# Some New Generalized Retarded Gronwall-Like Inequalities and Their Applications in Nonlinear Systems

Authors: Haiyong Qin; Xin Zuo; Jianwei Liu
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

The Gronwall inequalities are of significance in mathematics and engineering. This paper generalizes the Gronwall-like inequalities from different perspectives. Using the proposed inequalities, the difficulties to discuss the controllability of integrodifferential systems of mixed type can be solved. Meanwhile, two examples as their applications are also given to show the effectiveness of our main results.


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## Body

## 1. Introduction

Integral inequalities provide a powerful and important tool in the study of qualitative properties of solutions of nonlinear differential, integral, and integrodifferential equations, as well as in the modeling of science and engineering problems (see [1]). One of the most famous inequalities of this type is known as “Gronwall’s inequality,” “Bellman’s inequality,” or “Gronwall-Bellman’s inequality” (see [2, 3]). Recently, the celebrated Gronwall inequality and its generalizations play increasingly important roles in the qualitative analysis of differential, integral, and integrodifferential equations. Based on the different purposes, many researchers put their efforts in exploring new inequalities and their applications in many fields, and many useful Gronwall-like integral inequalities have been established in various problems (see [4–17]).Lipovan [18] proved a Gronwall-like inequality, and in order to show its applications, Lipovan applied his main results to the qualitative analysis of solutions to certain integral equations, functional differential equations, and retarded differential equations. Ye et al. [19] gave a generalized Gronwall inequality with singularity which can be applied to weakly singular Volterra integral equations and fractional integral and integrodifferential equations. Liu [20] proved a comparison result, which is widely known later and always used to provide explicit bounds on solutions and estimate on noncompactness. In addition, some existence theorems of solutions and iterative approximation of the unique solution for the nonlinear integrodifferential equations of mixed type are obtained. However, it is worth mentioning that it is difficult to deal with integrodifferential systems which include a Fredholm operator in nonlinearity unless powerful integral inequalities are established.In this paper, we prove a generalization of the Gronwall inequality. As an application, we show that the inequality can be applied to the controllability analysis of abstract control system and existence analysis. Sufficient conditions ensuring the controllability of certain impulsive integrodifferential system of mixed type are obtained. The main difficulties from the Fredholm operator can be overcome.The rest of the paper is organized as follows. In Section2, we present some preliminaries and lemmas and prove some generalized Gronwall-like inequalities. In Section 3, we discuss the controllability of impulsive integrodifferential systems of mixed type in a Banach space as an application. In Section 4, we give another example to illustrate the application of our main results vividly. Finally, conclusions are given in Section 5.


