# A Generalization of Linear and Nonlinear Retarded Integral Inequalities in Two Independent Variables

Authors: Ying Jiang; Guojing Xing; Chenghui Zhang
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Integral inequalities, which provide explicit bounds on unknown functions, are used to serve as handy tools in the study of the qualitative properties of solutions to differential and integral equations. By utilizing some analysis techniques, such as amplification method, differential, and integration, several new types of linear and nonlinear retarded integral inequalities in two independent variables are provided. These results generalize and complement previous ones. An illustrative example is given to support the obtained results. The study of the numerical example shows that the new results presented in this paper work well in the analysis of retarded integral inequalities in two independent variables.


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## Body

## 1. Introduction

With the development of science and technology, various inequalities have been paid more and more attention, and the generalization of inequalities has become one of the important research directions in modern mathematics. The integral inequality, which has integrals of unknown functions, is an important type of inequality. For nonlinear differential equations derived from the natural science and engineering technology, especially from various branches of mathematics, it is difficult or impossible to obtain explicit solutions in most cases. Therefore, it is of great significance to get the bounds of the solutions to those nonlinear differential equations. Integral inequalities just can provide the bounds of the solutions to the nonlinear differential equations and integral equations. Hence, integral inequalities are used to serve as handy tools in the study of the qualitative properties of solutions to differential and integral equations, such as existence, uniqueness, boundedness, oscillation, stability, and invariant manifold. For example, these inequalities have been widely employed to investigate the stability of switched systems which can be applied to modeling many engineering system problems in real world, such as traffic control, automobile engine control, switching power converters, and multiagent consensus [1–5]. For some related contributions on various classes of integral inequalities, we refer the reader to [1–20] and the references cited therein.For convenience, throughout this paper,R represents the set of real numbers, R+=0,∞, and C(A,B) signifies the class of all continuous functions defined on set A with range in the set B.In what follows, we provide some background details that motivated our study. One of the most famous and widespread integral inequalities in the study of differential and integral equations is Gronwall-Bellman-type inequality [6–8], which can be described as follows.Theorem 1.
Letu and f be nonnegative continuous functions on an interval a,b satisfying (1)ut≤c+∫atfsusds,t∈a,bfor some constant c≥0. Then(2)ut≤cexp⁡∫atfsds,t∈a,b.In recent years, many scholars have done a lot of researches and generalization of the above integral inequality, which make the integral inequalities develop continually and the application fields expand gradually. Pachpatte [9, 10] investigated the inequality (3)ut≤u1+∫0tfsus+psds+∫0tfs∫0sgσuσdσdsand the retarded inequality (4)ut≤u2+∫0tfsusds+∫0αtgsusds,where α∈C1I,I is nondecreasing with αt≤t on I=0,T and u1 and u2 are constants. Abdeldaim and El-Deeb [11] generalized [9] and analyzed the following retarded linear and nonlinear inequalities:(5)ut≤u0+∫0αtfsus+psds+∫0αtfs∫0sgσuσdσds,ut≤u0+∫0αtφusfsφus+psds+∫0αtφusfs∫0sgσφuσdσds,respectively. Tian et al. [16] introduced the retarded inequalities in two independent variables as follows.Theorem 2 (see [16, Theorem  1]).
Letu, f, and g∈C(R+×R+,R+), a(x)>0, b(y)>0, a′(x)≥0, b′(y)≥0, and α, β∈C1(R+,R+) be nondecreasing with α(x)≤x and β(y)≤y on R+. Moreover, let φ∈C1(R+,R+) be an increasing function with φ(∞)=∞ and let φ(x)>0 on (0,∞),ψ∈C1(R+,R+) be a nondecreasing function with ψ(x)>0 on (0,∞). If (6)φux,y≤ax+by+∫0αx∫0βyut,sft,sψut,s+gt,sdsdt,then, for 0≤x<ξ1,0≤y<η1, (7)ux,y≤φ-1Ω-1G-1GPx,y+∫0αx∫0βyft,sdsdt,where(8)Px,y=Ωa0+by+∫0xa′sφ-1as+b0ds+∫0αx∫0βygt,sdsdt.Ωx=∫x0xdsφ-1s,x>x0>0,Gz=∫z0zdsψφ-1Ω-1s,z>z0>0,Ω-1, φ-1, and G-1 are the inverses of Ω, φ, and G, respectively; (ξ1,η1)∈R+×R+ is chosen so that(9)GPx,y+∫0αx∫0βyft,sdsdt∈dom⁡G-1,t,s∈R+×R+,G-1GPx,y+∫0αx∫0βyft,sdsdt∈dom⁡Ω-1,t,s∈R+×R+with dom⁡· denoting the function domain.Theorem 3 (see [16, Corollary  1]).
Assume thatu,f,g,a,b,α, and β are defined as in Theorem 2. Let φ(u)=up and ψ(u)=uq-1 in Theorem 2, where p≥q>1 are positive constants. If (10)upx,y≤ax+by+∫0αx∫0βyut,sft,suq-1t,s+gt,sdsdt,then, for all (x,y)∈R+×R+,(11)ux,y≤p-1pλx,yp-q/p-1+p-qp∫0αx∫0βyft,sdsdt1/p-q,whenp>q,p-1pλx,y1/p-1exp⁡1p∫0αx∫0βyft,sdsdt,whenp=q,where(12)λx,y=pp-1a0+byp-1/p+∫0xa′sas+b01/pds+∫0αx∫0βygt,sdsdt.Motivated by the recent contributions of Abdeldaim and El-Deeb [11], Zhang and Meng [14], and Tian et al. [16], our principal goal is to extend the inequalities with one variable in [11] to those with two variables which include Theorems 2 and 3 as special cases.The rest of the work is organized as follows. A useful lemma that plays a fundamental role in the proofs of the main theorems is presented in Section2. In Section 3, we propose our main theorems and corollary on several new types of linear and nonlinear retarded integral inequalities in two independent variables. An illustrative example is given to indicate the usefulness of these inequalities in Section 4, which is followed by a short conclusion in Section 5.


