# Certain Nonlinear Integral Inequalities and Their Applications

Authors: Yazhou Tian; Min Fan; Yuangong Sun
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Several new retarded integral inequalities of Gronwall-Bellman-Pachpatte type are presented which generalize the inequalities to the more general nonlinear case in the literature and provide explicit bounds on unknown functions. These results include many existing ones as special cases and can be used as tools in the qualitative analysis of certain classes of integrodifferential equations.


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## Body

## 1. Introduction

It is well known that integral inequalities are very useful to investigate the existence, uniqueness, boundedness, oscillation, and other qualitative properties of solutions to differential equations and integrodifferential equations [1–19]. One of the fundamental inequalities is the Gronwall inequality, which was established in 1919 by Gronwall [1]. As a generalization of Gronwall inequality, Gronwall-Bellman inequality plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integrodifferential equations. An important nonlinear generalization of Gronwall-Bellman inequality is Bihari’s inequality [2]. In recent years, there has been much research activity concerning linear and nonlinear generalizations of Gronwall-Bellman-Bihari type inequalities. To mention a few, Pachpatte [3] presented a new integral inequality and studied the boundedness properties of some linear integrodifferential equations, one of which we give below for the convenience of the reader.Theorem 1 (see Pachpatte [3]).
Letu,f,g,h∈C(R+,R+), R+=0,∞, and u0 be a nonnegative constant. If (1)ut≤u0+∫0tfsus+hsds+∫0tfs∫0sgτuτdτds,then (2)ut≤u0+∫0ths+fsu0exp⁡∫0sfσ+gσdσ+∫0shσexp⁡∫σsfτ+gτdτdσds,for t∈R+.Very recently, Abdeldaim and El-Deeb [4, 5] investigated the following integral inequalities which generalized the main results of [3].Theorem 2 (see [4, Theorem  2.1]).
Letu,f,g,h∈C(R+,R+) and suppose that α∈C1(R+,R+) are increasing functions with α(t)≤t and α(0)=0, for t∈R+, and u0 is a nonnegative constant. If (3)ut≤u0+∫0αtfsus+hsds+∫0αtfs∫0sgτuτdτds,then(4)ut≤u0+∫0tα′shαs+α′sfαsexp∫0αsfτ+gτdτu0+∫0αshσexp⁡-∫0σfτ+gτdτdσds,∀t∈R+.Theorem 3 (see [4, Theorem 2.2] and [5, Theorem 2.1]).
Letu,f,g,h∈C(R+,R+) and assume that φ,φ′,α∈C1(R+,R+) are increasing functions with φ′≤k, φ>0, α(t)≤t, and α(0)=0, for t∈R+, and u0 and k are positive constants. If (5)ut≤u0+∫0αtφusfsφus+hsds+∫0αtφusfs∫0sgτφuτdτds,then (6)ut≤Φ-1Φu0+∫0αths+fsβsds,∀t∈R+,where (7)βt≤exp∫0αtkhs+gsdsφ-1u0-∫0αtkfsexp⁡∫0skhσ+gσdσds,Φr=∫1rdtφt,r>1,and Φ-1 and φ-1 stand for the inverses of the functions Φ and φ, respectively.Note that the functionφ of inequalities established in [4, 5] satisfies φ′≤k; that is, φ is global Lipchitz. However, in the real system there may exist the non-Lipschitz φ, such as φ(t)=t2 or φ(t)=et, and then the obtained results can not be applied to this kind of systems. The natural question now is as follows: is it possible to relax this condition? The aim of this paper is to give an affirmative answer to this question.In this paper, we study certain classes of integral inequalities which generalize the inequalities established in [3–5] to the more general nonlinear case. The obtained results can be used as tools in the study of qualitative theory of certain classes of integrodifferential equations with more general nonlinearities. At the end, two examples are provided to illustrate the main results.


