In this note we study the continuability of the solutions of a Liénard type equation with forcing term under suitable assumptions.Fil: Napoles Valdes, Juan Eduardo. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura. Departamento de Matemática; ArgentinaFil: Lugo, Luciano Miguel. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura. Departamento de Matemática; ArgentinaFil: Guzmán, Paulo Matias. Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Guzmán, Paulo Matias

Lugo, Luciano Miguel

Napoles Valdes, Juan Eduardo

English

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International Journal of Mathematics And its ApplicationsVolume 3, Issue 4–E (2015), 15–21.ISSN: 2347-1557Available Online: http://ijmaa.in/Applications•ISSN:2347-1557•InternationalJournal of MathematicsAnditsInternational Journal of Mathematics And its ApplicationsA Note on the Continuability of Solutions of a PerturbedSecond Order Nonlinear Differential Equation of LiénardTypeResearch ArticleJuan E. Nápoles Valdes1∗, Luciano M. Lugo Motta Bittencurt2 and Paulo M. Guzmán21 UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina and UTN, FRRE, French 414, (3500) Resistencia, Argentina.2 UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina.Abstract: In this note we study the continuability of the solutions of a Liénard type equation with forcing term under suitableassumptions.MSC: 34C11.Keywords: Continuability, second order, non-autonomous.c© JS Publication.1. IntroductionIn this note we consider the equationx′′ + f(x)x′ + a(t)g(x)h(x′) = b(t, x, x′), (1)where a : I → R+, b : I× R2 → R, f : R → R+, g : R → R and h : R → R+ are continuous functions in their argumentsand the function a ∈ C1 satisfying 0 < a ≤ a(t) < +∞, with R := (−∞,+∞), R+ := (0,+∞) and I := [t0,+∞) it is notexcluded t0 ≡ 0.We shall determine sufficient conditions for continuability of solutions of equation (1). Also we include some remarks andexamples to illustrate our results, which improve some of the previously obtained results, some preliminary work in thisdirection are [7], [16] and [30] and references cited therein.The solutions of equation (1) are bounded if there exists a constant k > 0 such that |x(t)| < k for all t≥t0 > 0 for some t0.By continuable we mean a solution which is defined on a half line I.In the last forty decades, many authors have investigated the Liénard equationx′′ + f(x)x′ + g(x) = 0. (2)They have examined some qualitative properties of the solutions. The book of Sansone and Conti [30] contains an almostcomplete list of papers dealing with these equation as well as a summary of the results published up to 1960. The book of∗ E-mail: jnapoles@exa.unne.edu.ar15A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Liénard TypeReissig, Sansone and Conti [25] updates this list and summary up to 1962. The list of the papers which appeared between1960 and 1970 is presented in the paper of John R. Graef [8]. Among the papers which were published in the last years werefer to the following ones [2–5, 10–14, 18, 19, 23, 25–27, 31–38].If in (1) we make a(t)≡1, b(t, x, x′)≡0, and h(x′) = 1, it is clear that equation (1) becomes equation (2) so, every qualitativeresult for the equation (1) produces a qualitative result for (2).2. ResultsLet F (x) =∫ x0f(s)ds, G(x) =∫ x0g(s)ds and H(y) =∫ y0rh(r)dr. Our main assumptions are:(i) xg(x) > 0, x 6= 0(ii) There exists a continuous function u : I → R such that |b(t, x, x́)| ≤ u(t), u ∈ L1(t0,+∞).(iii) There exists non negative constant M such that |y|h(y)≤ MH(y) for |y| ≥ 1 and Lim|y|→∞H(y) = ∞. We write theequation (1) as the systemx′ = y,y′ = −f(x)y − a(t)g(x)h(y) + b(t, x, y).