC:34B15;26D15;26D20

Keywords:Integral inequalities;Integro-sum inequalities;Impulsive integro-differential equations;Boundedness; Estimates

1.Introduction

The theory of integral inequalities[1,2,4,5,26,28,45]and its numerous linear,nonlinear generalizations for continuous,discontinuous functions of one and n independent variables

?Corresponding author.Tel.:+39089964268;fax:+39089964191.

E-mail addresses:iovane@diima.unisa.it(G.Iovane),borys@http://m.wendangku.net/doc/85e9ed270722192e4536f6fa.html.ua(S.D.Borysenko).

0362-546X/$-see front matter c 2006Elsevier Ltd.All rights reserved.

doi:10.1016/j.na.2006.03.006

Yu.A.Mitropolskiy et al./Nonlinear Analysis66(2007)2140–21652141 have been very important in investigating different qualitative characteristics of solutions, both for ordinary and partial differential equations(functional-differential equations,integro-differential equations,impulsive differential equations,etc.)such as:boundedness,existence, uniqueness,continuous dependence of parameters,stability,attraction,practical stability[4–22, 29–31,35–37,44–64].In the one-dimensional case,all the main results in the theory of integral inequalities for continuous functions are based on the solvability of Chaplygin’s problem [3,56]for the integral inequality

u(t)≤φ(t)+

t

t0

K(t,s,u(s))d s(1)

for establishing the estimate u(t)≤σφ(t),whereσφ(t)is the solution of V olterra’s integral equation

σ(t)=φ(t)+ t

t0

K(t,s,σ(s))d s

as mentioned in[45].

Similarly,for the discrete case like for the continuous case Chaplygin’s problem solvability was investigated for the inequality

u(t)≤φ(t)+

i

K(t,t i,u(t i)).(2)

The works[8–19,21,59–61,63]were dedicated precisely to this question(i.e.the solvability of Chaplygin’s problem).In detail there were investigated inequalities of a certain type:

u(t)≤φ(t)+

t

t0K(t,s,u(s))d s+

t0

Ψ(t,t k)βk(u(t k?0)),(3)

where u(t)is a nonnegative piecewise continuous function with?rst kind discontinuities at the points{t i},t0

The solution of Chaplygin’s problem for(3),described in[15],?nds its generalization in the multidimensional case in[63].

In the works[11,15,60],numerous generalizations of(3)for functional variables(integro-sum functional inequalities)were investigated and the papers[16–19]were devoted to the applications of the method of integro-sum inequalities for solving problems of boundedness, Lyapunov stability,Chetaev stability,attraction of the motion for perturbed impulsive systems of ordinary differential equations.

In Section2we describe some historical aspects of the theory of the integro-sum inequalities (3),dwelling only on the most important results.

In Section3we formulate the conditions of solvability for Chaplygin’s problem for the integro-sum inequality of Wendroff’s type:

u(x)≤φ(x)+

G n H(y,u(y))d y+

n?1

j=1

Γj∩G n

βj(x)u(x)dμφ

j

,(4)

where x=(x1,x2),G n?R2,μis some measure concentrated on the curves{Γj},j=1,∞.

Moreover,we introduce a new integro-sum inequality of Bellman–Bihari type(also with retardation).

2142Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

As an application,we investigate a hyperbolic differential equation with impulse in?uence on hypersurfaces,described by {Γj };it will be obtained in some exact estimates of solutions such as the equations and the boundedness conditions of solutions.All the results of this article use the results from [1–64].2.Integro-sum inequalities 2.1.Previous results

In this section,we formulate the main results on Gronwall–Bellman–Bihari type integral inequalities for discontinuous functions.

Proposition 2.1(Samoilenko and Perestyuk [64],1977).Let a nonnegative piecewise continuous function u (t )satisfy for t ≥t 0the inequality

u (t )≤C + t

t 0

V (τ)u (τ)d τ+

t 0<τi

βi u (τi ),?t ≥t 0,(5)

where C ≥0,βi ≥0,V (τ)>0,and τi are the ?rst kind discontinuity points of the function u (t ).

Then the following estimate holds for the function u (t ),

u (t )≤C

t 0<τi

(1+βi )exp t t 0

V (τ)d τ .(6)Remark 2.1.We formulate the result of Proposition 2.1as in the last work ([64],Lemma 1,

p.12).

Remark 2.2.As for applications of the results of Proposition 2.1see [15,61–64].

Proposition 2.2(Borysenko [8],1982).Let V (t )be a nonnegative piecewise continuous function at t ≥t 0,with ?rst kind discontinuities at the points t i ,and satisfying the integral inequality

V (t )≤C + t

t 0

P (τ)V m (τ)d τ+

t 0

βi V (t i ?0),m >0,m =1,(7)

where t 1

V (t )≤

t 0

(1+βi ) C 1?m +(1?m ) t

t 0

P (τ)d (τ) 1/(1?m ),(8)

if 0

t 0

(1+βi )×

1?(m ?1)C m ?1

t 0

m ?1

(1+βi )

t

t 0

P (τ)d (τ)

?1/(m ?1)

,

(9)

Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–21652143

if m >1,for arbitrary t ∈[t 0,∞[:

t

t 0

P (τ)d (τ)

(m ?1) t 0

m ?1

(1+βi ).Remark 2.3.As for applications of the results of Proposition 2.2see [8,9,15,63].

