# Some Nonlinear Delay Volterra–Fredholm Type Dynamic Integral Inequalities on Time Scales

Authors: Yazhou Tian; A. A. El-Deeb; Fanwei Meng
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We are devoted to studying a class of nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales, which can provide explicit bounds on unknown functions. The obtained results can be utilized to investigate the qualitative theory of nonlinear delay Volterra–Fredholm type dynamic equations. An example is also presented to illustrate the theoretical results.


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## Body

## 1. Introduction

Since Hilger established the theory of time scales [1], it has become the research focus of mathematics and engineering field [2]. Particularly, integral inequalities play an important role in studying the qualitative properties of dynamic equations on time scales. For example, the integral inequalities were employed to investigate the stability of switched systems or uncertain nonlinear systems [3, 4].In recent years, many authors have been devoted to studying different kinds of integral inequalities and their applications [5–24], especially the application of Volterra–Fredholm integrodifferential system [25–28]. To mention a few, in [8], Gu and Meng considered the nonlinear dynamic integral inequalities on time scales and applied the theoretical results to Volterra–Fredholm integrodifferential system, and Liu [9] investigated the linear delay Volterra–Fredholm type dynamic integral inequalities which generalized the main results of [8]. In [20], Xu and Ma considered Volterra–Fredholm type integral inequalities in two independent variables and their applications in partial differential equations. Very recently, in [22], Ding and Ahmad studied Volterra–Fredholm type integral inequalities and their applications to fractional differential equations. As is known to us, few authors pay attention to nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales. This is the main reason why we establish this topic.This paper investigates a class of nonlinear dynamic integral inequalities on time scales, which can be utilized as effective tools in the study of delay Volterra–Fredholm type dynamic equations. At the end, we provide an example to illustrate the main results.


