On Some Qualitiative Properties of Solutions to Certain Third Order Vector Differential Equations with Multiple Constant Deviating Arguments

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1 ISSN (print) (online) International Journal of Nonlinear Science Vol.6(8) No.3pp.8-9 On Some Qualitiative Properties of Solutions to Certain Third Order Vector Differential Equations with Multiple Constant Deviating Arguments Adeleke T. Ademola Ayman M. Mahmoud Department of Mathematics Obafemi Awolowo University Ile-Ife 5 Nigeria Faculty of Science New Valley Branch Assiut University New Valley El-Khargah 7 Egypt (Received 6 August 7 accepted September 8) Abstract: This paper consider third order vector differential equation with the multiple constant deviating arguments denoted by τ i >. The stability of the zero solution of this third order system with P (t) = is discussed. Sufficient conditions to ensure the boundedness of solutions with P (t) is also considered. By constructing Lyapunov functional we establish two new results on the stability and boundedness of solutions which include and improve some related results in the literature. An example is given to illustrate the importance of the theoretical analysis made in this work and to show the effectiveness of the method employed. Keywords: Vector differential equation; Stability; Boundedness; Lyapunov functional; Multiple constant deviating arguments Introduction Differential equations of second and third-order with and without delay are essential tools in scientific modeling of problems from many fields of sciences and technologies such as biology chemistry physics mechanics electronics engineering economy control theory medicine atomic energy and information theory. Many authors have proposed different methods in the literature to discuss qualitative behaviour of solutions to linear and nonlinear differential equations. In this direction we can mention Lyapunov s direct method which demands the construction of a suitable positive definite functional whose derivative is negative definite; that is it involves finding the system of closed surfaces that contained the origin and converging to it. Today this method is not only known as an unavoidable tool in the study of qualitative behaviour of solutions but also in the exploration of various other properties of solutions of ordinary delay and Stochastic-differential equations. In the applications the method is considered as overall technique and a functioning approach in studying the theory of dynamical system control systems power system analysis system with time-lag etc. The qualitative properties of solutions such as stability boundedness convergence existence of a unique periodic solutions etc of scalar third order ordinary or delay differential equation with deviating arguments have been rigorously studied and are still being under consideration in the literature by authors because of their practical applications. Many researchers investigated the asymptotic stability boundedness and periodicity of solutions of various third order differential equations with delay see the books of Burton [ ] Driver [3] Hale [4] Kolmanovskii and Myshkis [5] which contain wide-ranging consequences on the subject matters; also we can mention the outstanding papers of Abou-El-Ela et al. [6] Ademola and Arawomo [7] Ademola et al. [8] Afuwape and Omeike [9] Mahmoud [] Omeike [ ] Sadek [3 4] Shekhar et al. [5] Tẹjumọla and Tchegnani [6] Tunç([7]-[]) and the references therein. In and 993 Ezeilo and Tejumola [] Afuwape [3] and Meng [4] respectively discussed the ultimate boundedness and existence of periodic solutions of the nonlinear vector ordinary differential equation of the form X + AẌ + BẊ + H(X) = P (t X Ẋ Ẍ). Corresponding author. address: atademola@oauife.edu.ng Copyright c World Academic Press World Academic Union IJNS.8..5/8