## 2. Lemma

The subsequent lemma is helpful in proving our main theorems.Lemma 4.
Assume thatu, f, and g∈C(R+×R+,R+) and φ∈C(R+,R+) is an increasing function with φ(∞)=∞ and ψ∈C(R+,R+) is a nondecreasing function. Suppose that c is a nonnegative constant and α,β∈C1(R+,R+) are nondecreasing with α(x)≤x, β(y)≤y, α(0)=0, and β(0)=0 on R+. If(13)φux,y≤c+∫0αx∫0βyft,sψut,sdsdt+∫0αx∫0βyft,s∫0t∫0sgτ,ωψuτ,ωdωdτdsdt,then, for 0≤x<ξ,0≤y<η,(14)ux,y≤φ-1G-1Gc+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt,where (15)Gz=∫z0zdsψφ-1s,z>z0>0;φ-1 and G-1 are the inverses of φ and G, respectively; (ξ,η)∈R+×R+ is chosen so that(16)Gc+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt∈dom⁡G-1,0≤x<ξ,0≤y<η.Proof.
Define the nondecreasing positive functionz by(17)zx,y=c+ε+∫0αx∫0βyft,sψut,sdsdt+∫0αx∫0βyft,s∫0t∫0sgτ,ωψuτ,ωdωdτdsdt,where ε is an arbitrary small positive number. Utilizing inequality (13) and the monotonicity of φ-1, we get(18)ux,y≤φ-1zx,y.Differentiating (17) with respect to x and combining (18) and the monotonicities of φ-1, z, and ψ, we conclude that (19)zxx,y=α′x∫0βyfαx,sψuαx,sds+α′x∫0βyfαx,s∫0αx∫0sgτ,ωψuτ,ωdωdτds≤ψφ-1zx,yα′x∫0βyfαx,sds+ψφ-1zx,yα′x∫0βyfαx,s∫0αx∫0sgτ,ωdωdτds.On account of ψφ-1zx,y≥ψφ-1c+ε>0, we deduce that (20)zxx,yψφ-1zx,y≤α′x∫0βyfαx,s1+∫0αx∫0sgτ,ωdωdτds.Integrating the latter inequality on 0,x and letting ε→0, we have(21)Gzx,y≤Gc+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdtowing to (15). By virtue of (16), (18), and the last inequality, we obtain inequality (14). The proof is complete.Remark 5.
Assume that∫z0∞ψφ-1s-1ds=∞. Then G(∞)=∞ and (14) is valid on R+×R+; that is, one can select ξ=∞ and η=∞.


