# Oscillation Criteria for Some Higher Order Integrodynamic Equations on Timescales

Authors: Said R. Grace; Mohamed A. El-Beltagy
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We study the oscillation behavior for some higher order integrodynamic equations on timescales. We establish some new sufficient conditions guaranteeing that all solutions of theses equations are oscillatory. Some numerical examples in the continuous case are given to validate the theoretical results.


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## Body

## 1. Introduction

Integrodynamic equations on timescales are an important topic with applications in many physical systems. For general basic ideas and background, we refer to [1]. Oscillation results of integral equations of Volterra type are scant and only few results exist on this subject. Related studies can be found in [2–6]. In this paper, we investigate the oscillatory behavior of the solutions of some higher order integrodynamic equations on timescale in the form(1)rxΔΔn-1t=et-∫0tat,sFs,xsΔs.To the best of our knowledge, there appear to be no such results on the oscillation of (1). Therefore, our main goal here is to initiate such a study by establishing some new criteria for the oscillation of (1) and other related equations. This work is an extension to the analysis done in [7]. The nonoscillatory behavior for some higher order integrodynamic equations was studied recently in [8].We takeT⊆R to be an arbitrary timescale with 0∈T and Sup⁡T=∞.Whenever we writet≥s, we mean t∈[s,∞)∩T.We assume throughout the following:(I)
e, r:T→R and a:T×T→R are rd-continuous functions, r(t)>0, and a(t,s)≥0 for t≥s; and there exist rd-continuous functions p, q:T→R+ such that(2)at,s≤ptqsfort≥s,sup⁡pt≔k1<∞,(3)supt≥0∫0tqsΔs≔k2<∞;(II)
F:T×R→R is continuous and assume that there exist f1, f2:T×R→R continuous functions such that F(t,x)=f1(t,x)-f2(t,x) for t≥0 and xfi(t,x)>0 for x≠0,t≥0, and i=1,2;(III)
there exist constantsβ and γ which are the ratios of positive odd integers and pi:T→R+=(0,∞), i=1,2, such that(4)f1t,x≥p1txβ,f2t,x≤p2txγ,forx>0,t≥0,f1t,x≤p1txβ,f2t,x≥p2txγ,forx<0,t≥0.By a solution of (1) we mean a Δ-differential function defined on T that is nontrivial in every neighborhood of infinity. A solution x(t) of (1) is said to be oscillatory if there exists tn⊆T such that x(tn)=0 or x(t)xσ(t)<0; otherwise, it is called nonoscillatory.


## 2. Auxiliary Results

We employ the following lemmas.Lemma 1 (see [9]).
IfX and Y are nonnegative, then(5)Xλ+λ-1Yλ-λXYλ-1≥0;λ>1,(6)Xλ-1-λYλ-λXYλ-1≤0;λ<1,where equality holds if and only if X=Y.Lemma 2 (Young’s inequality).
LetX,Y≥0, n>1, and 1/n+1/m=1, then(7)XY≤1nXn+1mYm,and equality holds if and only if Y=Xn-1.Lemma 3 (see [3, corollary 1]).
Assume thatn∈N, s,t∈T, and f:T→R is rd-continuous function, and then(8)∫st∫ηnt⋯∫η2tfη1Δη1Δη2⋯Δηn=-1n-1∫sthn-1s,σηfηΔη.In Lemma3, hn stand for the Taylor monomials (see [1, section 1.6]) which are defined recursively by(9)h0t,s=1,hn+1t,s=∫sthnτ,sΔτfort,s∈T,n∈N.It follows that h1(t,s)=t-s for any timescale, but simple formulas, in general, do not hold for n≥2.Fort≥s∈T, we define(10)Hn-1t,s≥h0t,s+h1t,s+⋯+hn-1t,s,Ht,t0=∫t0t1rsHn-1s,t0Δs∀t0≥0.