2 A. Ademola and A. Mahmoud: On Some Qualitiative Properties of Solutions to Certain Third Order Vector 8 In 985 Abou-El-Ela [5] and Afuwape [6] studied criteria which guarantee ultimate boundedness of solutions for the nonlinear non autonomous vector ordinary differential equations X + F (X Ẋ)Ẍ + (Ẋ) + H(X) = P (t X Ẋ Ẍ) and X + AẌ + (Ẋ) + H(X) = P (t X Ẋ Ẍ) respectively. Also in 995 Feng [7] proved the existence of a unique periodic solution for the third order non autonomous vector ordinary differential equation X + A(t)Ẍ + B(t)Ẋ + H(X) = P (t X Ẋ Ẍ). Moreover in 999 Tiryaki [8] and Tunç [9] discussed conditions for ultimate boundedness and existence of periodic solutions for the third order vector ordinary differential equations X + AẌ + (Ẋ) + H(X) = P (t X Ẋ Ẍ) and X + F (X Ẋ)Ẍ + BẊ + H(X) = P (t X Ẋ Ẍ) respectively. In another development Afuwape and Omeike [3] in 4 gave criteria for ultimate boundedness of solutions of the third order nonlinear vector differential equation X + F (Ẍ) + (Ẋ) + H(X) = P (t X Ẋ Ẍ). In 5 Tunç and Ateş [3] investigated conditions for asymptotic stability of the zero and boundedness of solutions of the differential equation X + (X Ẋ Ẍ)Ẍ + B(t)Ẋ + F (X) = P (t X Ẋ Ẍ). Besides it is worth-mentioning that according to our observation there are only a few papers on fourth order vector delay differential equations for example Abou-El-Ela et al. [3 33]. On third order vector delay differential equations we have the following outstanding results by authors. In 5 Omeike [34] investigated criteria for the boundedness and stability of solutions of the nonlinear differential system of the form X + AẌ + BẊ + (X(t τ(t))) = P (t). Recently in 6 Mahmoud and Tunç [35] Fatmi and Remili [36] and Tunç [37] imposed criteria on the nonlinear functions which guaranteed boundedness and stability of trivial solution of the following system of third order vector delay differential equations X + AẌ + (Ẋ) + (X(t τ))) = P (t) and ((X)X ) + AX + BX + H (X(t r(t))) = P (t) X + AẌ + (Ẋ(t τ(t))) + (X(t τ(t))) = F (t X Ẋ Ẍ) respectively. Asymptotic stability boundedness existence and uniqueness of periodic solutions of differential equations for certain second third and fourth- order with multiple delays have been discussed by few authors see for example Ademola et al. [38] Ademola [39] Korkmaz and Tunç [4] Tunç [4] etc. This paper therefore consider the third order vector delay differential equations with multiple constant deviating arguments τ i > of the following form n X(t) + F (X(t) Ẋ(t))Ẍ(t) + (Ẋ(t)) + H i (X(t τ i )) = P (t) () where t R + t τ i > and X is defined in R n ; the function F is continuous symmetric and n n- matrix : R n R n H i : R n R n and P : R R n are continuous functions in addition and H i are differentiable functions with H i () = = (). IJNS homepage:

