Mathematica Bohemica

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1 Mathematica Bohemica Anthon Ui Afuwape; Mathew Omonigho Omeike Ultimate boundedness of some third order ordinar differential equations Mathematica Bohemica, Vol. 137 (01), No. 3, Persistent URL: Terms of use: Institute of Mathematics AS CR, 01 Institute of Mathematics of the Academ of Sciences of the Cech Republic provides access to digitied documents strictl for personal use. Each cop of an part of this document must contain these Terms of use. This paper has been digitied, optimied for electronic deliver and stamped with digital signature within the project DML-CZ: The Cech Digital Mathematics Librar

2 137(01) MATHEMATICA BOHEMICA No. 3, ULTIMATE BOUNDEDNESS OF SOME THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS Anthon Ui Afuwape, Medellín, M. O. Omeike, Abeokuta (Received Februar 15, 011) Abstract. We prove the ultimate boundedness of solutions of some third order nonlinear ordinar differential equations using the Lapunov method. The results obtained generalie earlier results of Eeilo, Tejumola, Reissig, Tunç and others. The Lapunov function used doesnotinvolvetheuseofsignumfunctionsasusedbothers. Kewords: ultimate boundedness, complete Lapunov function, differential equation of third-order MSC010:34K0 1. Introduction Motivation for this paper comes from the generaliation of the works b Reissig [18andTejumola[1bEeiloin[13,therecentworksofAfuwapeandOmeike [6andAdemolaetal[1. In exciting work, Eeilo[13 investigated the equation of the form x + ϕ 1 (ẋ) + ϕ (x, ẋ)}ẍ + g 1 (ẋ) + h(x) = p(t, x, ẋ, ẍ) for ultimate boundedness generaliing the works of Reissig[18 on andthatoftejumola[1on x + ϕ (x, ẋ)ẍ + g 1 (ẋ) + h(x) = p(t, x, ẋ, ẍ) x + ϕ 1 (ẋ)ẍ + g(ẋ) + h(x) = p(t, x, ẋ, ẍ). This research was supported b Universit of Antioquia Research Grant through SUI No. IN1013CE. 355

3 We shall consider here the equation (1.1) x + f 1 (ẍ) + f (ẋ, ẍ)} + g(x, ẋ) + h(x) = p(t, x, ẋ, ẍ) where f 1, f, g, h, pdependontheargumentsdisplaed. Ourassumptionson f 1, f, g, h,and pshallallowustogeneralietheresultsofademolaetal[1,afuwape [,[3 and a particular case of Afuwape and Omeike[4 concerning x + f 1 (ẍ) + g 1 (ẋ) + h(x) = p(t, x, ẋ, ẍ); andqian[17andaparticularcaseoftunç[concerning x + f (x, ẋ)ẍ + g(x, ẋ) = p(t). The assumptions will also give us an opportunit to discuss the ultimate boundedness results which generalie the earlier ones. A good record of ultimate boundedness resultsofthesetpesisrecordedinthebook[19,andthepapershara[16,afuwape and Omeike[6 and references therein. Also, the recent excellent book of Hadddad [15 includes a good summar of the theoretical works on the subject. Consider the third order nonlinear ordinar differential equation of the form(1.1), or its equivalent sstem form (1.) ẋ =, ẏ =, ż = f 1 () + f (, )} g(x, ) h(x) + p(t, x,, ), where f 1, f, g, hand parecontinuousintheirrespectivearguments,andthedots denote differentiation with respect to t. The object of this paper is to discuss the ultimate boundedness of solutions of Eq.(1.1).Itiswellknownthattheultimateboundednessisaverimportantproblem in the theor and applications of differential equations. An effective method for studing the ultimate boundedness of nonlinear differential equations is still Lapunov s direct method(see[1 [11). In[13, incomplete Lapunov functions augmented with signum functions and with certain restrictive conditions on the nonlinearfunctionswereused.ouraiminthispaperistostudamoregeneraleq.(1.1) for ultimate boundedness of solutions, using a complete Lapunov function with less restrictiveconditionsonthenonlinearfunctions f 1, f, g, hand p.intheprocess, weshallbeabletogeneralieearlierresultsof[1,[,[3andaparticularcaseof[4, [17,[. 356