## 3. Main Results

In this section we present the following main results.Theorem 4.
Let conditions (I) and (II) hold withf2=0. If(11)limsupt→∞⁡1Ht,t0∫t0t1ru∫t0uhn-2u,sesΔsΔu=∞,liminft→∞⁡1Ht,t0∫t0t1ru∫t0uhn-2u,sesΔsΔu=-∞,for all t0≥0, then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1). Hence either x(t) is eventually positive or x(t) is eventually negative.
First assumex(t) is eventually positive. Fix t0≥0 and suppose x(t)>0 for t≥t1 for some t1≥t0. From (1), we see that(12)rxΔΔn-1t=et-∫0t1at,sFs,xsΔs-∫t1tat,sf1s,xsΔs≤et-∫0t1at,sFs,xsΔs,fort≥t1.Let(13)b≔max0≤t≤t1⁡Ft,xt<∞.By assumption (3), we have(14)-∫0t1at,sFs,xsΔs≤∫0t1at,sFs,xsΔs≤b∫0t1ptqsΔs≤bk1∫0t1qsΔs≤bk1k2≔k.Hence, from (12), we get(15)rxΔΔn-1t≤et+k,∀t≥t1.Integrating (15) (n-1)-times from t1 to t and then using Lemma 3, we obtain(16)rxΔt≤h0t,t1rxΔt1+h1t,t1rxΔΔt1+⋯+hn-2t,t1rxΔΔn-2t1+khn-1t,t1+∫t1t∫t1ξn-1⋯∫t1ξ2eξ1Δξ1Δξ2⋯Δξn-1=khn-1t,t1+∑i=0n-2hit,t1rxΔΔit1+∫t1thn-2t,σsesΔs.From the properties of the functions hi and the definition of the function Hn-1(t,t1) for all t1≥0, we get (17)rxΔt≤cHn-1t,t1+∫t1thn-2t,σsesΔs,where(18)c≔max⁡k,max0≤i≤n-2⁡rxΔΔit1.Dividing (17) by r(t) and hence integrating from t1 to t we obtain(19)xt≤xt1+cHt,t1+∫t1t1ru∫t1uhn-2u,σsesΔsΔu≤xt1+cHt,t0+∫t0t1ru∫t0uhn-2u,σsesΔsΔu.Dividing (19) by H(t,t0) and taking liminf of both sides of (19) as t→∞, we obtain a contradiction to the fact that x(t)>0 for t≥t1. The proof of the case when x(t) is eventually negative is similar. This completes the proof.From the proof of Theorem4, one can easily extract the following result on the asymptotic behavior of the nonoscillatory solutions of (1).Theorem 5.
Let conditions (I) and (II) hold withf2=0 and suppose(20)limsupt→∞⁡1Ht,t0∫t0t1ru∫t0uesΔsΔu<∞,liminft→∞⁡1Ht,t0∫t0t1ru∫t0uesΔsΔu>-∞for all t0≥0. If x(t) is nonoscillatory solution of (1), then(21)xt=OHt,t0ast⟶∞.Next, we present the following result.Theorem 6.
Let conditions (I) and (II) hold withf1=0 and f2 is nondecreasing in the second variable. If conditions (11) hold for every constant c>0,(22)limt→∞⁡1Ht,t0∫t0t1ru∫t0uhn-2u,στ∫t0τaτ,sf2s,cΔsΔτΔu<∞,for any t0≥0, then every bounded solution of (1) is oscillatory.Proof.