Proposition 2.3(Perestyuk and Chernikova [62],1984).Let u (t )be a nonnegative piecewise

continuous function at t ≥t 0,with ?rst kind discontinuities at t =τi and satisfying the inequality

u (t )≤C + t

t 0

V (τ)Φ(u (τ))d (τ)+

t 0<τi

βi u (τi ),?t ≥t 0,(10)

where C ≥0,βi ≥0,V (t )is a positive continuous function,Φ(u )is a positive continuous nondecreasing function for 0

u (t )≤Ψ?1

i t τi

V (τ)d τ ,τi

if

t τi

V (τ)d τ<Ψ?1i (ˉu ?0),

where

Ψ?1i (u )=

u

c i

d u

Φ(u ),c i =(1+βi )Ψ?1i τ

τi ?1V (τ)d τ

Ψ0(u )=

u

c

d u

Φ(u ),i =1,2,...,τ0=t 0.Remark 2.4.As for the applications of Proposition 2.3with Φ(u )=u m ,m =1see [63,64].Proposition 2.4(Borysenko [10],1989).Let us consider the integro-sum equation of the

following form

σ(t )=φ(t )+

t

t 0

K (t ,s ,σ(s ))d s +

t 0

Ψ(t ,t k )μk (σ(t k ?0)),(12)where σ(t ),φ(t ),Ψ(t ,t k )are continuous nonnegative functions (k =1,2,...)for t ≥t 0,except

for σ(t ),which has ?rst kind discontinuities at the points t k and

t 0

lim i →∞

t i =∞.

The function K (t ,s ,u ),which is nonnegative at t ≥s ≥t 0,is determined in the domain t ≥s ≥t 0,|u |≤k and at ?xed t and s it is nondecreasing with respect to u;the functions μk (σ)are continuous nonnegative and nondecreasing with respect to σ.

Then,for an arbitrary t ∈[t 0,∞[the estimate u (t )≤σφ(t )exists where σφ(t )is some solution of Eq.(12),continuous in each interval [t k ,t k +1[,k =0,1,...;u (t )is a piecewise

2144Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

continuous function with ?rst kind discontinuities at t i points;this function satis?es the integro-sum inequality:

u (t )≤φ(t )+ t

t 0

K (t ,s ,σ(s ))d s +

t 0

Ψ(t ,t k )μk (σ(t k ?0)),(13)

where σ(t k ?0)=lim t →t k ?0σ(t ).

Corollary 2.1.Let us consider a nonnegative piecewise continuous function V (t ),at t ≥t 0with ?rst kind discontinuities at the points t i ,and satisfying the integro-sum inequality

V (t )≤Ψ(t )+ t

P (τ)V m (τ)d (τ)+

t 0

βi V (t i ?0),m >1,

where t 1

Then the following estimates hold

V (t )≤Ψ(t )

t 0

(1+βi ) 1+(1?m ) t

t 0

Ψm ?1(τ)P (τ)d τ 1/(1?m ),

for 0

V (t )≤Ψ(t )

t 0

(1+βi )exp

t

t 0

P (τ)d τ

for m =1,?t ≥t 0;

V (t )≤Ψ(t )

t 0

(1+βi )

×

1?(m ?1)

t 0

m ?1

(1+βi )

t t 0

Ψm ?1(τ)P (τ)d τ

?1/(m ?1)

,

if m >1,?t ≥t 0:

t t 0

Ψm ?1(τ)P (τ)d τ<

(m ?1)

t 0

(1+βi )m ?1

?1

.

Proposition 2.5(Borysenko [10],1989).Let u (t ,x )be a nonnegative function which is determined in the domain

D =

k ,j >1

D kj ,D kj =((t ,x ):t ∈[t k ?1,t k [,x ∈[x j ?1,x j [,k =1,2,...,j =1,2,...) .

Moreover let it be continuous in D,with the exception of the points {t i ,x i }of ?nite jumps:u (t i ?0,x i ?0)=u (t i +0,x i +0)and satisfy the integro-sum inequality

u (t ,x )≤ψ(t ,x )+q (t ,x ) t t 0 x

x 0

f (ξ,η)u m (ξ,η)d ξd η

+

(t 0,x 0)<(t i ,x i )<(t ,x )

βi u (t i ?0,x i ?0),(14)

q (t 0,x 0)=1,where ψ(t ,x )>0,?(t ,x )∈D it is nondecreasing with respect to (t ,x ):

Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–21652145

?p ≤P,q ≤Q,ψ(p ,q )≤ψ(P ,Q )at (p ,q )∈D,(P ,Q )∈D;q (t ,x )≥1,?(t ,x )∈D,the values βi ≥0?i ∈N,the function f is nonnegative,where f (ξ,η)=0,(ξ,η)∈D lp ,for l =p,for arbitrary l =1,2,...,p =1,2,....Here (t i ,x i )<(t i +1,x i +1),if t i

u (t ,x )≤ψ(t ,x )q (t ,x )Π(t 0,x )

× 1+(1?m )

t t 0

x

x 0

ψm ?1(ξ,η)q m (ξ,η)f (ξ,η)d ξd η 1/(1?m ),if 0

u (t ,x )≤ψ(t ,x )q (t ,x )

(t 0,x )exp

t t 0

x x 0

f (ξ,η)q (ξ,η)d ξd η

,

(15)

if m =1,?(t ,x )∈D;

u (t ,x )≤ψ(t ,x )q (t ,x )Π(t 0,x )

1?(m ?1)Πm ?1(t 0,x )

× t t 0

x x 0

ψm ?1(ξ,η)q m (ξ,η)f (ξ,η)d ξd η

?1/(m ?1)

,

if m >1,?(t ,x )∈D:

t t 0

x

x 0

ψm ?1(ξ,η)q m (ξ,η)f (ξ,η)d ξd η<[(m ?1)Πm ?1(t 0,x )]?1.