(3)For any function M : I→ R we let M(t)+ = max {M(t), 0} and M(t)− = max {−M(t), 0} so that M(t) = M(t)+−M(t)−.We first give a continuability result for (1).Theorem 2.1. Under the conditions i)-iii) above, any solution of equation of (1) is continuable to the right of its initialt-value, i.e. all solutions of (1) can be defined for all t.Proof. Let x(t) be a solution of (1) with initial t-value t0 in I. Suppose, on the contrary, that x(t) can not be continuedpast the finite time T > t0, T ∈ I. It suffices to show that x(t) remains bounded as t approaches T from the left. We defineV (t, x, y) = G(x) +H(y) + C,where C is a non negative constant. From definitions of G(x) and H(y) it follows thatV (t, 0, 0) = 0 and V (t, x, y) > 0 for all x, y 6= 0.Calculating the time derivative of Liapunov function V (t, x, y), we findV (t, x, y) = − 1a(t)y2h(y)f(x) +b(t, x, x́)a(t)yh(y)− á(t)a(t)H(y).Applying the assumptions i)-iii) we haveV (t, x, y) ≤ MB(t)aH(y) +a′+(t)aH(y) =[MB(t)a+a′+(t)a]H(y).Thus, we haveV (t, x, y) ≤ m(t)H(y),wherem(t) =MB(t)a(t)+a′+(t)a(t).16Juan E.Nápoles Valdes, L.M. Lugo Motta Bittencurt and P.M.GuzmánNow let K =∫ Tt0m(t)dt for some positive constant K, so by Gronwall’s inequality we haveH(y) ≤ V (t, x, y) ≤ [V (t0, x0, y0) +K] exp∫ tt0m(s)ds (4)Hence, we see H(y) remains bounded as t→ T− and so y is bounded on [t0, T ]. An integration of first equation of (3) showsthat x(t) is also bounded on [t0, T ] contradicting the assumption that x(t) is a solution of (1) with finite escape time. Thiscompletes the proof.Remark 2.2. It is clear that this result can be written in terms of the system (3).Remark 2.3. To obtain another boundedness result for solutions of (1) we impose additional conditions on∫ ∞t0B(t)a(t)dt, (5)and ∫ ∞t0a′+(t)a(t)dt, (6)and not use the boundedness of a(t).Remark 2.4. Notice that (6) implies that a(t) is bounded from above.Now we will prove a boundedness theorem for solution (x(t), y(t)) of system (3) by using a modification of the technique ofthe previous result. In addition to the given assumptions we suppose that:i´) f(x) ≥ f0 > 0 for all x ∈ R, xg(x) > 0, x 6= 0 and Lim|y|→∞G(x) =∞.ii´) y2h(y)≤MH(y) +N1, |y|h(y) ≤MH(y) +N2 for all y ∈ R.iii´) There are continuous functions ri : I → I, i = 1, 2 such that |b(t, x, y)| ≤ r1(t) + r2(t) |y| for all (t, x, y) ∈ IxR2.Theorem 2.5. If assumptions i´)-iii´) hold then for each solution x(t) of equation (1), with initial t-value t0 ∈ I, x́(t) isbounded on I, if in addition Lim|y|→∞H(y) =∞ holds then x(t) is bounded also on I.Proof. Let x(t) be a continuable solution of (1) with initial t-value t0. Multiplying equation (1) byx́(t)h(x́(t))and integratingon [t0, t] ⊂ [t0, T ), we getH(y)−H(y0) +∫ tt0[(f(x(s))x́(s))2 +(x́(s)h(x́(s)))2]dt++a [G(x)−G(x0)] ≤∫ tt0b(s, x(s), x́(s))x́(s)h(x́(s))dt.Using assumptions i ´)-iii ´) the above inequality can be written in the formH(y)−H(y0) + a [G(x)−G(x0)] ≤(7)≤∫ tt0[M(−f0 + r2) +Mr1]H(y(s))dt+∫ tt0[−f0N2 + r1N1 + r2N2] dt.define V as17A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Liénard TypeV (t, x, y) = aG(x(t)) +H(y(t))the inequality (7) takes the formV (t, x, y) ≤ V (t0) +m0 +∫ tt0M(s)V (s, x(s), y(s))dt.where m0 =∫ tt0[−f0N2 + r1N1 + r2N2] dt and M(t) = [M(−f0 + r2) +Mr1]. By using the Gronwall-Bellman inequalitythere is a constant C, depending on r1(t) and r2(t) but not on V (t, x(t), y(t)) such thatV (t, x(t), y(t)) ≤ (V (t0) +m0)C.From this it follow that G(x(t)) is bounded for t ≥ t0. The boundedness of x(t) follows from i ´) and the boundedness ofx́(t) is easily obtained from the first equation (3).Remark 2.6. It is very easy to give examples of equations in which the conclusions of Theorem are not valid, breached someof the conditions of the theorem. So we have that if f ≡ 0 and iii´) is violated, the equation x́ + e2tx = x́ has the boundedsolution x(t) = sin(et), with an unbounded derivative. Under assumptions f(x) ≥ f0 > 0 for some positive constant f0,the class of equation (1) is not very large, but if this condition is not fulfilled, we can exhibit equations that have unboundedsolutions, for example x́− xx́+ 2x = 2x(3− x) has the unbounded solution x(t) = e2t.Remark 2.7. If, instead of the assumptions iií) one assumes r1(t) and r2(t) integrable on I,the conclusion of Theorem 2.5is false, as examplex́+x2 + 2x2 + 1x́+x1 + x2=(t+ 1 + 2t)x1 + t2which has x = t as a solution.From Theorem 