Letx(t) be a bounded nonoscillatory solution of (1) and assume that x(t) is eventually positive. Fix t0≥0 and suppose x(t)>0 for t≥t1 for some t1≥t0 and x(t)≤c1 for some constant c1>0. From (1), we have(23)rxΔΔn-1t=et-∫0t1at,sFs,xsΔs+∫t1tat,sf2s,xsΔs.Proceeding as in the proof of Theorem 4, we get (14). Thus,(24)rxΔΔn-1t≤k+et+∫t1tat,sf2s,c1Δs.The rest of the proof is similar to that of Theorem 4 and hence it is omitted.Theorem 7.
Let conditions (I) and (II) hold withβ>1 and γ=1 and suppose that conditions (11) hold and(25)limt→∞⁡1Ht,t0∫t0t1ru∫t0uhn-2u,στ∫t0τaτ,sp11/β-1sp2β/β-1sΔsΔτΔu<∞for all t0≥0. Then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1), Hence either x(t) is eventually positive or x(t) is eventually negative. First, assume x(t) is eventually positive. Fix t0≥0 and assume x(t)>0 for t≥t1 for some t1≥t0.
Using conditions (I) and (II) withβ>1 and γ=1 in (1), we have(26)rxΔΔn-1t≤et-∫0t1at,sFs,xsΔs+∫t1tat,sp2sxs-p1sxβsΔsfor t≥t1. Proceeding as in the proof of Theorem 4, we get (14) and hence(27)rxΔΔn-1t≤k+et+∫t1tat,sp2sxs-p1sxβsΔs,fort≥t1.By applying (5) with(28)λ=β,X=p11/βtxt,Y=1βp2tp1-1/βt1/β-1,we obtain(29)p2txt-p1txβt≤β-1ββ/1-βp11/1-βtp2β/1-βt,fort≥t1.Using (29) in (26), we find (30)rxΔΔn-1t≤k+et+Gt,fort≥t1,where(31)Gt=β-1ββ/1-β∫t1tat,sp11/1-βsp2β/1-βsΔs.Integrating (30) n-times from t1 to t and then using Lemma 3, we have(32)xt≤xt1+cHt,t1+∫t1t1ru∫t1uhn-2u,σsesΔsΔu+∫t1t1ru∫t1uhn-2u,σsGsΔsΔu≤xt1+cHt,t0+∫t0t1ru∫t1uhn-2u,σsesΔsΔu+∫t0t1ru∫t0uhn-2u,σsGsΔsΔu,where c is given in (18). The rest of the proof is similar to that of the proof of Theorem 4 and hence is omitted.Theorem 8.
Let conditions (I) and (II) hold withβ=1 and γ<1. If, in addition to conditions (11), we suppose(33)limt→∞⁡1Ht,t0∫t0t1ru∫t0uhn-2u,στ∫t0τaτ,sp1γ/γ-1sp21/γ-1sΔsΔτΔu<∞,for any t0≥0, then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1). First, assume x(t) is eventually positive. Fix t0≥0 and suppose x(t)>0 for t≥t1 for some t1≥t0. Using conditions (I) and (II) with β=1 and γ<1 in (1), we find(34)rxΔΔn-1t≤et-∫0t1at,sFs,xsΔs+∫t1tat,sp2sxγs-p1sxsΔsfor t≥t1. Hence(35)rxΔΔn-1t≤k+et+∫t1tat,sp2sxγs-p1sxsΔs,fort≥t1,where k is defined as in the proof of Theorem 4. By applying (6) with(36)λ=γ,X=p21/γtxt,Y=1γp1tp2-1/γt1/γ-1,we obtain(37)p2txγt-p1txt≤1-γγγ/γ-1p1γ/γ-1tp21/γ-1t,fort≥t1.Using (37) in (35), we find(38)rxΔΔn-1t≤k+et+1-γγγ/1-γ∫t1tat,sp1γ/γ-1sp21/1-γsΔs.The rest of the proof is similar to the proof of Theorem 4 and hence is omitted.Next, we present the following result with different nonlinearities, that is, withβ>1 and γ<1.Theorem 9.
