Generalized integral inequalities for discontinuous functions with two independent variables and their applications

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R E S E A R C H

Open Access

Generalized integral inequalities for

discontinuous functions with two

independent variables and their applications

Yuzhen Mi

*

*_{Correspondence:}

[email protected] Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China

Abstract

This paper investigates integral inequalities for discontinuous functions with two independent variables involving two nonlinear terms. We do not require that

ω

(u) is in the class

℘

or the class

j

in Gallo and Piccirllo’s paper (Nonlinear Stud. 19:115-126, 2012). My main results can be applied to generalize Borysenko and Iovane’s results (Nonlinear Anal., Theory Methods Appl. 66:2190-2230, 2007) and to give results similar to Gallo-Piccirllo’s. Examples to show the bounds of solutions of a partial diﬀerential equation with impulsive terms are also given, which cannot be estimated by Gallo and Piccirllo’s results.

MSC: 26D15; 26D20

Keywords: integral inequalities; discontinuous functions; impulse; nonlinear

1 Introduction

Integral inequalities and their various linear and nonlinear generalizations involving con-tinuous or disconcon-tinuous functions play very important roles in investigating different qualitative characteristics of solutions for differential equations, partial differential equa-tions and impulsive differential equaequa-tions such as existence, uniqueness, continuation, boundedness, continuous dependence of parameters, stability, and attraction. The liter-ature on inequalities for continuous functions and their applications is vast (see [–]). In the one-dimensional case, all the main results in the theory of integral inequalities for continuous functions are almost based on the solvability of Chaplygin’s problem [] for the integral inequality

u(x)≤ϕ(x) + x

x

x,s,u(s)ds. (.)

Recently, more attention has been paid to generalizations of Gronwall-Bellman’s results for discontinuous functions and their applications (see [–]). One of the important things is that Samoilenko and Perestyuk [] studied the following inequality:

u(x)≤c+ x

x

f(s)u(s)ds+ x<xi<x

βiu(xi– ) (.)

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for the nonnegative piecewise continuous functionu(x), wherec,βiare nonnegative con-stants,f(x) is a positive function, andxiare the ﬁrst kind discontinuity points of the func-tionu(x). Then Borysenko [] investigated integral inequalities with two independent variables,

u(x,y)≤a(x,y) + x

x y

y

τ(s,t)u(s,t)ds dt

+

(x,y)<(xi,yi)<(x,y)

βiu(xi– ,yi– ). (.)

Hereu(x,y) is an unknown nonnegative continuous function with the exception of the points (xi,yi) where there is a ﬁnite jump:u(xi– ,yi– )=u(xi+ ,yi+ ),i= , , . . . .

In , Borysenko and Iovane [] considered the following inequalities:

u(x,y)≤a(x,y) + x

x y

yo

τ(s,t)u(s,t)ds dt

+

(x,y)<(xi,yi)<(x,y)

βium(xi– ,yi– ), m> , (.)

u(x,y)≤a(x,y) + x

x y

yo

τ(s,t)um(s,t)ds dt

+

(x,y)<(xi,yi)<(x,y)

βium(xi– ,yi– ), m> , (.)

u(x,y)≤a(x,y) +q(x,y) x

x y

yo

τ(s,t)umσ(s),σ(t)ds dt

+

(x,y)<(xi,yi)<(x,y)

βium(xi– ,yi– ), m> . (.)

Later, Gallo and Piccirllo [] studied the following inequalities:

u(x,y)≤a(x,y) +q(x,y) x

x y

yo

τ(s,t)ωuσ(s),τ(t)ds dt

+

(x,y)<(xi,yi)<(x,y)

βium(xi– ,yi– ), m> . (.)

In this paper, motivated by the work above, we will establish the following much more general integral inequality:

u(x,y)≤a(x,y) + 

n= bn(x)

bn(x)

cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt

+g(x,y) (x,y)<(xi,yi)<(x,y)

βium(xi– ,yi– ), m> , (.)

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applied to an estimation of the bounds of the solutions of a partial diﬀerential equation with impulsive terms.

