# Some Nonlinear Integral Inequalities Connected with Retarded Terms on Time Scales

Authors: Zareen A. Khan; Hijaz Ahmad; Saima Rashid; Kadir Kaynak; Miao-Kun Wang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The objective of this research is to formulate a specific class of integral inequalities of Gronwall kind concerning retarded term and nonlinear integrals with time scales theory. Our results generate several new inequalities that reflect continuous and discrete form, as well as giving the unknown function an upper bound estimate. The effectiveness of such inequalities arises from the belief that it is widely relevant in unique circumstances where there is no valid utilization of various available inequalities. Applications are additionally represented to display the legitimacy of built-up hypotheses.


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## Body

## 1. Introduction

Over the last decades, a variety of basic and critical inequalities have been inspected with improvement in the methods of differential and integral equations, which are anticipating a great deal of study in the analysis of boundedness, global existence, and stability of solutions of differential and integral equations as well as difference equations [1, 2]. Among others, the inequalities of the Gronwall-Bellman type are of unique significance for providing explicit estimates for unknown functions.Bellman [3] has proven the integral inequality
(1)Lℏ≤m+∫ℏ1ℏGPLPdP,ℏ∈ℏ1,ℏ,for some m≥0, which is significantly dedicated to evaluate the equilibrium and asymptotic behavior with a view to find solutions for integral equations. Such inequalities have been remarkably strengthened over a very long period of time by verifying their importance and intrinsic potential in the diverse fields of applied sciences.Afterward, Pachpatte [4] replaced the constant m from the prior integral inequality by a nondecreasing function mℏ and contemplated
(2)Lℏ≤mℏ+Hℏ∫0ℏGPLPdP,ℏ∈0,∞.The inequality (2) encourages fresh speculations and can be classified as incredible techniques in the understanding of specific differential and integral equations.El-Owaidy et al. [5] introduced yet another basic type of inequality
(3)Lℏ≤m0+∫0ℏGPLΨP+∫0PJxLxdxdP,ℏ∈0,∞,in which integrand accommodates the power and allows it more challenging to determine the unknown function where 0≤Ψ<1 is included.Several scholars researched linear and nonlinear modifications for booming such kinds of inequalities (see [6–12]). Most articles tend to do with retarded nonlinear integral inequalities. In this database, the retarded integral inequality was identified by Lipovan [13].
(4)Lℏ≤m+∫ℏ1ℏGPρLPdP+∫ωℏ1ωℏJPρLPdP,ℏ1≤ℏ<ℏ2,ρ∈Cℝ+,ℝ+, ρL>0 for L>0, and the integral consists of nonlinear equations in its more standard way. Over the span of late years, multiple retarded integral inequalities have been discussed by numerous researchers [14, 15] and the references therein.Remarkably, the dynamic inequalities predict a necessary position within the production of the basic principle of time-scale dynamic equations. Hilger [16] was named to be the primary researcher who started the growth of time scale analytics. The ultimate point is to study an equation or an inequality which can be dynamic such that a time scale T be a domain of an unknown function. The motivation that guides the time scale theory is to unify continuous and discrete inspection. Several authors have approached and finished proper assessment of the characterizations and utilization of different types of inequalities on a timely basis [17–21] and the references in that. Around the beginning, Bohner and Peterson [22] tested the integral inequality
(5)Lℏ≤mℏ+∫ℏ1ℏJPLPΔP,ℏ∈T.Later in 2010, Li [23] has assessed the consequent nonlinear integral inequality of one variable
(6)Lγℏ≤mℏ+Hℏ∫ℏ1ℏGPLρP+JPΔP,for ℏ∈ℏ1 with initial conditions Lℏ=Ωℏ, ℏ∈β,ℏ1∩T, Yρℏ≤mℏ1/γ for ℏ∈ℏ1, ρℏ≤ℏ1, where γ≥1 is a constant, ρℏ≤ℏ, −∞<β=infρℏ,ℏ∈T0≤ℏ1, and Ωℏ∈Crdβ,ℏ1∩T,ℝ+. Besides that, Pachpatte [24] has attempted to see the augmentation of the integral inequality
(7)Lℏ≤mℏ∫ℏ1ℏGPLP+∫ℏ1PJP,xLxΔxΔP,such that JP,x≥0, JΔP,x≥0 for ℏ,x∈T and σ≤ℏ. Further, in 2017, Haidong [25] suggested that nonlinear integral inequality be generalized as follows
(8)Lℏ≤mℏ+Gℏ∫ωℏ1ωℏG1PLP+G2P∫ωℏ1ℏJxLxΔxΔP+λmT∫ωℏ1ωTG1PLP+G2P∫ωℏ1ℏJxLxΔxΔP,where λ≥0. Although a lot of studies have been conducted on integral inequalities related to time scales, there is not that much research on retarded integral inequalities on time scales has been performed. For certain cases, however, specific types of integral and differential equations via power are required to investigate time scale delay inequalities in order to thrive and meet the desired targets.Our primary concern of this work is not only to analyze nonlinear integral inequalities with retarded term but also to explore the well-known existing results which determine the explicit bounds of the solutions of the unknown functions of the particular dynamic equations on time scales. The new speculations are used as supportive tools to exhibit the description of integral inequalities and equations. The offered dynamic integral strategy for acquiring new results is clear and effective. There are other benefits of this technique: it is fast and short. Moreover, the proposed procedure can be modified to solve various systems with nonlinear fractional partial differential equations.The rest of the manuscript is arranged as follows. In Section2, we have some preliminary data which is an essential element for our key studies. Section 3 is devoted to theoretical discussions on the immense solutions of the problem beneath consideration. The examples supporting the theoretical consequences are given in Section 4. Finally, a few concluding feedback and suggestions for future research are provided in Section 5 and thus completes this work.


