Bol. Mat. 16(1), 1–10 (2009)
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Boundedness in third order nonlinear differential equations with bounded delay Cemil Tun¸ c1 Faculty of Arts and Sciences, Department of Mathematics Y¨ uz¨ unc¨ u Yıl University, 65080, Van, Turkey
Se presenta un teorema que incluye condiciones suficientes para el acotamiento de soluciones de una cierta clase de ecuaciones diferenciales no lineales de tercer orden con un retardo constante. Nuestros resultados extienden un resultado reciente de Omeike (2009). Palabras Claves: Acotamiento, funcional de Lyapunov ecuaciones diferenciales no lineales, tercer orden, retardo. A theorem is presented which includes sufficient conditions for the boundedness of solutions of a certain class of third order nonlinear differential equations with a constant delay. Our result extends a recent result by Omeike (2009). Keywords: Boundedness, Lyapunov functional, nonlinear differential equations, third order, delay. MSC: 34K20.
1
Introduction
A problem of considerable interest in qualitative theory of ordinary differential equations of higher order, with or without delay, is the determination of the boundedness of solutions. In the last three decades, specially in recent years, some authors dealt with the problem for various nonlinear delay differential equations of third order. In particular, we refer readers to the papers of Afuwape and Omeike [2], Sinha [4], Tejumola and Tchegnani [5], Tun¸c [6, 7, 8, 9, 10, 11, 12, 13, 14], Zhu [15], and the references therein for some results achieved on the boundedness of solutions of certain nonlinear delay differential equations of third order. 1
[email protected]
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C. Tunc, Boundedness in third order
In the meantime Omeike, in 2009, [3] considered the following nonlinear differential equation of third order, with a constant deviating argument r, d2 x d3 x (t) + a(t) (t) + b(t) g dt3 dt2
µ
¶ dx (t) + c(t) h(x(t − r)) = p(t) . dt
Omeike discussed the stability and boundedness of solutions of this equation when p(t) = 0 and p(t) 6= 0. In this paper, instead of the above equation, we consider the following non–autonomous differential equation of third order with a constant deviating argument, r, given by: µ ¶ 2 µ ¶ d3 x d x dx dx (t) (t) (t) + a(t) ψ (t) + b(t) g dt3 dt dt2 dt +c(t) h(x(t − r) µ ¶ dx d2 x dx (t), (t − r), 2 (t) , = p t, x(t), x(t − r), dt dt dt
(1)
whose associated system is dx (t) = y(t) , dt dy (t) = z(t) , dt dz (t) = −a(t) ψ(y(t)) z(t) − b(t) g(y(t)) − c(t) h(x(t)) dt Z t dh +c(t) (x(s)) y(s) ds t−r dt +p(t, x(t), x(t − r), y(t), y(t − r), z(t)) ,
(2)
where r is a positive constant; the functions a(t), b(t), c(t), ψ(y), g(y), h(x) and p depend only on the arguments displayed explicitly; t ∈ R+ , R+ = [0, ∞). The functions a(t), b(t), c(t), ψ(y), g(y), h(x) and p are assumed to be continuous for their respective arguments on R+ , R+ , R+ , R , R, R and R+ × R5 , respectively; g(0) = 0, h(0) = 0; the derivatives da db dc dh dt (t), dt (t), dt (t) and dx (x) exist and are continuous. It should be noted that the continuity of the functions a(t), b(t), c(t), ψ(y), g(y), h(x) and p guarantees the existence of the solution of equation (1). Besides, it is assumed that the functions ψ(y), g(y), h(x) and p(t, x, x(t − r), y, y(t −
Bol. Mat. 16(1), 1–10 (2009)
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r), z) satisfy a Lipschitz condition in x, y, z, x(t − r) and y(t − r). Hence the solution is unique. Finally, all solutions considered are supposed to be real valued and throughout the paper x(t), y(t) and z(t) are abbreviated as x, y and z, respectively. The motivation of this paper has come from the result by Omeike [3, Theorem 2] and the papers mentioned above. Our purpose here is to extend the result established by Omeike [3, Theorem 2] to the preceding non–autonomous differential equation with the deviating argument r for the boundedness of all solutions. Clearly, the equation discussed in Omeike [3, Theorem 2] is a special case of our equation, equation (1).
