# Boundedness of the Solutions to Certain Delay Dynamic Integrodifferential Systems on Time Scales

Authors: Min Fan; Yazhou Tian
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

In this paper, the bounds on the solutions of certain delay dynamic integrodifferential systems on time scales are considered. Based on a new Gronwall-Bellman type delay integral inequality, we can estimate the boundedness of solutions to integrodifferential systems. At the end, an example is presented to state the main results.


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## Body

## 1. Introduction

The theory of time scales was established and developed by Hilger [1] and Bohner and Peterson [2, 3]. At present, different kinds of integral inequalities and their applications in differential, integral, and integrodifferential equations have become the research focus; see the papers [4–23]. To list a few, Ma and Pečarić [10] established an integral inequality on time scales to study the boundedness of solutions of the delay dynamic differential system. Wang and Xu [13] investigated some integral inequalities in two independent variables on time scales. In [15, 16], Ma et al. considered the generalized two-dimensional fractional differential system with Hadamard derivative. However, to the best of our knowledge, there are very little known results on discussing the bounds on the solutions of delay integrodifferential system on time scales.Motivated by the works in [10, 15, 16], we further investigate the delay dynamic integrodifferential system on time scales. By introducing a new Gronwall-Bellman type integral inequality, we obtain the bounds on the solutions of a class of delay dynamic integrodifferential system on time scales.In the following,R denotes the set of real numbers and R+=[0,+∞), C(M,S) represents the class of all continuous functions defined on set M with range in the set S. T denotes an arbitrary time scale, T0=[t0,+∞)∩T, t0∈T, and Crd denotes the set of rd-continuous functions.


## 2. Problem Description and Preliminaries

Consider the following delay integrodifferential system(1)xptΔ=G1t,xτ1t,yτ1t,∫t0tH1t,s,xτ2s,yτ2sΔs,yqtΔ=G2t,xτ1t,yτ1t,∫t0tH2t,s,xτ2s,yτ2sΔs,t∈T0with the initial conditionIxt=φt,yt=ψtfort∈α,t0∩Twithφτit≤C11/p,ψτit≤C21/q,foreveryt∈T0withτit≤t0i=1,2,where Gi:T0×R3→R, Hi:T0×T0×R2→R are continuous functions, τi:T0→T,τi(t)≤t(i=1,2), -∞<α=mint∈T0{inf⁡τ1(t),infτ2(t)}≤t0, C1=xp(t0),C2=yq(t0), p, q are some positive constants, and φ,ψ∈Crd([α,t0]∩T,R).Remark 1.
If we letHi≡0, the system considered in this paper reduces to the one in [10].
The following lemmas are useful in our main results.Lemma 2 (see [4], Lemma 2.1).
Assume thatc≥0 and k≥l>0. Then(2)cl/k≤lkml-k/kc+k-lkml/kfor any m>0.Lemma 3.
Assume thatu,a,b,c∈Crd(T0,R+), d∈Crd(T02,R+), and a(t) is nondecreasing. If (3)ut≤at+∫t0tbsusΔs+∫t0tcs∫t0sds,ξuξΔξΔs,t∈T0,then(4)ut≤ateft,t0,t∈T0,where(5)ft=bt+ct∫t0tdt,ξΔξ.Proof.
First we assume thata(t)>0; from (3), we have (6)utat≤1+∫t0tbsusas+cs∫t0sds,ξuξaξΔξΔs.Defining a function r(t) by right side of (6), we obtain (7)utat≤rt,rt0=1,r(t) is nondecreasing, and (8)rΔt=btutat+ct∫t0tdt,ξuξaξΔξ≤bt+ct∫t0tdt,ξΔξrt,which implies that (9)rt≤eft,t0,where f(t) is defined as in (5). Combining (7) and (9), we get the required inequality (4).
Ifa(t)=0 for t∈T0, we carry out the above procedure with a(t)+ε instead of a(t), where ε>0 is an arbitrary small constant, and subsequently pass to the limit as ε→0 to obtain (4). The proof is complete.