Here Π(t 0,x )def

=

(t 0,x 0)<(t i ,x i )<(t ,x )(1+βi q (t i ,x i )).

Let us consider the Euclidean space R n with points x =(x 1,...,x n ),x 0=(x 10,...,x n 0);with the order x 0≤x (x i 0≤x i ),?i =1,...,n .We de?ne x x 0...d u = x 1x 10... x n

x n 0

...d u 1...d u n ,

x 0

αk = x 0

αk .

Let us introduce a space of F -continuous functions F :R n →R n such that:(A)F (x )=(F 1(x ),F 2(x ),...,F n (x )),where F j :R n →R n j =1,2,...,n ;

(B)F (x )≤x ;

(C)lim |x |→∞F j (x )=∞,?j =1,2,...,n .We consider the domain D ?R n :

D =D k 1,...,k n ={x :x 1∈[x k 1?1,x k 1[,...x n ∈[x k n ?1,x k n [,k j =1,2,...,n };We denote by {x k }={x k 1,...,x k n }the points of ?nite jumps of the function u (x ):u (x i ?0)=u (x i +0),?i ∈N .Let us de?ne as F ?the space of functions f (x ):f ≥0;f =0,only if x ∈D k 1,...,k n at k i =k j ,i ,j =1,...,n .

2146Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

Proposition 2.6(Borysenko [11],1997).Let a nonnegative function u (x )be determined in the domain D and satisfy the inequality

u (x )≤ψ(x )+q (x ) x

x 0f (τ)u m (p (τ))d τ+ x

x 0

f (s ) s x 0

g (τ)u m

(σ(τ))d τ d s +

x 0

βi u (x i ?0),(16)

with m >0,and where p (t ),σ(t )∈F ,{x k }are the points of ?nite jumps of u (x ),ψ(x )is a

nondecreasing function,ψ(x )>0,f ∈F ?,q (x )≥1,g (x )≥0,βi ≥0.Then the following estimates are justi?ed:(A)

u (x )≤ψ(x )q (x )Π(x 0

,x ) 1+(1?m )

x

x 0

f (t )

× ψm ?1(t )q m

(p (t ))+ t

x 0

g (τ)ψm ?1(τ)q m (σ(τ))×

ψ(σ(τ))

ψ(τ)

m

d τ d t 1/(1?m )

,

if 0

(B)

u (x )≤ψ(x )q (x )Π(x 0,x )exp

x x 0

Q (τ)d τ

,

if m =1,?x ∈D ;

(C)

u (x )≤ψ(x )q (x )Π(x 0,x ) 1+(1?m )Πm ?1(x 0,x ) x

x 0f (t )

× ψm ?1(t )q m (p (t ))+ t

x

0g (τ)ψm ?1(τ)q m (σ(τ))

×

ψ(σ(τ))ψ(τ)

m d τ d t ?1/(m ?1),if m >1,?x ∈D :

x

x 0f (t ) ψm ?1(t )q m (p (t ))+ t t 0g (τ)ψm ?1(τ)q m (σ(τ)) ψ(σ(τ))

ψ(τ) m d τ d t

<

Π1?m (x 0,x )

m ?1

.Here Π(x 0

,x )=

x 0

(1+βi q (x i )),

Q (t )=

f (t )q (p (t ))ψ(p (t ))+

g (t )q (σ(t ))ψ(σ(t ))

ψ(p (t ))

.

Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–21652147

Proposition 2.7(Borysenko [11],1997).Let us consider a certain integro-sum functional inequality:

υ(x ) ?(x )+q (x ) x

x 0

f (τ)W (υ(p (τ)))d τ+

x 0

βi υ(x i ?0),(17)

where x x 0,q (x ) 1,?(x )is positive nondecreasing,βi =const 0;υ(x )is a piecewise continuous function with ?rst kind discontinuities at the points x i :x 0

=∞,f 0,where W (x )∈ˉF

:(a)W (αβ) W (α)W (β);(b)W :[0,∞[→[0,∞[,W (0)=0;(c)W is nondecreasing.

Let us suppose that p (s )∈F .Then for arbitrary x ∈[x 0,T ],T ∞,the following inequality is justi?ed:

υ(x ) ?(x )q (x )Φ?1

i x

x i f (t )

?(t )W ?(t )d t ,where x ∈]x i ,x i +1[,

x

x i

f (t )??1(t )W ?(t )d t ∈Dom (Φ?1i ),

Φ0(ξ)=

ξ

1

d η

W (η)

,Φi (ξ)=

ξ

l i

d η

W (η)

,i =1,2,...

l i =(1+βi q (x i ))Φ?1i ?1 x i

x i ?1

f (y )??1(y )W ?

(y )d y ,

i =1,2,....

Here W ?(υ)=W (g (p (υ)?(p (υ)))).