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Main result
We prove the following theorem. Theorem. In addition to the basic assumptions imposed on the functions a(t), b(t), c(t), ψ(y), g(y), h(x) and p, let us assume that there exist positive constants δ(0), δ1 , δ2 , a, b, c, α and L such that the following conditions hold: i. h(x) ≥ δ(0) , x 6= 0 , x dh (x) ≤ c , dx g(y) ≥ b , y 6= 0 , y ψ(y) ≥ 1 , ii. 0 < a ≤ a(t) ≤ L , 0 < δ1 ≤ c(t) ≤ b(t) ≤ L , dc db −L ≤ (t) ≤ (t) ≤ 0 , dt dt 1 da (t) ≤ δ2 < δ1 (b − αc) , 2 dt b 1 >α> , c a
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C. Tunc, Boundedness in third order iii. |p(t, x, x(t − r), y, y(t − r), z)| ≤ q(t) ,
where q ∈ L1 (0, ∞), L1 (0, ∞) is the space of Lebesgue integrable functions. Then, there exists a finite positive constant K such that the solution x(t) of equation (1) defined by the initial function x(t) = φ(t) , dx dφ (t) = (t) , dt dt d2 x d2 φ (t) = (t) , dt2 dt2 satisfies √ |x(t)| ≤ K, ¯ ¯ √ ¯ dx ¯ ¯ (t)¯ ≤ K, ¯ dt ¯ ¯ 2 ¯ √ ¯d x ¯ ¯ ¯ ≤ (t) K, ¯ dt2 ¯ for all t ≥ t(0), where φ ∈ C 2 ([t(0) − r, t(0)], R), provided that ¾ ½ 2(αa − 1) 2δ5 , . r < min (2 + α) L c Lαc Proof. We define a Lyapunov functional V = V (t, x(t), y(t), z(t)) by Z 2
y
2 V (t, x(t), y(t), z(t)) = α z + 2 y z + 2 α b(t) g(η) dη 0 Z y +2 a(t) ψ(η) η dη + 2 α c(t) h(x) 0 Z x +2 c(t) h(s) ds 0 Z 0 Z t +λ y 2 (θ) dθ ds , (3) −r
t+s
where λ is a positive constant which will be determined later in the proof. By the assumptions a(t) > 0 and ψ(y) ≥ 1, from (3) we have
Bol. Mat. 16(1), 1–10 (2009)
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2 V (t, x(t), y(t), z(t)) ≥ α z 2 + 2 y z Z +2 α b(t)
y
g(η) dη + a(t) y 2 Z x +2 α c(t) h(x) y + 2 c(t) h(s) ds 0 Z 0 Z t +λ y 2 (θ) dθ ds . (4) 0
−r
t+s
Taking into account the assumptions of the theorem and the discussion in Omeike [3, Theorem 2], from (4) one can conclude that δ0 δ1 δ4 2 δ3 2 δ3 2 x + y + z 2 4 4 Z 0 Z t y 2 (θ) dθ ds +λ −r t+s ¡ 2 ¢ ≥ D1 x + y 2 + z 2 ,
V (t, x(t), y(t), z(t)) ≥
(5)
for some positive constants δ0 , δ1 , δ3 and δ4 , where ½ ¾ 1 1 D1 = min δ0 δ1 δ4 , δ3 . 2 4 For the time derivative of the Lyapunov functional V (t, x(t), y(t), z(t)), along a trajectory of the system (2), we have dV dt
Z x Z y dc db = (t) h(s) ds + α (t) g(η) dη dt dt 0 0 Z y dc da +α (t) h(x) y + (t) ψ(η) η dη dt dt 0 · ¸ g(y) dh − b(t) − α c(t) (x) − λ r y 2 y dt Z t dh − [α a(t) ψ(y) − 1] z 2 + c(t) y (x(s)) y(s) ds t−r dx Z t Z t dh +α c(t) z (x(s)) y(s) ds − λ y 2 (s) ds dx t−r t−r +(y + α z) p(t, x, x(t − r), y, y(t − r), z) .
