This paper deals with the oscillatory behavior of forced second-order integrodynamic equations on time scales. The results are new for the continuous and discrete cases and can be applied to Volterra integral equation on time scale. We also provide a numerical example in the continuous case to illustrate the results.

1. Introduction

Dynamic equations on time scales is a fairly new topic and for general basic ideas and background, we refer the reader to [1]. Oscillation and nonoscillation results for integral equations of Volterra type have been discussed only in a few papers in the literature; see [2–5].

In this paper, we are concerned with the oscillatory behavior of solutions of the second-order integrodynamic equation on time scales of the form
(1)(r(t)xΔ(t))Δ+∫0ta(t,s)F(s,x(s))Δs=e(t).
We take T⊆R to be an arbitrary time scale with 0∈T and SupT=∞. Whenever we write t≥s, we mean t∈[s,∞)∩T. We assume throughout the following.

e,r:T→R and a:T×T→R are rd-continuous and r(t)>0 and a(t,s)≥0 for t>s and there exist rd-continuous functions k,m:T→R+ such that
(2)a(t,s)≤k(t)m(s)∀t≥s,supt≥0k(t)≕k1<∞,(3)supt≥0∫0tm(s)Δs=k2<∞.

F:T×R→R is continuous and assume that there exists rd-continuous function q:T→(0,∞) and a real number β with 0<β≤1 such that
(4)xF(t,x)≤q(t)|x|β+1forx≠0,t≥0.
We only consider those solutions of (1) which are nontrivial and delta-differentiable on [0,∞). The term solution henceforth applies to such solutions of (1).

A solution x(t) of (1) is said to be oscillatory if, for every t0>0, we have
(5)inft≥t0x(t)<0<supt≥t0x(t),
and it is said to be nonoscillatory otherwise.

To the best of our knowledge, there are no results on the oscillatory behavior of solutions of (1). Therefore, the main goal of this paper is to establish some new criteria for the oscillatory behavior of (1) and other related equations. Also we provide some numerical examples to illustrate the results when T=R.

2. Main Results

We shall employ the following lemmas.

Lemma 1 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

If X and Yare nonnegative, then
(6)Xλ+(λ-1)Yλ-λXYλ-1≥0;λ>1,Xλ-(1-λ)Yλ-λXYλ-1≤0;λ<1,
where equality holds if and only if X=Y.

This section is devoted to the study of the oscillatory properties of (1). In what follows, we let
(7)g±(t)=e(t)∓k1(1-β)ββ/(1-β)×∫0tpβ/(β-1)(s)q1/(1-β)(s)m1/(1-β)(s)Δs;sssssssssssssssssssssssssssssssssssss0<β<1.
Now we give sufficient conditions under which a nonoscillatory solution x(t) of (1) satisfies
(8)x(t)=O(t)ast⟶∞.

Theorem 2.

Let 0<β<1, conditions (I) and (II) hold, and for all t0>0, the function t/r(t) is bounded:
(9)∫t0∞sr(s)Δs<∞,(10)∫t0∞sp(s)Δs<∞.
If
(11)limsupt→∞1t∫t0t1r(u)∫t0ug-(s)ΔsΔu<∞,liminft→∞1t∫t0t1r(u)∫t0ug+(s)ΔsΔu>-∞,
then every nonoscillatory solution x(t) of (1) satisfies
(12)limsupt→∞|x(t)|t<∞.

Proof.

Let x(t) be a nonoscillatory solution of (1). First, assume x(t) is eventually positive, say x(t)>0 for t≥t1 for some t1≥0. Using condition (4) in (1) we have
(13)(r(t)xΔ(t))Δ≤e(t)-∫0t1a(t,s)F(s,x(s))Δs+∫t1ta(t,s)q(s)xβ(s)Δsfort≥t1.
Let
(14)c:=max0≤t≤t1|F(t,x(t))|<∞.
By assumption (3) we obtain
(15)|-∫0t1a(t,s)F(s,x(s))Δs|≤∫0t1a(t,s)|F(t,x(t))|Δs≤c∫0t1a(t,s)Δs≤ck1k2≕bfort≥t1.
Hence, from (13) we have
(16)(r(t)xΔ(t))Δ≤e(t)+b+k1×∫t1t[m(s)q(s)xβ(s)-p(s)x(s)]Δs+k1∫t1tp(s)x(s)Δsfort≥t1.