## 2. Integral Inequalities

In this section, we present a generalization of the Gronwall-like inequality which can be called a comparison result in many literatures (see [20]). Unless otherwise stated, we denote R+=(0,+∞), R+=[0,+∞) in this paper.Lemma 1.
Suppose thatu, f, g∈C([t0,T),R+). Let w∈C(R+,R+) be nondecreasing with w(u)>0 for u>0 and let α, β∈C1([t0,T),[t0,T)) be nondecreasing with α(t), β(t)≤t on [t0,T). If (1)ut≤m1t+m2t∫αt0αtfswusds+m3t∫βt0βtgswusds,where mi(t) is a nonnegative, continuous function defined on t0≤t<T, and there exist nonnegative constants Mi such that mi(t)≤Mi(i=1,2,3), then, for t0≤t<t1, one has (2)ut≤G-GM1+M2∫αt0αtfsds+M3∫βt0βtgsds,where G(r)=∫1r‍1/w(s)ds, r>0, t1∈(t0,T) is chosen so that (3)GM1+M2∫αt0αtfsds+M3∫βt0βtgsds∈DomG-,for all t lying in the interval [t0,t1).Proof.
Noting the conditions we imposed, we have(4)ut≤M1+M2∫αt0αtfswusds+M3∫βt0βtgswusds.Let us denote (5)Ut=M1+M2∫αt0αtfswusds+M3∫βt0βtgswusds.Obviously, we have U(t0)=M1 and (6)U′t=M2fαtwuαtα′t+M3gβtwuβtβ′t.Since α(t)≤t and β(t)≤t on [t0,T), then (7)U′t≤M2fαtwUtα′t+M3gβtwUtβ′t.By the definitions of G, we obtain that (8)ddtGUt≤M2fαtα′t+M3gβtβ′tintegrate both sides, and we conclude that (9)GUt≤GM1+M2∫αt0αtfsds+M3∫βt0βtgsds.Because G- is increasing on Dom(G-), we get (10)ut≤G-GM1+M2∫αt0αtfsds+M3∫βt0βtgsds.Remark 2.
Next, we shall show that Lemma1 generalizes some existing results:(1)
Form1(t)≡k, m2(t)≡1, and m3(t)≡0, we obtain theorem in [18]. Further supposing that α(t)≡t, we get the celebrated Bihari’s inequality.(2)
Setw(u)≡u. Note that G(u)=∫1∞‍1/sds=∞; then the previous result (2) holds.(3)
Compared with Theorem1 in [19], this lemma has a different range of applications.Corollary 3.
Suppose thatu, f, g∈C([t0,T),R+). Let α, β∈C1([t0,T),[t0,T)) be nondecreasing with α(t), β(t)≤t on [t0,T). If (11)ut≤M1+M2∫αt0αtfsusds+M3∫βt0βtgsusds,where Mi(i=1,2,3) is nonnegative constants, then, for t0≤t<T, one has (12)ut≤M1exp⁡M2∫αt0αtfsds+M3∫βt0βtgsds.Remark 4.
It is easy to get the following results.(1)
Assume thatM2=1 and M3=0; we know that corollary in [18] is valid.(2)
WithM3=0 and α(t)=t, we obtain the celebrated Gronwall-Bellman inequality.Theorem 5.
Suppose thatu, f, g∈C([t0,T),R+). Let w∈C(R+,R+) be nondecreasing with w(u)>0 for u>0 and let α, β∈C1([t0,T),[t0,T)) be nondecreasing with α(t), β(t)≤t on [t0,T). If (13)ut≤M1+M2∫αt0αtfswusds+M3∫βt0βtgswusds+M4∫t0Tusλds,where Mi(i=1,2,3,4) are nonnegative constants, 0≤λ<1, and ∫1∞‍1/w(s)ds=∞, then, for t0≤t<T, one has (14)ut≤m-0,where m(s)=(2s-MM1)1-λ-s1-λ-(1-λ)MM4T, M is a positive constant, and m-(·) represents the inverse of m(·).Proof.
By Lemma1 and Remark 2 (2), for t0≤t<T, there exists a constant M>0 such that (15)ut≤MM1+M4∫t0Tusλds.Define (16)pt=MM1+M4∫t0tusλds+M4∫t0Tusλds;we have (17)pt0=MM1+MM4∫t0Tusλds,p′t≤MM4ptλ.Integrating from t0 to t, we get (18)pt1-λ-pt01-λ≤1-λMM4t-t0;then (19)pt≤pt01-λ+1-λMM4t-t01/1-λ.Since (20)2pt0-MM1=pT≤pt01-λ+1-λMM4T-t01/1-λ,we can deduce that (21)2pt0-MM11-λ-pt01-λ≤1-λMM4T.Let (22)ms=2s-MM11-λ-s1-λ-1-λMM4T.Observe that m∈CMM1/2,∞,R+ and (23)mMM12=-MM121-λ-1-λMM4T<0,lims→0⁡mss1-λ=21-λ-1>0.It is easy to get that there exists a s0 such that m(s0)=0; then p(t0)≤s0. Therefore (24)ut≤pt0≤m-0,t∈t0,T.The proof is completed.Remark 6.
(i) IfM1=M4=0, we have m-(t0)=0; that is, u(t)≡0.
(ii) Generally speaking, the spectral radius of Fredholm operators should not be less than one. However, there is no doubt that here the above inequality is satisfied as a particular case.