2.1 and Theorem 2.5, we have the following result.Corollary 2.8. If, in addition to assumptions of Theorem 2 we consider that H(y)→∞ as |y| → ∞, then all solutions of(1) are bounded.3. Conclusion.In this section we present some observations that demonstrate the prominence of our results on several well-known resultsof the literature.Remark 3.1. In [23] the authors consider the following second order non-autonomous and non-linear differential equationx” + a(t)g(x,x’)x’ + b(t)h(x) = p(t,x,x’); where a, b, g, h and p are continuous functions. For the cases p(t,x,x’)=0 andp(t,x,x’)6= 0, respectively, the authors establish sucents conditions under which the zero solution is globally asymptoticallystable and every solution and its derivative are bounded. It is clear that these results are no contradicting with our results.Remark 3.2. Tunç studied the boundedness of the solutions of equation ((r(t)x’)’ + φ(t, x, x’)x’ + p(t)f(x) = q(t, x, x’);under assumption φ(t; x; x’)≥ 0 for all (t; x; x’)∈IxR2. It is clear that theabove remarks still valid.18Juan E.Nápoles Valdes, L.M. Lugo Motta Bittencurt and P.M.GuzmánRemark 3.3. Our results are consistent with those of Baker of [1], because him studied the (p(t)x’)’ + φ(t, x, x’)x’ +p(t)f(x) = 0 with φ(t, x, x’)≥0.Remark 3.4. Our results cover the Theorems I and III of [16] refer to the equation y” + c(t)f(y)y’ + a(t)b(y) = 0 obtainedfrom (1) if f(t, x, x’) = c(t)f(y) and h(x’)≡1. Kroopnick obtainedthe above results considering c(t) 0 and f(y) > 0. This remark is valid in the case of equation y” + c(t)f(y)y’ + a(t)b(y) =e(t); studied in [17] under the same assumptions on c and f.Remark 3.5. In [9] the authors obtained qualitative results of the solutions of equation (a(t)x’)’ + h(t, x, x’) + q(t)f(x)g(x’)= e(t, x, x’) when h(t, x, x’) = f(t, x, x’)x’, a(t) = p(t), f(x) = g(x) and g(x’) = h(x’), the Theorems 2.1 and 2.5 andCorollary 2.8 of this paper have contact points with our results under milder conditions on function e(t, x, x’).Remark 3.6. A. Castro and R. Alonso [6] considered the special casex́+ h(t)x́+ x = 0; (8)of equation (1) under condition h∈C1(I) and h(t)≥ b > 0. Further, they required that the condition ah´(t) + 2h(t) = 4a befulfilled, and obtained various results on the stability of the trivial solution of (17). Taking into account the previous resultswe can obtain the stability of trivial solution of (17) considering instead of (7) and (11) C1 ≤ 1 and∞∫t0w(t)dt < 1.Remark 3.7. Other results obtained for special cases of the equation (1) and which are improved by foregoing theorems andcorollary are [12, 18, 21, 22, 26–29, 32].References[1] J.T.Baker, Stability properties of a second order damped and forced nonlinear differential equation, SIAM J. Appl. Math.,27(1974), 159-166.[2] N.Boudonov, Qualitative theory of ordinary differential equations, Universidad de la Habana, Cuba, no dated (Spanish).[3] T.A.Burton, The generalized Liénard equation, SIAM J. Control, Ser A#(1965), 223-230.[4] T.A.Burton and R.Grimmer, On the asymptotic behaviour of solutions of x” + a(t)f(x) = 0, Proc. Camb. Phil. Soc.,70(1971), 77-88.[5] T.A.Burton and C.G.Townsend, On the generalized Liénard equation with forcing term, J. Differential Equations,4(1968), 620-633.[6] A.Castro and R.Alonso, Variants of two Salvadori’s results on asymptotic stability, Revista Ciencias Matemáticas,VIII(1987), 45-53.[7] H.El-Owaidy and A.S.Zagrout, A note on the second order nonlinear differential equations, Indian J. Pure Appl. Math.,15(4)(1984), 441-444.[8] J.R.Graef, On the generalized Liénard equation with negative damping, J. Differential Equations, 12(1972), 34-62.[9] J.R.Graef and Paul W. Spikes, Continuability, boundedness and convergence to zero of solutions of a perturbed nonlineardifferential equation, Czechoslovak Math. J., 45(120)(1995), 663-683.[10] John R.Graef and C.Tunc, Continuability and boundedness of multi-delay functional integro-differential equations of thesecond order, Rev. R. Acad. Cienc. Exactas F́ıs. Nat. Ser. A Math. RACSAM, 109(1)(2015), 169-173.