## 3. Main Results

The following are the main results of this paper.Theorem 6.
Letu,a,f,g, and h∈C(R+×R+,R+) and let α,β∈C1(R+,R+) be nondecreasing with α(x)≤x, β(y)≤y, α(0)=0, and β(0)=0 on R+. If the inequality (22)ux,y≤ax,y+∫0αx∫0βyft,sut,s+ht,sdsdt+∫0αx∫0βyft,s∫0t∫0sgτ,ωuτ,ωdωdτdsdtholds, for all (x,y)∈R+×R+, then(23)ux,y≤ax,y+exp⁡∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt×∫0xexp⁡-∫0αl∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt×∂∂l∫0αl∫0βyft,sat,s+ht,sdsdt+∫0αl∫0βyft,s∫0t∫0sgτ,ωaτ,ωdωdτdsdtdl.Proof.
Letting(24)zx,y=∫0αx∫0βyft,sut,s+ht,sdsdt+∫0αx∫0βyft,s∫0t∫0sgτ,ωuτ,ωdωdτdsdt,then z(0,y)=z(x,0)=0 and(25)ux,y≤ax,y+zx,y.Our assumptions on f,u,h,g,α, and β indicate that z is a positive function which is nondecreasing with respect to each of the two variables. Differentiating z with respect to x and using (25), we arrive at(26)zxx,y=α′x∫0βyfαx,suαx,s+hαx,sds+α′x∫0βyfαx,s∫0αx∫0sgτ,ωuτ,ωdωdτds≤α′x∫0βyfαx,saαx,s+zαx,s+hαx,sds+α′x∫0βyfαx,s∫0αx∫0sgτ,ωaτ,ω+zτ,ωdωdτds=α′x∫0βyfαx,szαx,sds+α′x∫0βyfαx,s∫0αx∫0sgτ,ωzτ,ωdωdτds+α′x∫0βyfαx,saαx,s+hαx,sds+α′x∫0βyfαx,s∫0αx∫0sgτ,ωaτ,ωdωdτds.By virtue of the monotonicity of z, we get (27)zxx,y-zx,yα′x∫0βyfαx,s1+∫0αx∫0sgτ,ωdωdτds≤α′x∫0βyfαx,saαx,s+hαx,sds+α′x∫0βyfαx,s∫0αx∫0sgτ,ωaτ,ωdωdτds.Multiplying the latter inequality by e-∫0α(x)∫0β(y)ft,s1+∫0t∫0sg(τ,ω)dωdτdsdt yields (28)∂∂xzx,ye-∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt≤exp⁡-∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt×α′x∫0βyfαx,saαx,s+hαx,sds+∫0βyfαx,s∫0αx∫0sgτ,ωaτ,ωdωdτds.Integrating this inequality on 0,x, we deduce that(29)zx,y≤exp⁡∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt×∫0xexp⁡-∫0αl∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt×α′l∫0βyfαl,saαl,s+hαl,sds+∫0βyfαl,s∫0αl∫0sgτ,ωaτ,ωdωdτdsdl.Combining (25) with (29), we get inequality (23). This completes the proof.Theorem 7.
Letu, a, f, g, and h∈C(R+×R+,R+), a(x,y)>0, ax≥0, ay≥0, and α,β∈C1(R+,R+) be nondecreasing with α(x)≤x, β(y)≤y, α(0)=0, and β(0)=0 on R+. Moreover, let γ and ψ∈C1(R+,R+) be nondecreasing function with γ>0 and ψ>0 on (0,∞). If (30)ux,y≤ax,y+∫0αx∫0βyγut,sft,sψut,s+ht,sdsdt+∫0αx∫0βyγut,sft,s∫0t∫0sgτ,ωψuτ,ωdωdτdsdt,then, for 0≤x<ξ,0≤y<η,(31)ux,y≤Ω-1G-1GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt,where(32)Px,y=Ωax,y+∫0αx∫0βyht,sdsdt,(33)Ωx=∫x0xdsγs,x>x0>0,(34)Gz=∫z0zdsψΩ-1s,z>z0>0.Ω-1 and G-1 are the inverses of Ω and G, respectively; (ξ,η)∈R+×R+ is chosen so that (35)GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt∈dom⁡G-1,G-1GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt∈dom⁡Ω-1 for 0≤x<ξ,0≤y<η.Proof.
Define the nondecreasing functionz by(36)zx,y=∫0αx∫0βyγut,sft,sψut,s+ht,sdsdt+∫0αx∫0βyγut,sft,s∫0t∫0sgτ,ωψuτ,ωdωdτdsdt.Then(37)ux,y≤ax,y+zx,y. Differentiating (36) and using (37) and the monotonicity of γ, we obtain(38)zxyx,y=α′xβ′yγuαx,βyfαx,βyψuαx,βy+hαx,βy+α′xβ′yγuαx,βyfαx,βy∫0αx∫0βygτ,ωψuτ,ωdωdτ≤α′xβ′yγaαx,βy+zαx,βyfαx,βyψuαx,βy+hαx,βy+fαx,βy∫0αx∫0βygτ,ωψuτ,ωdωdτ.Let T1≤ξ and T2≤η be arbitrary numbers. Utilizing (38) and the monotonicities of a,z, and γ, we get that, for 0≤x<T1 and 0≤y<T2, (39)zxyx,y≤α′xβ′yγaT1,T2+zx,yfαx,βyψuαx,βy+hαx,βy+fαx,βy∫0αx∫0βygτ,ωψuτ,ωdωdτ.For a(T1,T2)>0 and γaT1,T2+zx,y>0,(40)zxyx,yγaT1,T2+zx,y≤α′xβ′yfαx,βyψuαx,βy+hαx,βy+fαx,βy∫0αx∫0βygτ,ωψuτ,ωdωdτ.From another point of view,(41)∂∂yzxγaT1,T2+zx,y≤zxyγaT1,T2+zx,y. It follows from (40) and (41) that (42)∂∂yzxγaT1,T2+zx,y≤α′xβ′yfαx,βyψuαx,βy+hαx,βy+fαx,βy∫0αx∫0βygτ,ωψuτ,ωdωdτ.Integrating the above inequality on 0,y with respect to the second variable and taking zx(x,0)=0 into account, we have(43)zxx,yγaT1,T2+zx,y≤zxx,0γaT1,T2+zx,0+α′x∫0βyfαx,sψuαx,s+hαx,s+fαx,s∫0αx∫0sgτ,ωψuτ,ωdωdτds=α′x∫0βyfαx,sψuαx,s+hαx,s+fαx,s∫0αx∫0sgτ,ωψuτ,ωdωdτds.From (33), the latter relation gives(44)∂∂xΩaT1,T2+zx,y≤α′x∫0βyfαx,sψuαx,s+hαx,s+fαx,s∫0αx∫0sgτ,ωψuτ,ωdωdτds.Integrating the last inequality over 0,x, we get (45)ΩaT1,T2+zx,y≤ΩaT1,T2+∫0αx∫0βyft,sψut,s+ht,s+ft,s∫0t∫0sgτ,ωψuτ,ωdωdτdsdt=Px,y+∫0αx∫0βyft,sψut,sdsdt+∫0αx∫0βyft,s∫0t∫0sgτ,ωψuτ,ωdωdτdsdt≤PT1,T2+∫0αx∫0βyft,sψaT1,T2+zt,sdsdt+∫0αx∫0βyft,s∫0t∫0sgτ,ωψaT1,T2+zτ,ωdωdτdsdt,where P is defined as in (32). Combining (37) and the monotonicity of a and employing Lemma 4, we obtain(46)ux,y≤aT1,T2+zx,y≤Ω-1G-1GPT1,T2+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt,where G is defined as in (34). Taking x=T1 and y=T2, we conclude that(47)uT1,T2≤Ω-1G-1GPT1,T2+∫0αT1∫0βT2ft,s1+∫0t∫0sgτ,ωdωdτdsdt.As T1≤ξ and T2≤η are arbitrary, we get the desired inequality (31). The proof is complete.Theorem 8.
Assume thatu,a,f,g,h,α,β,γ, and ψ are defined as in Theorem 7. Moreover, let φ∈C1(R+,R+) be increasing function with φ(∞)=∞ and φ(x)>0 on (0,∞). If (48)φux,y≤ax,y+∫0αx∫0βyγut,sft,sψut,s+ht,sdsdt+∫0αx∫0βyγut,sft,s∫0t∫0sgτ,ωψuτ,ωdωdτdsdt,then, for 0≤x<ξ,0≤y<η,(49)ux,y≤φ-1Ω-1G-1GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt,where (50)Px,y=Ωax,y+∫0αx∫0βyht,sdsdt,Ωx=∫x0xdsγφ-1s,x>x0>0,Gz=∫z0zdsψφ-1Ω-1s,z>z0>0;φ-1, Ω-1, and G-1 are the inverses of φ, Ω, and G, respectively; (ξ,η)∈R+×R+ is chosen so that (51)GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt∈dom⁡G-1,G-1GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt∈dom⁡Ω-1for 0≤x<ξ,0≤y<η.Proof.
Define functionz by (36). Then(52)ux,y≤φ-1ax,y+zx,y.The rest of the proof is similar to that of Theorem 7 and hence is omitted.Remark 9.
Lettinga(x,y)=a(x)+b(y), γ(u(x,y))=u(x,y), and g(x,y)=0 in Theorem 8, Theorem 8 turns out to be Theorem 2. Therefore, the inequality established in Theorem 8 generalizes that of [16, Theorem  1].Ifφ(u)=up, γ(u)=uq, and ψ(u)=un in Theorem 8, where p≥q+n>1, and p,q, and n are positive constants, then we have the following corollary.Corollary 10.
Assume thatu,a,f,g,h,α, and β are defined as in Theorem 8. If(53)upx,y≤ax,y+∫0αx∫0βyuqt,sft,sunt,s+ht,sdsdt+∫0αx∫0βyuqt,sft,s∫0t∫0sgτ,ωunτ,ωdωdτdsdt,then, for all (x,y)∈R+×R+,(54)ux,y≤p-qpλx,yp-q-n/p-q+p-q-npθx,y1/p-q-n,whenp>q+n,npλx,y1/nexp⁡1pθx,y,whenp=q+n,where(55)λx,y=pp-qax,yp-q/p+∫0αx∫0βyht,sdsdt,θx,y=∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt.Proof.
Assume thatp>q+n and let φ(u)=up, γ(u)=uq, and ψ(u)=un. Then we have φ-1(u)=u1/p, and so(56)Ωx=∫x0xdsγφ-1s=∫x0xs-q/pds=pp-qxp-q/p-x0p-q/p,Ω-1x=p-qpx+x0p-q/pp/p-q,ψφ-1Ω-1x=p-qpx+x0p-q/pn/p-q,Gz=∫z0zdsψφ-1Ω-1s=∫z0zp-qps+x0p-q/p-n/p-qds=pp-q-np-qpz+x0p-q/pp-q-n/p-q-pp-q-np-qpz0+x0p-q/pp-q-n/p-q,G-1z=pp-qp-q-npz+p-qpz0+x0p-q/pp-q-n/p-qp-q/p-q-n-x0p-q/p,Ω-1G-1GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt=p-qpλx,yp-q-n/p-q+p-q-npθx,yp-q/p-q-n-x0p-q/p+x0p-q/pp/p-q=p-qpλx,yp-q-n/p-q+p-q-npθx,yp/p-q-n,where λ and θ are defined in (55). Using Theorem 8, one can easily obtain (57)ux,y≤φ-1Ω-1G-1GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt=p-qpλx,yp-q-n/p-q+p-q-npθx,y1/p-q-n.
Whenp=q+n, (58)ψφ-1Ω-1x=p-qpx+x0p-q/p,Gz=∫z0zdsψφ-1Ω-1s=∫z0zp-qps+x0p-q/p-1ds=pp-qln⁡z+pp-qx0p-q/p-ln⁡z0+pp-qx0p-q/p,G-1z=z0+pp-qx0p-q/pexp⁡p-qpz-pp-qx0p-q/p,Ω-1G-1GPx,y+∫0αx∫0βyft,s1+∫0t∫0sgτ,ωdωdτdsdt=p-qpλx,yexp⁡p-qpθx,y-x0p-q/p+x0p-q/pp/p-q=npλx,yexp⁡npθx,yp/n,where λ and θ are the same as in (55). By Theorem 8, similar discussions can give(59)ux,y≤npλx,y1/nexp⁡1pθx,y.This completes the proof.Remark 11.
Lettinga(x,y)=a(x)+b(y), q=1, n=q-1, and g(x,y)=0, Corollary 10 reduces to Theorem 3. Hence, the inequality established in Corollary 10 includes the result of [16, Corollary 1].