## 3. Controllability of Differential Systems of Mixed Type

In this section, we shall give an application to show that the proposed inequalities are useful in investigating the existence of mild solutions and controllability of differential systems of mixed type. Unfortunately, since the spectral radius of Fredholm operators should not be less than one, the inequality used in previous paper may be not suitable (see [21–26]). Therefore, more powerful integral inequalities should be established to solve the problem. In order to illustrate this problem, we consider the following impulsive integrodifferential system in a Banach space: (25)x′t=Atxt+ft,xt,Sxt,Txt+But,t∈I=0,b,Δxti=Iixti=xti+-xti-,i=1,2,…,s,x0=x0,where operators S and T are defined as follows: (26)Sxt=∫0tkt,s,xsds,Txt=∫0bht,s,xsds.A(t) is a family of linear operators which generates an evolution operator (27)G:Δ=t,s∈0,b×0,b:0≤s≤t≤b⟶LX,where L(X) is the space of all bounded linear operators in X and X is a Banach space. Assume that k∈C[Δ×X,X] and h∈C[I×I×X,X]. f is continuous. 0=t0<t1<t2<⋯<ts<ts+1=b. Ii∈C[X,X], (i=1,2,…,s) are impulsive functions, and x(ti+) and x(ti-) represent the right and the left limits of x(t) at t=ti, respectively. B∈L[U,X] is a bounded linear operator and the control function u(·) is given in L2[I,U] and U is a Banach space. Set Tr={x∈X∣x≤r} and Br={x∈PC[I,X]∣x≤r}. PC[I,X]={x∈CI,X∣x is continuous on (ti,ti+1), i=0,1,…,s, x(ti-)=x(ti), and the right limit x(ti+) exists, i=1,2,…,s}. Obviously, PC[I,X] is a Banach space with the norm xPC=supt∈I{x(t)}.Suppose that the following hypotheses are satisfied.(H1)
A(t):D(A)→X is a family of linear operators, generating an equicontinuous evolution system {G(t,s):(t,s)∈Δ}; that is, (t,s)→{G(t,s)x:x∈Br} is equicontinuous for t>0 and for all bounded subsets Br.(H2)
For anyr>0, f is uniformly continuous on I×Tr×Tr×Tr and Ii(i=1,2,…,s) are bounded on Tr. There exist functions bp∈C[I,R+](p=1,2,3,4) and k∗, h∗>0 such that (28)ft,x,y,z≤b1t+b2txλ+b3ty+b4tz,kt,s,x≤k∗xλ,ht,s,x≤h∗xλ,where0≤λ<1. Define bp=max{bpt∣t∈I}.(H3)
The linear operatorW:L2[I,U]→X is defined by (29)Wu=∫0bGb,sBusds.
(i)
W has an invertible operator W-1 which takes values in L2[I,U]/KerW and there exist positive constants LB and LW such that B≤LB and W-1≤LW;(ii)
there existsKW∈C[I,R+] such that, for any bounded set H⊂X, (30)αW-1Ht≤KWtαH.DefineKW=max{KW(t)∣t∈I}; α(·) represents the Kuratowski noncompactness measure.(H4)
There existlq∈C[I,R+](q=1,2) such that, for any equicontinuous set, D⊂Br, such that (31)αft,Ds,TDs,SDs≤l1tαDsλ+l2tαTDs.Definelq=max{lq(t)∣t∈I}.Theorem 7.
Assume that conditions(H1)–(H4) hold. Then the system (25) is controllable.Proof.
Using(H3) (i), for every x∈PC[I,X], without loss of generality, define the control (32)u0xt=W-1x1-Gb,0x0-∫0bGb,sfs,xs,Txs,Sxsdst,ujxt=W-1x1-Gb,0x0-∫0bGb,sfs,xs,Txs,Sxsds-∑i=1jGt,tiIixtit,where j=1,…,s. Define operator Q as follows: (33)Qxt=Gt,0x0+∫0tGt,sf+Busxsds+∑0<ti<tGt,tiIixti;clearly, using the control usx(t), the fixed point of operator Q is a solution of the system (25), and x1=(Qx)(b); that is, system (25) is controllable. From the conditions we imposed, it is easy to get that operator Q is continuous.