[11] T.Hara and T.Yoneyama, On the global center of generalized Liénard equation and its application to stability problems,Funkcial. Ekvac., 28(1985), 171-192.19A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Liénard Type[12] Y.Q.Hasan and L.M.Zhu, The bounded solutions of Liénard equation, J. Applied Sci., 7(8(2007), 1239-1240.[13] D.Hricisakova, Existence of positive solutions of the Liénard differential equation, J. Math. Anal. And Appl., 176(1993),545-553.[14] A.O.Ignatiev, Stability of a linear oscillator with variable parameters, Electronic J. Differential Equations, 17(1997),1-6.[15] J.Kato, Boundedness theorems on Liénard type differential equations with damping, J. Northeast Normal University,2(1988), 1-35.[16] A.Kroopnick, Note on bounded Lp-solutions of a generalized Liènard equation, Pacific J. Math., 94(1)(1981), 171-175.[17] A.Kroopnick, Properties of solutions to a generalized Liénard equation with forcing term, Applied Math. E-Notes,8(2008), 40-44.[18] F.Nakajima, Ultimate boundedness of solutions for a generalized Liénard equation with forcing term, Tôhoku Math. J.,46(1994), 295-310.[19] J.E.Nápoles, On the global stability of non-autonomous systems, Revista Colombiana de Matemáticas, 33(1999), 1-8.[20] J.E.Nápoles, On the boundedness and global asymptotic stability of the Liénard equation with restoring term, Revistade la UMA, 41(2000), 47-59.[21] J.E.Nápoles, A note on the qualitative behavior of some second order nonlinear differential equations, DivulgacionesMatemáticas, 10(2)(2002), 91-99.[22] J.E.Nápoles and C.Negrón, On the behaviour of solutions of second order linear differential equations, Revista Inte-gración, 14(1996), 49-56.[23] B.S.Ogundare, S.Ngcibi and V.Murali, Boundedness and stability properties of solutions to certain second-order differ-ential equation, Adv. Differ. Equ. Control Process, 5(2)(2010), 79-92.[24] P.Omari, G.Villari and F.Zanoĺın, Periodic solutions of the Liénard equation with one-side growth restrictions, J. Dif-ferential Equations, 67(1987), 278-293.[25] R.Reissig, G.Sansone and R.Conti, Qualitative theorie nichtlinearer differentialgleichungen, Edizioni Cremonese, Rome,(1963).[26] J.A.Repilado and A.I.Ruiz, On the behaviour of solutions of differential equation x´´ + g(x)x´ + a(t)f(x) = 0-(I),Revista Ciencias Matemáticas, VI(1985), 65-71.[27] J.A.Repilado and A.I.Ruiz, On the behaviour of solutions of differential equation x´´ + g(x)x´ + a(t)f(x) = 0-(II),Revista Ciencias Matemáticas, VII(1986), 35-39.[28] A.I.Ruiz, Behaviour of trajectories of nonautonomous systems of differential equations, Universidad de Oriente, Cuba,(1988).[29] Shao, Jing and Wei Song, Limit circle/limit point criteria for second order sublinear differential equations with dampingterm, Abstract and Applied Analysis, 2011(2011), Article ID 803137.[30] G.Sansone and R.Conti, Nonlinear differential equations, MacMillan, New York, (1964).[31] C.Tunc, A note on boundedness of solutions to a class of non-autonomous differential equations of second order, Appl.Anal. Discrete Math., 4(2)(2010), 361-372.[32] C.Tunc, On the boundedness of solutions of a non-autonomous differential equation of second order, Sarajevo J. Math.,7(19)(2011), 19-29.[33] C.Tunc, Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations, Appl.Comput. Math., 10(3)(2011), 449-462.[34] C.Tunc, On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order20Juan E.Nápoles Valdes, L.M. Lugo Motta Bittencurt and P.M.Guzmánwith multiple deviating arguments, Afr. Mat., 23(2)(2012), 249–259.[35] C.Tunc and O.Tunc, A note on certain qualitative properties of a second order linear differential system, Appl. Math.Inf. Sci., 9(2)(2015), 953-956.[36] W.R.Utz, Properties of solutions of certain second order nonlinear differential equations, Proc. Amer. Math. Soc.,8(1957), 1024-1028.[37] G.Villari, On the existence of periodic solutions for Liénard’s equation, Nonlin. Anal., 7(1983), 71-78.[38] G.Villari, On the qualitative behaviour of solutions of Liénard equation, J. Differential Equation, 67(1987), 269-277.21