## 3. Main Results

Theorem 4.
Assume that(10)G1t,x,y,u≤l11tN11t,xr11+l12tN12t,yr12+l13tu,G2t,x,y,u≤l21tN21t,xr21+l22tN22t,yr22+l23tu,H1t,s,x,y≤ω13t,sN13s,xr13+ω14t,sN14s,yr14,H2t,s,x,y≤ω23t,sN23s,xr23+ω24t,sN24s,yr24,t,s∈T0hold, where lij∈Crd(T0,R+) (i=1,2,j=1,2,3), ωij∈Crd(T0×T0,R+) (i=1,2,j=3,4), Nij:T0×R+→R+ is a continuous function satisfying (11)0≤Nijt,x-Nijt,y≤Mijt,yx-yfor x≥y>0, where Mij:T0×R+→R+ is continuous (i=1,2,j=1,2,3,4), p and q are constants, and rij is a constant with p≥rij>0(i=1,2,j=1,3), q≥rij>0(i=1,2,j=2,4). Furthermore, suppose that (x(t),y(t)) is a solution of system (1) satisfying the initial condition I. Then, for any constant m>0, (12)xt≤A~teKt,t01/p,yt≤Φ2t+h2tA~teKt,t01/qeA1/qt,t0,t∈T0,where (13)A~t=Φ1t+r12qmr12-q/q∫t0tl12sM12s,q-r12qmr12/qΦ2seAs,t0Δs+r14qmr14-q/q∫t0tl13s∫t0sω14s,ξM14ξ,q-r14qmr14/qΦ2ξeAξ,t0ΔξΔs,At=r22qmr22-q/ql22tM22t,q-r22qmr22/q+r24qmr24-q/ql23t∫t0tω24t,ξM24ξ,q-r24qmr24/qΔξ,Φ1t=C1+∫t0tl11sN11s,p-r11pmr11/pΔs+∫t0tl12sN12s,q-r12qmr12/qΔs+∫t0tl13s∫t0sω13s,ξN13ξ,p-r13pmr13/p+ω14s,ξN14ξ,q-r14qmr14/qΔξΔs,Φ2t=C2+∫t0tl21sN21s,p-r21pmr21/pΔs+∫t0tl22sN22s,q-r22qmr22/qΔs+∫t0tl23s∫t0sω23s,ξN23ξ,p-r23pmr23/p+ω24s,ξN24ξ,q-r24qmr24/qΔξΔs,Kt=r11pmr11-p/pl11tM11t,p-r11pmr11/p+r12qmr12-q/ql12tM12t,q-r12qmr12/qh2teAt,t0+l13t∫t0tr13pmr13-p/pω13t,ξM13ξ,p-r13pmr13/p+r14qmr14-q/qω14t,ξM14ξ,q-r14qmr14/qh2ξeAξ,t0Δξ,h2t=r21pmr21-p/p∫t0tl21sM21s,p-r21pmr21/pΔs+r23pmr23-p/p∫t0tl23s∫t0sω23s,ξM23ξ,p-r23pmr23/pΔξΔs.Proof.
The solution(x(t),y(t)) of system (1) satisfies (14)xpt=C1+∫t0tG1s,xτ1s,yτ1s,∫t0sH1s,ξ,xτ2ξ,yτ2ξΔξΔs,yqt=C2+∫t0tG2s,xτ1s,yτ1s,∫t0sH2s,ξ,xτ2ξ,yτ2ξΔξΔs.Combining (10) and (14), we have (15)xtp≤C1+∫t0tl11sN11s,xτ1sr11Δs+∫t0tl12sN12s,yτ1sr12Δs+∫t0tl13s∫t0sω13s,ξN13ξ,xτ2ξr13+ω14s,ξN14ξ,yτ2ξr14ΔξΔs,ytq≤C2+∫t0tl21sN21s,xτ1sr21Δs+∫t0tl22sN22s,yτ1sr22Δs+∫t0tl23s∫t0sω23s,ξN23ξ,xτ2ξr23+ω24s,ξN24ξ,yτ2ξr24ΔξΔs.Defining (16)z1pt=C1+∫t0tl11sN11s,xτ1sr11Δs+∫t0tl12sN12s,yτ1sr12Δs+∫t0tl13s∫t0sω13s,ξN13ξ,xτ2ξr13+ω14s,ξN14ξ,yτ2ξr14ΔξΔsand (17)z2qt=C2+∫t0tl21sN21s,xτ1sr21Δs+∫t0tl22sN22s,yτ1sr22Δs+∫t0tl23s∫t0sω23s,ξN23ξ,xτ2ξr23+ω24s,ξN24ξ,yτ2ξr24ΔξΔs,we can obtain that z1(t) and z2(t) are nondecreasing, and (18)xt≤z1t,yt≤z2t,t∈T0.Case 1: for t∈T0 with τi(t)>t0, we obtain (19)xτit≤z1τit≤z1t,yτit≤z2τit≤z2t.Case 2: for t∈T0 with τi(t)≤t0, by the initial condition I, we have (20)xτit=φτit≤C11/p≤z1t,yτit=ψτit≤C21/q≤z2t,t∈T0.Both (19) and (20) imply that (21)xτit≤z1t,yτit≤z2ti=1,2,t∈T0.This together with (16) and (17) yields (22)z1pt≤C1+∫t0tl11sN11s,z1r11sΔs+∫t0tl12sN12s,z2r12sΔs+∫t0tl13s∫t0sω13s,ξN13ξ,z1r13ξ+ω14s,ξN14ξ,z2r14ξΔξΔs,z2qt≤C2+∫t0tl21sN21s,z1r21sΔs+∫t0tl22sN22s,z2r22sΔs+∫t0tl23s∫t0sω23s,ξN23ξ,z1r23ξ+ω24s,ξN24ξ,z2r24ξΔξΔs.Define (23)ut=C1+∫t0tl11sN11s,z1r11sΔs+∫t0tl12sN12s,z2r12sΔs+∫t0tl13s∫t0sω13s,ξN13ξ,z1r13ξ+ω14s,ξN14ξ,z2r14ξΔξΔs,vt=C2+∫t0tl21sN21s,z1r21sΔs+∫t0tl22sN22s,z2r22sΔs+∫t0tl23s∫t0sω23s,ξN23ξ,z1r23ξ+ω24s,ξN24ξ,z2r24ξΔξΔs.Combining (22) and (23), we have (24)z1t≤u1/pt,z2t≤v1/qt,t∈T0.By Lemma 2 and (24), for any real number m>0, we get (25)z1rijt≤rijpmrij-p/put+p-rijpmrij/pi=1,2,j=1,3and(26)z2rijt≤rijqmrij-q/qvt+q-rijqmrij/qi=1,2,j=2,4.By assumption (11) and the last inequalities, it follows that (27)Nijs,z1rijs≤rijpmrij-p/pMijs,p-rijpmrij/pus+Nijs,p-rijpmrij/pi=1,2,j=1,3and(28)Nijs,z2rijs≤rijqmrij-q/qMijs,q-rijqmrij/qvs+Nijs,q-rijqmrij/qi=1,2,j=2,4.Substituting the last inequalities into (23), we can obtain (29)ut≤Φ1t+r11pmr11-p/p∫t0tl11sM11s,p-r11pmr11/pusΔs+r12qmr12-q/q∫t0tl12sM12s,q-r12qmr12/qvsΔs+r13pmr13-p/p∫t0tl13s∫t0sω13s,ξM13ξ,p-r13pmr13/puξΔξΔs+r14qmr14-q/q∫t0tl13s∫t0sω14s,ξM14ξ,q-r14qmr14/qvξΔξΔsand(30)vt≤Φ2t+r21pmr21-p/p∫t0tl21sM21s,p-r21pmr21/pusΔs+r22qmr22-q/q∫t0tl22sM22s,q-r22qmr22/qvsΔs+r23pmr23-p/p∫t0tl23s∫t0sω23s,ξM23ξ,p-r23pmr23/puξΔξΔs+r24qmr24-q/q∫t0tl23s∫t0sω24s,ξM24ξ,q-r24qmr24/qvξΔξΔs,where Φ1(t) and Φ2(t) are defined as in (13).
Let(31)f2t=Φ2t+r21pmr21-p/p∫t0tl21sM21s,p-r21pmr21/pusΔs+r23pmr23-p/p∫t0tl23s∫t0sω23s,ξM23ξ,p-r23pmr23/puξΔξΔs.From (30), we have (32)vt≤f2t+r22qmr22-q/q∫t0tl22sM22s,q-r22qmr22/qvsΔs+r24qmr24-q/q∫t0tl23s∫t0sω24s,ξM24ξ,q-r24qmr24/qvξΔξΔs.Since f2(t) is nondecreasing, by Lemma 3, we have (33)vt≤f2teAt,t0,where A(t) is defined as in (13). Since u(t) is nondecreasing, it follows from (33) that (34)vt≤Φ2t+h2tuteAt,t0,where h2(t) is defined as in (13). Substituting (34) into (29), we can obtain (35)ut≤A~t+∫t0tr11pmr11-p/pl11sM11s,p-r11pmr11/p+r12qmr12-q/ql12sM12s,q-r12qmr12/qh2seAs,t0usΔs+∫t0tl13s∫t0sr13pmr13-p/pω13s,ξM13ξ,p-r13pmr13/p+r14qmr14-q/qω14s,ξM14ξ,q-r14qmr14/qh2ξeAξ,t0uξΔξΔs.Using Lemma 3, we get (36)ut≤A~teKt,t0,where A~(t) and K(t) are defined as in (13). From (34), we have (37)vt≤Φ2t+h2tA~teKt,t0eAt,t0.Combining (36) and (37) with (24) and (18), we obtain the desired inequality (12). This completes the proof.Remark 5.
Theorem4 generalizes [10, Theorem 3.1].Remark 6.
Assumptions (10) and (11) are easily satisfied. For example, we can choose N(t,x)=t2x1/3, and it is not difficult to verify that (38)0≤Nt,x-Nt,y=t2x1/3-y1/3≤Mt,yx-y,where M(t,y)=t2y-2/3/3.