SetΩ0={x∈PC[I,X]∣x=λQx,0≤λ≤1}. Assume that there exists λ0∈[0,1] such that x-(t)=λ0(Qx-)(t). Next, we shall use the method of piecewise discussion.(i)
Whent∈[0,t1], (34)x-t=λ0Gt,0x0+λ0∫0tGt,sf+Bu0x-sds.From Ji et al. [23], we know that there exists LG>0 such that G(t,s)≤LG for any (t,s)∈I×I. Thus (35)x-t≤LGx0+LG∫0tf+Bu0x-sds≤LGx0+LG∫0tfds+LGLB∫0tu0x-sds,where (36)∫0tfds≤∫0bb1sds+b2∫0tx-sλds+b3k∗t1∫0tx-sλds+b4h∗t1∫0bx-sλds,u0x-L2=W-1x1-Gb,0x0-∫0bGb,sfdsL2≤LWx1+LGx0+LG∫0bfds.Let u(t)=x-(t); then u(t)∈C[[0,t1],R+]; we have (37)ut≤LGx0+LGLBbLWx1+LWLGx0+LG∫0bb1sds+b2∫0tusλds+b3k∗t1∫0tusλds+b4h∗t1∫0busλds+LG2LBLWb∫0bb1sds+b2∫0busλds+b3k∗t1∫0busλds+b4h∗t1∫0busλds.Since ∫1∞‍1/sλds=∞, then by Theorem 5, there exists a constant C0 such that u(t)≤C0, t∈[0,t1]; that is, there exists a constant C0 independent of u such that u(t)≤C0, t∈[0,t1]. The above inequality implies that x-(t)≤C0. From (H2), there also exists a constant M0>0 independent of x- such that (38)ft,x-t,x-′t,Tx-t,Sx-t≤M0,I1x-t1≤M0,∀t∈0,t1,x-t≤C0.Thus x-(t1+)=x-(t1)+I1(x-(t1))≤C0+M0.(ii)
Whent∈(t1,t2],(39)v-t=x-t,t∈t1,t2,x-t1+,t=t1.Then v-(t)∈C[[t1,t2],X] and (40)v-t=λ0Gt,0x0+λ0∫0t1Gt,sf+Bu1v-sds+λ0∫t1tGt,sf+Bu1v-sds+λ0Gt,t1I1xt1.From results (38) and similar to the proof of (i), we can know that there exists C1>0 that does not depend on x- such that v-(t)≤C1,t∈[t1,t2]. So x-(t)≤C1, t∈(t1,t2].
By the same method as above, we can prove that there exists a constantCs>0 that does not depend on x- such that (41)x-t≤Cs,t∈ts,b.Let C=max{Ci∣0≤i≤s}; then x-(t)≤C, t∈I. Thus Ω0 is a bounded set in PC[I,X]. Take R>C; let Ω={x∈PCI,X∣xPC<R}; obviously Ω is a bounded open set in PC[I,X] and θ∈Ω. From the choice of R, we know that if x∈∂Ω and λ∈[0,1], we have x≠λQx.
LetH⊂Ω¯ be a countable set and H⊂co¯({θ}⋃‍Q(H)). By (H1) and (H2), it is easy to see that Q(H) is equicontinuous on each [ti,ti+1], (i=0,1,2,…,s).
Next, we shall prove that(QH)(t) is relatively compact for each [ti,ti+1]. In the same way, we discuss step by step as follows. (i)
Whent∈[t0,t1](42)αHt≤αQHt≤αGt,0x0∣x∈H+α∫0tGt,sf+Bu0xsds∣x∈H≤2LGl1∫0tαHsλds+2LGl2k∗b∫0tαHsds+LGLB∫0tαu0x∣x∈Hds,αu0x∣x∈H≤KW2LGl1∫0bαHsλds+2LGl2k∗b∫0tαHsds.Let m(t)=α(H(t)), t∈[0,t1]; then m(t)∈C[[0,t1],R+]. Thus(43)mt≤2LGl1∫0tαHsλds+2LGl2k∗b∫0tαHsds+2KWLGl1b∫0tαHsλds+2KWLGl2bk∗t∫0tαHsds.By Remark 6 (i), we have m(t)≡0, t∈[0,t1]. Thus α(H(t))=0; that is, H is a relatively compact set in PC[[0,t1],X]. Since I1∈C[X,X], α(I1(H(t1)))=0, then H(t1) is a relatively compact set in X.(ii)
Fort∈(t1,t2], we know (44)αHt≤αGt,0x0∣x∈H+α∫0tGt,sf+Bu1xsds∣x∈H+αI1xt1∣x∈H≤2LGl1∫0tαHsλds+2LGl2k∗b∫0tαHsds+LGLB∫0tαu1x∣x∈Hds.Similar to the proof of (i), we can deduce that m(t)≡0, t∈[t1,t2]. Therefore α(H(t))=0, t∈[t1,t2]. In particular, α(H(t2))=0, so H is a relatively compact set in C[[t1,t2],X]. Similarly, we can show that H is a relatively compact set in C[[ti,ti+1],X](i=2,3,…,s). Thus H is a relatively compact set in PC[I,X].
In conclusion, we deduce thatQ has at least one fixed point in Ω by the Mönch fixed point theorem; that is, system (25) has at least one mild solution in PC[I,X]. Thus system (25) is controllable on I.


