In this paper, a complete Lyapunov functional was con-
structed and used to obtain criteria (when p = 0) for uniform asymptotic
stability of the zero solution of the nonlinear delay differential equation
(1.1). When p ≠ 0, sufficient conditions are also established for uni-
form boundedness and uniform ultimate boundedness of solutions of
this equation. Our results improve and extend some well known results
in the literature

Adeleke Timothy, Ademora

Peter Olutola, Arawamo

Okayama University Scientific Achievement Repository

Math. J. Okayama Univ. 55 (2013), 157–166UNIFORM STABILITY AND BOUNDEDNESSOF SOLUTIONS OF NONLINEAR DELAYDIFFERENTIAL EQUATIONS OF THE THIRD ORDERADEMOLA, A.T. and ARAWOMO, P.O.Abstract. In this paper, a complete Lyapunov functional was con-structed and used to obtain criteria (when p = 0) for uniform asymptoticstability of the zero solution of the nonlinear delay differential equation(1.1). When p 6= 0, sufficient conditions are also established for uni-form boundedness and uniform ultimate boundedness of solutions ofthis equation. Our results improve and extend some well known resultsin the literature.1. IntroductionMany works have been done by several authors on the properties of so-lutions of ordinary differential equations of the second, third, fourth, fifthand higher order in which the unknown functions and its derivatives areall evaluated at the same instant, t, see for instance Reissig et. al., [10], asurvey book and Ademola et. al., [1], Ezeilo [6], Omeike [8], Tejumola [12],Tunc¸ [17] and the references cited therein to mention few. A more generaltype of differential equation often called a functional differential equation isone in which the unknown function occurs with various different arguments.This means an equation expressing some derivatives of x at time t in termof x (and its lower order derivative if any) at time t and at earlier instants.With respect to our observation in the relevant literature, interesting re-sults have been obtained on the properties of solutions of these classes ofequations see for examples Burton [4, 5], Hale [7] and Yoshizawa [18] whichcontain general results on the subject matter. Notable authors that havecontributed immensely to the qualitative properties of solutions (in partic-ular stability and boundedness of solutions) of nonlinear third order delaydifferential equations include Afuwape and Omeike [2], Omeike [9], Sadek[11], Tunc¸ [13, 14, 15, 16, 17] and Zhu [19]. These authors dealt with theproblems by constructing Lyapunov functionals and obtain criteria for sta-bility and boundedness of solutions.The purpose of this paper is to obtain criteria for uniform stability, uni-form boundedness and uniform ultimate boundedness of solutions for a moreMathematics Subject Classification. Primary 34K12; Secondary 34K20.Key words and phrases. Uniform stability, Uniform boundedness, Uniform ultimateboundedness, Lyapunov functional, Delay differential equation.157158 A.T. ADEMOLA AND P.O. ARAWOMOgeneral third order nonlinear delay differential