A note on the continuability of solutions of a perturbed second order nonlinear differential equation of lienard type

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International Journal of Mathematics And its ApplicationsVolume 3, Issue 4–E (2015), 15–21.ISSN: 2347-1557Available Online: http://ijmaa.in/Applications•ISSN:2347-1557•InternationalJournal of MathematicsAnditsInternational Journal of Mathematics And its ApplicationsA Note on the Continuability of Solutions of a PerturbedSecond Order Nonlinear Differential Equation of Lie´nardTypeResearch ArticleJuan E. Na´poles Valdes1∗, Luciano M. Lugo Motta Bittencurt2 and Paulo M. Guzma´n21 UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina and UTN, FRRE, French 414, (3500) Resistencia, Argentina.2 UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina.Abstract: In this note we study the continuability of the solutions of a Lie´nard type equation with forcing term under suitableassumptions.MSC: 34C11.Keywords: Continuability, second order, non-autonomous.c© JS Publication.1. IntroductionIn this note we consider the equationx′′ + f(x)x′ + a(t)g(x)h(x′) = b(t, x, x′), (1)where a : I → R+, b : I× R2 → R, f : R → R+, g : R → R and h : R → R+ are continuous functions in their argumentsand the function a ∈ C1 satisfying 0 < a ≤ a(t) < +∞, with R := (−∞,+∞), R+ := (0,+∞) and I := [t0,+∞) it is notexcluded t0 ≡ 0.We shall determine sufficient conditions for continuability of solutions of equation (1). Also we include some remarks andexamples to illustrate our results, which improve some of the previously obtained results, some preliminary work in thisdirection are [7], [16] and [30] and references cited therein.The solutions of equation (1) are bounded if there exists a constant k > 0 such that |x(t)| < k for all t≥t0 > 0 for some t0.By continuable we mean a solution which is defined on a half line I.In the last forty decades, many authors have investigated the Lie´nard equationx′′ + f(x)x′ + g(x) = 0. (2)They have examined some qualitative properties of the solutions. The book of Sansone and Conti [30] contains an almostcomplete list of papers dealing with these equation as well as a summary of the results published up to 1960. The book of∗ E-mail: jnapoles@exa.unne.edu.ar15A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Lie´nard TypeReissig, Sansone and Conti [25] updates this list and summary up to 1962. The list of the papers which appeared between1960 and 1970 is presented in the paper of John R. Graef [8]. Among the papers which were published in the last years werefer to the following ones [2–5, 10–14, 18, 19, 23, 25–27, 31–38].If in (1) we make a(t)≡1, b(t, x, x′)≡0, and h(x′) = 1, it is clear that equation (1) becomes equation (2) so, every qualitativeresult for the equation (1) produces a qualitative result for (2).2. ResultsLet F (x) =∫ x0f(s)ds, G(x) =∫ x0g(s)ds and H(y) =∫ y0rh(r)dr. Our main assumptions are:(i) xg(x) > 0, x 6= 0(ii) There exists a continuous function u : I → R such that |b(t, x, x´)| ≤ u(t), u ∈ L1(t0,+∞).(iii) There exists non negative constant M such that |y|h(y)≤ MH(y) for |y| ≥ 1 and Lim|y|→∞H(y) = ∞. We write theequation (1) as the systemx′ = y,y′ = −f(x)y − a(t)g(x)h(y) + b(t, x, y).(3)For any function M : I→ R we let M(t)+ = max {M(t), 0} and M(t)− = max {−M(t), 0} so that M(t) = M(t)+−M(t)−.We first give a continuability result for (1).Theorem 2.1. Under the conditions i)-iii) above, any solution of equation of (1) is continuable to the right of its initialt-value, i.e. all solutions of (1) can be defined for all t.Proof. Let x(t) be a solution of (1) with initial t-value t0 in I. Suppose, on the contrary, that x(t) can not be continuedpast the finite time T > t0, T ∈ I. It suffices to show that x(t) remains bounded as t approaches T from the left. We defineV (t, x, y) = G(x) +H(y) + C,where C is a non negative constant. From definitions of G(x) and H(y) it follows thatV (t, 0, 