## 2. Basic Material on Time Scales

Below, we interpret some basic definitions and valuable theorem regarding time scale calculus.A time scaleT is an arbitrary nonempty closed subset of the real numbers ℝ.Definition 1 (see [22]).
The forward jump operatorσ on T be defined by σℏ=infP∈T:P>ℏ for all ℏ∈T. If σℏ=ℏ, then ℏ is said to be right-dense and right-scattered if σℏ>ℏ. The backward jump operator and left-scattered and left-dense points are defined in a similar way.Definition 2 (see [22]).
ξ:T→ℝ+ is called the graininess function if ξℏ=σℏ−ℏ. Also, ξℏ=0 for T=ℝ and ξℏ=P, with a positive constant P for T=Pℤ.Definition 3 (see [22]).
The setTk is derived from T as follows: if T has a left-scattered maximum m, then Tk=T−m; otherwise, Tk=T.Definition 4 (see [22]).
For someℏ∈Tk and a function L∈T,ℝ, the delta derivative of L is denoted by LΔℏ and satisfies
(9)Lσℏ−LP−LΔℏσℏ−P≤εσℏ−P,∀ε>0,P∈ℵ,
and a neighborhood ofℏ is ℵ. Also, L is the delta differential function at ℏ.Definition 5 (see [22]).
An antiderivative ofL is L1 if L1Δℏ=Lℏ, ℏ∈Tk, whereas
(10)∫ς1ςLℏΔℏ=L1ς−L1ς1,ς1,ς∈T,
is the Cauchy integral ofL.Definition 6 (see [22]).
The set of all regressive and rd-continuous functionsG:T⟶ℝ will be represented by R provided 1+ξℏGℏ≠0 for all ℏ∈Tk holds.Definition 7.
IfT=ℝ and G:ℝ⟶ℝ are a continuous function, then the exponential function is given by
(11)eGℏ,P=e∫PℏGτ∗dτ∗,eαℏ,P=eαℏ−P,eαℏ,0=eαℏ,P,ℏ,α∈ℝ.Theorem 8 (see [22]).
IfG∈R, then
(i)
eGℏ,ℏ≡1 and e0ℏ,r≡1(12)eGσℏ,P≡1+ξℏGℏeGℏ,P,1eGℏ,P=e⊖Gℏ,P=eGP,ℏ.(ii)
IfJ∈R+, then ⊖J∈R+(iii)
IfJ∈R+, then eJℏ,ℏ0>0, for all ℏ0,ℏ∈T, where ⊖J=−J/1+ξJ.To more descriptions of the study of the time scale, we direct the reader to Bohner and Peterson [22] excellent monograph, which describes and organizes most of the time scale logic.