equation(1.1)...x+f(x, x˙, x¨)x¨+g(x(t−r(t)), x˙(t−r(t)))+h(x(t−r(t))) = p(t, x, x˙, x¨)or its equivalent system given byx˙ = y, y˙ = z, z˙ = p(t, x, y, z) − f(x, y, z)z − g(x, y) − h(x)+t∫t−r(t)gx(x(s), y(s))y(s)ds +t∫t−r(t)gy(x(s), y(s))z(s)ds+t∫t−r(t)h′(x(s))y(s)ds(1.2)where 0 ≤ r(t) ≤ γ, γ > 0 is a constant which will be determined later,the functions f, g, h and p are continuous in their respective arguments andthe derivatives fx(x, y, z) fz(x, y, z), gx(x, y), gy(x, y), h′(x) exist and arecontinuous for all x, y, z with h(0) = g(0, 0) = g(x, 0) = 0 for all x. Thedots, as elsewhere, stands for differentiation with respect to the independentvariable t. Also, as usual, condition for uniqueness of solutions of (1.2) will beassumed. The results obtained in this investigation improve, generalize andcomplement existing results on the third order nonlinear delay differentialequation in the literature.2. PreliminariesConsider the general autonomous delay differential system(2.1) X˙ = F (Xt), Xt(θ) = X(t+ θ), −r ≤ θ ≤ 0. t ≥ 0where F : CH → Rn is a continuous mapping , F (0) = 0, we suppose thatF takes closed bounded set of Rn. Here (C, ‖ · ‖) is the Banach space ofcontinuous function φ : [−r, 0]→ Rn with supremum norm, r > 0, CH is anopen ball of radius H in C; CH := {φ ∈ (C[−r, 0],Rn) : ‖φ‖ < H}. It hasbeen shown by Burton [5], that if φ ∈ CH , and t ≥ 0, then there is at leastone continuous solution X(t, t0, φ) satisfying (2.1) for t > t0 on the interval[t0, t0 +α), such that Xt(t, φ) = φ and α is a positive constant. If there is aclosed subset B ⊂ CH such that the solution remain in B, then α =∞.Definition 1. [5]. A continuous function W : R+ → R+ with W (0) = 0,W (s) > 0 if s 6= 0, and W strictly increasing is a wedge. (We denote wedgesby W or Wi, where i is an integer).Definition 2. [5]. The zero solution of (2.1) is asymptotically stable if itis stable and if for each t0 ≥ 0 there is an η > 0 such that ‖φ‖ ≤ η impliesUNIFORM STABILITY AND BOUNDEDNESS OF SOLUTIONS 159thatX(t, t0, φ)→ 0 as t→∞.Definition 3. [4]. An element ψ ∈ CH is in the ω−limit set of φ, sayΩ(φ), if X(t, 0, φ) is defined on R+ and there is a sequence {tn}, tn →∞ asn→ +∞, with ‖Xtn(φ)−ψ‖ → 0 as n→∞, where Xtn(φ) = X(tn+θ, 0, φ)for −r ≤ θ < 0.Definition 4. [19]. A set Q ⊂ CH is an invariant set if for any φ ∈ Q, thesolution X(t, 0, φ) of (2.1) is defined on R+ and Xt(φ) ∈ Q for t ∈ R+.Lemma 2.1. [5, 7, 19]. If φ ∈ CH is such that the solution xt(φ) of (2.1)with x0(φ) = φ is defined on R+ and ‖Xt(φ)‖ ≤ H1 < H for t ∈ R+, thenΩ(φ) is nonempty, compact, invariant set, anddist(Xt(φ),Ω(φ)) → 0, as t→∞.Lemma 2.2. [4, 19].) Let V (φ) : CH → R be a continuous functionalsatisfying a local Lipschitz condition. V (0) = 0 and such that(i) W (|φ(0)|) ≤ V (φ) ≤W2(‖φ‖) where W1(r) and W2(r) are wedges;(ii) V˙(2.1)(φ) ≤ 0 for φ ∈ CH .Then the zero solution of (2.1) is uniformly stable.If we define Z = {φ ∈ CH : V˙(2.1)(φ) = 0}, then Xt = 0 of (2.1) is asymp-totically stable, provided that the largest invariant set in Z is M = {0}.Next, consider the system(2.2) X˙ = F (t,Xt), Xt = X(t+ θ), −r ≤ θ ≤ 0, t ≥ 