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R E S E A R C H

Open Access

On some new weakly singular Volterra

integral inequalities with maxima and their

applications

Yong Yan

*

*Correspondence: kdyan698@163.com Department of Mathematics, Sichuan University for Nationalities, Kangding, Sichuan 626001, P.R. China

Abstract

In this paper, we consider a general form of nonlinear integral inequalities with the unknown function composed with a given function on the left hand side, more than one distinct nonlinear integrals on its right-hand side, and weakly singular kernels, and involving maxima of unknown function. Requiring neither monotonicity nor separability of given functions, we apply monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply the obtained result to a boundary value problem of integro-differential equations with maxima for uniqueness.

MSC: 26D10; 26D15

Keywords: integral inequalities; weakly singular; integro-differential equations with maxima; uniqueness

1 Introduction

The Gronwall-Bellman inequality [, ] is an important tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equations. There can be found a lot of its generalizations in various cases in the literature (e.g.[–]). Lipovan [] investigated the retarded integral inequality

u(t)≤c+

t

t

f(su(s)ds+

α(t)

α(t)

g(su(s)ds, t≤t<t.

Their results were further generalized by Agarwalet al.[] in  to the inequality

u(t)≤a(t) +

n

i=

αi(t) αi(t)

fi(t,si

u(s)ds, t≤t<t.

Another aspect of integral inequalities is to consider the unknownucomposed with a given function on the left hand side, which has been developed (see [–]). On the basis of discussion (see [–]) on integral inequalities in multi-variables.

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In recent years, many researchers have devoted many efforts to investigating weakly sin-gular integral inequalities and their applications (see [–]). In  McKee [] con-sidered the following initial value problem:

y(t) =f(t,y) +c

t

y(s)

(ts)αds+q(t), ≤tT, y() =y, (.)

whenα=for the diffusion of discrete particle in a turbulent fluid. Henry [] used inte-gral inequality with singular kernel to prove global existence and exponential decay results for a parabolic differential equation. Medved [] presented a new method to discuss non-linear singular integral inequalities of Henry type

u(t)≤a(t) +b(t)

t

t

(ts)β––F(s)u(s)ds, t≥. (.)

In  Ma and Pečairé [] considered the following nonlinear singular inequalities with power nonlinearity:

up(t)≤a(t) +b(t)

t

t

β––f(s)uq(s)ds, t≥. (.)

Along with the development of automatic control theory and its applications to compu-tational mathematics and modeling, attention was also put to integral inequalities with the maxima of the unknown function. Actually, many problems in the control theory can be modeled in the form of differential equations with the maxima of the unknown func-tion [, ]. For example, the equafunc-tion describing the work of the regulator [] can be presented as

Tu(t) +u(t) +q max

s∈[th,t]u(s) =f(t), (.)

whereTandqare constants. Equations involving maxima of an unknown function are

called differential equations with maxima [, ]. Such a problem again requires a new type of integral inequalities as a tool to investigate its qualitative properties. There have been given some results for integral inequalities containing the maxima of the unknown function [–].

In  Thiramanuset al.[] considered the following system of integral inequalities:

u(t)≤r(t) +

t

t

(ts)α–p(s)u(s) +q(s) max

ξ∈[βs,s]u(ξ)

ds, t∈[t,T),

u(t)≤φ(t), t∈[βt,t],

(.)

whereα> ,  <β< ,r,p,q, andφare nonnegative continuous functions. In this paper we generally consider the system of integral inequalities

ϕu(t)≤a(t) +

m

i=

bi(t) bi(t)

tαisαiki(βi–)

sqi(γi–)g i(t,si

u(s)ds

+

m+n

j=m+

bj(t) bj(t)

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×ωj

max ξ∈[cj(s)–h,cj(s)]

fu(ξ) ds, t∈[t,t), (.)

u(t)≤ψ(t), tb∗(t) –h,t

,

wherea,f,gi’s, andωi’s are nonnegative continuous functions,bi’s are nonnegative

con-tinuously differentiable and nondecreasing functions and b∗(t) :=min{min≤imbi(t),

minm+≤jmcj(bj(t))}. As required in previous work [, ], we suppose that ≤bi(t)≤t,

h> , is a constant and theωi’s are definite positive,i.e.,ωi(s) >  fors> . In this paper

we require neither monotonicity ofa,ωi’s,gi’s andg nora(t)≥. We monotonize those

ωi’s to make a sequence of functions in which each possesses stronger monotonicity than

the previous one so as to give an estimation for the unknown function. Finally, we ap-ply the obtained result to a boundary value problem of integro-differential equations with maxima for uniqueness.