## 2. Main Results

Throughout the paper, letR be the set of real numbers, R+=[0,∞), C(M,S) be the class of all continuous functions defined on set M with range in the set S, and T be an arbitrary time scale. The set Tk is derived from T as follows: if T has a left-scattered maximum n, then Tk=T-{n}; otherwise, Tk=T. I=[t0,T1]∩Tk, where t0∈Tk, T1∈Tk, and T1>t0. Crd denotes the set of rd-continuous functions. R represents the set of all regressive and rd-continuous functions, and R+={p∈R:1+μ(t)p(t)>0,t∈T}. The graininess function μ:T→[0,∞) is defined by μ(t):=σ(t)-t, the forward jump operator σ:T→T by σ(t)=inf{s∈T:s>t}, and the circle plus addition ⊕ is defined by (p⊕q)(t)=p(t)+q(t)+μ(t)p(t)q(t),t∈Tk.Next, we introduce some lemmas to establish the main results.Lemma 1 (comparison theorem [2]).
Letu,b∈Crd and a∈R+. If (1)uΔt≤atut+bt,t≥t0,t∈Tk,then (2)ut≤ut0eat,t0+∫t0tbτeat,στΔτ,t≥t0,t∈Tk.Lemma 2 (see [2]).
Assume thatw:T×Tk→R+ is continuous at (t,t) with t>t0 and t0,t∈Tk, and w(t,·) is rd-continuous on [t0,σ(t)]. For any ε>0, if there exists a neighborhood U of t, independent of ξ∈[t0,σ(t)], such that (3)wσt,ξ-ws,ξ-wtΔt,ξσt-s≤εσt-s,s∈U,where wtΔ represents the derivative of ω with respect to t, then (4)vt=∫t0twt,ξΔξimplies (5)vΔt=∫t0twtΔt,ξΔξ+wσt,t.Lemma 3 (see [6]).
Leta≥0 and p≥q>0. For any K>0, we have (6)aq/p≤qpKq-p/pa+p-qpKq/p.Theorem 4.
Assume thatp≥q>0, p≥r>0, and λ≥0 are positive constants, u,a,b,f1,f2,g:I→R+ are rd-continuous functions, a is nondecreasing, τ:I→I,τ(t)≤t, -∞<α=inf{τ(t),t∈I}≤t0, and φ∈Crd(α,t0∩T,R+). If u satisfies (7)upt≤at+bt∫t0tf1suqτs+f2s∫t0sgξurξΔξΔs+λbT1∫t0T1f1suqτs+f2s∫t0sgξurξΔξΔs,t∈Iwith the initial condition (8)ut=φt,fort∈α,t0,φτt≤a1/ptforeveryt∈Iwithτt≤t0,then (9)ut≤Dλ,T1eA⊕Bt,t0+∫t0teA⊕Bt,σsCsΔs1/pforanyK>0,t∈Iunder the condition that eA⊕B(T1,t0)<(λ+1)/λ and μ(t)B1(t)<1, where (10)At=btqpKq-p/pf1t+rpKr-p/pf2t∫t0tgξΔξ,Bt=B1t1-μtB1t,Ct=1+μtBtC1t,B1t=bΔt∫t0σtqpKq-p/pf1s+rpKr-p/pf2s∫t0sgξΔξΔs,C1t=bΔt∫t0σtp-qpKq/pf1s+p-rpKr/pf2s∫t0sgξΔξΔs+btp-qpKq/pf1t+p-rpKr/pf2t∫t0tgξΔξ,Dλ,T1=aT1+λ∫t0T1eA⊕BT1,σsCsΔsλ+1-λeA⊕BT1,t0.Proof.
Definey(t) by (11)yt=aT1+bt∫t0tf1suqτs+f2s∫t0sgξurξΔξΔs+λbT1∫t0T1f1suqτs+f2s∫t0sgξurξΔξΔs,t∈I.Then, y(t)≥0, y(t) is nondecreasing, (12)ut≤y1/pt,t∈I,and (13)yt0=aT1+λbT1∫t0T1f1suqτs+f2s∫t0sgξurξΔξΔs.Next, we will prove that, for every t∈I, u(τ(t))≤y1/p(t).
Case 1. When τ(t)>t0,t∈I, we obtain (14)uτt≤y1/pτt≤y1/pt.
Case 2. For t∈I with τ(t)≤t0, by the initial condition (8), we have (15)uτt=φτt≤a1/pt≤a1/pT1≤y1/pt.From (14) and (15), we always have the relation (16)uτt≤y1/pt.By simple computation, it follows from (12) and (16) that (17)yΔt=bΔt∫t0σtf1suqτs+f2s∫t0sgξurξΔξΔs+btf1tuqτt+f2t∫t0tgξurξΔξ≤bΔt∫t0σtf1syq/ps+f2s∫t0sgξyr/pξΔξΔs+btf1tyq/pt+f2t∫t0tgξyr/pξΔξ.For any K>0, it follows from Lemma 3 that (18)yq/pt≤qpKq-p/pyt+p-qpKq/p,yr/pt≤rpKr-p/pyt+p-rpKr/p.This together with (17) implies (19)yΔt≤bΔt∫t0σtf1syq/ps+f2s∫t0sgξyr/pξΔξΔs+btf1tyq/pt+f2t∫t0tgξyr/pξΔξ≤yσtbΔt∫t0σtqpKq-p/pf1s+rpKr-p/pf2s∫t0sgξΔξΔs+ytbtqpKq-p/pf1t+rpKr-p/pf2t∫t0tgξΔξ+bΔt∫t0σtp-qpKq/pf1s+p-rpKr/pf2s∫t0sgξΔξΔs+btp-qpKq/pf1t+p-rpKr/pf2t∫t0tgξΔξ=Atyt+B1tyσt+C1t=Atyt+Bt1+μtBtyt+μtyΔt+C1t, which yields (20)11+μtBtyΔt≤At+Bt1+μtBtyt+C1t,i.e., (21)yΔt≤A⊕Btyt+Ct,t∈I; where C(t)=[1+μ(t)B(t)]C1(t). Note that y,C∈Crd and A⊕B∈R+. By Lemma 1, we get (22)yt≤yt0eA⊕Bt,t0+∫t0teA⊕Bt,σsCsΔs,t∈I. From (11) and (13), we get (23)λ+1λyt0-1λaT1=yT1≤yt0eA⊕BT1,t0+∫t0T1eA⊕BT1,σsCsΔs; i.e., (24)yt0≤Dλ,T1.Therefore, (25)yt≤Dλ,T1eA⊕Bt,t0+∫t0teA⊕Bt,σsCsΔs,t∈I. Noting that u(t)≤y1/p(t), we obtain the required inequality (9). The proof is complete.Remark 5.