0,Where F : [0,∞) × C → R is continuous and takes bounded sets intobounded sets. The following lemma is a well-known result obtained byBurton [4].Lemma 2.3. [4]. Let V : R+ × C → R be continuous and locally Lipschitzin φ. If(i) W0(|Xt|) ≤ V (t,Xt) ≤W1(|Xt|) +W2(t∫t−r(t)W3(Xt(s))ds)and(ii) V˙(2.2)(t,Xt) ≤ −W4(|Xt|) + N, for some N > 0 where Wi (i =0, 1, 2, 3, 4) are wedges.Then Xt of (2.2) is uniformly bounded and uniformly ultimately boundedfor bound B.160 A.T. ADEMOLA AND P.O. ARAWOMO3. Main ResultsIn the case when p(t, x, y, z) ≡ 0. (1.2) becomesx˙ = y, y˙ = z, z˙ = −f(x, y, z)z − g(x, y) − h(x)+t∫t−r(t)gx(x(s), y(s))y(s)ds +t∫t−r(t)gy(x(s), y(s))z(s)ds+t∫t−r(t)h′(x(s))y(s)ds(3.1)where f, g, h and r are the functions defined in Section 1. Let (xt, yt, zt)be any solution of (3.1), the main tool in the proofs of our results is thecontinuously differentiable functional V ≡ V (xt, yt, zt), defined by2V = 2(α + a)∫ x0h(ξ)dξ + 4∫ y0g(x, τ)dτ + 4yh(x) + bβx2+ 2(α+ a)∫ y0τf(x, τ, 0)dτ + βy2 + 2z2 + 2aβxy + 2βxz+ 2(α+ a)yz +∫ 0−r(t)∫ tt+s[λ1y2(θ) + λ2z2(θ)]dθds(3.2a)where α, β are positive fixed constants satisfying(3.2b) b−1c < α < a;(3.2c) 0 < β < min{(ab− c)a−1, (ab− c)A−11 ,12(a− α)A−12},where A1 := 1+ a+ δ−1(g(x, y)y− b)2and A2 := 1+ δ−1(f(x, y, z)− a)2;λ1 and λ2 are also positive constants which will be determined later.We havethe following result.Theorem 3.1. Further to the fundamental assumptions on f, g, h and rappearing in (3.1), suppose that a, a1, b, c, δ, γ and M are positive constantsand that:(i) a ≤ f(x, y, z) ≤ a1, yfx(x, y, 0) ≤ 0, yfz(x, y, z) ≥ 0 for all x, y, z;(ii) g(0, 0) = 0 = g(x, 0), b ≤g(x, y)yfor all x, y 6= 0, |gx(x, y)| ≤ L forsome L ≥ 0 and |gy(x, y)| ≤M for all x, y;(iii) h(0) = 0,h(x)x≥ δ (x 6= 0), h′(x) ≤ c for all x and ab > c;UNIFORM STABILITY AND BOUNDEDNESS OF SOLUTIONS 161(iv) 0 ≤ r(t) ≤ γ, r′(t) ≤ β0 0 < β0 < 1.Then the trivial solution of (3.1) is uniformly asymptotically stable, providedthat(3.3) γ < min{δA−13 , 2(1 − β0)(αb− c)A−14 , (a− α)(1 − β0)A−15}where A3 := c+L+M, A4 := (α+ a)(1− β0) + (c+L)(2 + α+ β + a) andA5 := 2(1 − β0)(c+ L+M) +M(2 + α+ β + a).Remark. Whenever f(x, x˙, x¨) ≡ a (a > 0 is a constant) and g(x(t −r(t)), x˙(t − r(t))) ≡ g(x˙(t − r(t))), Eq. (1.1) reduces to that discussed bySadek [11] and Tunc¸ [14]. Also, if f(x, x˙, x¨) ≡ f(x˙) (1.1) specializes to thatdiscussed by Afuwape and Omeike [2]. Besides, an incomplete Lyapunovfunctionals were constructed and used to obtain stability results in [2, 11]and [14] compared with a complete Lyapunov functional used in this inves-tigation. Hence our result includes and generalizes theirs.Proof of Theorem 3.1. Let (xt, yt, zt) be any solution of (3.1), the timederivative of the functional defined in (3.2) along a solution of (3.1) is sim-plified to giveV˙(3.1) = (α + a)y∫ y0τfx(x, τ, 0)dτ + aβy2 + [βx+ (α+ a)y + 2z]×∫ tt−r(t)(gx(x(s), y(s))y(s) + gy(x(s), y(s))z(s) + h′(x(s))y(s))ds+ r(t)(λ1y2 + λ2z2)+ 2βyz + 2y∫ y0gx(x, τ)dτ − β(g(x, y)y− b)xy− β(f(x, y, z)− a)xz − βh(x)xx2 −[(α+ a)g(x, y)y− 2h′(x)]y2−[2f(x, y, z) − (α+ a)]z2 − (α+ a)yz[f(x, y, z)− f(x, y, 0)]− (1− r′(t))∫ tt−r(t)(λ1y2 + λ2z2)ds.162 A.T. ADEMOLA AND P.O. ARAWOMOOn applying the hypotheses of the theorem and the fact that 2pq ≤ p2+ q2,we obtainV˙(3.1) ≤12(c+ L+M)(βx2 + (α + a)y2 + 2z2)r(t) + λ1y2r(t)+ λ2z2r(t)−12βδx2 − (αb− c)y2 −12(a− α)z2−{ab− c− β[1 + a+ δ−1(g(x, y)y− b)2]}y2−{12(a− α)− β[1 + δ−1(f(x, y, z)− a)2]}z2−β4δ[δx+ 2(g(x, y)y− b)y]2−β4δ[δx+ 2(f(x, y, z)− a)z]2−[λ1(1− r′(t))−12(c+ L)(2 + α+ β + a)] ∫ tt−r(t)y2(s)ds−[λ2(1− r′(t))−M2(2 + α+ β + a)] ∫ tt−r(t)z2(s)dsNow, in view of estimates (3.2b), (3.2c), the fact that 0 ≤ r(t) ≤ γ andr′(t) ≤ β0 0 < β0 < 1, this inequality becomesV˙(3.1) ≤ −β2[δ − (c+ L+M)γ]x2−{ab− c−[12(α+ a)(c+ L+M) + λ1]γ}y2−{12(a− α)− (c+ L+M + λ2)γ}z2−[λ1(1− β0)−12(c+ L)(2 + α+ β + a)] ∫ tt−r(t)y2(s)ds−[λ2(1− β0)−M2(2 + α+ β + a)] ∫ tt−r(t)z2(s)ds.Choosing λ1 := 2−1(1− β0)−1(c+L)(2 + α+ β + a) > 0 and λ2 := 2−1(1−β0)−1M(2 + α+ β + a) > 0, it follows thatV˙(3.1) ≤ −β2[δ − (c+ L+M)γ]x2−{ab− c−[12(α+ a)(c+ L+M) +(c+ L)(2 + α+ β + a)2(1− β0)]γ}y2{12(a− α)−[c+ L+M +M(2 + α+ β + a)2(1− β0)]γ}z2.UNIFORM STABILITY AND BOUNDEDNESS OF SOLUTIONS 163Applying estimate (3.3), there exists a constant δ0 > 0 such that(3.4) V˙(3.1) ≤ −δ0(x2 + y2 + z2)for all x, y, z where δ0 = min{β2[δ − (c+ L+M)γ], ab− c−[12(α+ a)(c+L+M) +(c+ L)(2 + α+ β + a)2(1− β0)]γ,12(a− α)−[c+ L+M +M(2 + α+ β + a)2(1− β0)]γ}.By (3.4) V˙(3.1) = 0 and the system (3.1), we can easily obtain x = y = z = 0.Hence, condition (ii) of Lemma 2.2 is satisfied.Next, since h(0) = 0, the functional V defined in (3.2) can be recast in theformV =1b∫ x0[(α+ a)b− 2h′(ξ)]h(ξ)dξ + 2∫ y0(g(x, τ)τ− b)τdτ+1b(h(x) + by)2+∫ y0[(α+ a)f(x, τ, 0) − (α2 + a2)]τdτ+12(αy + z)2 +12(βx+ ay + z)2 +12β(b− β)x2 +12βy2+12∫ 0−r(t)∫ tt+s[λ1y2(θ) + λ2z2(θ)]dθds.Now, by hypotheses of the theorem and the fact that the double integrals∫ 0−r(t)∫ tt+s[λ1y2(θ) + λ2z2(θ)]dθdsare non negative, it follows thatV ≥12b[δ(αb− c+ ab− c) + bβ(b− β)]x2 +12[α(a− α) + β]y2+12(αy + z)2 +12(βx+ ay + z)2.(3.5)By estimates (3.2b) and (3.2c), the quadratic in the right hand side of (3.5) ispositive definite, hence there exists a positive constant δ1 = δ1(a, b, c, α, β, δ)such that(3.6) V ≥ δ1(x2 + y2 + z2)for all x, y and z. Estimate (3.6) establishes the lower inequality in condition(i) of Lemma 2.2.Furthermore, since g(x, 0) = 0 for all x implies that g(x, y) ≤ My for allx, y 6= 0 and h(0) = 0 implies h(x) ≤ cx (x 6= 0). These inequalities and the164 A.T. ADEMOLA AND P.O. ARAWOMOfact that 2pq ≤ p2 + q2, the functional V defined in (3.2) yieldsV ≤ η1(x2 + y2 + z2) +∫ 0−r(t)∫ tt+sη2[x2(θ) + y2(θ) + z2(θ)]dθds(3.7)where η1 :=12 max{(1+a+b)β+(2+α+a)c, (1+a1)(α+a)+2(c+M)+(1+a)β, 2 + α + β} and η2 :=12 max{1, λ1, λ2}. From estimate (3.7) the upperinequality in codition (i) of Lemma 2.2 follows. By estimates (3.4), (3.6)and (3.7) the hypotheses of Lemma 2.2 are satisfied. Hence, by Lemma 2.2the trivial solution of (3.1) is uniformly stable and uniformly asymptoticallystable. Next, if p(t, x, y, z) 6= 0, we have the following results.Theorem 3.2. If the assumptions of Theorem 3.1 