Let conditions (I) and (II) hold withβ>1 and γ<1 and suppose that there exists a positive rd-continuous function g:T→R+ such that(39)limt→∞1Ht,t0∫t0t1ru∫t0uhn-2u,στ∫t0τaτ,sc1gβ/β-1sp11/1-βs+c2gγ/γ-1sp21/1-γsΔsΔτΔu<∞for all t0≥0, where(40)c1=β-1ββ/1-β,c2=1-γγγ/1-γ.If conditions (11) hold for all t0≥0, then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1). First, assume x(t) is eventually positive. Fix t0≥0 and suppose x(t)>0 for t≥t1 for some t1≥t0. Using conditions (I) and (II) in (1), we have(41)rxΔΔn-1t≤et-∫0t1at,sFs,xsΔs+∫t1tat,sgsxs-p1sxβsΔs+∫t1tat,sp2sxγs-gsxsΔsfor t≥t1.
As in the proof of Theorems7 and 8, one can easily find(42)rxΔΔn-1t≤k+et+∫t1tat,sβ-1ββ/1-βgβ/β-1sp11/1-βs+1-γγγ/1-γgγ/γ-1sp21/1-γsΔs.The rest of the proof is similar to that of Theorem 4 and hence is omitted.For the cases when both functionsf1 and f2 are superlinear, that is, β>γ>1, or sublinear, that is, 1>β>γ>0, we present the following result.Theorem 10.
Let conditions (I) and (II) hold withβ>γ. If, in addition to conditions (11), we suppose(43)limt→∞⁡1Ht,t0∫t0t1ru∫t0uhn-2u,στ∫t0τaτ,sp1γ/γ-βsp2β/β-γsΔsΔτΔu<∞,for all t0≥0, then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1). First, assume x(t) is eventually positive. Fix t0≥0 and suppose x(t)>0 for t≥t1 for some t1≥t0. Using conditions (I) and (II) in (1) with γ<1 and β>1, we have(44)rxΔΔn-1t≤et-∫0t1at,sFs,xsΔs+∫t1tat,sp2sxγs-p1sxβsΔsfor t≥t1. By applying Lemma 2 with(45)n=βγ>1,X=xγs,Y=γβp2sp1s,m=ββ-γ,we find(46)p2sxγs-p1sxβs=βγp1sxγsγβp2sp1s-γβxγsβ/γ=βγp1sXY-1nXn≤βγp1s1mYm=β-γγγβp2sβ/β-γp1sγ/γ-β.Using (46) in (44), we have(47)rxΔΔn-1t≤k+et+β-γγγββ/β-γ∫t1tat,sp2β/β-γsp1γ/γ-βsΔs,for t≥t1. The rest of the proof is similar to the proof of Theorem 4 and hence is omitted.Remark 11.
The results of this section will remain the same if we replace condition (3) of assumption (I) by(48)supt≥t0⁡∫0t0at,sΔs≔k-<∞,∀t0≥0,with k-=k1k2.Remark 12.
We note that we can obtain criteria on the asymptotic behavior of the nonoscillatory solutions of (1) similar to Theorem 5. The details are left to the reader.