2 Main results

Let

=

i,j≥

ij, ij=

(x,y) :xi–≤x<xi,yj–≤y<yj

,

fori,j= , , . . . ,x>  andy> , and letDz(x,y) denote the ﬁrst-order partial derivative ofz(x,y) with respect toxand _k=uk(x,y) = .

Consider (.) and assume that

(H) a(x,y)is deﬁned onanda(x,y)= ;βiis a nonnegative constant for any positive integeri;

(H) fn(x,y,s,t)(n= , ) are continuous and nonnegative functions on×and satisfy a certain condition:fn(x,y,s,t) = (n= , ) if(s,t)∈ij,i=jfor arbitraryi,j= , , . . .; (H) ω(u)andω(u)are continuous and nonnegative functions on[,∞)and are positive

on(,∞)such thatω_ω(u)_(u) is nondecreasing; (H) g(x,y)is continuous and nonnegative on;

(H) u(x,y)is nonnegative and continuous onwith the exception of the points(xi,yi) where there is a ﬁnite jump:u(xi– ,yi– )=u(xi+ ,yi+ ),i= , , . . .. Here(xi,yi) < (xi+,yi+)ifxi<xi+,yi<yi+,i= , , , . . ., andlimi→∞xi=∞,limi→∞yi=∞; (H) bn(x)andcn(y)(n= , ) are continuously diﬀerentiable and nondecreasing such that

x≤bn(x)≤xon[x,∞)andy≤cn(y)≤yon[y,∞). LetWj(u) =_uu

j

dz

ωj(z) foru≥ujandj= ,  whereujis a given positive constant. Clearly,

Wjis strictly increasing so its inverseWj–is well deﬁned, continuous, and increasing in its corresponding domain.

Theorem . Suppose that(Hk) (k= , . . . , )hold and u(x,y)satisﬁes(.)for a positive constant m.If we let ui(x,y) =u(x,y)for(x,y)∈ii,then the estimate of u(x,y)is recursively

given,for(x,y)∈ii,i= , , . . . ,by

ui(x,y)≤W–

W◦W–

W

ri(x,y)

+ b(x)

b(xi–)

c(y)

c(yi–)

f(x,y,s,t)ds dt

+ b(x)

b(xi–)

c(y)

c(yi–)

f(x,y,s,t)ds dt

, (.)

where

r(x,y) = max

x≤ξ≤x,y≤η≤ya(ξ,η), fn(x,y,s,t) =x≤ξ≤maxx,y≤η≤yfn(ξ,η,s,t),

ri(x,y) =r(x,y) + i–

k= 

n= bn(xk)

bn(xk–)

cn(yk)

cn(yk–)

fn(x,y,s,t)ωn

uk(s,t)ds dt

+g(x,y) i–

k=

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provided that

W

ri(x,y)+ b(x)

b(xi–)

c(y)

c(yi–)

f(x,y,s,t)ds dt≤

∞

u

dz

ω(z) ,

W◦W–

W

ri(x,y)

+ b(x)

b(xi–) c(y)

c(yi–)

+ b(x)

b(xi–)

c(y)

c(yi–)

f(x,y,s,t)ds dt≤

∞

u

dz

ω(z). (.)

The proof is given in Section .

Remark . Ifωjsatisﬁes u

uj

dz

ωj(z) =∞forj= , , theniin Theorem . can be any nonzero

integer. Reference [] pointed out that different choices ofujinWjdo not affect our results forj= , . Ifa(x,y)≡, then defineW() =  and (.) is still true.

Remark . Ifa(x,y) is nondecreasing, Theorem . generalizes many known results. For example:

() If we takef(x,y,s,t) =τ(s,t),f(x,y,s,t) = ,ω(u) =u,m= ,b(x) =x,c(y) =yand

g(x,y) = , then (.) reduces to (.). It is easy to check thatW(u) =ln_uu andW–  (u) = ueu. From Theorem ., we know that for (x,y)∈ii

ui(x,y)≤ri(x,y)e

x xi–

y

yi–τ(s,t)ds dt

with

ri(x,y) =a(x,y) + i–

k= xk

xk– yk

yk–

τ(s,t)uk(s,t)ds dt+ i–

k=

βkuk(xk– ,yk– ).