3 8 International Journal of Nonlinear ScienceVol.6(8)No.3pp. 8-9 Moreover in this work the matrices (Y ) and H i (X) are the Jacobian defined by J (Y ) = ( gi y j ) J H (X) = ( h i x j ) J Hn (X) = ( h ni x j ) (i j = n) respectively where (x i )(y i ) (g i ) and (h i h i h ni ) are the constituent of variables X Y and H i respectively. Also it is assume that the Jacobian matrices J (Y ) and J Hi (X) exist and are continuous. We need the following notations and definitions:. λ i (M) (i = n) is the eigenvalues of the square matrix M;. X Y conform to any pair of vectors X Y R n defined as n x iy j. The Euclidean length in R n will be denoted by. so that in particular X X = X for arbitrary X R n ; and 3. The n n matrix M is negative-definite if MX X < for all X R n. Now we consider the stability criteria for the general non-autonomous delay differential system x(t) = f(t x t ) x t (s) = x(t + s) h s t () where f : R + C H R n is a continuous mapping f(t ) = and suppose that f takes closed bounded sets into bounded sets of R +. Here (C ) is the Banach space of continuous functions ϕ : [ h ] R n with supremum norm h > ; C H is the open H-ball in C; C H := {ϕ C([ h ] R n ) : ϕ < H}. Let S be the set of ϕ C such that ϕ H we shall represents S as the set of all functions ϕ C satisfying ϕ() H where H is large enough. Definition [] Let W : R n R + be a continuous satisfying W () = W (s) > if s > and W strictly increasing is a wedge. Definition [] Let D R n be an open set with D. Then a function V : [ ) D [ ) is positive definite if V (t ) = and if there is a wedge W with V (t x) W ( x ) and the function V (t x) is said to be decrescent if V (t x) W ( x ) where W is a wedge. Theorem [] If there is a Lyapunov functional V (t x) for () and the wedges W i (i = ) such that (i) W ( ϕ() ) V (t ϕ) W ( ϕ ); and (ii) V() (t ϕ). Then the zero solution x = of () is uniformly stable. Theorem [4] Suppose that there exists a continuous Lyapunov functional V (t ϕ) defined for all t R + and ϕ S which satisfies the following conditions (i) a( ϕ() ) V (t ϕ) b ( ϕ() )+b ( ϕ ) where a(r) b (r) b (r) CI (CI denotes the families of continuous increasing functions) and are positive for r > H and a(r) as r ; and (ii) V() (t ϕ). Then the solutions of () is uniformly bounded. Main results Now we present the first main result for stability of solution of equation () with P (t) =. Theorem 3 In addition to the fundamental assumptions on the functions F and H i suppose that there exist positive constants a a a 3 b i and c i. Assume also for (i = n) the following conditions are satisfied: (i) The matrix F is symmetric and a λ i (F (X Y )) a J(F (X Y )Y X) is negative definite and J(F (X Y )Y Y ) is symmetric for all X Y R n ; (ii) () = J (Y ) is symmetric and a 3 λ i (J (Y )) a 4 for all Y R n ; IJNS for contribution: editor@nonlinearscience.org.uk

4 A. Ademola and A. Mahmoud: On Some Qualitiative Properties of Solutions to Certain Third Order Vector 83 (iii) H i () = H i (X) ; (X ) J Hi (X) are symmetric and c i λ i (J Hi (X)) b i for all X R n ; and (iv) a a 3 n b i >. Then the trivial solution X of equation () with P = is uniformly stable if { µa3 n τ < min b i (µ + ) n n b i where µ = a a 3 + n b i a 3 >. Our second main result of this paper is the following boundedness theorem. Theorem 4 Further to hypotheses (i) (iii) of Theorem 3 suppose that: P (t) Q(t) a µ n n b i where Q(t) L ( ) L ( ) is the space of Lebesgue integrable functions. If { µa3 n τ < min b } i (µ + ) n a µ n b n i n b i then there exists a constant K > such that the solution X(t) of equation () defined by the initial function satisfies the estimates where ϕ C ([t τ t ] R). } X(t) = ϕ(t) Ẋ(t) = ϕ(t) Ẍ(t) = ϕ(t) t τ t t X(t) K Ẋ(t) K and Ẍ(t) K for all t t The following two Lemmas are important for the proof of the main results. Lemma 5 Suppose that M is a real n n symmetric matrix. If ā λ i (M) a > (i = n) then ā X X M X X a X X Lemma 6 Assume that Ẋ = Y Ẏ = Z. Then d () dt H(σX) X dσ = H(X) Y d () dt (σy ) Y dσ = (Y ) Z d (3) dt σf (X σy )Y Y dσ F (X Y )Y Z. Proof. ā X X MX MX a X X. () d dt H(σX) X dσ = = = σ J H (σx)y X dσ + σ J H (σx)x Y dσ + σ H(σX) Y dσ + σ H(σX) Y dσ H(σX) Y dσ H(σX) Y dσ = σ H(σX) Y = H(X) Y. IJNS homepage:

5 84 International Journal of Nonlinear ScienceVol.6(8)No.3pp. 8-9 The proof of () is similar to that of (). (3) d dt σf (X σy )Y Y dσ = + σf (X σy )Y Z dσ σj(f (X σy ) Y X)Y Y dσ + σf (X σy )Y Z dσ + σ σj(f (X σy ) Y Y )Z Y dσ σ σj(f (X σy ) Y Y )Z Y dσ. Since J(F Y X) is negative-definite and J(F Y Y ) is symmetric because of assumption (i) of Theorem 3. It follows that d dt σf (X σy )Y Y dσ σf (X σy )Y Z dσ + = σ F (X σy ) Z = F (X Y )Y Z. σ F (X σy )σy Z dσ σ 3 Proof of theorem 3 We can write equation () with P (t) = in the following equivalent system Ẋ = Y Ẏ = Z Ż = F (X Y )Z (Y ) H i (X) + Let V = V (X t Y t Y t ) be a Lyapunov functional defined by V (X t Y t Z t ) =µ + Z Z + + δ i where s is a real variable such that the integrals H i (σx) X dσ + τ i τ i t+s J Hi (X(s))Y (s)ds. H i (X) Y + µ Y Z (σy ) Y dσ + µ t+s Y (θ) dθds Y (θ) dθds and δ i > are constants to be determined later in the proof. Therefore we find V (X t Y t Z t ) µ H i (σx) X dσ + σf (X σy )Y Y dσ H i (X) Y + µ Y Z + Z Z + (σy ) Y dσ + µ Since λ i (F (X Y )) a because of condition (i) we get from Lemma 5 V (X t Y t Z t ) µ H i (σx) X dσ + H i (X) Y + + Z + µy + µ(a µ) Y. σf (X σy )Y Y dσ. (σy ) Y dσ (3) (4) IJNS for contribution: editor@nonlinearscience.org.uk

6 A. Ademola and A. Mahmoud: On Some Qualitiative Properties of Solutions to Certain Third Order Vector 85 Let V (X t Y t Z t ) Z + µy + µ(a µ) Y + V where Since Thus V = µ H i (σx) X dσ + H i (X) Y + σ (σy ) = J (σ Y )Y and () = it follows that Then we have (σy ) Y dσ + (Y ) = (σy ) Y dσ = H i (X) Y = J (σ Y )Y dσ. (σy ) Y dσ. σ J (σ σ Y )Y Y dσ dσ. σ { J (σ σ Y )Y Y + H i (X) Y }dσ dσ. (5) Since J (Y ) is symmetric and the eigenvalues are positive. Consequently the square root J (Y ) exists this is again symmetric and non-singular. Therefore we have J Y Y + H i Y = J Y J Y + H i Y = J Y + J J H i. H i J H i (6) Therefore we get J σ 3 H i = J Hi (σ 3 X)J (σ σ Y )H i (σ 3 X) X. Since H i () = and from (6) it follows by integrating both sides from σ 3 = to σ 3 = that J Y Y + H i Y Thus we can write (5) as the following (σy ) Y dσ + Then we find V = H i (X) Y J Hi (σ 3 X)J (σ σ Y )H i (σ 3 X) X dσ 3. σ J Hi (σ 3 X)J (σ σ Y )H i (σ 3 X) X dσ dσ dσ 3. σ {µi J Hi (σ 3 X)J (σ σ Y )}H i (σ 3 X) X dσ dσ dσ 3 σ σ 3 {µi J Hi (σ 3 X)J (σ σ Y )}J Hi (σ 3 σ 4 X) X dσ dσ dσ 3 dσ 4. IJNS homepage:

7 86 International Journal of Nonlinear ScienceVol.6(8)No.3pp. 8-9 Now we let M = {µi J Hi (σ 3 X)J (σ σ Y )}J Hi (σ 3 σ 4 X). Since λ i (J ) a 3 λ i (J ) a 3 and λ i (J Hi ) b i it follows that λ i (J J H i ) a 3 b i. Thus we have λ i (µi J J H i ) = λ i (µi) λ i (J J H i ) µ b i. a 3 Since µ = a a 3 + n b i a 3 > therefore from condition (iv) we obtain Then from Lemma 5 we get λ i (M) (µ a 3 V 4 n b i )c i = (a a 3 b i )a 3 c i =: α i >. σ σ 3 α i X X dσ dσ dσ 3 dσ 4. Therefore from the estimate of V we have V α i X + Z + µy + µ(a µ) Y. But from the definition of µ and the condition (iv) in Theorem 3 we find a µ = a a a 3 + n b i = a n a 3 b i > a 3 a 3 then there exists a positive constant D such that V (X t Y t Z t ) D ( X + Y + Z ). (7) From (7) there exists a continuous function W ( ϕ() ) > such that W ( ϕ() ) V (ϕ). Now we shall prove the existence of a continuous function W ( ϕ ) satisfying the inequality V (ϕ) W ( ϕ ). By using the hypotheses of Theorem 3 we find F (X Y ) a n; by (i). Since σ H i(σx) = J Hi (σx)x and H i () = then from by (iii) we have H i (X) = Furthermore since σ (σy ) = J (σy )Y But δ i τ i t+s Y (θ) dθds = J Hi (σx)xdσ (Y ) J Hi (σx) X dσ n b i X. and () = therefore from condition (ii) we get δ i J (σy ) Y dσ n a 4 Y. τ i (θ t + τ i ) Y (θ) dθ δ i Y τi = δ i τ i Y dθ. Then by substituting in (4) and by using the inequality u v ( u + v ) there exists a positive constant D satisfying V (X t Y t Z t ) D ( X + Y + Z ). (8) From (4) (3) and by using a basic calculation of Lemma 6 we have the time derivative of the functional V = V (X t Y t Z t ) along the solution of (3) as the following V (3) (X t Y t Z t ) J Hi (X)Y Y + µ Z Z µ (Y Y ) Z F (X Y )Z + µy + Z J Hi (X(s))Y (s)ds + (δ i τ i )Y Y δ i Y (θ) dθ. (9) IJNS for contribution: editor@nonlinearscience.org.uk

8 A. Ademola and A. Mahmoud: On Some Qualitiative Properties of Solutions to Certain Third Order Vector 87 On applying the hypotheses of Theorem 3 λ i (F (X Y )) a λ i (J (Y )) a 3 λ i (J Hi (X)) b i and the inequality u v ( u + v ) it follows that Z F (X Y )Z a Z (Y ) Y a 3 Y By similar we get Y J Hi (X(s))Y (s)ds Y J Hi (X(s))Y (s)ds n b i Y Y (s) ds t n bi ( Y (t) + Y (s) )ds n bi τ i Y + n bi Y (s) ds. Z J Hi (X(s))Y (s)ds n bi τ i Z + t n bi Y (s) ds. The preceding inequalities lead to (9) as the following V (3) (X t Y t Z t ) ( Let δ i = (µ + ) n b i. Then we get { V (3) µa 3 Let τ = max τ i hence we obtain { V (3) µa 3 Now the last inequality implies b i ) Y + µ Z µa 3 Y a Z + + µ n (b i τ i ) Y + n n (b i τ i ) Z (δ i n bi µ n b i ) b i (µ + ) n for some positive constant α provided that b i τ(µ + ) n (δ i τ i ) Y Y (θ) dθ. ( b i τ i } Y a µ n n b i τ i ) Z. ( b i } Y a µ τ n b i ) Z. V (3)(X t Y t Z t ) α( Y + Z ) () { µa3 n τ < min b i (µ + ) n n b i } a µ n n b. i Therefore from (7) (8) and () all the assumptions of Theorem are satisfied so the zero solution of () with P (t) = is uniformly stable. Thus this completes the proof of Theorem 3. IJNS homepage:

9 88 International Journal of Nonlinear ScienceVol.6(8)No.3pp Proof of theorem 4 Next we reconsider the functional V (X t Y t Z t ) defined in (4) using the hypotheses of Theorem 3 and (7) we find that V (X t Y t Z t ) D ( X + Y + Z ) D ( Y + Z ). () Since P (t) then the time derivative of the functional V (X t Y t Z t ) satisfies the following by using (9) and () Since P (t) Q(t) we have V (X t Y t Z t ) α( Y + Z ) + µy + Z P (t). V (X t Y t Z t ) µy + Z P (t) (µ Y + Z )Q(t). Using the following inequalities Y + Y Z + Z and let D 3 = max{µ } it follows that V (X t Y t Z t ) D 3 ( + Y + + Z )Q(t) = D 3 Q(t) + D 3 ( Y + Z )Q(t). Therefore from () we obtain V (X t Y t Z t ) D 3 Q(t) + D 3 D Q(t)V (x t y t z t ) = D 3 Q(t) + D 4 Q(t)V (X t Y t Z t ) () where D 4 = D 3 D. Integrating both sides of the inequality () with respect to s from to t it follows that V (X t Y t Z t ) V (X Y Z ) D 3 Q(s)ds + D 4 V (X s Y s Z s )Q(s)ds. Let D 5 = D 3 Q(s)ds + V (X Y Z ). Therefore we have By using ronwall-bellman inequality we get V (X t Y t Z t ) D 5 + D 4 V (X s Y s Z s )Q(s)ds. In view of the above discussion thus V (X t Y t Z t ) D 5 exp( Q(s)ds). D ( X + Y + Z ) V (X t Y t Z t ) D 5 exp( Let Q(s)ds = K > since Q L ( ). Hence X + Y + Z D 5 D exp(d 4 K ). Q(s)ds). Then X + Y + Z K where K = D 5 D exp(d 4 K ). Hence all solutions of () are uniformly bounded. This completes the proof of Theorem 4. IJNS for contribution: editor@nonlinearscience.org.uk

10 A. Ademola and A. Mahmoud: On Some Qualitiative Properties of Solutions to Certain Third Order Vector 5 89 Example In this section we provide an example to illustrate the application of the results we obtained in the previous sections. As a special case of () for the case n = let us choose F H and P that appeared in () as follows: x +y 8 + +exp(x +y ) F (X Y ) = x +y 8 + +exp(x +y ) it follows that λ (F (X Y )) = 8 + x + y x + y λ (F (X Y )) = exp(x + y ) + exp(x + y ) Then we get λi (F (X Y )) 8 = a >. Different view of the function 8 + Figure : Different Views of the function 8 + can choose x +y +exp(x +y ) x +y +exp(x +y ) F (X Y ). [ ] y + tan y (Y ) = y + tan y The path of the function y + arctan y is shown in Fig.. Therefore we find [ ] + +y J (Y ) = + +y is shown in Fig.3. Also choose + y [ ] x (t τ ) + tan x (t τ ) H (X(t τ )) = x (t τ ) + tan x (t τ ) λi (J (Y )) = a3 >. The behaviour of + [ ] x (t τ ) + tan x (t τ ) H (X(t τ )) =. x (t τ ) + tan x (t τ ) It follows that JH (X(t τ )) = [ + +x (t τ ) + +x (t τ ) IJNS homepage: are shown in Fig.. Also we ]

11 9 International Journal of Nonlinear ScienceVol.6(8)No.3pp. 8-9 J H (X(t τ )) = λ i (J Hi (X)). Therefore we obtain c i = and b i =. Let it tends to and P (t) = [ ] + +x (t τ ) + +x (t τ ) P (t) = sin t +t cos t +t Q(s)ds = [ sin t +t cos t +t ] + t = Q(t) ds = π + s that is Q L ( ). The behaviour of the function P (t) and the bounds +t on P (t) are shown in Fig.4. Figure : The behaviour of the function y + arctan y (Y ). Figure 4: The behaviour of the function P (t) and the bounds Figure 3: The behaviour of the function + +t +y J (Y ).. on P (t) We can see that Thus from the above estimates we get µ = a a 3 + b i a 3 = 5 >. µa 3 b i (µ + ) n b = 3 i a µ n b = 3 i 4. Thus all conditions of Theorem 3 and Theorem 4 hold. This shows that the zero solution of () is uniformly stable and all solutions of the same equation are uniformly bounded for the special case of () when n = provided that 6 Conclusion 3 τ < min{ 3 4 } = 3. A vector differential equation of third-order with multiple constant deviating arguments is considered. Sufficient conditions are established for the uniformly stable of the solutions for P (t) = and also the boundedness of solutions of this equation. In the proofs of the main results we employ the Lyapunov functional. An example is also constructed to illustrate our theoretical analysis. IJNS for contribution: editor@nonlinearscience.org.uk