## 4. Further Oscillation Results

This section is devoted to the study of the oscillatory properties of (1) with f1=0.Theorem 13.
Let conditions (I) and (II) hold withf1=0 and γ<1. Assume that there exists a rd-continuous function p:T→R+ such that(49)limt→∞⁡1t∫t0t1ru∫t0uhn-2u,στ∫t0τpγ/γ-1sp21/1-γsq1/1-γsΔsΔτΔu<∞,(50)Hn-1t,t0rtisboundedfort≥t0,limt→∞⁡Ht,t0<∞,(51)∫∞spsΔs<∞.If(52)limt→∞⁡sup⁡1t∫t0t1ru∫t0uhn-2u,σsesΔsΔu<∞,(53)limt→∞⁡inf⁡1t∫t0t1ru∫t0uhn-2u,σsesΔsΔu>-∞,for all t0≥0, then every nonoscillatory solution x(t) of (1) satisfies(54)limt→∞⁡sup⁡xtt<∞.Proof.
Letx(t) be a nonoscillatory solution of (1). First, assume x(t) is eventually positive. Fix t0≥0 and suppose x(t)>0 for t≥t1 for some t1≥t0. Using conditions (I) and (II) in (1), we have(55)rxΔΔn-1t≤et+∫0t1at,sFs,xsΔs+∫t1tat,sp2sxγsΔs,fort≥t1.As in the proof of Theorem 4, we obtain (14) and hence (55) becomes(56)rxΔΔn-1t≤k+et+k1∫t1tp2sqsxγs-psxsΔs+k1∫t1tpsxsΔs,fort≥t1.By applying (6) with(57)λ=γ,X=p2q1/γx,Y=1γpp2q1/γ1/γ-1,we have(58)p2tqtxγt-ptxt≤1-γγγ/1-γpγ/γ-1tp21/1-γtq1/1-γt.Using (58) in (56), we find(59)rxΔΔn-1t≤k+et+k11-βββ/β-1∫t1tpγ/γ-1sp21/1-γsq1/1-γsΔs+k1∫t1tpsxsΔs,∀t≥t1.Integrating this inequality (n-1)-times from t1 to t and using Lemma 3, we have(60)rxΔt≤khn-1t,t1+∑i=0n-2hit,t1rxΔΔit1+∫t1thn-2t,σsesΔs+k1∫t1thn-2t,σu∫t1upsxsΔsΔu+k11-γγγ/1-γ∫t1thn-2t,σu∫t1upγ/γ-1sp21/1-γsq1/1-γsΔsΔu.As in the proof of Theorem 4, one can easily find(61)xΔt≤crtHn-1t,t1+1rt∫t1thn-2t,σsesΔs+k1thn-2t,t1rt∫t1tpsxsΔs+k11-γγγ/1-γ1rt∫t1thn-2t,σu∫t1upγ/γ-1sp21/1-γsq1/1-γsΔsΔu,where c is given by (18). Using condition (50), we see that thn-2(t,t1)/r(t) is bounded for t≥t1; say by k3>0, and we see that(62)xΔt≤crtHn-1t,t1+1rt∫t1thn-2t,σsesΔs+k1k3∫t1tpsxsΔs+k11-γγγ/1-γ1rt∫t1thn-2t,σu∫t1upγ/γ-1sp21/1-γsq1/1-γsΔsΔu.Integrating this inequality from t1 to t and employing Lemma 3 in [10] to interchange the order of integration we obtain(63)xt≤xt1+cHt,t1+∫t1t1ru∫t1uhn-2u,σsesΔsΔu+k1k3t∫t1tpsxsΔs+k11-γγγ/1-γ∫t1t1ru∫t1uhn-2t,στ∫t1τpγ/γ-1sp21/1-γsq1/1-γsΔsΔτΔu.Using conditions (49), (50), and (53), there exist positive constants c- and c̿ such that(64)xtt≤c-+c̿∫t1tspsxssΔs.Applying Gronwall’s inequality [1, Corollary 6.7] to inequality (64) and then using condition (51), we have(65)limt→∞⁡sup⁡xtt<∞.If x(t) is eventually negative, we set y(t)=-x(t) to see that y(t) satisfies (1) with e(t) replaced by -e(t) and f1(t,x) by f1(t,-y).
It follows in a similar manner that(66)limt→∞⁡sup⁡-xtt<∞.We conclude from (65) and (66) that (54) holds.