## 4. Application

In this section, we present an example to illustrate the main results.Example 7.
Consider system (1) with the initial condition I and p=q=2, G1, G2, H1, and H2 satisfy(39)G1t,x,y,u≤l11tx1/3+l12ty1/2+l13tu,G2t,x,y,u≤l21tx1/3+l22ty1/2+l23tu,H1t,s,x,y≤ω13t,sx1/2+ω14t,sy1/4,H2t,s,x,y≤ω23t,sx1/2+ω24t,sy1/4,t,s∈T0 with rij=1,i=1,2,j=1,2,3,4. Then the solution of system (1) satisfies (40)xt≤A~teKt,t01/2,yt≤Φ2t+h2tA~teKt,t01/2eA1/2t,t0,t∈T0,where (41)A~t=Φ1t+2-3/2m-1/4∫t0tl12sΦ2seAs,t0Δs+2-5/4m-3/8∫t0tl13s∫t0sω14s,ξΦ2ξeAξ,t0ΔξΔs,At=2-5/4m-3/8l23t∫t0tω24t,ξΔξ+2-3/2m-1/4l22t,Φ1t=C1+2-1/3m1/6∫t0tl11sΔs+2-1/2m1/4∫t0tl12sΔs+∫t0tl13s∫t0s2-1/2m1/4ω13s,ξ+2-1/4m1/8ω14s,ξΔξΔs,Φ2t=C2+2-1/3m1/6∫t0tl21sΔs+2-1/2m1/4∫t0tl22sΔs+∫t0tl23s∫t0s2-1/2m1/4ω23s,ξ+2-1/4m1/8ω24s,ξΔξΔs,Kt=132-1/3m-5/6l11t+2-3/2m-3/4l12th2teAt,t0+l13t∫t0t2-3/2m-3/4ω13t,ξ+2-9/4m-7/8ω14t,ξh2ξeAξ,t0Δξ,h2t=132-1/3m-5/6∫t0tl21sΔs+2-3/2m-3/4∫t0tl23s∫t0sω23s,ξΔξΔs.In fact, the solution of system (1) satisfies the following integral equation (42)x2t=x2t0+∫t0tG1s,xτ1s,yτ1s,∫t0sH1s,ξ,xτ2ξ,yτ2ξΔξΔs,y2t=y2t0+∫t0tG2s,xτ1s,yτ1s,∫t0sH2s,ξ,xτ2ξ,yτ2ξΔξΔs.Therefore, (43)xt2≤C1+∫t0tl11sxτ1s1/3Δs+∫t0tl12syτ1s1/2Δs+∫t0tl13s∫t0sω13s,ξxτ2ξ1/2+ω14s,ξyτ2ξ1/4ΔξΔs,yt2≤C2+∫t0tl21sxτ1s1/3Δs+∫t0tl22syτ1s1/2Δs+∫t0tl23s∫t0sω23s,ξxτ2ξ1/2+ω24s,ξyτ2ξ1/4ΔξΔs.We can choose (44)N11t,x=x1/3,N12t,x=x1/2,N13t,x=x1/2,N14t,x=x1/4and (45)N21t,x=x1/3,N22t,x=x1/2,N23t,x=x1/2,N24t,x=x1/4.Since (46)N11t,x-N11t,y=x1/3-y1/3≤13y-2/3x-y,M11(t,y) can be selected as y-2/3/3. Similarly, we choose (47)M12t,y=12y-1/2,M13t,y=12y-1/2,M14t,y=14y-3/4,M21t,y=13y-2/3,M23t,y=12y-1/2. Applying Theorem 4 to (42) yields (40).



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