0) = 0 and V (t, x, y) > 0 for all x, y 6= 0.Calculating the time derivative of Liapunov function V (t, x, y), we findV (t, x, y) = − 1a(t)y2h(y)f(x) +b(t, x, x´)a(t)yh(y)− a´(t)a(t)H(y).Applying the assumptions i)-iii) we haveV (t, x, y) ≤ MB(t)aH(y) +a′+(t)aH(y) =[MB(t)a+a′+(t)a]H(y).Thus, we haveV (t, x, y) ≤ m(t)H(y),wherem(t) =MB(t)a(t)+a′+(t)a(t).16Juan E.Na´poles Valdes, L.M. Lugo Motta Bittencurt and P.M.Guzma´nNow let K =∫ Tt0m(t)dt for some positive constant K, so by Gronwall’s inequality we haveH(y) ≤ V (t, x, y) ≤ [V (t0, x0, y0) +K] exp∫ tt0m(s)ds (4)Hence, we see H(y) remains bounded as t→ T− and so y is bounded on [t0, T ]. An integration of first equation of (3) showsthat x(t) is also bounded on [t0, T ] contradicting the assumption that x(t) is a solution of (1) with finite escape time. Thiscompletes the proof.Remark 2.2. It is clear that this result can be written in terms of the system (3).Remark 2.3. To obtain another boundedness result for solutions of (1) we impose additional conditions on∫ ∞t0B(t)a(t)dt, (5)and ∫ ∞t0a′+(t)a(t)dt, (6)and not use the boundedness of a(t).Remark 2.4. Notice that (6) implies that a(t) is bounded from above.Now we will prove a boundedness theorem for solution (x(t), y(t)) of system (3) by using a modification of the technique ofthe previous result. In addition to the given assumptions we suppose that:i´) f(x) ≥ f0 > 0 for all x ∈ R, xg(x) > 0, x 6= 0 and Lim|y|→∞ G(x) =∞.ii´) y2h(y)≤MH(y) +N1, |y|h(y) ≤MH(y) +N2 for all y ∈ R.iii´) There are continuous functions ri : I → I, i = 1, 2 such that |b(t, x, y)| ≤ r1(t) + r2(t) |y| for all (t, x, y) ∈ IxR2.Theorem 2.5. If assumptions i´)-iii´) hold then for each solution x(t) of equation (1), with initial t-value t0 ∈ I, x´(t) isbounded on I, if in addition Lim|y|→∞H(y) =∞ holds then x(t) is bounded also on I.Proof. Let x(t) be a continuable solution of (1) with initial t-value t0. Multiplying equation (1) byx´(t)h(x´(t))and integratingon [t0, t] ⊂ [t0, T ), we getH(y)−H(y0) +∫ tt0[(f(x(s))x´(s))2 +(x´(s)h(x´(s)))2]dt++a [G(x)−G(x0)] ≤∫ tt0b(s, x(s), x´(s))x´(s)h(x´(s))dt.Using assumptions i ´)-iii ´) the above inequality can be written in the formH(y)−H(y0) + a [G(x)−G(x0)] ≤(7)≤∫ tt0[M(−f0 + r2) +Mr1]H(y(s))dt+∫ tt0[−f0N2 + r1N1 + r2N2] dt.define V as17A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Lie´nard TypeV (t, x, y) = aG(x(t)) +H(y(t))the inequality (7) takes the formV (t, x, y) ≤ V (t0) +m0 +∫ tt0M(s)V (s, x(s), y(s))dt.where m0 =∫ tt0[−f0N2 + r1N1 + r2N2] dt and M(t) = [M(−f0 + r2) +Mr1]. By using the Gronwall-Bellman inequalitythere is a constant C, depending on r1(t) and r2(t) but not on V (t, x(t), y(t)) such thatV (t, x(t), y(t)) ≤ (V (t0) +m0)C.From this it follow that G(x(t)) is bounded for t ≥ t0. The boundedness of x(t) follows from i ´) and the boundedness ofx´(t) is easily obtained from the first equation (3).Remark 2.6. It is very easy to give examples of equations in which the conclusions of Theorem are not valid, breached someof the conditions of the theorem. So we have that if f ≡ 0 and iii´) is violated, the equation x´ + e2tx = x´ has the boundedsolution x(t) = sin(et), with an unbounded derivative. Under assumptions f(x) ≥ f0 > 0 for some positive constant f0,the class of equation (1) is not very large, but if this condition is not fulfilled, we can exhibit equations that have unboundedsolutions, for example x´− xx´+ 2x = 2x(3− x) has the unbounded solution x(t) = e2t.Remark 2.7. If, instead of the assumptions iii´) one assumes r1(t) and r2(t) integrable on I,the conclusion of Theorem 2.5is false, as examplex´+x2 + 2x2 + 1x´+x1 + x2=(t+ 1 + 2t)x1 + t2which has x = t as a solution.From Theorem 2.1 and Theorem 2.5, we have the following result.Corollary 2.8. If, in addition to assumptions of Theorem 2 we consider that H(y)→∞ as |y| → ∞, then all solutions of(1) are bounded.3. Conclusion.In this section we present some observations that demonstrate the prominence of our results on several well-known resultsof the literature.Remark 