[8, Theorem 3] is the special case of Theorem 4 with a(t)≡u0, b(t)≡1, λ=p=q=r=1, and τ(t)=t.Remark 6.
Theorem4 generalizes [9, Theorem 3.1] to the nonlinear case.Remark 7.
From (9), we can obtain that the upper bound of u(t) depends on parameters p,q,r, and λ. For practical system, the above parameters are easily to be obtained; therefore the theoretical upper bound can be computed by (9).If we takeλ=1/b(T1),b(T1)≠0, then the following result can be obtained.Corollary 8.
Letp,q,r,u,a,b,f1,f2,g,τ,φ be defined as in Theorem 4 and u satisfy (26)upt≤at+bt∫t0tf1suqτs+f2s∫t0sgξurξΔξΔs+∫t0T1f1suqτs+f2s∫t0sgξurξΔξΔs,t∈Iand (8). If eA⊕B(T1,t0)<1+b(T1) and μ(t)B1(t)<1, then (27)ut≤D¯λ,T1eA⊕Bt,t0+∫t0teA⊕Bt,σsCsΔs1/pforanyK>0,t∈I,where A(t),B(t),C(t),B1(t),C1(t) are defined as in Theorem 4 and (28)D¯λ,T1=aT1bT1+∫t0T1eA⊕BT1,σsCsΔs1+bT1-eA⊕BT1,t0.Theorem 9.
Suppose thatp≥q>0, p≥r>0, and λ≥0 are positive constants, u,a,b,g:I→R+ are rd-continuous functions, a is nondecreasing, v(t,s),w(t,s) are defined as in Lemma 2 such that vtΔ(t,s)≥0 and wtΔ(t,s)≥0 for t≥s, τ:I→I,τ(t)≤t, -∞<α=inf{τ(t),t∈I}≤t0, and φ∈Crd(α,t0∩T,R+). If u satisfies (29)upt≤at+bt∫t0tvt,suqτs+wt,s∫t0sgξurξΔξΔs+λbT1∫t0T1vT1,suqτs+wT1,s∫t0sgξurξΔξΔs,t∈Iand (8), then (30)ut≤D~λ,T1eA~⊕B~t,t0+∫t0teA~⊕B~t,σsC~sΔs1/pforanyK>0,t∈Iunder the condition that eA~⊕B~(T1,t0)<(λ+1)/λ and μ(t)B2(t)<1, where (31)A~t=btqpKq-p/pvσt,t+rpKr-p/pwσt,t∫t0tgξΔξ+∫t0tvtΔt,s+wtΔt,s∫t0sgξΔξΔs,B~t=B2t1-μtB2t,C~t=1+μtB~tC2t,B2t=bΔt∫t0σtqpKq-p/pvσt,s+rpKr-p/pwσt,s∫t0sgξΔξΔs,C2t=bΔt∫t0σtp-qpKq/pvσt,s+p-rpKr/pwσt,s∫t0sgξΔξΔs+btp-qpKq/pvσt,t+p-rpKr/pwσt,t∫t0tgξΔξ+∫t0tp-qpKq/pvtΔt,s+p-rpKr/pwtΔt,s∫t0sgξΔξΔs,D~λ,T1=aT1+λ∫t0T1eA~⊕B~T1,σsC~sΔsλ+1-λeA~⊕B~T1,t0.Proof.
Definez(t) by (32)zt=aT1+bt∫t0tvt,suqτs+wt,s∫t0sgξurξΔξΔs+λbT1∫t0T1vT1,suqτs+wT1,s∫t0sgξurξΔξΔs,t∈I.Then, z(t)≥0 for t∈I, z(t) is nondecreasing, (33)ut≤z1/pt,and (34)zt0=aT1+λbT1∫t0T1vT1,suqτs+wT1,s∫t0sgξurξΔξΔs.It is not difficult to obtain that (35)uτt≤z1/pt.Taking the derivative of z(t) and by Lemma 2, we get (36)zΔt=bΔt∫t0σtvσt,suqs+wσt,s∫t0sgξurξΔξΔs+btvσt,tuqτt+wσt,t∫t0tgξurξΔξ+bt∫t0tvtΔt,suqτs+wtΔt,s∫t0sgξurξΔξΔs.
It follows from (33) and (35) that (37)zΔt≤bΔt∫t0σtvσt,szq/ps+wσt,s∫t0sgξzr/pξΔξΔs+btvσt,tzq/pt+wσt,t∫t0tgξzr/pξΔξ+bt∫t0tvtΔt,szq/ps+wtΔt,s∫t0sgξzr/pξΔξΔs.By Lemma 3, we get (38)zq/pt≤qpKq-p/pzt+p-qpKq/p,zr/pt≤rpKr-p/pzt+p-rpKr/pfor any K>0. Combining (37) with (38) yields (39)zΔt≤zσtbΔt∫t0σtqpKq-p/pvσt,s+rpKr-p/pwσt,s∫t0sgξΔξΔs+ztbtqpKq-p/pvσt,t+rpKr-p/pwσt,t∫t0tgξΔξ+∫t0tvtΔt,s+wtΔt,s∫t0sgξΔξΔs+bΔt∫t0σtp-qpKq/pvσt,s+p-rpKr/pwσt,s∫t0sgξΔξΔs+btp-qpKq/pvσt,t+p-rpKr/pwσt,t∫t0tgξΔξ+∫t0tp-qpKq/pvtΔt,s+p-rpKr/pwtΔt,s∫t0sgξΔξΔs=A~tzt+B2tzσt+C2t.The remainder of the proof is similar to that of Theorem 4, and hence we omit it here.Remark 10.
In [14], we investigated the integral inequalities with mixed nonlinearities on time scales; however, the delay terms are not considered; furthermore, the inequalities considered in this paper are Volterra–Fredholm type. The results established in Theorems 4 and 9 generalize [14, Theorems 2.1 and 2.2].