hold true and(3.8a) |p(t, x, y, z)| ≤ p1(t) + p2(t)(|x| + |y|+ |z|)where p1(t) and p2(t) are continuous functions satisfying(3.8b) p1(t) ≤ P00 < P0 <∞ and there exists ǫ > 0 such that(3.8c) 0 ≤ p2(t) ≤ ǫ.Then the solutions of the system (1.2) are uniformly bounded and uniformlyultimately bounded, provided that the inequality in (3.3) holds true.If p(t, x, y, z) ≡ p(t) 6= 0, Eq. (1.2) becomesx˙ = y, y˙ = z, z˙ = p(t)− f(x, y, z)z − g(x, y) − h(x)+t∫t−r(t)gx(x(s), y(s))y(s)ds +t∫t−r(t)gy(x(s), y(s))z(s)ds+t∫t−r(t)h′(x(s))y(s)ds(3.9)Corollary 1. If the assumptions of Theorem 3.1 are satisfied and|p(t)| ≤ P1, 0 < P1 < ∞ then the solutions of (3.9) are uniformly boundedand uniformly ultimately bounded, provided that the inequality in (3.3)holds.Remark. If f(x, x˙, x¨) ≡ a, a > 0 is a constant, g(x(t− r(t)), x˙(t− r(t))) ≡g(x˙(t − r(t))) and p(t, x, x˙, x¨) ≡ p(t), (1.1) specializes to that discussedby Sadek [11], our assumptions and conclusion coincide with his. Also, ifp2(t) ≡ 0, (1.1) reduces to the one discussed by Tunc¸ [13]. Finally, wheneverUNIFORM STABILITY AND BOUNDEDNESS OF SOLUTIONS 165f(x, x˙, x¨) = h(x˙) and p1(t) ≡ m, m > 0 is a constant, then the hypothesesand conclusion of Theorem 3.2 coincide with that of Afuwape and Omeikein [2] except f(x, y, z) ≤ a1 a necessary and sufficient condition for uni-form boundedness and uniform ultimate boundedness of solutions of (1.1)which could not be found in [2]. Hence, our result revises and generalizesthe situation given in [2, 11] and [14].Proof of Theorem 3.2. Let (xt, yt, zt) be any solution of (1.2), the deriv-ative of the functional V defined in (3.2) along a solution of (1.2) isV˙(1.2) = V˙(3.1) + (βx+ (α+ a)y + 2z)p(t, x, y, z).Now, from estimate (3.4) and the fact that q ≤ |q|, this equation becomesV˙(1.2) ≤ −δ0(x2 + y2 + z2) + η3(|x|+ |y|+ |z|)|p(t, x, y, z)|where η3 = max{β, α + a, 2}. In view of (3.8) and choosing ǫ < 3−1η−13 δ0there exists η4 = δ0 − 3η3ǫ > 0 such that(3.10) V˙(1.2) ≤ −η42(x2 + y2 + z2) +32η23P20 η−14 ,sinceη42[(|x| − η3P0η−14)2+(|y| − η3P0η−14)2+(|z| − η3P0η−14)2]≥ 0for all x, y and z. From estimate (3.10), hypothesis (ii) of Lemma 2.3 issatisfied. Also from estimates (3.6) and (3.7), condition (i) of Lemma 2.3follows. This completes the proof of the theorem. Acknowledgment: The authors would like to thank the referee(s) fortheir several helpful suggestions.References[1] A.T. Ademola, M.O. Ogundiran, P.O. Arawomo and O.A. Adesina, Boundedness re-sults for a certain third order nonlinear differential equations. 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Differential Equation 8 (2) (1992),249-259.ADEMOLA, Adeleke TimothyDepartment of MathematicsUniversity of IbadanIbadan, Nigeriae-mail address: ademola672000@yahoo.com;at.ademola@mail.ui.edu.ngARAWOMO, Peter OlutolaDepartment of MathematicsUniversity of IbadanIbadan, Nigeriae-mail address: womopeter@gmail.com(Received November 4, 2010 )

UNIFORM STABILITY AND BOUNDEDNESS OF SOLUTIONS OF NONLINEAR DELAY DIFFERENTIAL EQUATIONS OF THE THIRD ORDER

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