2 Main result

Consider system (.) of integral inequalities witht<tinR+:= [,∞).C(M,S) denotes

the class of all continuous functions defined on set Mwith range in the setS.B(ξ,η) =

 s

ξ–( –s)η–ds,ηC,Reξ> ,Reη> ) is the well-known beta function. As in [],

we sayμ∝μforμ,μ:A⊂R→R\ {}ifμ(s)/μ(s) is nondecreasing onA.

Suppose that

(H) allbi: [t,t)→R+(i= , , . . . ,m+n),cj: [t,t)→R+(j=m+ ,m+ , . . . ,m+n)

are continuously differentiable and nondecreasing such thatbi(t)≤t,cj(t)≤ton

[t,t);

(H) f,ϕ:R+→R+, andψ: [b∗(t) –h,t]→R+are continuous functions,ϕis strictly

increasing such thatlimt→+∞ϕ(t) = +∞;

(H) allgi(t,s)(i= , , . . . ,m+n) are continuous and nonnegative functions on

[t,t)×[b∗(t),t);

(H) allωi(i= , , . . . ,m+n) are continuous onR+and positive on(, +∞);

(H) a(t)is continuous and nonnegative function on[t,t);

(H) ki,qi∈[, ],αi∈(, ],βi∈(, ),pqii– ) +  > ,pkii– ) +  > such that

p +kiαii– ) +qii– )≥(p> ,i= , , . . . ,m+n).

For thoseωi’s given in (H), defineω˜i(t) inductively by

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

˜

ω(t) :=maxτ∈[,t]{ ¯ω(τ)}, t≥,

˜

ωi+(t) :=maxτ∈[,t]{ω¯ωi˜+(τ)

i(τ) } ˜ωi(t), t≥,i= , , . . . ,m– ,

˜

ωm+(t) :=maxτ∈[,t]{ˆ

ωm+(maxs∈[,τ]{f(s)})

˜

ωm(τ) } ˜ωm(t), t≥,

˜

ωj+(t) :=maxτ∈[,t]{

ˆ

ωj+(maxs∈[,τ]{f(s)})

˜

ωj(τ) } ˜ωj(t), t≥,j=m+ , . . . ,m+n– ,

(.)

whereωˆj(t) :=maxτ∈[,t]{ ¯ωj(τ)}forj=m+ , . . . ,m+n,ω¯i(t) :=ωi(t) +εifort≥,i:=εif

ωi() =  or :=  ifωi()=  fori= , , . . . ,m+n, andε>  be a given very small constant.

Remark  Iff andωi(u) (i= , . . . ,m) are continuous and nondecreasing functions onR+

and are positive on (,∞) such thatω∝· · ·∝ωmωm+◦f ∝ · · ·∝ωm+nf, then define

functionω˜i(u) :=ωi(u) (i= , . . . ,m),ω˜j(u) :=ωi(f(u)) (j=m+ , . . . ,m+n).

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Then

u(t)≤ϕ–

Wm–+n

Wm+n

rm+n(t)

+

bm+n(t) bm+n(t)

˜

gm+n(t,s)ds

q

(.)

for all t∈[t,T],where Wi–is the inverse of the function

Wi(u) := u

ui

dx ˜

ωqi(ϕ–(xq))

, uui> ,i= , . . . ,m+n, (.)

ui> is a given constant, ω˜i(i= , , . . . ,m+n)are defined by(.),ri(t)is defined by

r(t) :=aˆ(t)and

ri+(t) :=Wi–

Wi

ri(t)

+

bi(t) bi(t) ˜ gi(t,s)ds

, i= , , . . . ,m+n– , (.)

˜

gi(t,s) := ( +m+n)q–dqi(t)

max ι∈[t,t]

gi(ι,s) q

, tt, (.)

ˆ

a(t) := (+m+n)q–(max

τ∈[t,t]{a(τ)})q,di(t) :=α

–p

i tαiki(βi–)+qi(γi–)+/pB(

pqi(γi–)+

αi ,pkii

) + )p,p+

q= ,and T<tis the largest number such that

Wi

ri(T)

+

αi(T) αi(t)

max ι∈[t,T]

f(ι,s)ds

ui

dz ˜

ωqi(ϕ–(zq))

, i= , , , . . . ,m+n. (.)

In order to prove the theorem, we need the following lemma.

Lemma . Letα,β,γ,and p be positive constants,a and b be nonnegative constants. Then

t

pa(β–)spb(γ–)ds=t

δ

αB

pb(γ – ) + 

α ,pa(β– ) + 

, t∈R+,

whereδ:=p[aα(β– ) +b(γ– )] + ≥.