## 3. Application

In this part, we present an example to illustrate the theoretical results.Example 11.
Consider the delay Volterra–Fredholm type integral equation on time scales(40)upt=at+bt∫t0tFs,uτs,∫t0sGξ,uξΔξΔs+∫t0T1Hs,uτs,∫t0sLξ,uξΔξΔs,t∈I=t0,T1with the initial condition (41)ut=φt,fort∈α,t0,φτt≤at1/pforeveryt∈Iwithτt≤t0,where u(t) and u(τ(t)) defined on the interval I are the state and state delay variables, respectively. a,b:I→R, |a| is nondecreasing, b(T1)≠0, T1∈T, and T1>t0. τ:I→I satisfying τ(t)≤t, -∞<α=inf{τ(t),t∈I}≤t0, and φ∈Crd(α,t0∩T,R), F,H:I×R×R→R, and G,L:I×R→R are continuous functions.Suppose that(42)Ft,u,v≤f1tuq+f2tv,Gt,u≤gtur,Ht,u,v≤f1tuq+f2tv,Lt,u≤gtur,where p≥q>0 and p≥r>0 are real constants and f1, f2, and g are nonnegative rd-continuous functions on I. If eA⊕B(T1,t0)<1+|b(T1)| and μ(t)B1(t)<1, then the solution u(t) of (40) satisfies (43)ut≤Dλ,T1eA⊕Bt,t0+∫t0teA⊕Bt,σsCsΔs1/pforanyK>0,t∈I,where (44)At=btqpKq-p/pf1t+rpKr-p/pf2t∫t0tgξΔξ,Bt=B1t1-μtB1t,Ct=1+μtBtC1t,B1t=bΔt∫t0σtqpKq-p/pf1s+rpKr-p/pf2s∫t0sgξΔξΔs,C1t=bΔt∫t0σtp-qpKq/pf1s+p-rpKr/pf2s∫t0sgξΔξΔs+btp-qpKq/pf1t+p-rpKr/pf2t∫t0tgξΔξ,Dλ,T1=aT1bT1+∫t0T1eA⊕BT1,σsCsΔs1+bT1-eA⊕BT1,t0.Actually, by (40), u(t) satisfies (45)utp≤at+bt∫t0tFs,uτs,∫t0sGξ,uξΔξΔs+∫t0T1Hs,uτs,∫t0sLξ,uξΔξΔs≤at+bt∫t0tf1suτsq+f2s∫t0sgξuξrΔξΔs+∫t0T1f1suτsq+f2s∫t0sgξuξrΔξΔs,t∈I.It is not difficult to verify (43) is satisfied by Corollary 8.



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