Hence

r(x,y) =a(x,y), u(x,y) =a(x,y)e

x x_

y

y_τ(s,t)ds dt_,

r(x,y) =a(x,y)( +β)e

x_ x

y_ yτ(s,t)ds dt_,

u(x,y) =a(x,y)( +β)e

x x_

y

y_τ(s,t)ds dt_.

After recursive calculations, we have

u(x,y)≤a(x,y)(x,y)<(xi,yi)<(x,y)( +βi)e x

x

y

yτ(s,t)ds dt_,

which is the same as the expression in [];

() If we takef(x,y,s,t) =τ(s,t),f(x,y,s,t) = ,ω(u) =u,m> ,b(x) =x,c(y) =yand

g(x,y) = , then (.) reduces to (.) and Theorem . becomes Theorem . in []; () If we takef(x,y,s,t) =τ(s,t),f(x,y,s,t) = ,ω(u) =um_,_m_{> ,}_b

(x) =x,c(y) =yand

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() Ifσ(t) >  on [t,∞) wheret=min{x,y}, then (.) can be rewritten as

u(x,y)≤a(x,y) +q(x,y) σ(x)

σ(x) σ(y)

σ(yo)

τ(σ–₍_s_),_σ–₍_t₎₎

σ(σ–₍_s₎₎_σ₍_σ–₍_t₎₎u

m₍_s_,_t₎_{ds dt}

+

(x,y)<(xi,yi)<(x,y)

βium(xi– ,yi– ). (.)

If we letf(x,y,s,t) =q(x,y) τ(σ

–_(s),_σ–_(t))

σ(σ–_(s))_σ₍σ–_(t)),f(x,y,s,t) = , andω(u) =um, the above in-equality is the same as (.).

Consider the inequality

ϕu(x,y)≤a(x,y) + 

n= bn(x)

bn(x) cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt

+g(x,y) (x,y)<(xi,yi)<(x,y)

βiψ

u(xi– ,yi– )

, (.)

which looks much more complicated than (.).

Corollary . Suppose that(Hk) (k= , . . . , )hold,ψ(u)is positive on(,∞),ϕ(u)is pos-itive and strictly increasing on(,∞)and u(x,y)satisﬁes(.).If we let ui(x,y) =u(x,y)

for(x,y)∈ii,then the estimate of u(x,y)is recursively given,for(x,y)∈ii,i= , , . . . ,

by

ui(x,y)≤ϕ–

W_–

W◦W–

W

ri(x,y)+ b(x)

b(xi–)

c(y)

c(yi–)

+ b(x)

b(xi–)

c(y)

c(yi–)

, (.)

where Wj(u) =_uu

j

dz

ωj(ϕ–(z)),r(x,y)andfn(x,y,s,t)are given in Theorem.,ri(x,y)is deﬁned

as follows:

ri(x,y) =r(x,y) + i–

k= 

n= bn(xk)

bn(xk–)

cn(yk)

cn(yk–)

fn(x,y,s,t)ωn

uk(s,t)

ds dt

+g(x,y) i–

k=

βkψ

uk(xk– ,yk– )

.

Proof Letϕ(u(x,y)) =h(x,y). Since the functionϕis strictly increasing on [,∞), its in-verseϕ–is well deﬁned. Equation (.) becomes

h(x,y)≤a(x,y) + 

n= bn(x)

bn(x)

cn(y)

cn(yo)

fn(x,y,s,t)ωn

ϕ–h(s,t)ds dt

+g(x,y) (x,y)<(xi,yi)<(x,y)

βiψ

ϕ–h(xi– ,yi– )

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Letωn=ωn◦ϕ–andψ=ψ◦ϕ–. Equation (.) becomes

h(x,y)≤a(x,y) + 

n= bn(x)

bn(x) cn(y)

cn(yo)

fn(x,y,s,t)ωn

h(s,t)ds dt

+g(x,y) (x,y)<(xi,yi)<(x,y)

βiψ

h(xi– ,yi– )

. (.)