Consider a class of functions F 1[2,16,17]:f ∈F 1?:(a)f (x )is positive,continuous,and nondecreasing for x >0;(b)?t >1,u ≥0 ?t ?1f (u )≤f (t ?1u );(c)f (0)=0.

The following statement is valid.

Proposition 2.8(Borysenko [11],1997).Let the piecewise continuous nonnegative function ?(x )with ?rst kind discontinuities at the points {x i }satisfy the inequality (17),where ?,q ,p,satisfy the conditions of Proposition 2.7,function W belongs to the class of functions F 1.Then,for an arbitrary x ∈[x 0,x ?]a certain estimate for υ(x )is justi?ed:

υ(x )≤?(x )q (x )ˉΦ?1i

x x i

G (τ)d τ ,x ∈]x i ,x i +1[,i =0,1,2,...;where

ˉΦ

0(ξ)=

ξ

1

W

?1

(σ)d σ,ˉΦ

i (ξ)=

ξ

l i

W ?1(σ)d σ,

i =1,2,...;

l i =(1+βi q (x i ))Φ?1

i ?1

x i x i ?1

G (τ)d τ

,G (t )=f (t )q (p (t )),x ?

=sup x :

x

x i ?1

G (τ)d τ∈

Dom (ˉΦ?1i ?1

),i =1,2,...

.

2148Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

Proposition 2.9(Samoilenko and Borysenko [60],1998).Let us consider the integro-sum inequality in the following form:

u (x )≤u 0+q 1(x ) x x 0f (s )W 1(u (p (s )))d s +q 2(x ) x

x 0

g (s )W 2(u (σ(s )))d s + x 0

βi u (x i ?0),(18)

where f (x ),g (x ),p (x ),σ(x )are nonnegative continuous functions:

p (s )≤x,?x >x 0,σ(x )≤x ,?x ≥x 0,q 1(x )≥1,?x ≥x 0,q 2(x )≥1,?x ≥x 0;W 1(x )∈F ,W 2(x )∈F 1;u 0=const ≥1,βi =const ≥0,u (x )a piecewise continuous nonnegative function with ?rst kind discontinuities at the points {x i },which satis?es conditions of Proposition 2.7.Then for an arbitrary x ≥x 0a certain estimate holds for u (x ):

u (x )≤q 1(x )q 2(x )S i (x )F ?1

x

x 0

Ψ(ˉx )d ˉx ,(19)

where S i (x )=G ??1i

[ x

x i f (s )q 1(p (s ))d s ].G ?i (ξ)= ξl i W ?11(σ)d σ,i =1,2,...;G ?0(ξ)= ξ

u 0

W ?11(v)d v ;l i =(1+βi )S i ?1(x i ),F (η)=

η

u 0

W ?12(s )d s ,F ?1,G ??1

i are the inverses of the functions F and G ?i respectively and

x

x i f (τ)q 1(p (τ))d τ∈Dom (G ??1

i ),i =1,2,..., x

x 0

Ψ(τ)d τ∈Dom (F ?1),?i =1,2....

Here Ψ(x )=g (x )W 2[q 1(σ(x ))q 2(σ(x ))]S i (σ(x )).

Proposition 2.10(Borysenko [14],2004).Let a nonnegative piecewise continuous on J =[t 0,∞]function V (t ),with ?rst kind discontinuities at points {t i }:t 1

V (t )≤ψ(t )+

t

t 0

q (τ)V (τ)d τ+

t 0

a i V m (t i ?0),where ψ(t )is a positive monotonously nondecreasing on J function,q (t )≥0,a i ≥0,m >0.

Then for V (t )the following estimates hold:

V (t )≤ψ(t )

t 0

(1+a i ψm ?1(t i ))exp t t 0

q (s )d s ,0

t 0

(1+a i ψm ?1(t i ))exp m t t 0

q (s )d s ,

m ≥1,?t ≥t 0.

Remark 2.5.Proposition 2.10generalizes the fundamental results for discontinuous functions reached by Bellman.

Yu.A.Mitropolskiy et al./Nonlinear Analysis66(2007)2140–21652149 Proposition2.11(Borysenko[14],2004).Let the nonnegative piecewise continuous on J =[t0,∞]function V with?rst kind discontinuities at the points{t i}satisfy the inequality:

V(t)≤ψ(t)+

t

t0q(τ)V m(τ)dτ+

t0

a i V m(t i?0),(21)

whereψ(t)>0,q(t)≥0,a i≥0,m>0,m=1,ψ(t)is nondecreasing on all J.

Then V(t)satis?es certain estimates:

V(t)≤ψ(t)

t0

1+(1?m)

t

t0

ψm?1(τ)q(τ)dτ

1/(1?m)

,

(22)

?t≥t0for0

V(t)≤ψ(t)

t0

?

?1?(m?1)

t0

(1+a i mψm?1(t i))

m?1

× t

t0

q(τ)ψm?1(τ)dτ

?

?

?1/(m?1)

,(23)

?t≥t0:

t

t0q(τ)ψm?1(τ)dτ≤

1

m

,m>1,(24)

t0

m

m?1

1/(m?1)

.(25)

Remark2.6.Proposition2.11generalizes the result for discontinuous functions obtained by Bihari.