## 3. Nonlinear Powered Integral Inequalities via Retarded Term

Given the documentations all through the content for the simplicity of perusing:ℝ stands for the set of real numbers, ℂrd is the set of rd continuous functions, ℝ+=0,∞, ℏ1∈T, ℏ1≥0, T0=ℏ1,∞∩T, and(H1).
The continuous functionLℏ∈ℂrdT0,ℝ+ is a nonnegative.(H2).
ξℏ∈CrdT0,ℝ+ is a function which is increasing and differentiable on ℏ1,∞ so that ξℏ≤ℏ, ξℏ1=ℏ1.(H3).
Gℏ, Jℏ, Bℏ∈ℂrdT0,ℝ+.(H4).
mℏ∈ℂrdT0,0,∞ is nondecreasing.(H5).
μ,κ∈ℂrdℝ+,ℝ+ is a nondecreasing function and μJ>0, κJ>0 for J>0.Specifically, the fundamental lemma to be used afterwards is presented below:Lemma 9 (see [26]).
Letψ≥0, j≥ϑ≥0, and j≠0, then
(13)ψϑ/j≤ϑjψ+j−ϑj.Proof.
Forϑ=0, inequality (13) is accurate, unless δ=1 if ϑ>0, δ=ϑ/j, and
(14)ψϑ/j≤ϑjYϑj/jψ+j−ϑjYϑ/j,Y>0.
We obtain (13) for Y=1.Theorem 10.
Suppose that (H1), (H2), (H3), and (H4) be satisfied. Moreover
(15)Lℏ≤mℏ+∫ℏ1ξℏGPLPLP+∫ℏ1PJxLxΔxℵΔP+∫ℏ1ξℏGPLP∫ℏ1PBxLxΔxΔP.
∀ℏ∈T0. Then
(16)Lℏ≤mℏ+H1ℏe⊖−Yσℏ,ℏ11−H1ℏ∫ℏ1ξℏχPe⊖−YP,P1ΔP,where ℵ∈0,1 is a positive constant,
(17)H1ℏ=∫ℏ1ξℏGPHPℵHP+∫ℏ1PJxHxΔx+1−ℵ+∫ℏ1PBxHxΔxΔP,(18)Yℏ=Gℏ2ℵHℏ1+∫ℏ1ℏJPΔP+1−ℵ+2Hℏ∫ℏ1ℏBPΔP,(19)χℏ=Gℏℵ1+∫ℏ1ℏJPΔP+∫ℏ1ℏBPΔP.Proof.
Let us define(20)Qℏ=∫ℏ1ξℏGPLPLP+∫ℏ1PJxLxΔxℵΔP+∫ℏ1ξℏGPLP∫ℏ1PBxLxΔxΔP,therefore, (15) reaches to
(21)Lℏ≤mℏ+Qℏ,Qℏ1=0,by executing Lemma 9 and (21) into (20), we deduce
(22)Qℏ≤∫ℏ1ξℏGPLPℵLP+∫ℏ1PJxLxΔx+1−ℵΔP+∫ℏ1ξℏGPLP∫ℏ1PBxLxΔxΔP≤∫ℏ1ξℏGPmP+QPℵmP+QP+∫ℏ1PJxmx+QxΔx+1−ℵΔP+∫ℏ1ξℏGPmP+QP∫ℏ1PBxmx+QxΔxΔP≤∫ℏ1ξℏGPmPℵmP+∫ℏ1PJxmxΔx+1−ℵGPmP+GPmP∫ℏ1PBxmxΔxΔP+∫ℏ1ξℏGP2ℵmP1+∫ℏ1PJxΔx+1−ℵGP+2GPmP∫ℏ1PBxΔxQPΔP+∫ℏ1ξℏGPℵ1+∫ℏ1PJxΔx+GP∫ℏ1PBxΔxQ2PΔP≤H1ℏ+∫ℏ1ξℏYPQPΔP+∫ℏ1ξℏχPQ2PΔP,where H1ℏ, Yℏ, and χℏ are quoted in (17), (18), and (19), respectively. Fixing ℏ2∈T0 for an arbitrary ℏ∈ℏ1,ℏ2∩T and taking Q1ℏ by