It is easy to see thatψ(u) > ,ω(u) andω(u) are continuous and nonnegative functions on [,∞), andω(u)

ω(u) is nondecreasing on (,∞). Even thoughψ(u) is much more general, in the same way as in Theorem ., for (x,y)∈ii,i= , , . . . , we can obtain the estimate ofu(x,y),

ui(x,y)≤ϕ–

W_–

W◦W–

W

ri(x,y)+ b(x)

b(xi–)

c(y)

c(yi–)

+ b(x)

b(xi–)

c(y)

c(yi–)

. (.)

This completes the proof of Corollary ..

Ifϕ(u) =uλ_where_λ_{>  is a constant, we can study the inequality}

uλ₍_x_,_y₎_≤_a₍_x_,_y_{) +}



n= bn(x)

bn(x) cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt

+g(x,y) (x,y)<(xi,yi)<(x,y)

βiψ

u(xi– ,yi– )

. (.)

According to Corollary ., we have the following result.

Corollary . Suppose that(Hk) (k= , . . . , )hold,ψ(u) > ,and u(x,y)satisﬁes(.).

If we let ui(x,y) =u(x,y)for(x,y)∈ii,then the estimate of u(x,y)is recursively given,for

(x,y)∈ii,i= , , . . . ,by

ui(x,y)≤

W_–

W◦W–

W

ri(x,y)+ b(x)

b(xi–)

c(y)

c(yi–)

+ b(x)

b(xi–)

c(y)

c(yi–)



λ

, (.)

where Wj(u) =_uu

j

dz

ω(zλ)

,r(x,y),ri(x,y)andfn(x,y,s,t)are given in Corollary..

3 Proof of Theorem 2.1

Obviously, for any (x,y)∈,r(x,y) is positive and nondecreasing with respect toxand

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We ﬁrst consider (x,y)∈={(x,y) :x≤x<x,y≤y<y}and have from (.)

u(x,y)≤a(x,y) + 

n= bn(x)

bn(x) cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt

≤r(x,y) + 

n= bn(x)

bn(x) cn(y)

cn(yo)

_f_n(x,y,s,t)ωn

u(s,t)ds dt. (.)

Take any ﬁxedx∈(x,x),y∈(y,y), and for arbitraryx∈[x,x],y∈[y,y] we have

u(x,y)≤r(x,y) + 

n= bn(x)

bn(x) cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt. (.)

Let

z(x,y) =r(x,y) + 

n= bn(x)

bn(x) cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt (.)

andz(x,y) =r(x,y). Hence,u(x,y)≤z(x,y). Clearly,z(x,y) is a nonnegative, nondecreas-ing and diﬀerentiable function forx∈[x,x] andy∈[y,y]. Moreover,bn(x) (orcn(y)) is diﬀerentiable and nondecreasing inx∈[x,x] (ory∈[y,y]) forn= , . Thus,bn(x)≥ (orc_n(y)≥) forx∈[x,x] (ory∈[y,y]). Sincer(x,y) >  andω(z(x,y)) > , we have

Dz(x,y)

ω(z(x,y))

≤

c(y)

c(yo)f(x,y,b(x),t)b(x)ω(u(b(x),t))dt

ω(z(x,y)) +

c(y)

c(yo)f(x,y,b(x),t)b(x)ω(u(b(x),t))dt

ω(z(x,y))

≤

c(y)

c(yo)f(x,y,b(x),t)b

(x)ω(z(b(x),t))dt

ω(z(x,y)) +

c(y)

c(yo)f(x,y,b(x),t)b

(x)ω(z(b(x),t))dt

ω(z(x,y))

≤

c(y)

c(yo)f(x,y,b(x),t)b

(x)ω(z(x,t))dt

ω(z(x,y)) +

c(y)

c(yo)f(x,y,b(x),t)b(x)ω(z(b(x),t))dt

ω(z(x,y))

≤

c(y)

c(y) f

x,y,b(x),tb_(x)dt

+ c(y)

c(y) f

x,y,b(x),tb_(x)ω(z(b(x),t))

ω(z(b(x),t))

dt. (.)