Proof Change the variablev:v=sα, thends=t

αv

(–α)/αdv, and we have t

pa(β–)spb(γ–)ds=

 

tαvpa(β–)tv/αpb(γ–)t αv

(–α)/α

dv

=  αt

p[(β–)+b(γ–)]+

 

vpb(γα–)+–( –v)pa(β–)dv

=t

δ

αB

pb(γ – ) + 

α ,pa(β– ) + 

, t≥.

This completes the proof.

Lemma .(Discrete Jensen inequality) Let If A, . . . ,Anbe nonnegative for real numbers

and r> .Then

(A+· · ·+An)rnr–

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Lemma .(see []) Suppose that

(C) allhi(i= , , . . . ,n)are continuous and nondecreasing onR+and are positive on

(,∞)such thath∝h∝ · · · ∝hm+n;

(C) a(t)is continuously differentiable intand nonnegative on[t,t)wheret,tare

constants andt<t;

(C) allbi: [t,t)→R+(i= , , . . . ,n)are continuously differentiable and

nondecreasing such thatbi(t)≤ton[t,t);

(C) allfi(t,s),i= , . . . ,n,are continuous and nonnegative functions on[t,t)×[t,t).

If u(t)is a continuous and nonnegative function on[t,t)satisfying

u(t)≤a(t) +

n

i=

bi(t) bi(t)

fi(t,s)hi

u(s)ds, t≤t<t,

then

u(t)≤Wn–

Wn

rn(t)

+

bn(t) bn(t)

max

t≤τtfi(τ,s)ds

, t≤tT,

where for all t∈[t,T],where Hi–is the inverse of the function

Hi(u) := u

ui

dx hi(x)

, uui> ,i= , , . . . ,n,

ˆ

rn(t)is defined byrˆ(t) :=a(t) +

t t|a

(s)|ds,and

ˆ

ri+(t) :=Hi–

Hi

ˆ ri(t)

+

αi+(t)

αi+(t)

max t≤τt

fi(τ,s)ds

, i= , , . . . ,n– ,

and T<tis the largest number such that

Hi

ˆ ri(T)

+

αi(T)

αi(t)

max

t≤τtfi(τ,s)ds≤ ∞

ui

dz hi(z)

, i= , , , . . . ,n.

Proof of Theorem. First of all, we monotonize some given functionsf, ωi, and ain

system (.) of integral inequalities. Let

˜

f(t) :=max τ∈[,t]

f(τ), t≥, a˜(t) := max

τ∈[t,t]

a(τ), tt. (.)

From (.) we see that the functionWiis strictly increasing and therefore its inverseWi–

is well defined, continuous, and increasing in its domain. The sequence{ ˜ωi(t)}, defined by

ωi(s), consists of nondecreasing nonnegative functions onR+and satisfies

ωi(t)≤ ˜ωi(t), i= , , . . . ,m,

ωi(t)≤ ˆωi(t), i=m+ , . . . ,m+n,

ˆ ωi

˜

f(t)≤ ˜ωi(t), i=m+ , . . . ,m+n.

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Moreover,

˜

ωiω˜i+, i= , , . . . ,m+n, (.)

because the ratiosω˜i+(t)/ω˜i(t),i= , , . . . ,m+n, are all nondecreasing. Furthermore, let

ˆ

gi(t,s) := max ι∈[t,t]

gi(ι,s), (.)

which is nondecreasing intfor each fixed sand satisfiesgˆi(t,s)≥gi(t,s)≥ for alli=

, , . . . ,m+n. We note thata˜(t)≥a(t) andgˆi(t,s)≥fi(t,s) and they are continuous and

nondecreasing int. From the monotonicity off˜(t) we obtain the inequality

max ξ∈[ci(s)–h,ci(s)]

fu(ξ)≤ max

ξ∈[ci(s)–h,ci(s)]

˜ fu(ξ)

≤ ˜f

max ξ∈[ci(s)–h,ci(s)]

u(ξ) , ∀sb∗(t),t

. (.)

From (.), (.), (.), and the definition ofgˆi(t,s), we obtain

ϕu(t)≤ ˜a(t) +

m

i=

bi(t) bi(t)

tαisαiki(βi–)

sqi(γi–)gˆ i(t,s)ω˜i

u(s)ds

+

m+n

j=m+

bj(t) bj(t)

tαjsαjkj(βj–)

sqj(γj–)gˆ j(t,s)

× ˆωj

˜ f

max ξ∈[cj(s)–h,cj(s)]

u(ξ)

ds

≤ ˜a(t) +

m

i=

bi(t) bi(t)

tαisαiki(βi–)

sqi(γi–)gˆ i(t,s)ω˜i

u(s)ds

+

m+n

j=m+

bj(t) bj(t)

tαjsαjkj(βj–)sqj(γj–)gˆ j(t,s)

× ˜ωj

max ξ∈[cj(s)–h,cj(s)]

u(ξ) ds, tbj(t),t

,

u(t)≤ψ(t), tb∗(t) –h,t

.