Next by employing Theorem13, we present the following oscillation result for (1) with f2=0.Theorem 14.
Let conditions (I) and (II) hold withf1=0 and γ<1. Suppose that conditions (50), (52), and (53) hold. If(67)limt→∞⁡sup⁡t∫t0tspsΔs<∞,(68)limt→∞⁡∫t0t1ru∫t0uhn-2u,στ∫t0τpγ/γ-1sp21/1-γsq1/1-γsΔsΔτΔu<∞,(69)limt→∞⁡sup⁡∫t0t1ru∫t0uhn-2u,στesΔsΔu=+∞,(70)limt→∞⁡inf⁡∫t0t1ru∫t0uhn-2u,σsesΔsΔu=-∞,for all t0≥0, then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1). First, assume x(t) is eventually positive. Fix t0≥0 and suppose x(t)>0 for t≥t1 for some t1≥t0. The proof when x(t) is eventually negative is similar.
Proceeding as in the proof of Theorem13, we arrive at (58). Therefore(71)xt≤xt1+cHt,t1+k1k3t∫t1tspsxssΔs+∫t1t1ru∫t1uhn-2u,σsesΔsΔu+k11-γγγ/1-γ∫t1t1ru∫t1uhn-2t,στ∫t1τpγ/γ-1sp21/1-γsq1/1-γsΔsΔτΔu.Conditions (67) and (68) imply conditions (50) and (49), respectively, and so the conclusion of Theorem 13 holds. This together with condition (50) and (68) shows that the second term and last integral are bounded.
Takingliminf for both sides of (71) as t→∞ and using (67) and (70) result in a contradiction with the fact that x(t) is eventually positive.
Similar to the sublinear case one can easily prove the following theorems for the integrodynamic equation (1) when γ=1.Theorem 15.
Let conditions (I) and (II) hold withf1=0 and γ=1. Moreover, assume(72)∫∞sp2sΔs<∞and conditions (50), (52), and (53) hold for any t0≥0. Then every nonoscillatory solution of (1) satisfies (54).Theorem 16.
Let conditions (I) and (II) hold withf1=0 and γ=1. In addition, suppose conditions (50), (52), (53), (69), and (70) hold. If(73)limt→∞⁡sup⁡t∫t0tsp2sΔs<∞,for all t0≥0, then (1) is oscillatory.Similar to the above results, one can easily prove the following theorems for the integrodynamic equation (1) with f2=0.Theorem 17.
Let conditions (I) and (II) hold withf2=0 and β>1. Assume that there exists a rd-continuous function p:T→R+ such that(74)limt→∞⁡1t∫t0t1ru∫t0uhn-2u,στ∫t0τp1/1-βsp1β/β-1sqβ/β-1sΔsΔτΔu<∞.If conditions (50)–(53) hold for all t0≥0, then every nonoscillatory solution of (1) satisfies (54).Theorem 18.
Let conditions (I) and (II) hold withf2=0 and β>1. Assume that there exists a rd-continuous function p:T→R+ such that(75)limt→∞∫t0t1ru∫t0uhn-2u,στ∫t0τp1/1-βsp1β/β-1sqβ/β-1sΔsΔτΔu<∞,limt→∞⁡sup⁡t∫t0tspsΔs<∞.If conditions (50), (52), (53), (68), and (69) hold for all t0≥0, then (1) is oscillatory.