3.1. In [23] the authors consider the following second order non-autonomous and non-linear differential equationx” + a(t)g(x,x’)x’ + b(t)h(x) = p(t,x,x’); where a, b, g, h and p are continuous functions. For the cases p(t,x,x’)=0 andp(t,x,x’)6= 0, respectively, the authors establish sucents conditions under which the zero solution is globally asymptoticallystable and every solution and its derivative are bounded. It is clear that these results are no contradicting with our results.Remark 3.2. Tunc¸ studied the boundedness of the solutions of equation ((r(t)x’)’ + φ(t, x, x’)x’ + p(t)f(x) = q(t, x, x’);under assumption φ(t; x; x’)≥ 0 for all (t; x; x’)∈IxR2. It is clear that theabove remarks still valid.18Juan E.Na´poles Valdes, L.M. Lugo Motta Bittencurt and P.M.Guzma´nRemark 3.3. Our results are consistent with those of Baker of [1], because him studied the (p(t)x’)’ + φ(t, x, x’)x’ +p(t)f(x) = 0 with φ(t, x, x’)≥0.Remark 3.4. Our results cover the Theorems I and III of [16] refer to the equation y” + c(t)f(y)y’ + a(t)b(y) = 0 obtainedfrom (1) if f(t, x, x’) = c(t)f(y) and h(x’)≡1. Kroopnick obtainedthe above results considering c(t) 0 and f(y) > 0. This remark is valid in the case of equation y” + c(t)f(y)y’ + a(t)b(y) =e(t); studied in [17] under the same assumptions on c and f.Remark 3.5. In [9] the authors obtained qualitative results of the solutions of equation (a(t)x’)’ + h(t, x, x’) + q(t)f(x)g(x’)= e(t, x, x’) when h(t, x, x’) = f(t, x, x’)x’, a(t) = p(t), f(x) = g(x) and g(x’) = h(x’), the Theorems 2.1 and 2.5 andCorollary 2.8 of this paper have contact points with our results under milder conditions on function e(t, x, x’).Remark 3.6. A. Castro and R. Alonso [6] considered the special casex´+ h(t)x´+ x = 0; (8)of equation (1) under condition h∈C1(I) and h(t)≥ b > 0. Further, they required that the condition ah´(t) + 2h(t) = 4a befulfilled, and obtained various results on the stability of the trivial solution of (17). Taking into account the previous resultswe can obtain the stability of trivial solution of (17) considering instead of (7) and (11) C1 ≤ 1 and∞∫t0w(t)dt < 1.Remark 3.7. Other results obtained for special cases of the equation (1) and which are improved by foregoing theorems andcorollary are [12, 18, 21, 22, 26–29, 32].References[1] J.T.Baker, Stability properties of a second order damped and forced nonlinear differential equation, SIAM J. Appl. Math.,27(1974), 159-166.[2] N.Boudonov, Qualitative theory of ordinary differential equations, Universidad de la Habana, Cuba, no dated (Spanish).[3] T.A.Burton, The generalized Lie´nard equation, SIAM J. Control, Ser A#(1965), 223-230.[4] T.A.Burton and R.Grimmer, On the asymptotic behaviour of solutions of x” + a(t)f(x) = 0, Proc. Camb. Phil. Soc.,70(1971), 77-88.[5] T.A.Burton and C.G.Townsend, On the generalized Lie´nard equation with forcing term, J. Differential Equations,4(1968), 620-633.[6] A.Castro and R.Alonso, Variants of two Salvadori’s results on asymptotic stability, Revista Ciencias Matema´ticas,VIII(1987), 45-53.[7] H.El-Owaidy and A.S.Zagrout, A note on the second order nonlinear differential equations, Indian J. Pure Appl. Math.,15(4)(1984), 441-444.[8] J.R.Graef, On the generalized Lie´nard equation with negative damping, J. Differential Equations, 12(1972), 34-62.[9] J.R.Graef and Paul W. Spikes, Continuability, boundedness and convergence to zero of solutions of a perturbed nonlineardifferential equation, Czechoslovak Math. J., 45(120)(1995), 663-683.[10] John R.Graef and C.Tunc, Continuability and boundedness of multi-delay functional integro-differential equations of thesecond order, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Math. RACSAM, 109(1)(2015), 169-173.[11] T.Hara and T.Yoneyama, On the global center of generalized Lie´nard equation and its application to stability problems,Funkcial. Ekvac., 28(1985), 171-192.19A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Lie´nard Type[12] Y.Q.Hasan and L.M.Zhu, The bounded solutions of Lie´nard equation, J. Applied Sci., 7(8(2007), 1239-1240.