(.)

Letp+q= ,p> , thenq> . Sincepqii– ) +  > ,pkii– ) +  > , andp+kiαii

) +qii– )≥ fori= , . . . ,m+n. By Lemma ., Hölder’s inequality, and (.) we get

fort∈[t,t)

ϕu(t)≤ ˜a(t) +

m

i=

bi(t) bi(t)

tαisαipki(βi–)

spqi(γi–)ds

p bi(t) bi(t) ˆ

giq(t,s)ω˜iqu(s)ds

q

+

m+n

j=m+

bj(t) bj(t)

tαjsαjpkj(βj–)

spqj(γj–)ds

p

× bj(t)

bj(t) ˆ gjq(t,s)ω˜jq

max ξ∈[cj(s)–h,cj(s)]

u(ξ) ds

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≤ ˜a(t) + m i= t

tαisαipki(βi–)

spqi((γi–))ds

p bi(t) bi(t) ˆ

giq(t,s)ω˜qiu(s)ds

q

+

m+n

j=m+

t

tαjsαjpkj(βj–)spqj(γj–)ds

p

× bj(t)

bj(t) ˆ gjq(t,s)ω˜jq

max ξ∈[cj(s)–h,cj(s)]

u(ξ) ds

q

≤ ˜a(t) +

m

i=

di(t) bi(t)

bi(t) ˆ

giq(t,s)ω˜qiu(s)ds

q

+

m+n

j=m+

dj(t) bj(t)

bj(t) ˆ gqj(t,s)ω˜qj

max ξ∈[cj(s)–h,cj(s)]

u(ξ) ds

q

, (.)

where we use ≤bi(t)≤tand the definition ofdi(t).

By Lemma . and (.), we get fort∈[t,t)

ϕqu(t)≤( +m+n)q–

˜ aq(t) +

m

i=

dqi(t)

bi(t) bi(t) ˆ

gqi(t,s)ω˜qiu(s)ds

+

m+n

j=m+

djq(t)

bj(t) bj(t) ˆ gqj(t,s)ω˜qj

max ξ∈[cj(s)–h,cj(s)]

u(ξ) ds

. (.)

Then from (.), we see thatrˆ(t) is nondecreasing on [t,t). By the definition ofg˜i(t,s)

andˆr(t), and (.), we have

ϕqu(t)≤ ˆr(t) +

m

i=

bi(t) bi(t) ˜ gi(t,s)ω˜qi

u(s)ds

+

m+n

j=m+

bj(t) bj(t) ˜ gj(t,s)ω˜qj

max ξ∈[cj(s)–h,cj(s)]

u(ξ) ds, t∈[t,t),

u(t)≤ψ(t), tb∗(t) –h,t

.

(.)

Consider the auxiliary system of inequalities with (.)

ϕqu(t)≤ ˆr(σ) +

m

i=

bi(t) bi(t)

˜ gi(σ,s)ω˜qi

u(s)ds

+

m+n

j=m+

bj(t) bj(t) ˜ gj(σ,s)ω˜jq

max ξ∈[cj(s)–h,cj(s)]

u(ξ) ds, (.)

for allt∈[t,σ], whereσis chosen arbitrarily such thatt≤σT.

Notice that maxs∈[b∗(t)–h,t]ψ(s)≤ϕ–(ˆr

/q

 (σ)) becausemaxs∈[J(t)–h,t]ψ(s)≤ϕ–(( +

m+n)

p–

p aq(t

))≤ϕ–(ˆr(σ)). Define a functionz(t) : [B∗(t) –h,σ]→R+such that

z(t) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˆ r(σ) +

m i=

bi(t)

bi(t)g˜i(σ,s)ω˜

q i(u(s))ds

+mj=+mn+bbjj((tt))g˜j(σ,s)ω˜qj(maxξ∈[cj(s)–h,cj(s)]u(ξ))ds, t∈[t,σ],

ˆ

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Clearly,z(t) is nondecreasing. By (.) and the definition ofz(t) we have

u(t)≤ϕ–zq(t), tb(t

) –h,σ

. (.)