## 2. Main Results

In what follows,R denotes the set of real numbers, C(M,S) and C1(M,S) denote the sets of all continuous functions and all continuously differentiable functions defined on set M with range in the set S, respectively, and f-1 stands for the inverse to f.The following lemmas are very useful in the proof of our main results.Lemma 4 (see [6, Lemma 2.1]).
Assume thata≥0 and m≥n>0. Then (8)an/m≤nmKn-m/ma+m-nmKn/m,for any K>0.Lemma 5 (see [7, Corollary 1]).
Assume thatu,ki∈C(R+,R+), i=1,2,3, and a0 is a positive constant. If (9)ut≤a0+∫0tk1susds+∫0tk2su2sds+∫0tk3su3sds,then(10)ut≤a0exp⁡∫0tk1sds1-a0exp⁡∫0tk1sds∫0tk2sds2-2a02exp⁡∫0t2k1sds∫0tk3sds,t∈0,T,where T is the largest number such that(11)exp∫0Tk1sds∫0Tk2sds<1a0,exp∫0T2k1sds∫0Tk3sds1-a0exp⁡∫0Tk1sds∫0Tk2sds2<12a02.Theorem 6.
Letu0, p, q, and r be nonnegative constants satisfying p≥q>0, p≥r>0, u,f,g,h∈C(R+,R+), α∈C1(R+,R+), α′≥0, α(t)≤t, and α(0)=0. If (12)upt≤u0+∫0αtfsuqs+hsds+∫0αtfs∫0sgτurτdτds,then, for any K>0, (13)ut≤u0+∫0tHs+qpKq-p/pα′sfαsexp⁡∫0αsJτdτu0+∫0sHσexp⁡-∫0ασJτdτdσds1/p,t∈R+,where (14)Ht=α′thαt+α′tfαtp-qpKq/p+p-rpKr/p∫0αtgτdτ,Jt=qpKq-p/pft+rqKr-q/pgt.Proof.
Define a functiony by (15)yt=u0+∫0αtfsuqs+hsds+∫0αtfs∫0sgτurτdτds.Then, y(0)=u0, y is nondecreasing, and (16)ut≤y1/pt,uαt≤y1/pαt≤y1/pt,t∈R+.Differentiating y and using (16), we conclude that (17)y′t=α′tfαtuqαt+hαt+α′tfαt∫0αtgτurτdτ≤α′thαt+α′tfαtyq/pt+∫0αtgτyr/pτdτ.By virtue of Lemma 4, for any K>0, (18)yq/pt≤qpKq-p/pyt+p-qpKq/pyr/pt≤rpKr-p/pyt+p-rpKr/p.Combining (17) and (18), we deduce that (19)y′t≤Ht+α′tfαtqpKq-p/pyt+rpKr-p/p∫0αtgτyτdτ.Define another function v by (20)vt=qpKq-p/pyt+rpKr-p/p∫0αtgτyτdτ.Then v is nondecreasing, v(0)=qu0K(q-p)/p/p, and (21)yt≤pqKp-q/pvt,yαt≤pqKp-q/pvαt≤pqKp-q/pvt,t∈R+.It follows from (19) that(22)y′t≤Ht+α′tfαtvt.Differentiating v and using (22), we obtain (23)v′t=qpKq-p/py′t+rpKr-p/pα′tgαtyαt≤qpKq-p/pHt+α′tfαtvt+rqKr-q/pα′tgαtvt=qpKq-p/pHt+qpKq-p/pα′tfαt+rqKr-q/pα′tgαtvt.Integrating the latter inequality from 0 to t, we conclude that (24)vt≤qpKq-p/pexp⁡∫0αtJτdτu0+∫0tHσexp-∫0ασJτdτdσ.Therefore, (25)y′t≤Ht+qpKq-p/pα′tfαtexp⁡∫0αtJτdτu0+∫0tHσexp-∫0ασJτdτdσ,which yields (26)yt≤u0+∫0tHs+qpKq-p/pα′sfαsexp⁡∫0αsJτdτu0+∫0sHσexp-∫0ασJτdτdσds.By virtue of u(t)≤y1/p(t), (13) holds. This completes the proof.Remark 7.
Ifp=q=r=1, then Theorem 6 reduces to Theorem 2.Remark 8.
Ifp=q=r=1 and α(t)=t, then Theorem 6 reduces to Theorem 1.Theorem 9.