Applying Lemma 1 to m(s)q(s)xβ(s)-p(s)x(s) with
(17)λ=β,X=(qm)1/βx,Y=(1βp(qm)-1/β)1/(β-1),
we have
(18)m(s)q(s)xβ(s)-p(s)x(s)≤(1-β)ββ/(β-1)×pβ(β-1)(s)m1/(1-β)(s)q1/(1-β)(s).
Thus, we get
(19)(r(t)xΔ(t))Δ≤g+(t)+b+k1∫t1tp(s)x(s)Δsfort≥t1.
Integrating (19) from t1 to t we have
(20)r(t)xΔ(t)≤r(t1)xΔ(t1)+∫t1tg+(s)Δs+b(t-t1)+k1∫t1t∫t1up(s)x(s)ΔsΔufort≥t1.
Employing Lemma 3 in [7] to interchange the order of integration, we obtain
(21)r(t)xΔ(t)≤r(t1)xΔ(t1)+∫t1tg+(s)Δs+b(t-t1)+k1t∫t1tp(s)x(s)Δsfort≥t1,
and so
(22)xΔ(t)≤r(t1)xΔ(t1)r(t)+1r(t)∫t1tg+(s)Δs+b(t-t1)r(t)+k1tr(t)∫t1tp(s)x(s)Δsfort≥t1.
Integrating this inequality from t1 to t and using (9) and the fact that the function t/r(t) is bounded for t≥t1, say by k3, we see that
(23)x(t)≤x(t1)+r(t1)xΔ(t1)∫t1t1r(s)Δs+∫t1t1r(u)∫t1ug+(s)ΔsΔu+b∫t1tsr(s)Δs+k1k3×∫t1t∫t1up(s)x(s)ΔsΔufort≥t1.
Once again, using Lemma 3 in [7] we have
(24)x(t)≤x(t1)+r(t1)xΔ(t1)∫t1t1r(s)Δs+∫t1t1r(u)∫t1ug+(s)ΔsΔu+b∫t1tsr(s)Δs+k1k3t×∫t1tp(s)x(s)Δsfort≥t1,
or
(25)x(t)t≤1t[x(t1)+r(t1)xΔ(t1)∫t1t1r(s)Δsssssss+∫t1t1r(u)∫t1ug+(s)ΔsΔu+b∫t1tsr(s)Δs]+k1k3∫t1tp(s)x(s)Δsfort≥t1,
and so
(26)x(t)t≤c1+c2∫t1tsp(s)(x(s)s)Δsfort≥t1,
where c2=k1k3 and c1 is an upper bound for the expression
(27)1t[x(t1)+r(t1)xΔ(t1)∫t1t1r(s)Δssss+∫t1t1r(u)∫t1ug+(s)ΔsΔu+b∫t1tsr(s)Δs]fort≥t1.

Applying Gronwall’s inequality [1, Corollary 6.7] to inequality (26) and then using condition (10) we have
(28)limsupt→∞x(t)t<∞.
If x(t) is eventually negative, we can set y(t)=-x(t) to see that y(t) satisfies (1) with e(t) replaced by -e(t) and F(t,x) replaced by -F(t,-y). It follows in a similar manner that
(29)limsupt→∞-x(t)t<∞.
We conclude from (28) and (29) that (12) holds.

Next, by employing Theorem 2 we present the following oscillation result for (1).

Theorem 3.

Let 0<β<1, conditions (I), (II), (9), and (11) hold, and
(30)limsupt→∞t∫t0tsp(s)Δs<∞.
If
(31)limsupt→∞∫t0t1r(u)∫t0ug-(s)ΔsΔu=∞,liminft→∞∫t0t1r(u)∫t0ug+(s)ΔsΔu=-∞,
for all t0≥0, then (1) is oscillatory.

Proof.

Let x(t) be a nonoscillatory solution of (1), say x(t)>0 for t≥t1 for some t1≥0. The proof when x(t) is eventually negative is similar. Proceeding as in the proof of Theorem 2 we arrive at
(32)x(t)≤x(t1)+r(t1)xΔ(t1)∫t1t1r(s)Δs+∫t1t1r(u)∫t1ug+(s)ΔsΔu+b∫t1tsr(s)Δs+k1k3t×∫t1tsp(s)(x(s)s)Δsfort≥t1.
Condition (30) implies condition (10) and so the conclusion of Theorem 2 holds. This, together with (9), shows that the second and the last two integrals in (32) are bounded. Finally, taking limsupt→∞ and using conditions (30) and (31) result in a contradiction with the fact that x(t) is eventually positive.