## 4. Example

Example 1.
Consider the integral equation(60)upx,y=kx,y+∫0αx∫0βyFt,s,ut,s,∫0t∫0sHτ,ω,uτ,ωdωdτdsdt,where k:R+×R+→R and F:R+×R+×R×R→R are continuous functions, α,β∈C1(R+,R+) is nondecreasing with α(x)≤x, β(y)≤y, α(0)=0, and β(0)=0 on R+, and p≥4 is a constant. Suppose that(61)kx,y≤ax,y,Ft,s,u,v≤ft,su4+ht,su2+ft,su2v,Ht,s,u≤gt,su2,where a,f,h, and g are defined as in Corollary 10. Combining (60)-(61) yields(62)ux,yp≤ax,y+∫0αx∫0βyft,sut,s4+ht,sut,s2dsdt+∫0αx∫0βyft,sut,s2∫0t∫0sgτ,ωuτ,ω2dωdτdsdt.Exploiting Corollary 10, we obtain an explicit bound to the solutions of (60): (63)ux,y≤p-2pλx,yp-4/p-2+p-4pθx,y1/p-4,whenp>4,12λx,y1/2exp⁡14θx,y,whenp=4,where λ and θ are defined as in Corollary 10.


## 5. Conclusions

This paper investigates some new types of linear and nonlinear retarded integral inequalities in two independent variables. Several theorems and a corollary of these inequalities are obtained based on some analysis techniques, such as amplification method, differential, and integration. An illustrative example is studied to demonstrate the effectiveness of the new results.



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