(23)Q1ℏ≤H1ℏ2+∫ℏ1ξℏYPQPΔP+∫ℏ1ξℏχPQ2PΔP,
(22) can be carried out as
(24)Qℏ≤Q1ℏ,Qξℏ≤Q1ξℏ≤Q1ℏ,Q1ℏ1=H1ℏ2,since Q1ℏ is nondecreasing, hence we can compose (23) and (24) as
(25)Q1Δℏ≤YξℏξΔℏQ1ℏ+χξℏξΔℏQ12ℏ,take Zℏ=Q1−1ℏ, then ZΔℏ=−Q1−2ℏQ1Δℏ; therefore, the last inequality with Theorem 8 yields
(26)ZΔℏ+YξℏξΔℏZℏ≥−χξℏωΔℏ,or
(27)Zℏe⊝Yσℏ,ℏ1Δ=ZΔℏ+YξℏξΔℏZℏe⊝Yσℏ,ℏ1≥−χξℏξΔℏe!−Yσℏ,ℏ1.
Integrating (27) and applying Zℏ1=H1−1ℏ2, we attain
(28)Zℏ≥1−H1ℏ2∫ℏ1ℏχξPξΔPe⊝−Yσℏ,ℏ1ΔPH1ℏ2e⊝−Yσℏ,ℏ1.Substituting Zℏ=Q1−1ℏ in (28) and from (24), we get
(29)Qℏ≤H1ℏ2e⊝−Yσℏ,ℏ11−H1ℏ2∫ℏ1ℏχξPξΔPe⊝−Yσℏ,ℏ1ΔP.Substituting the previous value in (21), we have
(30)Lℏ2≤mℏ2+H1ℏ2e⊝−Yσℏ,ℏ11−H1ℏ2∫ℏ1ξℏχPe⊝−YP,P1ΔP,the arbitrariness of ℏ2 in the previous inequality claims the optimal bound in (16).Remark 11.
Choosemℏ=u0, γ=1, Lℏ≤ϕut, Jℏ=gt, and Gℏ=ft, then Theorem 10 will become a small deviation of Theorem 2.2 studied by Abdeldaim and EI-Deeb [14], if T=ℝ and qt=0.Theorem 12.
The presumption (H1), (H2), (H4), and (H5), Gℏ, Jℏ∈ℂrdT0,ℝ+, and
(31)Lℏ≤mℏ+∫ℏ1ξℏGPμLPLλP+∫ℏ1PJxκLxΔxℵΔP,∀ℏ∈T0,satisfy, then
(32)Lℏ≤Σ−1M¯ℏ,
∀ℏ∈ℏ1,ℏ2, where ℵ and λ are constants, ℵ>0, λ≥1, ℵ+λ>1, and
(33)M¯ℏ=Σmℏ+∫ℏ1ξℏGPFPΔP,Fℏ=G−1P−1PGmλℏ+∫ℏ1ℏJPΔP+λ∫ℏ1ωℏGPΔPℵ,(34)Gℓ=∫1ℓΔPκP,ℓ>0,(35)Pℓ=∫1ℓϕG−1PΔPμGPG−1Pℵ+λ−1,ℓ>0,(36)Σℓ=∫1ℓΔPμP,ℓ>0,the inverses of G, P, and Σ are G−1, P−1, and Σ−1, and select ℏ3 in such a way that