Corollary 4.

Let 0<β<1 and condition (I), (II), (9), and (30) hold. In addition, assume that
(33)limsupt→∞1t∫t0t1r(u)∫t0ue(s)ΔsΔu<∞,liminft→∞1t∫t0t1r(u)∫t0ue(s)ΔsΔu>-∞,(34)limt→∞1t∫t0t1r(u)×∫t0upβ/(β-1)(s)q(s)1/(1-β)m1/(1-β)(s)ΔsΔu<∞.
If
(35)limsupt→∞1t∫t0t1r(u)∫t0ue(s)ΔsΔu=∞,liminft→∞1t∫t0t1r(u)∫t0ue(s)ΔsΔu=-∞,
for all t0≥0, then (1) is oscillatory. Similar reasoning to that in the sublinear case guarantees the following theorems for the integrodynamic (1) when β=1.

Theorem 5.

Let β=1, and conditions (I) and (II) hold. In addition, suppose (9) and (33) and for all t0≥0 assume that
(36)limsupt→∞∫t0tsm(s)q(s)Δs<∞.
Then every nonoscillatory solution of (1) satisfies (12).

Theorem 6.

Let β=1, and conditions (I) and (II) hold. In addition, suppose (9) and (33) hold and assume that (35) is satisfied and for all t0≥0,
(37)limsupt→∞t∫t0tsm(s)q(s)Δs<∞.
Then (1) is oscillatory.

Remark 7.

The technique in this paper can be employed to Volterra integral equation on time scales of the form
(38)x(t)+∫0ta(t,s)F(s,x(s))Δs=e(t).

Remark 8.

We note that the results in this section can be obtained by using the additional assumption that the function a(t,s) is nonincreasing with respect to the first variable. In this case the function k1m(t) which appeared in the proofs and the function m(t) which appeared in the statements of the theorems are replaced by a(t,t). The details are left to the reader.

3. Illustrative Examples

As a numerical illustration of our results of Section 2, we consider the following equation:
(39)((1+t)3x′(t))′+∫0txα(s)(t2+1)(s2+1)ds=e(t),hhhhhhhhhhhhhhhhhhhlhhhhhhhl0<t≤Tf
with initial conditions x(0)=x0 and x′(0)=x0′. Equation (39) can be converted into two first order ordinary differential equations by substituting (1+t)3x′(t)=y(t). This will lead to the following system:
(40)x′(t)=y(t)(1+t)3;x(0)=x0,y′(t)=e(t)-∫0txα(s)(t2+1)(s2+1)ds;y(0)=x0′.
Many numerical techniques can be used to solve (40). In the current work, the second-order accurate modified Euler technique is considered. The time interval [0,Tf] will be subdivided into N equal subdivisions with Δt width for each subdivision. The prediction and correction steps of the modified Euler technique will be
(41)x-i+1=xi+Δtg(ti,xi,yi),y-i+1=yi+Δth(ti,xi,yi),xi+1=xi+0.5Δt[g(ti,xi,yi)+g(ti+1,x-i+1,y-i+1)],yi+1=yi+0.5Δt[h(ti,xi,yi)+h(ti+1,x-i+1,y-i+1)],
where
(42)g(t,x,y)=y(t)(1+t)3,(43)h(t,x,y)=e(t)-∫0txα(s)(t2+1)(s2+1)ds.
The integral (43) can be approximated numerically at each time instant ti using the trapezoidal rule, which has accuracy of O(Δt)2.

Let Tf=50, x(0)=0.1 and x′(0)=0, with α=1/3. For e(t)=t4sin(t), all conditions of Theorems 3 and 6 are satisfied and hence (39) is oscillatory (see Figure 1). In Figure 2, the initial conditions are changed to be x(0)=0.1 and x′(0)=0.1 with α=1; the solution is also oscillatory.

The numerical solution x(t) for e(t)=t4sin(t), α=1/3 and initial conditions x(0)=0.1, x′(0)=0.

The numerical solution x(t) for e(t)=t4sin(t), α=1 and initial conditions x(0)=0.1, x′(0)=0.1.

4. Conclusions

The oscillatory behavior of a second-order integrodynamic equation on time scales is considered. The results in the paper establish new criteria and it is shown that these results can be employed for Volterra integral equation on time scale. A numerical example in the continuous case is solved to show the validity of the results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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