Integrating both sides of the above inequality fromxtox, we obtain

W

z(x,y)–W

z(x,y)

≤

x

x c(y)

c(y) f

x,y,b(s),t

b_(s)ds dt

+ x

x c(y)

c(y) f

x,y,b(s),tb_(s)ω(z(b(s),t))

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Thus,

W

z(x,y)≤W

r(x,y)

+ b(x)

b(x) c(y)

c(y)

+ b(x)

b(x) c(y)

c(y)

f(x,y,s,t)φz(s,t)ds dt

forx≤x≤xandy≤y≤y, whereφ(u) =ωω(u)(u), or equivalently

ξ(x,y)≤W

r(x,y)+ b(x)

b(x) c(y)

c(y)

+ b(x)

b(x) c(y)

c(y)

f(x,y,s,t)φ

W_–ξ(s,t)ds dtz(x,y), (.)

where

ξ(x,y) =W

z(x,y).

It is easy to check thatξ(x,y)≤z(x,y),z(x,y) =W(r(x,y)) andz(x,y) is diﬀerentiable, positive and nondecreasing on (x,x] and (y,y]. Sinceφ(W–(u)) is nondecreasing from the assumption (H), we have

Dz(x,y)

φ(W_–(z(x,y)))≤ c(y)

c(y)f(x,y,b(x),t)b(x)dt

φ(W_–(z(x,y))) +

c(y)

c(y)f(x,y,b(x),t)b(x)φ(W–(ξ(b(x),t)))dt

φ(W–

 (z(x,y)))

≤

c(y)

φ[W_–(W(r(x,y)) +_b(x)b(x) _c(y)c(y)f(x,y,s,t)ds dt)]

+ c(y)

c(y)f(x,y,b(x),t)b(x)φ(W–(z(x,t)))dt

φ(W–

 (z(x,y)))

≤

c(y)

φ[W_–(W(r(x,y)) +_b(x)b(x) _c(y)c(y)f(x,y,s,t)ds dt)] +

c(y)

c(y) f

x,y,b(x),tb_(x)dt. (.)

Note that

x

x

Dz(s,y)

φ(W–

 (z(s,y)))

ds= x

x

Dz(s,y)ω(W–(z(s,y)))

ω(W–

 (z(s,y)))

ds= W–

 (z(x,y))

W–  (z(x,y))

du

ω(u) =W◦W–

z(x,y)

–W◦W–

z(x,y)

=W◦W–

z(x,y)

–W◦W–

W

r(x,y) =W◦W–

z(x,y)

–W

(9)

Integrating both sides of the inequality (.) fromxtox, we obtain

W◦W–

z(x,y)–W

r(x,y) =

x

x

Dz(s,y)

φ(W–

 (z(s,y)))

ds

≤

x

x

c(y)

c(y)f(x,y,b(s),t)b(s)dt

φ[W_–(W(r(x,y)) +_b(x)b(s) _c(y)c(y)f(x,y,τ,t)dτdt)]ds +

x

x c(y)

c(y) f

x,y,b(s),tb_(s)ds dt

≤W◦W–

W

r(x,y)

+ b(x)

b(x) c(y)

c(y)

–W

r(x,y)

+ b(x)

b(x) c(y)

c(y)

f(x,y,s,t)ds dt.

Thus,

W◦W–

z(x,y)

≤W◦W–

W

r(x,y)

+ b(x)

b(x) c(y)

c(y)

+ b(x)

b(x) c(y)

c(y)

_f_(_x_,_y_,_s_,_t₎_{ds dt}_.

Hence

u(x,y)≤z(x,y)≤W_–ξ(x,y)≤W_–z(x,y)

≤W_–

W◦W–

W

r(x,y)+ b(x)

b(x) c(y)

c(y)

+ b(x)

b(x) c(y)

c(y)

.

Since the above inequality is true for anyx∈[x,x],y∈[y,y], we obtain

u(x,y)≤W_–

W◦W–

W

r(x,y)

+ b(x)

b(x) c(y)

c(y)

+ b(x)

b(x) c(y)

c(y)

. (.)

Replacingxbyxandybyyyields

u(x,y)≤W_–

W◦W–

W

r(x,y)+ b(x)

b(x) c(y)

c(y)

+ b(x)

b(x) c(y)

c(y)

. (.)