(37)PGmλℏ+∫ℏ1ℏJPΔP+λ∫ℏ1ξℏGPΔP≤∫1ℓϕG−1PΔPμG−1PG−1Pℵ+λ−1,ξmℏ+∫ℏ1ξℏGPFPΔP≤ΔPμP,P−1PGmλℏ+∫ℏ1ℏJPΔP+λ∫ℏ1ξℏGPΔP≤ΔPϕP.Proof.
Letℏ3∈T0 be fixed for ℏ∈ℏ1,ℏ3∩T and letting
(38)Q2ℏ=mℏ3+∫ℏ1ξℏGPμLPLλP+∫ℏ1PJxκLxΔxℵΔP,
(31) and (38) imply that
(39)Lℏ≤Q2ℏ,Q2ℏ1=mℏ3,since Q2ℏ is nondecreasing, then (38) equals to
(40)Q2Δℏ=GξℏξΔℏμLξℏLλξℏ+∫ℏ1ξℏJPκLPΔPℵ≤GξℏξΔℏμLξℏQ2λξℏ+∫ℏ1ξℏJPκQ2PΔPγ≤GξℏξΔℏμLξℏZ1ℵξℏ,where
(41)Z1ξℏ=Q2λξℏ+∫ℏ1ξℏJPκQ2PΔP,(42)Z1ξℏ1=mλℏ3,Q2ξℏ≤Z1ξℏ.By the definition of Z1ξℏ, utilizing (34), (42), and Z1ξℏ>0, we have
(43)Z1ΔξℏξΔℏ=λQ2λ−1Q2ΔξℏξΔℏ+JξℏξΔℏκQ2ξℏ,so that
(44)Z1Δξℏ≤λGξℏξΔℏμZ1ξℏZ1ℵ+λ−1ξℏ+JξℏκZ1ξℏZ1ΔξℏκZ1ξℏ≤λGξℏξΔℏμWξℏZ1ℵ+λ−1ξℏκZ1ξℏ+Jξℏ.Integrating the above inequality, using (34) and (42), we get
(45)GZ1ξℏ≤Gmλℏ3+∫ℏ1ℏJξPΔP+∫ℏ1ℏλGξPξΔPμZ1ξPZ1ℵ+λ−1ξPκZ1ξPΔP≤Gmλℏ3+∫ℏ1ℏ2JξPΔP+∫ℏ1ℏλGξPξΔPμZ1ωPZ1ℵ+λ−1ξPκZ1ξPΔP,for ℏ<ℏ3. Denoting
(46)Q3ℏ=∫ℏ1ℏλGξPξΔPμZ1ξPZ1ℵ+λ−1ξPκZ1ξPΔP,
(45) and (46) give
(47)Z1ξℏ≤G−1Q3ℏ,Q3ℏ1=Gmλℏ3+∫ℏ1ℏ3JξPΔP.Differentiating (46) and applying (47), we observe that
(48)Q3Δℏ=λGξℏξΔℏμZ1ξℏZ1ℵ+λ−1ξℏκZ1ξℏ≤λGξℏξΔℏμG−1Q3ℏG−1Q3ℵ+λ−1ℏκG−1Q3ℏ,implies
(49)κG−1Q3ℏQ3ΔℏμG−1Q3ℏG−1Q3ℵ+λ−1ℏ≤λGξℏξΔℏ. Inequality (49) with integration and (35) and (47) generate the approximation
(50)PQ3ℏ≤PGmλℏ3+∫ℏ1ℏ3JξPΔP+λ∫ℏ1ℏGξPξΔPΔP,(51)Q3ℏ≤P−1PGmλℏ3+∫ℏ1ℏ2JξPΔP+λ∫ℏ1ξℏGPΔP.In view of (47) and (51), we derive
(52)Z1ℵξℏ≤G−1P−1PGmλℏ3+∫ℏ1ℏ2JξPΔP+λ∫ℏ1ξℏGPΔPℵ=Fℏ.Substitute (52) in (40), integrate from ℏ1 to ℏ in the resultant inequality, employ (36), and put ℏ=ℏ3, we attain
(53)Q2ℏ≤Σ−1Σmℏ3+∫ℏ1ξℏ2GPFPΔP.The acquired inequality in (32) can be produced by the arbitrary ℏ3 to the last inequality and from (39).Remark 13.