Proposition2.12(Borysenko,2005).Let the nonnegative function?(t),with?rst kind discontinuities at points{t i}:t1

?(t)≤C+ t

t0

q(s)?(s)d s+

t

t0

q(s)

s

t0

g(σ)?m(σ)dσ

d s

+

t0

βi?(t i?0),if m>0,(26)

where C≥0,q(t)≥0,g(t)≥0,βi=const≥0.Then for the function?(t)certain estimates are obtained:

2150Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

?(t )≤exp

t t 0

q (τ)d τ ?? C

t 0

(1+βi )

1?m +(1?m )

t t 0

g (s )exp (m ?1)

s

t 0

g (σ)d σ

d s ??

1/(1?m )

,

(27)

0

?(t )≤C

t 0

(1+βi )exp t t 0

(q (τ)+g (τ))d τ ,m =1(28)

?t ≥t 0.

?(t )≤C

t 0

(1+βi )exp

t

t 0

q (τ)d τ

·

1?(m ?1)

t 0

(1+βi )m ?1C m ?1

×

t t 0

g (s ) exp (m ?1)

s

t 0

g (σ)d σ

d s

?1/(m ?1)

,

(29)

m >1,?t >t 0:

t t 0

g (s ) exp (m ?1)

s

t 0

g (σ)d σ

d s < (m ?1)

t 0

(1+βi )m ?1C m ?1

?1

.(30)

Remark 2.7.All statements 2.1–2.12can be proved,using the inductive method and the methodology of the integral inequalities theory.We will illustrate this fact by proving Proposition 2.12.

Proof of Proposition 2.12.Suppose,that t ∈[t 0,t 1[.Then

?(t )≤C + t t 0

q (s )?(s )d s + t

t 0

q (s ) s t 0

g (σ)?m

(σ)d σ d s .

De?ne V (t )def

=C + t t 0q (s )?(s )d s + t t 0q (s )( s t 0g (σ)?m (σ)d σ)d s .Obviously,?(t 0)=V (t 0)=C ,?(t )≤V (t ),?t ≥t 0.Then

d V

d t =q (t )?(t )+q (t ) t

t 0

g (σ)?m (σ)d σ≤q (t ) V (t )+ t

t 0

g (σ)V m (σ)d σ .Let W (t )=V (t )+

t

t 0

g (σ)V m (σ)d σ.Then W (t 0)=V (t 0)=C ,V (t )≤W (t ),?t ≥t 0.

It is easy to see that

d W

d t

≤q (t )W (t )+g (t )W m (t ).

Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–21652151

From last differential inequality certain estimates follow for ?(t ):

?(t )≤exp

t t 0

q (τ)d τ C 1?m

+(1?m ) t

t 0

g (τ)×exp

(m ?1)

τ

t 0

q (σ)d σ

d τ

1

1?m

,for 0

?(t )≤C exp

t t 0

(q (τ)+g (τ))d τ

,

for m =1,t ≥t 0

?(t )≤C exp

t

t 0

q (τ)d τ 1?(m ?1)C m ?1

t

t 0

g (τ)

×exp

(m ?1)

t

t 0

q (σ)d σ

d τ

?1m ?

1

,for m >1,and ?t ≥t 0:(31)

t t 0

g (τ)exp (m ?1) t

t 0

g (σ)d σ d τ<[(m ?1)C m ?1]?1.

From (31)?t ∈[t 0,t 1[,?(t )satis?es the inequalities (27)–(29).

Applying the scheme described in [8,63,64]for interval [t k ,t k +1[,k =1,2,...,and the estimates for the function ?(t )on the interval [t k ?1,t k [,we obtained the estimates (by using inductive method)(27)–(29)on all of interval J .

2.2.Some new integro-functional inequalities of Bellman–Bihari type

Suppose that τ(s )∈ is a class of continuous functions τ:R →R ,such that τ(s )≤s ,lim |s |→∞τ(s )=∞.Certain results hold.

Proposition 2.13(Iovane,2005).Let a nonnegative,at t ≥t 0,piecewise continuous function ?(t ),with ?rst kind discontinuities at the points {t i }(t 0

?(t )≤n (t )+ t

t 0

g (s )?(τ(s ))d s +

t 0

βi ?m (t i ?0),?t ≥t 0,(32)

where n (t )is a positive nondecreasing function for t ≥t 0,g (s )≥0,parameter m >0,βi =const ≥0,then ?t ≥t 0the function ?(t )will satisfy a certain estimate:

?(t )≤n (t ) t 0

t 0g (s )

n (τ(s ))

n (s )d s ,(33)if 0

?(t )≤n (t ) t 0

t 0g (s )

n (τ(s ))

n (s )d s ,(34)

if m ≥1,?t ≥t 0.Here ?(t i ?0)=lim t →t i ?0?(t ).

Remark 2.8.From the estimates (33),(34)in some particular cases,well known results in the

theory of integral inequalities for continuous and discontinuous functions can be obtained.If n (t )=c =const >0,τ(s )=s ,βi =0,a classical result of Gronwall and Bellman follows

2152Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

from (33),(34)[4].If βi =0,the results in [2]are obtained;if m =1,n (t )=c =const,τ(s )=s ,the results in [64]can be reached,if m =1,the result in [63]is obtained;if τ(s )=s ,the results in [14]are obtained.For the discrete case,when n (t )=c ,g (s )=0,m =1,the results in [1,p.183]are obtained.