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Fori=  and (x,y)∈={(x,y) :x≤x<x,y≤y<y}, (.) becomes

u(x,y)≤r(x,y) + 

n= bn(x)

bn(x)

cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt

+g(x,y)βum(x– ,y– )

+ 

n= bn(x)

bn(x)

cn(y)

cn(y)

fn(x,y,s,t)ωn

u(s,t)ds dt

≤r(x,y) + 

n= bn(x)

bn(x)

cn(y)

cn(y)

fn(x,y,s,t)ωn

u(s,t)ds dt, (.)

where the deﬁnition ofr(x,y) is given in (.). Note that the estimate ofu(x,y) is known. Clearly, (.) is the same as (.) if replacer(x,y) and (x,y) byr(x,y) and (x,y). Thus, by (.) we have

u(x,y)≤W_–

W◦W–

W

r(x,y)+ b(x)

b(x) c(y)

c(y)

+ b(x)

b(x) c(y)

c(y)

.

This implies that (.) is true for (x,y)∈andi=  if replaceu(x,y) byu(x,y). Assume that (.) is true for (x,y)∈ii={(x,y) :xi–≤x<xi,yi–≤y<yi},i.e.,

ui(x,y)≤W_–

W◦W–

W

ri(x,y)+ b(x)

b(xi–)

c(y)

c(yi–)

+ b(x)

b(xi–)

c(y)

c(yi–)

. (.)

For (x,y)∈i+,i+={(x,y) :xi≤x<xi+,yi≤y<yi+}, (.) becomes

u(x,y)≤a(x,y) + 

n= bn(x)

bn(x)

cn(y)

cn(yo)

fn(x,y,s,t)ωn

u(s,t)ds dt

+g(x,y) (x,y)<(xi,yi)<(x,y)

βium(xi– ,yi– )

≤r(x,y) + i

k= 

n= bn(xk)

bn(xk–)

cn(yk)

cn(yk–)

fn(x,y,s,t)ωn

uk(s,t)ds dt

+g(x,y) i

k=

βkumk(xk– ,yk– )

+ 

n= bn(x)

bn(xi)

cn(y)

cn(yi)

fn(x,y,s,t)ωn

u(s,t)ds dt

≤ri+(x,y) + 

n= bn(x)

bn(xi)

cn(y)

cn(yi)

fn(x,y,s,t)ωn

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where we use the fact that the estimate of u(x,y) is already known for (x,y)∈ii (i= , , . . .). Again, (.) is the same as (.) if replacer(x,y) and (x,y) byri+(x,y) and (xi,yi). Thus, by (.) we have

u(x,y)≤W_–

W◦W–

W

ri+(x,y)

+ b(x)

b(xi)

c(y)

c(yi)

+ b(x)

b(xi)

c(y)

c(yi)

.

This shows that (.) is true for (x,y)∈i+,i+if replaceu(x,y) byui+(x,y). By induction, we know that (.) holds for (x,y)∈i+,i+for any nonnegative integeri. This completes the proof of Theorem ..

4 Applications

Consider the following partial diﬀerential equation with an impulsive term:

⎧ ⎪ ⎨ ⎪ ⎩

∂v(x,y)

∂x∂y =H(x,y,v(x,y)), (x,y)∈ii,x=xi,y=yi,

v|x=xi,y=yi=Ii(v),

v(x,y) =φ(x), v(x,y) =φ(y), φ(x) =φ(y)= ,

(.)

wherev∈R,H∈R,Ii∈R, andi= , , . . . . Assume that

(C) |H(x,y,v(x,y))| ≤h(x,y)e|v(x,y)|+h(x,y)e|v(x,y)|whereh,hare nonnegative and con-tinuous on,h(x,y) = ,h(x,y) = for(x,y)∈ij,i=j,i,j= , , . . .;

(C) |Ii(v)| ≤βi|v|mwhereβiandmare nonnegative constants.