As a special case onT=ℝ, if we take Lℏ=ux, mℏ=gx, Gℏ=kx,t, J=0, ℵ=1, μLℏ=ϕut, ℏ1=x0, and ξℏ=x, then Theorem 12 changes to Theorem 2.1 of [27] due to Oguntuase.Theorem 14.
IfGℏ∈ℂrdT0,ℝ+, the conditions (H1), (H2), (H4) and
(54)Lℵ+1ℏ≤mℏ+∫ℏ1ξℏGPLℵPΔP2+2∫ℏ1ξℏGPLℵPLℵP+∫ℏ1PGxLℵxΔxΔP,∀ℏ∈T0,hold. Then
(55)Lℏ≤m1/γ+1ℏ+2ℵ+1∫ℏ1ξℏGPΠPe⊝−GσP,P1ΔP,provided with ℵ∈0,1 is a constant and
(56)Πℏ=m1−ℵ/ℵ+1ℏ+21−ℵℵ+1∫ℏ1ξℏGPe⊝−G1−ℵℵP,P1ΔPℵ/1−ℵ,such that
(57)m1−ℵ/ℵ+1ℏ+21−ℵℵ+1∫ℏ1ξℏGPe⊝−G1−ℵ/ℵP,P1ΔP>0,∀ℏ∈T.Proof.
Fixℏ4∈T0 an arbitrary ℏ∈ℏ1,ℏ4∩T and denote the positive and nondecreasing function Q4ℏ by
(58)Q4ℵ+1ℏ=mℏ4+∫ℏ1ξℏGPLℵPΔP2+2∫ℏ1ξℏGPLℵPLℵP+∫ℏ1PGxLℵxΔxΔP.
(54) restates as
(59)Lℏ≤Q4ℏ,Lξℏ≤Q4ξℏ≤Q4ℏ,Q4ℏ1=m1/1+ℵℏ4.Differentiating (58) and employing (59), we get
(60)ℵ+1Q4ℵℏQ4Δℏ=2∫ℏ1ξℏGPLℵPΔPGξℏξΔℏLℵξℏ+2GξℏξΔℏLℵξℏLℵξℏ+∫ℏ1ξℏGPLℵPΔP,which becomes
(61)Q4Δℏ≤2ℵ+1GξℏξΔℏQ4ℵℏ+2∫ℏ1ξℏGPQ4ℵPΔP≤2ℵ+1GξℏξΔℏZ2ℏ,Z2ℏ1=mℵ/1+ℵℏ4,so that
(62)Z2ℏ=Q4ℵℏ+2∫ℏ1ξℏGPQ4ℵPΔP,(63)Q4ℏ≤Z21/ℵℏ,Q4ξℏ≤Z21/ℵξℏ≤Z21/ℵℏ.
Taking derivative (62) and from (63), we deduce
(64)Z2Δℏ=ℵQ4ℵ−1ℏQ4Δℏ+2GξℏξΔℏQ4ℵξℏ≤2ℵℵ+1GξℏξΔℏZ2ℵ−1/ℵℏZ2ℏ+2GξℏξΔℏZ2ℏ≤2ℵℵ+1GξℏξΔℏZ22ℵ−1/ℵℏ+2GξℏξΔℏZ2ℏ,or equivalently,
(65)Z21−2ℵ/ℵℏZ2Δℏ−2GξℏξΔℏZ21−ℵ/ℵℏ≤2ℵℵ+1GξℏξΔℏ,which by using Theorem 8, (61), and (63) leads to bound
(66)Z2ℏ≤e⊝−Gσℏ,ℏ1m1−ℵ/ℵ+1ℏ4+21−ℵℵ+1∫ℏ1ℏGξPξΔPe⊝−G1−ℵ/ℵσP,P1ΔPℵ/1−ℵ,where G=−2Gℏ. From (61) and (66), it is noticed that
(67)Q4Δℏ≤2ℵ+1GξℏξΔℏe⊝−Gσℏ,ℏ1m1−ℵ/ℵ+1ℏ4+21−ℵℵ+1∫ℏ1ξℏGPe⊝−G1−ℵ/ℵP,P1ΔPℵ/1−ℵ,
ℏ4∈T0 is chosen; therefore, the required estimation in (55) can be obtained by integrating the above inequality from ℏ1 to ℏ and then combining the obtained inequality with (59).