## 4. The Uniqueness and Global Existence of Solutions

Consider the following integral equation:(45)ut=kt+∫0αtfswusds+∫0βtgswusds,where k, f, g∈C(R+,R+) with w(0)=0 and α, β∈C1(R+,R+) are nondecreasing with α(t), β(t)≤t on RR+. Assume that (45) has a solution u∈C([0,T],R+) on some maximal interval of existence [0,T]. Moreover, if T<∞, (46)limsupt→T⁡ut=∞.Theorem 8.
Assume that(47)wx-wy≤Zx-ywith Z∈C(R+,R+) nondecreasing, Z(x)>0 for x>0. If (48)∫011Zsds=∫1∞1Zsds=∞,then (45) has a unique solution defined on R+. Moreover, if k is bounded on R+ and if either α, β is bounded on R+ or ∫0∞‍f(s)ds, ∫0∞‍g(s)ds<∞, then its solution is bounded on R+.Proof.
Suppose that, on some interval[0,t0], (45) has two solutions u1, u2∈C([0,t0],R+); we obtain (49)u1t-u2t=∫0αtfswu1s-wu2sds+∫0βtgswu1s-wu2sds.Denote u(t)=u1(t)-u2(t); we have (50)ut≤∫0αtfsZusds+∫0βtgsZusds.Set (51)Gr=∫1r1Zsds,r>0.Then G(0)=-∞ and G(∞)=∞. There exists ϵ>0, where(52)ut≤ϵ+∫0αtfsZusds+∫0βtgsZusds,0≤t≤t0.From Theorem 5, we know that (53)ut≤G-Gϵ+∫0αtfsZusds+∫0βtgsZusds,0≤t≤t0.From Remark 6, u(t)→0, ϵ→0; then u1(t)=u2(t), and the uniqueness of the solution can be obtained.
Next, we will show that the solution is global; that is,T=∞, where T is the maximal time of existence. If T<∞, we set k0=max0≤t≤T{k(t)}, and we obtain that (54)ut≤k0+∫0αtfsZusds+∫0βtgsZusds,0≤t<T,as w(u(s))=w(u(s))-w(0)≤Z(u(s)) for 0≤t<T. By Lemma 1 and u∈C[[0,T],R+], we deduce that (55)ut≤G-Gk0+∫0αtfsds+∫0βtgsds,0≤t<T.Since k is bounded on R+ and either α, β is bounded on R+ or ∫0∞‍f(s)ds, ∫0∞‍g(s)ds<∞ satisfies; then u(t) is bounded on R+. Thus the previous inequality (55) contradicts (46). Thus the global existence is proved.


## 5. Conclusions

This paper generalizes a more general Gronwall-like inequality with a Fredholm operator. Using the proposed inequality, we solve a difficult problem in the research of the controllability of integrodifferential systems of mixed type in Banach space. Meanwhile, we also prove the uniqueness and global existence of solutions for a class of integral equations. Therefore, the results we obtained are very important and powerful tools. However, it should be more useful than we can imagine in qualitative properties of many other nonlinear problems, such as existence, estimation of solutions, dependence of solutions on parameters in nonlinear analysis, and control.



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