Corollary . Suppose that(C)and(C)hold.If we let vi(x,y) =v(x,y)for(x,y)∈ii,then the solution of system(.)has an estimate for(x,y)∈ii

vi(x,y)≤– ln

e–ri(x,y)_–

x

xi– y

yi–

h(s,t)ds dt 

– x

xi– y

yi–

h(s,t)ds dt

, (.)

where

r(x,y) = max x≤ξ≤x,y≤η≤y

φ(ξ) +φ(η) –φ(x)> ,

ri(x,y) =r(x,y) + i–

k= xk

xk– yk

yk–

h(s,t)e|vk(s,t)|ds dt

+ i–

k= xk

xk– yk

yk–

h(s,t)e|vk(s,t)|ds dt+ i–

k=

βkvk(xk– ,yk– ) m

,

e–ri(x,y)_–

x

xi– y

yi–

h(s,t)ds dt 

– x

xi– y

yi–

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Proof The solution of (.) with an initial value is given by

v(x,y) =v(x,y) +v(x,y) –v(x,y) + x

x y

y

Hs,t,v(s,t)ds dt

+

(x,y)<(xi,yi)<(x,y)

Ii

v(xi– ,yi– )

=φ(x) +φ(y) –φ(x) + x

x y

y

Hs,t,v(s,t)ds dt

+

(x,y)<(xi,yi)<(x,y)

Ii

v(xi– ,yi– )

, (.)

which implies

v(x,y)≤φ(x) +φ(y) –φ(x)+ x

x y

y

h(s,t)e|v(s,t)|ds dt

+ x

x y

y

h(s,t)e|v(s,t)|ds dt+ (x,y)<(xi,yi)<(x,y)

βiv(xi– ,yi– ) m

. (.)

Let

u(x,y) =v(x,y), a(x,y) =φ(x) +φ(y) –φ(x), b(x) =b(x) =x,

c(y) =c(y) =y, g(x,y) = , ω(u) =eu, ω(u) =eu,

f(x,y,s,t) =h(s,t), f(x,y,s,t) =h(s,t).

Thus, (.) is the same as (.). It is easy to see that for any positive constantsuandu

r(x,y) = max

x≤ξ≤x,y≤η≤y

a(ξ,η)> ,

f(x,y,s,t) =h(s,t), f(x,y,s,t) =h(s,t),

W(u) = u

u

dz

ω(z)= u

u

e–zdz=e–u–e–u, W_–(u) = –lne–u–u,

W(u) = u

u

dz

ω(z) =

u

u

e–zdz= 

e–u–e–u,

W_–(u) = – ln

e–u– u,

ri(x,y) =r(x,y) + i–

k= xk

xk– yk

yk–

h(s,t)e|vk(s,t)|_{ds dt}

+ i–

k= xk

xk– yk

yk–

h(s,t)e|vk(s,t)|_{ds dt}₊

i–

k=

βkvk(xk– ,yk– ) m

.

Therefore, for any nonnegativeiand (x,y)∈ii vi(x,y)≤–

ln

e–ri(x,y)_–

x

xi– y

yi–

h(s,t)ds dt

 –

x

xi– y

yi–

h(s,t)ds dt

(13)

provided that

e–ri(x,y)_–

x

xi– y

yi–

h(s,t)ds dt

 –

x

xi– y

yi–

h(s,t)ds dt> .

Remark . From (.), we knowω(u) =eu_{. Clearly,} _ω

(u) =eu≤ω()ω(u) =eeu does not hold for largeu> . Thus,ω(u) =eu does not belong to the class℘ in []. Againω(u_) =eu ≥

ω(u) =  e

u_{does not hold for large}_u_{>  so}_ω_(_u_{) does not belong to} the classjin []. Hence, the results in [] cannot be applied to the inequality (.).

Competing interests

The author declares that she has no competing interests.

Acknowledgements

The author is grateful to Dr. Shengfu Deng for many helpful conversations during the course of this work. This research was supported by National Natural Science Foundation of China (No. 11371314), Guangdong Natural Science Foundation (No. S2013010015957), and the Project of Department of Education of Guangdong Province, China (No. 2012KJCX0074).

Received: 29 October 2013 Accepted: 8 May 2014 Published: 15 December 2014 References

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doi:10.1186/1029-242X-2014-524