## 4. Enforcement on Theoretical Results

This segment is about to discuss the immediate utilization of Theorem14 by assessing the boundedness and uniqueness of the retarded nonlinear integrodifferential equation on time scales. For this, let us consider
(68)γ+1LγℏLΔℏ=Hξℏ,Lξℏ+Aξℏ,Lξℏ,∫ℏ1ℏIP,LPΔP,Lℏ1=ℏ1,(69)∀ℏ∈T0,whereH,I:T0×ℝ⟶ℝ,A:T0×ℝ×ℝ⟶ℝandωℏ≤ℏ,0<ωΔℏ≤ℏ,ωℏ1=ℏ1.We suggest the boundedness on the solution of (68) in the first illustration.For example, ifLℏ is the solution of (68) with conditions
(70)Hℏ,Lℏ≤G2ℏL2ℵℏ,(71)Aℏ,Lℏ,∫ℏ1ℏIP,LPΔP≤IP,LPLℵℏ+∫ℏ1ℏIP,LPΔP,then
(72)∣Lℏ∣≤ℏ1+2ℵ+1∫ℏ1ξℏGPΠ1Pe⊝−GσP,P1ΔP,where G, ℵ, and ξ are the same as in Theorem 14 and
(73)Π1ℏ=ℏ11−ℵ+21−ℵℵ+1∫ℏ1ξℏGPe!−G1−ℵ/ℵP,P1ΔPℵ/1−ℵ.Proof.
Keepingℏ=P in (68) and integrating from ℏ1 to ℏ, we have
(74)Lℵ+1ℏ=Lℵ+1ℏ+∫ℏ1ℏHξP,LξPΔP+∫ℏ1ℏAξP,LξP,∫ℏ1PIx,LxΔxΔP,which with the help of (70) and (71) takes the form
(75)Lℵ+1ℏ=ℏ1ℵ+1+∫ℏ1ℏHξP,LξPΔP+∫ℏ1ℏAωP,LξP,∫ℏ1PIx,LxΔxΔP≤ℏ1ℵ+1+∫ℏ1ℏHξP,LξPΔP+∫ℏ1ℏAξP,LξP,∫ℏ1PIx,LxΔxΔP≤ℏ1ℵ+1+∫ℏ1ℏGξPLℵξPΔP2+2∫ℏ1ℏGξPLℵξPLℵξP+∫ℏ1PGx∣LℵxΔx.Let G¯ℏ=Gℏ/ξΔξ−1ℏ, then Gℏ≤G¯ℏ for ℏ∈T0; therefore, (75) yields
(76)Lℵ+1ℏ≤ℏ1ℵ+1+∫ℏ1ξℏG¯PLℵPΔP2+2∫ℏ1ξℏG¯PLℵPLℵP+∫ℏ1PG¯xLℵxΔx.
The requisite bound (72) can be accomplished by the reasonable implementation of Theorem 14 with some alterations in the previous inequality; hence, here, we remove the information.The second example is based on the uniqueness on the solution of (66).For example, we list the following hypotheses as below(77)Hℏ,L1ℏ−Hℏ,L2ℏ≤G2ℏL12ℵℏ−L22ℵℏ,(78)Aℏ,L1ℏ,∫ℏ1ℏIP,L1PΔP−Aℏ,L2ℏ,∫ℏ1ℏIP,L2PΔP≤IP,L1P−IP,L2PL1ℵℏ−L2ℵℏ+∫ℏ1ℏIP,L1P−IP,L2PΔP,then (68) has at most one solution.Proof.
The solutionsL1ℏ and L2ℏ of (68) transform into
(79)L1ℵ+1ℏ−L2ℵ+1ℏ=∫ℏ1ℏHξP,L1ξPΔP+∫ℏ1ℏAξP,L1ξP,∫ℏ1PIx,L1xΔxΔP−∫ℏ1ℏHξP,L2ξPΔP+∫ℏ1ℏAξP,L2ξP,∫ℏ1PIx,L2xΔxΔP.
Equivalently(80)L1ℵ+1ℏ−L2ℵ+1ℏ≤∫ℏ1ℏHξP,L1ξPΔP+∫ℏ1ℏAξP,L1ξP,∫ℏ1PIx,L1xΔxΔP−∫ℏ1ℏHξP,L2ξPΔP+∫ℏ1ℏAξP,L2ξP,∫ℏ1PIx,L2xΔxΔP≤∫ℏ1ℏHξP,L1ξP−HξP,L2ξPΔP+∫ℏ1ℏAξP,L1ξP,∫ℏ1PIx,L1xΔx−AξP,L2ξP,∫ℏ1PIx,L2xΔxΔP.Applying (77) and (78) in the above inequality, we get
(81)L1ℵ+1ℏ−L2ℵ+1ℏ≤∫ℏ1ξℏG¯PL1ℵP−L2ℵPΔP2+2∫ℏ1ξℏG¯PL1ℵP−L2ℵP×L1ℵP−L2ℵP+∫ℏ1PG¯xL1ℵx−L2ℵxΔx.
The last inequality by making few modifications to the Theorem14 for the function
(82)L1ℵ+1ℏ−L2ℵ+1ℏ,
induces(83)L1ℵ+1ℏ−L2ℵ+1ℏ≤0.Thus,L1ℏ=L2ℏ, and there exists at least one solution of (68).


## 5. Conclusion

Unlike some proven and defined inequalities in the literature, Theorem10, Theorem 12, and Theorem 14 have examined some dynamic integral inequalities of the Gronwall-Bellman form in a single independent variable with a retarded and nonlinear term that can be used to overcome the qualitative properties of integral equations. Our observations may be used to solve the difficulty of measuring the explicit bounds of undefined functions and to expand and unify continuous inequalities by the use of basic technologies. We believe that the findings obtained here are of a general kind and offer many contributions to statistical data and are useful to identify the existence and uniqueness of the integrodifferential equations. As should be obvious, the provided results present a helpful resource in the study of solutions of certain delay dynamic equations on time scales. The inequalities that have been created unify some known continuous and discrete inequalities. One can say that it will be attractive for the researchers to generalize our results for further exploration.



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*Source: 6639743-2020-12-09.xml*