## 5. Illustrative Examples

As we already mentioned, the results of the present paper are new for the cases whenT=R, that is, the continuous case, or when T=Z, that is, the discrete case. As a numerical illustration of our results in Section 3 with T=R, we consider the following equation:(76)1+t3x′t′+∫0txβs1+t21+s2ds=tpsin⁡twith initial conditions x(t0)=x0 and x′(t0)=x0′. Compare (76) with (1) to get that r(t)=1+t3, n=2, a(t,s)=1/1+t21+s2, F(t,s)=xβ(s), and e(t)=tpsin(t). We can easily show that conditions (I) and (II) are satisfied. Conditions (11) are satisfied only for p≥3.Equation (76) can be converted to two simultaneous first-order ODEs by substituting 1+t3x′(t)=y. This will lead to the following system:(77)x′t=yt1+t3;x0=x0,y′t=tpsint-∫0txβs1+t21+s2ds;y0=x0′.Many numerical techniques can be used to solve (77). In the current work, the second-order accurate modified Euler technique is considered. The time interval t0,Tf will be divided into M equal subdivisions with Δt width for each one. The prediction and correction steps of the modified Euler technique will be(78)x-i+1=xi+Δtgti,xi,yi,y-i+1=yi+Δthti,xi,yi,xi+1=xi+0.5Δtgti,xi,yi+gti+1,x-i+1,y-i+1,yi+1=yi+0.5Δthti,xi,yi+hti+1,x-i+1,y-i+1,where (79)gt,x,y=yt1+t3,ht,x,y=tpsin⁡t-∫0txβs1+t21+s2ds.The integral in (79) can be approximated numerically at each time instant ti using the trapezoidal rule which has accuracy of OΔt2. Solving (76) with p=1,2,3, and 4 and β=1/3 with initial conditions x(0)=0.1 and x′(0)=0.0, to get Figures 1 and 2. In Figure 1, p=1,2, the solution is not oscillatory as conditions (11) are not satisfied. In Figure 2, p=3,4, we get oscillatory solution and this example validates numerically Theorem 4. Similar results are obtained for β=1 and β=3.Figure 1
Solutionx(t) of (76) for β=1/3 and p=1,2.Figure 2
Solutionx(t) of (76) for β=1/3 and p=3,4.As another example, consider the following equation:(80)1+t4x′t′′+∫0txγs-xβs1+t21+s2ds=tpsin⁡twith initial conditions x(t0)=x0, x′(t0)=x0′, and x′′(t0)=x0′′. Follow the same procedure above and substitute 1+t4x′(t)=y to get the following system:(81)x′t=y1+t4;xt0=x0,y′t=z;yt0=1+t04x′t0,z′t=tpsin⁡t-∫0txγs-xβs1+t21+s2ds;zt0=y′t0=1+t04x′′t0+41+t03x′t0.The prediction and correction steps of the modified Euler technique will be(82)x-i+1=xi+Δtfti,xi,yi,zi,y-i+1=yi+Δtgti,xi,yi,zi,z-i+1=zi+Δthti,xi,yi,zi,xi+1=xi+0.5Δtfti,xi,yi,zi+fti+1,x-i+1,y-i+1,z-i+1,yi+1=yi+0.5Δtgti,xi,yi,zi+gti+1,x-i+1,y-i+1,z-i+1,zi+1=zi+0.5Δthti,xi,yi,zi+hti+1,x-i+1,y-i+1,z-i+1,where (83)ft,x,y,z=y1+t4,gt,x,y,z=z,ht,x,y,z=tpsin⁡t-∫0txγs-xβs1+t21+s2ds.We will consider the numerical solution of (80) with p=6 and with the initial conditions: x(0)=x′(0)=x′′(0)=0.Forγ=1 and β=5/3, the oscillatory solution in Figure 3 is obtained and this validates Theorem 7. Similar results are obtained to validate Theorem 8 (with β=1 and γ=1/3) and Theorems 9 and 10 (with β=5/3 and γ=1/3). Theorems 13–18 can be validated in a similar way.Figure 3
Solutionx(t) of (80) for γ=1 and β=5/3 and p=6.General Remarks
(1)
The results presented in this paper are new forT=R and T=Z.
(2)
The results of this paper are presented in a form which is essentially new for (1) with different nonlinearities.
(3)
The techniques offered in this paper can be employed to Volterra integral equations on timescales of the form(84)xt+∫0tat,sFs,xsΔs=et.
The formulation of our results to the above equation is left to the reader.
(4)
The results of this paper can be extended easily to delay integrodynamic equations of the form(85)rxΔΔn-1t=et-∫0tat,sFs,xgsΔs,
whereg:T→T is rd-continuous such that g(t)≤t and gΔ(t)≥0 for t≥0 and limt→∞⁡g(t)=∞. The formulation of the results is left to the reader.
(5)
We note that we can formulate the obtained results for the timescalesT=R (the continuous case), T=Z (the discrete case), T=qN0 with q>1 (the quantum calculus case), T=hZ with h>0, T=N02, and so forth. The details are left to the reader.



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