Sincez(t) is nondecreasing, from (.) we obtain

max ξ∈[cj(s)–h,cj(s)]

u(ξ)≤ max

ξ∈[cj(s)–h,cj(s)]

ϕ–zq)

ϕ–zqc j(s)

ϕ–zq(s), sb

j(t),bj(σ)

. (.)

It follows from (.), (.), and the definition ofz(t) that

z(t)≤ ˆr(σ) +

m

i=

bi(t) bi(t) ˜ gi(σ,s)ω˜iq

ϕ–zq(s)ds

+

m+n

j=m+

bj(t) bj(t) ˜ gj(σ,s)ω˜qj

ϕ–zq(s)ds, t[t

,σ]. (.)

In order to demonstrate the basic condition of monotonicity, lete(t) :=ϕ–(tq), which is

clearly a continuous and nondecreasing function onR+. Thus, for eachi,ω˜i(e(t)) is

contin-uous and nondecreasing onR+andω˜i(e(t)) >  fort> . Moreover, sinceω˜i(t)∝ ˜ωi+(t),

we see that the ratioω˜i+(b(t))/ω˜i(e(t)) is also a continuous and nondecreasing function on R+and satisfiesω˜i(e(t)) >  fort> , implying thatω˜qi(e(t))∝ω˜

q

i+(e(t)),i= , . . . ,m+n– .

Applying Lemma . to the case thatfi(t,s) =g˜i(σ,s),a(t) =rˆ(σ), andωi(t) =ω˜qi(ϕ–(t

q)),

i= , , . . . ,m+n, from (.) we obtain

z(t)≤Wm–+n

Wm+n

ˆ

rm+n(σ,t)

+

bm+n(t)

bm+n(t) ˜

gm+n(σ,s)ds

(.)

for allt≤t≤min{σ,T}, where

˜

r(σ,t) :=rˆ(σ),

˜

ri+(σ,t) :=Wi–

Wi

˜ ri(σ,t)

+

bi(t) bi(t) ˜ gi(σ,s)ds

, i= , , . . . ,m+n– , (.)

andT<tis the largest number such that

Wi

˜ ri(σ,T)

+

bi(T)

bi(t) ˜

gi(σ,s)ds

ui

dz ˜

ωiq(ϕ–(zq))

(.)

fori= , , , . . . ,m+n. Notice thatTT. In fact,Wiis strictly increasing by (.), so its

inverseWi–is continuous and increasing in its corresponding domain by (.). It follows from (.) and the definition ofg˜i(σ,s) that˜ri(σ,t) andg˜i(σ,s) are nondecreasing inσ.

Thus,Tsatisfying (.) gets smaller asσ is chosen larger. In particular,Tsatisfies the

same equation (.) asT whenσ=T. It follows from (.) and (.) that

u(t)≤ϕ–

Wm–+n

Wm+n

˜

rm+n(σ,t)

+

bm+n(t) bm+n(t) ˜

gm+n(σ,s)ds /q

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Takingt=σin (.), we have

u(σ)≤ϕ–

Wm–+n

Wm+n

˜

rm+n(σ,σ)

+

bm+n(σ) bm+n(t)

˜

gm+n(σ,s)ds /q

(.)

for ≤σT. It is easy to verifyr˜i(σ,σ) =ˆri(σ).

Thus, (.) can be written

u(σ)≤ϕ–

Wm–+n

Wm+n

˜ rm+n(σ)

+

bm+n(σ)

bm+n(t) ˜

gm+n(σ,s)ds /q

(.)

for ≤σT. Sinceσis arbitrary, replacingσwitht, we get

u(t)≤ϕ–

W–

m+n

Wm+n

˜ rm+n(t)

+

bm+n(t)

bm+n(t) ˜

gm+n(t,s)ds /q

(.)

fort≤tT. This completes the proof.

Corollary . Suppose that(H)-(H)hold,and uC((b∗(t) –h,t),R+)satisfies

ϕu(t)≤c+

m

i=

bi(t) bi(t)

tαisαiki(βi–)

sqi(γi–)g i(t,si

u(s)ds

+

m+n

j=m+

bj(t) bj(t)

tαjsαjkj(βj–)

sqj(γj–)g j(t,s)

×ωj

max ξ∈[cj(s)–h,cj(s)]

fu(ξ) ds, t∈[t,t),

u(t)≤ψ(t), tb∗(t) –h,t

,

(.)

where c≥is a constant.Then

u(t)≤ϕ–

Wm–+n

Wm+n

¯ rm+n(t)

+

bm+n(t)

bm+n(t) ˜

gm+n(t,s)ds /q

(.)

for all t∈[t,t),where Wi–is the inverse of Wi,Wiis defined in(.),¯ri(t)is defined by

¯

r(t) :=ϕq(M)and

¯

ri+(t) :=Wi–

Wi

¯ ri(t)

+

bi(t) bi(t) ˜ gi(t,s)ds

, i= , , . . . ,m+n– , (.)