4 . Main results Our main result is the following theorem. Theorem.1. Inadditiontothebasicassumptionsonthefunctions f 1, f, g, hand p,assumethatthefollowingconditionsaresatisfied(a, b, c, νand Abeing some positive constants): (i) (f 1 () + f (, ))/ aforall, 0, (ii) g(x, )/ bforall x, 0, (iii) h(x)/x νforall x 0, (iv) h (x) c, (v) ab > c, (vi) p(t, x,, ) A < forall t 0andforall x,,. Then ever solution x(t) of (1.1) ultimatel satisfies (.1) x(t) D, ẋ(t) D, ẍ(t) D where Disaconstantdependingonlon a, b, c, νand A. Remark.1.Theorem.1generaliestheresultsofAdemolaetal[1,ifweset f (ẋ, ẍ) = 0. Remark..InusingLapunov stheor,theorem.1givesadifferentmethod of discussing the works of Afuwape[,[3 who used the frequenc domain methods, with f (ẋ, ẍ) 0. Remark.3. Theorem.1generaliestheresultsofQian[17andTunç[ if f 1 (ẍ) 0.Thisbecomesobviousifwecarroutsomedifferentiationsof f (ẋ, ẍ) and g(x, ẋ)withrespecttotheirvariables x, ẋ, ẍtoobtaintheequivalentequation tothatof[17and[. However,the p(t)willbereplacedb p(t, x,, ),withthe appropriate conditions on it. 3. Preliminaries Itisconvenientheretoconsider,inplaceofEq.(1.1),thesstem(1.).Inorder toprovetheorem.1,weneedtoshowthateversolution (x(t), (t), (t))of(1.) satisfies (3.1) x(t) D, (t) D, (t) D 357

5 forallsufficientllarge t,where D isasuitableconstant. Set (x,, ) (x(t), (t), (t)). Ourproofof(3.1)restsentirelontwoproperties(statedinthelemmabelow)of thefunction V (t) V (x,, )definedb (3.) V (x,, ) = β(1 β)b x + b(β + αa 1 ) + αa 1 + [ + a + (1 β)bx with 0 < β < 1,and α > 0. Lemma3.1. SubjecttotheconditionsofTheorem.1, V (0, 0, 0) = 0andthere isapositiveconstant D 1 dependingonlon a, b, c, αand δsuchthat (3.3) V (t) V (x,, ) D 1 (x + + ) forall x,,. Letusset V (t) V (x(t), (t), (t)). Furthermore,therearefiniteconstants D, D 3 dependentonlon a, b, c, A, ν, δ and α such that for an solution (x(t), (t), (t)) of (1.), (3.4) providedthat x + + D 3. d dt V d dt V (x(t), (t), (t)) D ProofofLemma3.1. Clearl, V (0, 0, 0) = 0. Also,brearranging(3.)and choosing D 1 minβ(1 β)b ; b(β + αa 1 ); αa 1 } we have(3.3). Toprove(3.4),wefindthatthederivativeof Vwithrespectto talongthesolution path of(1.),(after simplifications) gives (3.5) d dt V (t) (1.) = b(1 β)xh(x) abβ + a[g(x, ) b } (αa 1 + 1)(f 1 + f ) a } + b (1 β)x ah(x) b(1 β)xg(x, )} + [b(αa 1 + 1) + a (αa 1 + 1)g(x, ) a(f 1 + f )} + ab(1 β)x (αa 1 + 1)h(x) b(1 β)x(f 1 + f )} + [b(1 β)x + a + (αa 1 + 1)p(t, x,, )}. Arearrangementofthisshowsthatfor x 0; 0; 0wehave (3.6) d dt V (t) (1.) = W 1 W W 3 W 4 W 5 W 6 W 7 + W p 358

6 where with W 1 = γ 1 b(1 β)x [ h(x) [ g(x, ) + δ 1 a b + η 1 abβ } x [ + ξ 1 (αa 1 f1 + f + 1) a + µ 1 α }, W = γ b(1 β)x [ h(x) [ h(x) + ax + η abβ }, x x W 3 = γ 3 b(1 β)x [ h(x) [ g(x, ) [ g(x, ) + b(1 β) b x + δ a b }, x [ g(x, ) [ g(x, ) W 4 = δ 3 a b + (αa 1 + 1) b + µ α }, [ W 5 = η 3 abβ f1 + f [ + a a + ξ (αa 1 f1 + f + 1) a }, W 6 = γ 4 b(1 β)x [ h(x) [ f1 + f + b(1 β) a x x [ + ξ (αa 1 f1 + f + 1) a }, W 7 = γ 5 b(1 β)x [ h(x) [ h(x) + (αa 1 + 1)x + µ 3 α }, x x W p = [b(1 β)x + a + (αa 1 + 1)p(t, x,, )} 5 γ i = 1; i=1 3 δ i = 1; i=1 3 η i = 1; i=1 3 ξ i = 1; γ i > 0, δ j > 0, η j > 0, ξ j > 0, i = 1,, 3, 4, 5; j = 1,, 3. Wenotethatforantworealnumbers, u, v,andfor s 1, +1}thereexistsa constant k > 0suchthat (su)v = i=1 ( ksu + 1 ) ( k v k u + 1 ( 4k v) > k u + 1 4k v). Moreover, this inequalit is retained when multiplied b an positive term. Bvirtueoftheconditionsofthetheoremonthenonlinearfunctions,wehavea constant k > 0suchthat W = γ b(1 β)x [ h(x) [ h(x) + ax + η abβ } x x γ b(1 β)νx + aνx + η abβ } [ ν[γ b(1 β) kax + a η bβ ν } 0, 4k ν x,, whenever 4η bβ k < γ b(1 β). a 359