Letu,f,g,h∈C(R+,R+). Assume further that ψ,ψ′∈C1(R+,R+), ψ′>0, ψ′′≥0, φ∈C1(R+,R+), φ>0, 0≤φ′≤aφ+b, a,b≥0, α∈C1(R+,R+), α′≥0, α(t)≤t, α(0)=0, and u0 is a positive constant. If, for t∈R+, (27)ψut≤u0+∫0αtφusfsφus+hsds+∫0αtfsφus∫0sgτφuτdτds,then (28)ut≤ψ-1G-1Gu0+∫0αths+fsβα-1sds,t∈0,T,where (29)βt=b0exp⁡∫0tk1sds1-b0exp⁡∫0tk1sds∫0tk2sds2-2b02exp⁡∫0t2k1sds∫0tk3sds,t∈0,T,k1t=α′tgαt+bψ′ψ-1u0α′thαt,k2t=1ψ′ψ-1u0α′tahαt+bfαt,k3t=aψ′ψ-1u0α′tfαt,b0=φψ-1u0,T is the largest number such that (30)exp⁡∫0Tk1sds∫0Tk2sds<1b0,exp⁡∫0T2k1sds∫0Tk3sds1-b0exp⁡∫0Tk1sds∫0Tk2sds2<12b02,Gr=∫r0r1φψ-1tdt,r>r0>0.Proof.
Define a functionz by (31)zt=u0+∫0αtφusfsφus+hsds+∫0αtfsφus∫0sgτφuτdτds.Then, z(0)=u0, z is nondecreasing, and (32)ut≤ψ-1zt,uαt≤ψ-1zαt≤ψ-1zt,t∈R+.It is not difficult to obtain (33)z′t=α′tφuαtfαtφuαt+hαt+α′tfαtφuαt∫0αtgτφuτdτ≤α′thαtφψ-1zt+α′tfαtφψ-1zt×φψ-1zt+∫0αtgτφψ-1zτdτ=α′thαtφψ-1zt+α′tfαtφψ-1ztwt,where (34)wt=φψ-1zt+∫0αtgτφψ-1zτdτ,w is nondecreasing, w(0)=b0, and φ(ψ-1(z(t)))≤w(t). Differentiating w, we have (35)w′t=φ′ψ-1ztψ-1zt′z′t+α′tgαtφψ-1zαt=φ′ψ-1zt1ψ′ψ-1ztz′t+α′tgαtφψ-1zαt.Noting that ψ is increasing and ψ′ is nondecreasing, we get ψ′(ψ-1(z(t)))≥ψ′(ψ-1(u0)), and so (36)w′t≤1ψ′ψ-1u0aφψ-1zt+bα′thαtφψ-1zt+α′tfαtφψ-1ztwt+α′tgαtφψ-1zt≤1ψ′ψ-1u0awt+bα′thαtwt+α′tfαtw2t+α′tgαtwt=k1twt+k2tw2t+k3tw3t.Integrating the latter inequality from 0 to t, we have (37)wt≤b0+∫0tk1swsds+∫0tk2sw2sds+∫0tk3sw3sds.By virtue of Lemma 5, (38)wt≤βt,t∈0,T.This and (33) imply that (39)z′t≤α′thαtφψ-1zt+α′tfαtφψ-1ztβt,and thus (40)z′tφψ-1zt≤α′thαt+α′tfαtβt.Integrating (40) from 0 to t, (41)zt≤G-1Gu0+∫0αths+fsβα-1sds,which yields (28) due to (32). The proof is complete.Remark 10.
The conditionφ′≤k in [4, 5] is relaxed to φ′≤aφ+b, where a and b are nonnegative constants. On the basis of this new assumption, the classes of φ are enlarged. For example, φ(t)=et satisfies φ′≤aφ+b when choosing a=1 and b=0.If we chooseψ(u)=up, φ(u)=uq, and p>q≥1 in Theorem 9, then φ′(u)=quq-1. Letting Φ(u)=uq-1-uq, u>0, we have uq-1≤uq+(1-1/q)q-1/q, and hence φ′(u)=quq-1≤quq+(1-1/q)q-1=qφ(u)+(1-1/q)q-1. Therefore, φ satisfies φ′(u)≤aφ(u)+b with a=q and b=(1-1/q)q-1, and then we can obtain the following result.Corollary 11.
Assume thatu,f,g,h∈C(R+,R+), α∈C1(R+,R+), α′≥0, α(t)≤t, α(0)=0, u0 is a positive constant, and, for t∈R+, (42)upt≤u0+∫0αtuqsfsuqs+hsds+∫0αtfsuqs∫0sgτuqτdτds.Then (43)ut≤u0p-q/p+p-qp∫0αths+fsβα-1sds1/p-q,t∈0,T,where p>q≥1, (44)βt=u0q/pexp⁡∫0tk1sds1-u0q/pexp⁡∫0tk1sds∫0tk2sds2-2u02q/pexp⁡∫0t2k1sds∫0tk3sds,k1t=α′tgαt+1p1-1qq-1u01/p-1α′thαt,k2t=1pu01/p-1α′tqhαt+1-1qq-1fαt,k3t=qpu01/p-1α′tfαt,and T is the largest number such that (45)exp⁡∫0Tk1sds∫0Tk2sds<1u0q/p,exp⁡∫0T2k1sds∫0Tk3sds1-u0q/pexp⁡∫0Tk1sds∫0Tk2sds2<12u02q/p.Proof.
By virtue ofφ(u)=uq and ψ(u)=up, we obtain φ(ψ-1(u))=uq/p and (46)Gr=∫r0r1φψ-1tdt=∫r0rt-q/pdt=pp-qrp-q/p-r0p-q/p,G-1r=p-qpr+r0p-q/pp/p-q.An application of Theorem 9 implies that (47)ut≤ψ-1G-1Gu0+∫0αths+fsβα-1sds=u0p-q/p+p-qp∫0αths+fsβα-1sds1/p-q.This completes the proof.