M:=max(maxs∈[b∗(t)–h,t]ψ(s),ϕ

–(( +m+n)–/qc)),

p+

q= ,t<tis the largest number

such that

Wi

¯ ri(t)

+

bi(t)

bi(t) ˜

gi(t,s)ds

ui

dz ˜

ωqi(ϕ–(zq))

, i= , , , . . . ,m+n, (.)

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Proof From (.) and the definition ofMwe get

ϕu(t)≤( +m+n)/q–ϕ(M) +

m

i=

bi(t) bi(t)

tαisαiki(βi–)sqi(γi–)g i(t,si

u(s)ds

+

m+n

j=m+

bj(t) bj(t)

tαjsαjkj(βj–)

sqj(γj–)g j(t,s)

×ωj

max ξ∈[cj(s)–h,cj(s)]

fu(ξ) ds, t∈[t,t),

u(t)≤M, tb∗(t) –h,t

.

(.)

Then from (.) we obtain (.) by Theorem ., where we choose a(t)( +m+

n)/q–ϕ(M). This completes the proof.

3 Applications

In this section, we apply our result to estimate solutions for the nonlinear integral equation and integral equation with a weakly singular kernel and maxima separately.

3.1 Differential equation with the maxima

Consider a system of differential equations with maxima

x(t) =F(t,x(t),maxs∈[β(t),α(t)]x(s)) +c

t

(ts)–

λx(s)ds, tt

,

x(t) =ψ(t), t∈[α(t) –h,t],

(.)

wherec,λ( <λ< ),t≥, andh>  are constants,ψ∈C([α(t) –h,t],R),FC(R+×

R,R),α,βC([t

,∞),R+),α(t) is a nondecreasing function,β(t)≤t,α(t)≤t, and  <

α(t) –β(t)≤hfortt.

Equation (.) is more general than the equation considered in Section  of [] so that the results of the integral inequalities obtained in [] do not work. We will give an esti-mate for solutions of system (.).

Corollary . Suppose in system(.)that

F(t,x,y)≤h(t)|x|μ

|x|+h(t)|y|μ

|y|+h(t), t≥,x,y∈R, (.)

where hiC(R+,R+) (i= , , ),allμi(i= , )are continuous and nondecreasing onR+

and are positive on(,∞)such thatμ∝μ,  <p< /λ, p+q = .For given u> and

u> ,let

Q(u) :=

u

uds sqμq

(s/q)

ds, uu> ,

Q(u) :=

u

uds sqμq

(s/q)

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Then every solution x(t,t,ψ)of system(.)has the estimate

x(t,t,ψ)≤

Q–

Q

γ(t)

+ q–tq/p

t

thq(s)ds

/q

, ∀tt,t

, (.)

whereγi(t)are defined byγ(t) := q–Nq(t)exp(q–tq(–)/p+|c|qB(,  –pλ)q/p)and

γ(t) :=Q–

Q

γ(t)+ q–tq/p

t

thq(s)ds

,

γ(t) :=Q–

Q

γ(t)

+ q–tq/p

t

thq(s)ds

,

N(t) :=maxs∈[α(t)–h,t]|ψ(s)|+|ψ(t)|( +|c|t–

λ/( –λ)) +t

t|h(s)|ds for tt,and t

is

the largest number such that

Q

γ

t∗+ q–tq/p

t

t

hq(s)ds

uds sqμq

(s/q)

,

Q

r

t∗+ q–tq/p

t

t

hq(s)ds

uds sqμq

(s/q)

.

(.)

Proof LetM=maxs∈[α(t)–h,t]|ψ(s)|andx(t) =x(t,t,ψ), the solution of system (.) de-fined for alltα(t) –h. The functionx(t) satisfies the following integral equation:

x(t) =ψ(t)

 –c(tt)–λ/( –λ)

+

t

t

Fs,x(s), max

ξ∈[β(s),α(s)]x(ξ) ds

+c

t

t

(ts)–λx(s)ds, tt,

x(t) =ψ(t), t

α(t) –h,t

.

(.)