(6)
Hence, subject to the assumptions of the theorem, from Omeike [3, Theorem 1], it follows that
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C. Tunc, Boundedness in third order
Z
dc (t) dt m2
x
0
db h(s) ds + α (t) dt
Z
y
g(η) dη + α 0
dc (t) h(x) y ≤ 0 . dt
Applying the assumptions of the theorem and the inequality 2 |mn| ≤ + n2 , we obtain the following inequalities: ·
¸ ¡ ¢ g(y) dh b(t) − α c(t) (x) − λ r y 2 ≥ b2 (t) − α c2 (t) − λ r y 2 y dx ¶ µ 2 b (t) = c(t) − α c y2 c(t) −λ r y 2 ≥ δ1 (b − α c) y 2 − λ r y 2 µ ¶ 1 = δ1 b − α c − λ r y 2 , δ1 µ ¶ Z y da g(y) dh (t) ψ(η) η dη − b(t) − α c(t) (x) − λ r y 2 dt y dx 0 µ ¶ 1 da g(y) dh ≤ (t) y 2 − b(t) − α c(t) (x) − λ r y 2 2 dt y dx ¶ µ 1 ≤ δ2 y 2 − δ1 b − α c − λ r y 2 δ1 = − [−δ2 + δ1 (b − α c) − λ r] y 2 = − (δ5 − λ r) y 2 , δ5 = −δ2 + δ1 (b − α c) > 0 , (α a(t) ψ(y) − 1) z 2 ≥ (α a − 1) z 2 , Z
t
c(t) y t−r
Z
t
α c(t) z t−r
1 1 dh (x(s)) y(s) ds ≤ L c r y 2 + L c dx 2 2
Z
t
y 2 (s) ds ,
t−r
dh 1 1 (x(s)) y(s) ds ≤ L α c r z 2 + L α c dx 2 2
Z
t
t−r
y 2 (s) ds ,
Bol. Mat. 16(1), 1–10 (2009)
(y + α z) p(t, x, x(t − r), y, y(t − r), z) ≤ |y + α z| |p(t, x, x(t − r), y, y(t − r), z)| ≤ (|y| + α |z|) q(t) ¡ ¢ ≤ 1 + α + y 2 + α z 2 q(t) ¡ ¢ ≤ (1 + α) q(t) + D2 y 2 + z 2 q(t) D2 ≤ (1 + α) q(t) + V (t, x(t), y(t), z(t)) q(t) , D1 where D2 = max{1, α}. The replacement of the preceding inequalities in (6) gives · ¸ 1 dV (t, x(t), y(t), z(t)) ≤ − δ5 − (λ + L c) r y 2 dt 2 · ¸ 1 − (αa − 1) − L α c r z 2 2 +(1 + α) q(t) D2 + V (t, x(t), y(t), z(t)) q(t) D1 · ¸Z t 1 y 2 (s) ds . + (1 + α) L c − λ 2 t−r Choose λ = 12 (1 + α)L c. Hence, · ¸ 1 dV (t, x(t), y(t), z(t)) ≤ − δ5 − (2 + α) L c r y 2 dt 2 · ¸ 1 − (α a − 1) − L α c r z 2 2 +(1 + α) q(t) D2 + V (t, x(t), y(t), z(t)) q(t) . D1 The last inequality implies dV (t, x(t), y(t), z(t)) ≤ −D3 y 2 − D4 z 2 + (1 + α) q(t) dt D2 + V (t, x(t), y(t), z(t)) q(t) , D1
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C. Tunc, Boundedness in third order
for some positive constants D3 and D4 , provided that ½ r < min
2(α a − 1) 2δ5 , (2 + α)L c Lα c
¾ .