12 A. Ademola and A. Mahmoud: On Some Qualitiative Properties of Solutions to Certain Third Order Vector 9 Acknowledgments The authors would like to thank the referee(s) for the several helpful remarks and suggestions. References [] T. A. Burton. Volterra Integral and Differential Equations. Academic Press New York 983. [] T. A. Burton. Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press 985. [3] R. D. Driver. Ordinary and Delay Differential Equations. Springer-Verlag New York 977. [4] J. K. Hale. Theory of Functional Differential Equations. Springer-Verlag New York 977. [5] V. Kolmanovskii and A. Myshkis. Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers Dordrecht 999. [6] A. M. A. Abou-El-Ela A. I. Sadek and A. M. Mahmoud. Stability and boundedness of solutions of a certain thirdorder nonlinear delay differential equation. ICST-ACSE Journal 9(9)(): 9-5. [7] A. T. Ademola and P. O. Arawomo. Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third-order. Math. J. Okayama Univ. 55(3): [8] A. T. Ademola P. O. Arawomo M. O. Ogunlaran and E. A. Oyekan. Uniform stability boundedness and asymptotic behaviour of solutions of some third-order nonlinear delay differential equations. Differential Equations and Control Processes 4(3): [9] 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. (8)(): [] A. M. Mahmoud. On the asymptotic stability of solutions for a certain non-autonomous third-order delay differential equation. British Journal of Mathematics and Computer Science 6(6)(3): -. [] M. O. Omeike. Stability and boundedness of solutions of some non-autonomous delay differential equations of the third-order. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S) 55(9)(): [] M. O. Omeike. New results on the stability of solutions of some non-autonomous delay differential equations of the third-order. Differntial Equations and Control Processes (): 8-9. [3] A. I. Sadek. Stability and boundedness of a kind of third-order delay differential system. Appl. Math. Lett. 6(3)(5): [4] A. I. Sadek. On the stability of solutions of some non-autonomous delay differential equations of the third-order. Asymptot. Anal. 43(5): -7. [5] P. Shekhar V. Dharmaiah and. Mahadevi. Stability and boundedness of solutions of delay differential equations of third-order. IOSR J. Math. 5(3)(6): 9-3. [6] H. O. Tẹjumọla and B. Tchegnani. Stability boundedness and existence of periodic solutions of some third and fourth-order nonlinear delay differential equations. J. Nigerian Math. Soc. 9(): 9-9. [7] C. Tunç. New results about stability and boundedness of solutions of certain non-linear third-order delay differential equations. Arab. J. Sci. Eng. 3(6)(A): [8] C. Tunç. Stability and boundedness of solutions of nonlinear differential equations of third-order with delay. Differential Equations and Control Processes 3(7): -3. [9] C. Tunç. On the stability of solutions for non-autonomous delay differential equations of third-order. Iran J. Sci. Technol. A 3(8)(A4): [] C. Tunç. On the stability and boundedness of solutions to third-order nonlinear differential equations with retarded argument. Nonlinear Dynam. 57(9)(-): [] C. Tunç. Stability and bounded of solutions to non-autonomous delay differential equations of third-order. Nonlinear Dynam. 6()(4): [] J. O. C. Ezeilo and H. O. Tejumola. Boundedness and periodicty of solutions of a certain system of third-order non-linear differential equations. Annali di Matematica Pura ed Applicata. 74(966)(4): [3] A. U. Afuwape. Ultimate boundedness results for a certain system of third-order nonlinear differential equations. J. Math. Anal. App. 97(983)(): 4-5. [4] F. W. Meng. Ultimate boundedness results for a certain system of third order nonlinear differential equations. J. Math. Anal. Appl. 77(993)(): IJNS homepage:

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