By (.) and the definition ofN(t), we get from (.)

x(t)≤ψ(t) +|c|t–λ/( –λ)

+ t tF

t,x(s), max

ξ∈[β(s),α(s)]x(ξ) ds

ψ(t) +|c|t–λ/( –λ)

+

t

t

h(s)ds+M+|c|

t

t

(ts)–λx(s)ds

+

t

t

h(s)μx(s)x(s)ds

+

t

th(s)μ

max ξ∈[β(s),α(s)]

x(ξ) max ξ∈[β(s),α(s)]

x(ξ) ds

N(t) +|c|

t

t

(ts)–λx(s)ds+

t

t

h(s)μx(s)x(s)ds

+

t

th(s)μ

max ξ∈[β(s),α(s)]

x(ξ) max ξ∈[β(s),α(s)]

x(ξ) ds, tt,

x(t)≤ψ(t)≤M, t

α(t) –h,t

.

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Set u(t) :=|x(t)| fort∈[α(t) –h,∞). Then, using the inequalitymaxξ∈[β(s),α(s)]u(ξ)≤

maxξ∈[α(s)–h,α(s)]u(ξ), we obtain

u(t)≤N(t) +

t

t

(ts)–λu(s)ds+

t

th(s)μ

u(s)u(s)ds

+

t

th(s)μ

max ξ∈[α(s)–h,α(s)]u(ξ)

max

ξ∈[α(s)–h,α(s)]u(ξ) ds, tt,

u(t)≤M, tα(t) –h,t

.

(.)

Using our Theorem . to the specifiedm= ,n= ,ϕ(u) =u,a(t) =N(t),αi=  (i= , , ),

g(t,s) = ,gi(t,s) =hi–(s) (i= , ),bi(t) =t(i= , , ),αi=  (i= , , ),k= ,q= ,

qi=ki=  (i= , ),β=  –λ,ω(s) =s,ω(s) =(s),ω(s) =(s), sinceμ∝μ, we see

that the ratioω∝ω∝ω, and from (.) we obtain

u(t)≤

Q–

Q

γ(t)

+ q–tqp t

th(s)ds

/q

(.)

for allt∈[t,t∗], wheret∗is given in (.). Inequality (.) proves the validity of inequality

(.).

Next, we discuss the uniqueness of solutions for system (.).

Corollary . Suppose that

F(t,x,y) –F(t,x,y)≤h(t)|x–x|+h(t)|y–y| (.)

for all ttand all xi,yi∈R(i= , ),where hiC([t,∞),R+).Then system(.)has at

most one solution on[t,t).

Proof Assume that (.) has two different solutionsu(t) =u(t,t,ψ) andv(t) =v(t,t,ψ),

defined fortα(t) –h. Thenu(t) andv(t) satisfy the integral equations defined for all

tα(t) –h. The two functionsu(t) andv(t) satisfy the integral equations

u(t) =ψ(t)

 –ct–λ/( –λ)+

t

tF

t,u(s), max

ξ∈[β(s),α(s)]u(ξ) ds

+

t

t

(ts)–λu(s)ds, tt,

v(t) =ψ(t)

 –ct–λ/( –λ)+ t

t

Ft,v(s), max

ξ∈[β(s),α(s)]v(ξ) ds

+

t

t

(ts)–λv(s)ds, tt,

(.)

andu(t) =v(t) =ψ(t) fort∈[α(t) –h,t]. It implies that

u(t) –v(t)≤

t

t

F

t,u(s), max

ξ∈[β(s),α(s)]u(ξ) –F

t,v(s), max

ξ∈[β(s),α(s)]v(ξ) Ds

+

t

t

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t

t

(ts)–λu(s) –v(s)ds+

t

t

h(s)u(s) –v(s)ds

+

t

t

h(s) max

ξ∈[β(s),α(s)]u(ξ) –ξ∈[maxβ(s),α(s)]v(ξ)

ds

t

t

(ts)–λu(s) –v(s)ds+

t

t

h(s)u(s) –v(s)ds

+

t

t

h(s) max

ξ∈[β(s),α(s)]

u(ξ) –v(ξ)ds, ∀tt. (.)

Letφ(t) :=|u(t) –v(t)|fortα(t) –h. Noting that

max

ξ∈[β(s),α(s)]u(ξ)≤ξ∈[αmax(s)–h,α(s)]u(ξ),

from (.) we obtain

φ(t)≤ε+

t

t

(ts)–λφ(s)ds+

t

t

h(s)φ(s)ds

+

t

t

h(s) max

ξ∈[α(s)–h,α(s)]φ(ξ)ds, tt,

φ(t)≤, tα(t) –h,t

.

(.)

Here, εis an arbitrary positive number, which is the formula of the system of integral inequalities (.). Applying our Corollary . to (.), we have

φ(t)≤–/qεexp

q

q–

Bq/p(,  –pλ)tq(–)/p++tq/p t

thq(s)ds

+tq/p

t

thq(s)ds

, (.)