Proposition 2.14(Iovane,2005).Let τ(s )∈ and the nonnegative function ?(t )satisfy the inequality

?(t )≤ψ(t )+q (t ) t

t 0

g (s )?m

(τ(s ))d s +

t 0

βi ?m (t i ?0),?t ≥t 0,(35)

where {t i },satisfying the conditions of Proposition 2.13,are ?rst kind discontinuity points of the function ?(t );ψ(t )is a positive nondecreasing function at t ≥t 0,q (t )≥1,g (t )≥0,?t ≥t 0,the parameter m >0,m =1,and βi =const ≥0.The function ?(t )will satisfy such an estimate:

?(t )≤ψ(t )

t 0

(1+βi ψm ?1(t i )q m (t i ))

× 1+(1?m ) t

t 0

g (s )ψm ?1(s )q m (τ(s ))d s ψ(τ(s ))

ψ(s ) m d s

11?m (36)

?t ≥t 0,0

?(t )≤ψ(t )

t 0

(1+βi m ψm ?1(t i )q m (t i ))

×???1?(m ?1) t 0

(1+βi m ψm ?1(t i )q m (t i )) m ?1

×

t

t 0

g (s )ψm ?1(s )q m (τ(s ))

ψ(τ(s ))ψ(s )

m

d s

???

?1m ?1,if m >1(37)

?t ≥t 0:

t

t 0

g (s )ψm ?1(s )q m (τ(s ))

ψ(τ(s ))ψ(s ) m

d s ≤1

m ,

t 0

(1+βi m ψm ?1(t i )q m (t i ))< 1+1

m ?1 1m ?1.

(38)

Remark 2.9.If βi =0,ψ(t )=c =const >0,q (t )=1,τ(s )=s ,from Proposition 2.14the

result presented by Bihari in [5]follows;if βi =0the result of the theorem coincides with the result given by Akinyele in [2];if q (t )=1,τ(s )=s from the result of the theorem the result presented by Borysenko in [14]follows.

Proposition 2.15(Iovane,2005).Let ?(t )be a nonnegative piecewise continuous function,with ?rst kind discontinuities at the points {t i }:t 1

Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–21652153

following integro-sum inequality:

?(t )≤n (t )+q (t ) t

t 0

f (s )?(σ(s ))d s +

t

t 0f (s )

s

t 0

g (t )?(τ(t ))d t

d s

+

t 0

βi ?m (t i ?0)

,

if m >0,where n (t )is a nondecreasing function,n (t )>0,q (t )≥1,f (s )≥0,σ(t )∈ ,g (t )≥0,βi ≥0.Then the following estimates hold:

?(t )≤n (t )q (t )

t 0

(1+βi q m ?1(t i )n m ?1(t i ))

×exp

t t 0

f (ξ)

g (σ(ξ))n (σ(ξ))+g (ξ)q (τ(ξ))n (τ(ξ))

n (σ(ξ))

d ξ

,

(39)

if 0

?(t )≤n (t )q (t )

t 0

(1+q m (t i )n m ?1(t i ))

× m

t t 0

f (ξ)q (σ(ξ))n (σ(ξ))

+g (ξ)q (τ(ξ))n (τ(ξ))

n (σ(ξ))

d ξ

(40)

if m ≥1.

3.Wendroff type inequalities for discontinuous functions 3.1.Lipschitz type discontinuity

We consider some set D ??R 2,where D ?=D \Γ,D =

j D j ,j =1,2,...;

Γ= j

Γj ,Γj ={(x ,y ):?j (x ,y )=0,j =1,2...},Γk ∩Γk +1=?,k =1,2,...,

?j (x ,y )are real-valued continuously differentiable functions such that grad ?j (x ,y )>0,for all j =1,2,...;

D 1={(x ,y ):x ≥0,y ≥0,?1(x ,y )<0};

D k ={(x ,y ):x ≥0,y ≥0,?k ?1(x ,y )>0,?k (x ,y )<0,?k >2,k ∈N };

G p ={(u ,v):(x ,y )∈D p ,0≤u ≤x ,0≤v ≤y ,p ∈N };μ?n is the Lebesgue Stiltjes measure concentrated on the curves {Γn }.

Let us consider a real-valued nonnegative,discontinuous,nondecreasing function u (x ,y )in D ?,which has ?nite jumps on the curves {Γj }.

Let g (x ,y )be a positive nondecreasing continuous function in R 2+,and let us assume that

u (x ,y )satis?es the following integro-sum inequality in D ?:

u (x ,y )≤g (x ,y )+

G n

Φ(τ,s ,u (τ,s ))d τd s +

n ?1 j =1 Γj ∩G n

W (x ,y ,u (x ,y ))d μφj ;(41)

2154Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

where Φ,W are de?ned in D ?,are nonnegative,nondecreasing functions for a 3D argument,with ?xed ?rst and second argument.

Theorem 3.1.Let the integro-sum equation of the following form be set

σ(x ,y )=g (x ,y )+

G n

Φ(τ,s ,σ(τ,s ))d τd s

+

n ?1 j =1 Γj ∩G n

W (x ,y ,σ(x ,y ))d μ?j ;(42)

where σ(x ,y )is a nonnegative discontinuous function,which has ?nite jumps on the curves {Γj }.The functions g,Φ,W are identities as in (41).Then,for x ≥0,y ≥0we have the estimate

u (x ,y )≤σg (x ,y ),

(43)

where σg (x ,y )is some solution of integro-sum equation (42),continuous in domain D ?,u (x ,y )satis?es inequality (41).