## 3. Applications

In this section, we will use the inequalities established in Theorems6 and 9 to study the boundedness of certain integrodifferential equations.Example 1.
Consider the following integral equation:(48)upt=at+∫0tGs,uαs,∫0sNτ,uατdτds,t∈R+,where a(t) is a continuous function defined on R+, at≤A, A is a nonnegative constant, and G∈C(R+×R×R,R) and N∈C(R+×R,R) satisfy (49)Gt,x,y≤btxq+y+ct,Nt,z≤dtzr,t∈R+,where p, q, and r are nonnegative constants satisfying p≥q>0, p≥r>0, and b,c,d∈C(R+,R+). Then every solution u of (48) satisfies (50)utp≤A+∫0tbsuαsq+csds+∫0tbs∫0sdτuατrdτds=A+∫0αt1α′α-1sbα-1susq+cα-1sds+∫0αtbα-1sα′α-1s∫0sdα-1τα′α-1τuτrdτds.Now, using Theorem 6, we obtain that, for any K>0, (51)ut≤A+∫0tHs+qpKq-p/pbsexp⁡∫0sJ~τdτA+∫0sHσexp⁡-∫0σJ~τdτdσds1/p,t∈R+,where (52)Ht=ct+btp-qpKq/p+p-rpKr/p∫0tdτdτ,J~t=qpKq-p/pbt+rqKr-q/pdt,which illustrates that the solution of (48) is global boundedness.Example 2.
Consider the integrodifferential equation(53)upt′=Ft,uαt,∫0tHτ,uατdτ,up0=C,t∈R+,where C≠0 is a constant, F∈C(R+×R×R,R), and H∈C(R+×R,R).
Assume that(54)Ft,U,V≤UqftUq+V+ht,Ht,W≤gtWq,t∈R+,where f,g,h∈C(R+,R+). Then the solution u of (53) satisfies (55)upt=C+∫0tFs,uαs,∫0sHτ,uατdτds,and hence (56)utp≤C+∫0tuαsqfsuαsq+hsds+∫0tfsuαsq∫0sgτuατqdτds=C+∫0αtusqα′α-1sfα-1susq+hα-1sds+∫0αtfα-1sα′α-1susq∫0sgα-1τα′α-1τuτqdτds.A suitable application of Corollary 11 to (56) yields (57)ut≤Cp-q/p+p-qp∫0αt1α′α-1shα-1s+fα-1sβα-1sds1/p-q=Cp-q/p+p-qp∫0ths+fsβsds1/p-q,t∈0,T,where (58)βt=Cq/pexp⁡∫0tk1sds1-Cq/pexp⁡∫0tk1sds∫0tk2sds2-2C2q/pexp⁡∫0t2k1sds∫0tk3sds,k1t=gt+1p1-1qq-1C1/p-1ht,k2t=1pC1/p-1qht+1-1qq-1ft,k3t=qpC1/p-1ftand T is the largest number such that (59)exp⁡∫0Tk1sds∫0Tk2sds<1Cq/p,exp⁡∫0T2k1sds∫0Tk3sds1-Cq/pexp⁡∫0Tk1sds∫0Tk2sds2<12C2q/p.



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