[13] D.Hricisakova, Existence of positive solutions of the Lie´nard differential equation, J. Math. Anal. And Appl., 176(1993),545-553.[14] A.O.Ignatiev, Stability of a linear oscillator with variable parameters, Electronic J. Differential Equations, 17(1997),1-6.[15] J.Kato, Boundedness theorems on Lie´nard type differential equations with damping, J. Northeast Normal University,2(1988), 1-35.[16] A.Kroopnick, Note on bounded Lp-solutions of a generalized Lie`nard equation, Pacific J. Math., 94(1)(1981), 171-175.[17] A.Kroopnick, Properties of solutions to a generalized Lie´nard equation with forcing term, Applied Math. E-Notes,8(2008), 40-44.[18] F.Nakajima, Ultimate boundedness of solutions for a generalized Lie´nard equation with forcing term, Toˆhoku Math. J.,46(1994), 295-310.[19] J.E.Na´poles, On the global stability of non-autonomous systems, Revista Colombiana de Matema´ticas, 33(1999), 1-8.[20] J.E.Na´poles, On the boundedness and global asymptotic stability of the Lie´nard equation with restoring term, Revistade la UMA, 41(2000), 47-59.[21] J.E.Na´poles, A note on the qualitative behavior of some second order nonlinear differential equations, DivulgacionesMatema´ticas, 10(2)(2002), 91-99.[22] J.E.Na´poles and C.Negro´n, On the behaviour of solutions of second order linear differential equations, Revista Inte-gracio´n, 14(1996), 49-56.[23] B.S.Ogundare, S.Ngcibi and V.Murali, Boundedness and stability properties of solutions to certain second-order differ-ential equation, Adv. Differ. Equ. Control Process, 5(2)(2010), 79-92.[24] P.Omari, G.Villari and F.Zanol´ın, Periodic solutions of the Lie´nard equation with one-side growth restrictions, J. Dif-ferential Equations, 67(1987), 278-293.[25] R.Reissig, G.Sansone and R.Conti, Qualitative theorie nichtlinearer differentialgleichungen, Edizioni Cremonese, Rome,(1963).[26] J.A.Repilado and A.I.Ruiz, On the behaviour of solutions of differential equation x´´ + g(x)x´ + a(t)f(x) = 0-(I),Revista Ciencias Matema´ticas, VI(1985), 65-71.[27] J.A.Repilado and A.I.Ruiz, On the behaviour of solutions of differential equation x´´ + g(x)x´ + a(t)f(x) = 0-(II),Revista Ciencias Matema´ticas, VII(1986), 35-39.[28] A.I.Ruiz, Behaviour of trajectories of nonautonomous systems of differential equations, Universidad de Oriente, Cuba,(1988).[29] Shao, Jing and Wei Song, Limit circle/limit point criteria for second order sublinear differential equations with dampingterm, Abstract and Applied Analysis, 2011(2011), Article ID 803137.[30] G.Sansone and R.Conti, Nonlinear differential equations, MacMillan, New York, (1964).[31] C.Tunc, A note on boundedness of solutions to a class of non-autonomous differential equations of second order, Appl.Anal. Discrete Math., 4(2)(2010), 361-372.[32] C.Tunc, On the boundedness of solutions of a non-autonomous differential equation of second order, Sarajevo J. Math.,7(19)(2011), 19-29.[33] C.Tunc, Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations, Appl.Comput. Math., 10(3)(2011), 449-462.[34] C.Tunc, On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order20Juan E.Na´poles Valdes, L.M. Lugo Motta Bittencurt and P.M.Guzma´nwith multiple deviating arguments, Afr. Mat., 23(2)(2012), 249–259.[35] C.Tunc and O.Tunc, A note on certain qualitative properties of a second order linear differential system, Appl. Math.Inf. Sci., 9(2)(2015), 953-956.[36] W.R.Utz, Properties of solutions of certain second order nonlinear differential equations, Proc. Amer. Math. Soc.,8(1957), 1024-1028.[37] G.Villari, On the existence of periodic solutions for Lie´nard’s equation, Nonlin. Anal., 7(1983), 71-78.[38] G.Villari, On the qualitative behaviour of solutions of Lie´nard equation, J. Differential Equation, 67(1987), 269-277.21

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A note on the continuability of solutions of a perturbed second order nonlinear differential equation of lienard type

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