The next results follow from Theorem 3.1.Proposition 3.1(Samoilenko et al.,2002).(A)The estimate

u (x ,y )≤g (x ,y )exp [F 1(x ,y )]

(βj (x ,y )),

(44)

if Φ=f 1(x ,y )u (x ,y ),f 1≥0:f 1∈C (R 2+),W =βj (x ,y )u (x ,y ),βj ∈C (R 2+),

?j =1,2,...,holds for all (x ,y )∈D ?.(B)The estimate

u (x ,y )≤g (x ,y )exp [F 2(x ,y )]

(βj (x ,y ))

· 1+ x 0

y 0

f 3(τ,s )

g ?1(τ,s )exp [?F 2(τ,s )]d τd s (45)

holds,if Φ=f 2(x ,y )u (x ,y )+f 3(x ,y ),with W as in part (A).

(C)The following assertions hold:

(i)The estimate u (x ,y )≤g (x ,y )

(βj (x ,y )) 1+(1?α) x 0

y

f 4(τ,s )

·g α?1(τ,s )d τd s

1/(1?α)

(46)

is true,if 0≤α<1,Φ=f 4(x ,y )u α(x ,y ),α=const >0,α=1,W =βj (x ,y )u (x ,y ).(ii)The estimate u (x ,y )≤g (x ,y ) (βj (x ,y ))

1+(1?α)α?1

(βj (x ,y ))

·

x

y

f 4(τ,s )

g α?1(τ,s )d τd s

?1/(1?α)

Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–21652155

holds for α>1and for an arbitrary (x ,y )∈D ?such that

x 0

y 0

f 4(τ,s )

g α?1(τ,s )d τd s < (α?1)α?1

(βj (x ,y ))

?1.(D)The following assertions hold.

(i)The estimate u (x ,y )≤g (x ,y )

(βj (x ,y ))exp [F 5(x ,y )]· 1+(1?α)

×

x

y

f 6(τ,s )

g α?1(τ,s )exp [(α?1)F 5(τ,s )]d τd s

1/(1?α)

(47)

holds for 0<α<1,Φ=f 5(x ,y )u (x ,y )+f 6(x ,y )u α(x ,y ).(ii)The estimate u (x ,y )≤g (x ,y )

(βj (x ,y ))exp [F 5(x ,y )]

· 1+(1?α)α?1

(βj (x ,y ))

x 0

y 0

f 6(τ,s )

g α?1(τ,s )·exp [(α?1)F 5(τ,s )]d τd s

?1/(α?1)

(48)

is true for α>1and arbitrary (x ,y )∈D ?such that

x 0

y 0

f 6

g α?1exp [(α?1)F 5]d τd s < (α?1)α?1

(βj (x ,y ))

?1.Here

F i (x ,y )def

=

x

y

f i (τ,s )d τd s ,i =1,3,5

(49) (βj (x ,y ))def

=

n ?1 j =1

1+ Γj ∩G n

βj (x ,y )d μ?j .(50)

3.2.Integral inequalities for discontinuous functions with discontinuities of non-Lipschitz type Proposition 3.2.Let a nonnegative function ?(t ,x ),determined in the domain

Ω=[∪k ,j ≥1Ωkj ={(t ,x ):t ∈[t k ?1,t k [,x ∈[x k ?1,x k [},

k =1,2,...,j =1,2,...]

be continuous in Ω,with the exception of the points {t i ,x i }where there is a ?nite jump:

?(t i ?0,x i ?0)=?(t i +0,x i +0),

?i =1,2,...

and satisfying in Ωa certain integro-sum inequality:

?(t ,x )≤a (t ,x )+ t t 0 x

x 0b (ξ,η)?(ξ,η)d ξd η

+

(t 0,x 0)<(t i ,x i )<(t ,x )

γi ?m (t i ?0,x i ?0),

(51)

2156Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

with m >0,t 0≥0,x 0≥0,where a (t ,x )>0,?(t ,x )∈Ωand nondecreasing with respect to (t ,x ):

?p ≤P ,

q ≤Q ?a (p ,q )≤a (P ,Q ),

?(p ,q )∈Ω,(P ,Q )∈Ω;γi =const ≥0,

?i ∈N ,b ≥0

satisfying a certain condition:

b (ξ,η)=0,

if (ξ,η)∈Ωij ,i =j

for arbitrary i ,j =1,2,....Here (t k ,x k )<(t k +1,t k +1),if t k

lim i →∞

t i =∞,

lim i →∞

x i =∞.

Then the function ?(t ,x )satis?es certain estimates:

?(t ,x )≤a (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )

[1+γi a m ?1(t i ,x i )]

×exp

t t 0 x

x 0

b (ξ,η)d ξd η ,if 0

?(t ,x )≤a (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )

[1+γi a m ?1(t i ,x i )]

×exp m

t t 0

x

x 0

b (ξ,η)d ξd η ,if ?m ≥1.

Remark 3.1.If γi =0,a (t ,x )=c =const >0,from the result of Proposition 3.2the classical

result of Wendroff follows (see [26,45]).If m =1,in that particular case we obtain the result in [10].For the one-dimensional case (?(t ,x )=?(t ))the result of the theorem coincides with the result which was presented in [14];if also a (t ,x )=a (t ),b (u ,v)=0,the result of the theorem coincides with the discrete analogous case (for m =1,Corollary 4.12,p.183[1],similar Theorem 4.21,p.190,if m =1).