Thus, we have dV (t, x(t), y(t), z(t)) ≤ (1 + α) q(t) dt D2 V (t, x(t), y(t), z(t)) q(t) . + D1 Integrating the last inequality from 0 to t, using the assumption q ∈ and the Gronwall–Reid–Bellman inequality (see Ahmad and Rama Mohana Rao [1]), we obtain L1 (0, ∞)
V (t, x(t), y(t), z(t)) ≤ V (0, x(0), y(0), z(0)) Z t +(1 + α) q(s) ds 0 Z D2 t + V (s, xs , ys , zs ) q(s) ds D1 0 ≤ {V (0, x(0), y(0), z(0)) + (1 + α) A} µ ¶ Z D2 t × exp q(s) ds D1 0 = K1 < ∞ , (7) where µ K1 = {V (0, x(0), y(0), z(0)) + (1 + α) A} exp
¶ D2 A , D1
R∞ is a constant and A = 0 q(s)ds. Hence, the inequalities (5) and (7) together imply that x2 (t) + y 2 (t) + z 2 (t) ≤
1 V (t, x(t), y(t), z(t)) ≤ K , D1
where K = D11 K1 . As a result, for solutions of the system (2), we can conclude that
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√ K, √ |y(t)| ≤ K, √ K, |z(t)| ≤
|x(t)| ≤
for all t ≥ 0. That is, √ |x(t)| ≤ K, ¯ ¯ √ ¯ dx ¯ ¯ (t)¯ ≤ K, ¯ dt ¯ ¯ 2 ¯ √ ¯d x ¯ ¯ ¯ K, ¯ dt2 (t)¯ ≤ for all t ≥ 0. This shows boundedness of all solutions of equation (1). The proof of the theorem is now complete.
References [1] S. Ahmad and M. Rama Mohana Rao, Theory of ordinary differential equations. With applications in biology and engineering (Affiliated East–West Press Pvt. Ltd., New Delhi, 1999). [2] A. U. Afuwape and M. O. Omeike, On the stability and boundedness of solutions of a kind of third order delay differential equations, Appl .Math. Comput. 200, 444 (2008). [3] M. O. Omeike, Stability and boundedness of solutions of some non–autonomous delay differential equation of the third order, An.S.tiint.Univ. Al. I. Cuza Ias.i.Mat. (N.S.) 55, 49 (2009). [4] A. S. C. Sinha, On stability of solutions of some third and fourth order delay–differential equations, Information and Control 23, 165 (1973). [5] H. O. Tejumola and B. Tchegnani, Stability, boundedness and existence of periodic solutions of some third and fourth order nonlinear delay differential equations J. Nigerian Math. Soc. 19, 9 (2000). [6] C. Tun¸c, New results about stability and boundedness of solutions of certain non–linear third–order delay differential equations, Arab. J. Sci.Eng. Sect. A Sci. 31, 185 (2006).
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C. Tunc, Boundedness in third order
[7] C. Tun¸c, On asymptotic stability of solutions to third order nonlinear differential equations with retarded argument, Commun. Appl. Anal. 11, 515 (2007). [8] C. Tun¸c, Stability and boundedness of solutions of nonlinear differential equations of third–order with delay, Differ. Uravn. Protsessy Upr. 43, 1 (2007). [9] C. Tun¸c, On the boundedness of solutions of third–order differential equations with delay, Differ. Uravn. 44, 446 (2008), in Russian; translation in Differ. equation 44, 464 (2008). [10] C. Tun¸c, On the boundedness of solutions of delay differential equations of third order, Arab. J. Sci. Eng. Sect. A Sci. 34, 227 (2009). [11] C. Tun¸c, Stability criteria for certain third order nonlinear delay differential equations, Port. Math. 66, 71 (2009). [12] C. Tun¸c, Boundedness criteria for certain third order nonlinear delay differential equations, J. Concr. Appl. Math. 7, 126 (2009). [13] C. Tun¸c, A new boundedness result to nonlinear differential equations of third order with finite lag, Commun. Appl. Anal. 13, 1 (2009). [14] C. Tun¸c, On the stability and boundedness of solutions to third order nonlinear differential equations with retarded argument, Nonlinear Dynam. 57, 97 (2009). [15] Y. F. Zhu, On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system, Ann. Differential Equations 8, 249 (1992).