 <p< /λ,p+q= , lettingε→, we obtain|u(t) –v(t)| ≤, which implies thatu(t) =v(t)

for allt∈[t,t). The uniqueness is proved.

3.2 Integral equation with maxima

Consider the system of integral equations with maxima

⎧ ⎪ ⎨ ⎪ ⎩

x(t) =a(t) +tt (ts)

β–sγ–f

(t,s,x(s))ds

+tt (ts)

β–sγ–f

(t,s,maxξ∈[β(s),α(s)]x(ξ)))ds, tt,

x(t) =ψ(t), t∈[α(t) –h,t],

(.)

whereψC([α(t) –h,t],R),fC(R+×R,R),t≥, andh>  are constants. Suppose

that

(a) |fi(t,s,u)| ≤pi(t,s)hi(|u|), wherepiC([t,∞)×[t,∞),R+),hiC(R+,R+)is

con-tinuous and nondecreasing onR+and is positive on(,∞)such thath∝h,pi(t,s)is

nondecreasing intfor each fixeds,βi∈(, ),γi>  –p and p +βi+γi– ≥(p> ,

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(a) α,βC([t,∞),R+), α(t)is a nondecreasing function,β(t)≤t, α(t)≤t, and  <

α(t) –β(t)≤hfortt;

(a) a(t)is continuous[t,∞).

First of all, we give an estimate for the solutions of (.).

Corollary . Suppose that(a)-(a)hold,andmaxs∈[α(t)–h,t]|ψ(s)| ≤–/q|a(t)|.Then

any solution x(t)of (.)has the estimate

u(t)≤

Q–

Q

η(t)

+ q–cq(t)

t

t

pq(t,s)ds

q

(.)

for t≤tt∗,where

η(t) :=Q–

Q

q–

max τ∈[t,t]

a(τ)

q

+ q–cq(t)

t

t

pq(t,s)ds

,

ci(t) =t/p+αi+βi–B(pi– ) + ,pi– ) + )

p,p+

q= ,Q–i is the inverse of the function

Qi(u) := u

ui

ds

hqi(sq)

, uui> ,i= , ,

and tis the largest number such that

Q

q–

max τ∈[t,t∗]

a(τ)

q

+ q–cqt

t

t

pqt∗,sds

uds

hq(s

q)

,

Q

η

t∗+ q–cqt

t

t

pqt∗,sds

uds

hq(sq)

.

Proof Leta˜(t) :=maxτ∈[,t]{|a(τ)|}. Thena˜(t) is a continuous and nondecreasing function

on [t,∞). From (.) and condition (a) we obtain

x(t)≤ ˜a(t) +

t

t

(ts)β–sγ–p

(t,s)hx(s)ds

+

t

t

(ts)β–sγ–p

(t,s)h max

ξ∈[β(s),α(s)]x(ξ)

ds

≤ ˜a(t) +

t

t

(ts)β–sγ–p

(t,s)hx(s)ds

+

t

t

(ts)β–sγ–p

(t,s)h

max ξ∈[β(s),α(s)]

x(ξ) ds (.)

for alltt. Letu(t) =|x(t)|fort∈[α(t) –h,∞). Then

u(t)≤ ˜a(t) +

t

t

(ts)β–sγ–p

(t,s)h

u(s)ds

+

t

t

(ts)β–sγ–p

(t,s)h

max

ξ∈[β(s),α(s)]u(ξ) ds, tt,

u(t) =ψ(t), t

α(t) –h,t

.

(15)

Using the inequalitymaxξ∈[β(s),α(s)]u(ξ)≤maxξ∈[α(s)–h,α(s)]u(ξ), which follows from

condi-tion (a), we obtain

u(t)≤ ˜a(t) +

t

t

(ts)β–sγ–p

(t,s)h

u(s)ds

+

t

t

(ts)β–sγ–p

(t,s)h

max

ξ∈[α(s)–h,α(s)]u(ξ) ds, tt,

u(t) =ψ(t), t

α(t) –h,t

.

(.)

Notice thatmaxs[α(t)–h,t]|ψ(s)| ≤–/q(a˜(t)) becausemaxs∈[α(t)–h,t]|ψ(s)| ≤–/q×

|a(t)|= –/q(a˜(t)). From (.) and Theorem ., we obtain (.). This completes the

proof.

Competing interests

The author declares to have no competing interests.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11461058) and Scientific Research Fund of SiChuan Provincial Education Department (No. 14ZA0296).

Received: 29 May 2015 Accepted: 26 October 2015

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