Proposition 3.3.Let a nonnegative function ?(t ,z ),determined in the domain Ω,be continuous in Ωexcept at the points {t i ,x i }—points of ?nite jumps —and satisfy the following integro-sum inequality

?(t ,x )≤a (t ,x )+

t t 0 x

x 0b (ξ,η)?m (ξ,η)d ξd η+

(t 0,x 0)<(t i ,x i )<(t ,x )

γi ?m (t i ?0,x i ?0),(53)

m >0,m =1,where a,b,γi satisfy the conditions of Proposition 3.2.Then for (t ,x )∈Ω,the

function ?(t ,x )satis?es the inequalities:

?(t ,x )≤a (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )(1+γi a m ?1

(t i ,x i )) 1+(1?m )× t t 0

x

x 0

a

m ?1

(ξ,η)b (ξ,η)d ξd η

1/1?m

,

(54)

Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–21652157

if 0

?(t ,x )≤a (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+m γi a m ?1(t i ,x i ))×??

?1?(m ?1)

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+m γi a m ?1(t i ,x i )) m ?1

× t t 0

x x 0

a m ?1(ξ,η)

b (ξ,η)d ξd η

???

?1m ?1,(55)

if m >1,for an arbitrary (t ,x )∈Ω:

t t 0 x

x 0

a m ?1(ξ,η)

b (ξ,η)d ξd η≤1

m ,

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+m γi a m ?1(t i ,x i ))< 1+

1m ?1

1/m ?1

.

(56)

Remark 3.2.Proposition 3.3gives a new analogy for the generalization of the result of Wendroff on discontinuous functions,which is independent on the result in [10](see also Theorem 3.4.1,p.200[63]).

Proposition 3.4.Let a nonnegative function ?(t ,x )satisfy the conditions of Proposition 3.3and the following inequality hold

?(t ,x )≤a (t ,x )+g (t ,x )

t t 0 x

x 0

b (ξ,η)?m (ξ,η)d ξd η+

(t 0,x 0)<(t i ,x i )<(t ,x )

γi ?m (t i ?0,x i ?0),(57)with m >0,where g (t ,x )≥1,a (t ,x ),b (t ,x ),γi satis?es the conditions of Proposition 3.2.

Then the following estimates are obtained:

?(t ,x )≤a (t ,x )g (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+γi a m ?1(t i ,x i )g m (t i ,x i ))× 1+(1?m )

t t 0

x

x 0

b (ξ,η)a

m ?1

(ξ,η)g m

(ξ,η)d ξd η

1/1?m

,

(58)

if 0

?(t ,x )≤a (t ,x )g (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )(1+γi g (t i ,x i ))

×exp

t t 0

x x 0

b (ξ,η)g (ξ,η)d ξd η

,

(59)

if m =1;

?(t ,x )≤a (t ,x )g (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+m γi a m ?1(t i ,x i )g m (t i ,x i ))

2158Yu.A.Mitropolskiy et al./Nonlinear Analysis 66(2007)2140–2165

×?

?1?(m ?1)

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+m γi a m ?1(t i ,x i )g m (t i ,x i )) m ?1

×

t t 0

x x 0

b (ξ,η)g m (ξ,η)a m ?1d ξd η??

?1/m ?1

,

(60)

if m >1,?(t ,x )∈Ω:

t t 0

x

x 0

b (ξ,η)g m (ξ,η)a m ?1d ξd η≤1/m ,

(61)

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+m γi a m ?1(t i ,x i )g m (t i ,x i ))<

1+

1m ?1

1/m ?1

.

(62)

3.3.Inequalities with retardation

Consider the class of functions of the -class of continuous functions τ(s ):R →R ,such that τ(s )≤s ,lim |s |→∞τ(s )≤∞.Certain results hold.

Proposition 3.5.Let σ∈ .Suppose that the functions ?,a,q,b satisfy the conditions of Proposition 3.4,γi =const ≥0and also ?(t ,x )satis?es a certain inequality

?(t ,x )≤a (t ,x )+q (t ,x ) t

t 0

b (ξ,η)?(σ(ξ),σ(η))d ξd η

+

(t 0,x 0)<(t i ,x i )<(t ,x )

γi ?m (t i ?0,x i ?0),(63)

with m >0.Then for ?(t ,x )the following estimates hold:

?(t ,x )≤a (t ,x )q (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+γi a m ?1(t i ,x i )q m (t i ,x i ))

×exp

t t 0

x x 0

b (ξ,η)a (σ(ξ),σ(η))q (σ(ξ),σ(ξ))a (ξ,η)

d ξd η ,

if m ∈]0,1];

?(t ,x )≤a (t ,x )q (t ,x )

(t 0,x 0)<(t i ,x i )<(t ,x )

(1+γi a m ?1(t i ,x i )q m (t i ,x i ))

×exp m

t t 0

x x 0

b (ξ,η)a (σ(ξ),σ(η))q (σ(ξ),σ(η))a (ξ,η)

d ξd η ,

(64)

if m ≥1.

Remark 3.3.From the result of Proposition 3.5,the results of the investigations in [63]as a particular case follow.If m =1the result of the theorem coincides with Theorem 3.4.3,p.207in [63].

相关推荐