7 Inasimilarmanner,wehaveconstants k i > 0, i = 3, 4,...,7,suchthat W 3 [ [ g(x, ) [ g(x, ) [ b(1 β) γ 3 ν k3 b x + b δ a b(1 β) 0, x,,,whenever k3 < 4δ a [ g(x, ) W 4 b [δ 3 a k4 (αa 1 + 1) + W 5 b(1 β) } 4k3 γ 3 ν [g(x, )/ b ; [ µ α (αa 1 + 1)[g(x, )/ b 4k4 0, x,,,whenever (αa 1 + 1)[g(x, )/ b k4 4µ α < [ [ a η 3 bβ k5 f1 + f [ a f1 + f + a 0, x,,,whenever 4ξ (αa 1 + 1) k 5 < [ [ W 6 b(1 β) γ 4 ν k6 f1 + f a x [ f1 + f [ + a ξ 3 (αa 1 b(1 β) + 1) } 4k6 0, Moreover, x,,,whenever b(1 β) 4ξ 3 (αa 1 + 1) k 6 < δ 3 a (αa 1 + 1) ; [ a ξ (αa 1 + 1) η 3 bβ [(f 1 + f )/ a ; γ 4 ν [(f 1 + f )/ a. [ W 7 ν[γ 5 b(1 β) k7(αa 1 + 1)x + µ 3 α (αa 1 + 1)ν } 4k7 0, x,,,whenever (αa 1 + 1)ν k6 < γ 5b(1 β) 4µ 3 α (αa 1 + 1). Also,wehavethat W 1 γ 1 b(1 β)νx + δ 1 abβ + µ 1 α D 4 (x + + ) where 0 < D 4 minγ 1 b(1 β)ν; δ 1 abβ; µ 1 α}. Moreover, a 4k 5 } W p [b(1 β) x + a + (αa 1 + 1) p(t, x,, ) } D 5 ( x + + ) where D 5 = Amaxb(1 β); a; (αa 1 + 1)}. Hence, using(3.6) we have } (3.7) V D4 (x + + ) + D 6 (x + + ) 1 where D 6 = 3 1 D

8 Ifwechoose (x + + ) 1 D 7 = D 6 D 1 4,inequalit(3.7)impliesthat V 1 D 4(x + + ). Weseeatoncethat V D 8, providedthat x + + D 8 D 1 4 ;andthiscompletestheverificationof(3.4), (with D D 8 ). Remark3.1.WenotethatintheworkofTunç[,usingtheLapunovmethod onlendedupwith V D( + ) + D( + ) p(t) which gave an incomplete nature of the function. 4. Proofoftheorem.1 Let (x(t), (t), (t))beansolutionof(1.).thenthereisevidentlat 0 0such that x (t 0 ) + (t 0 ) + (t 0 ) < D 3, where D 3 istheconstantinthelemma;forotherwise,thatisif x (t) + (t) + (t) D 3, t 0, then, b(3.4), V (t) D < 0, t 0, andthisinturnimpliesthat V (t) as t,whichcontradicts(3.3).hence toprove(3.4)itwillsufficetoshowthatif (4.1) x (t) + (t) + (t) < D 9 for t = T, where D 9 D 3 isafiniteconstant,thenthereisaconstant D 10 > 0,dependingon a, b, c, δ, α, ξand D 9,suchthat (4.) x (t) + (t) + (t) D 10 for t T. Ourproofof(4.)isbasedessentiallonanextensionofanargumentinthe proofof[8;lemma1. Forangivenconstant d > 0,let S(d)denotethesurface 361

9 x + + = d.because Viscontinuousin x,, andtendsto + as x + +,thereisevidentlaconstant D 11 > 0,dependingon D 9 aswellason a, b, c, δ, ξand α,suchthat (4.3) min V (x,, ) > max V (x,, ). (x,,) S(D 11) (x,,) S(D 9) Itiseastoseefrom(4.1)and(4.3)that (4.4) x (t) + (t) + (t) < D 11 for t T. Forsupposeonthecontrarthatthereisat > Tsuchthat x (t) + (t) + (t) D 11. Then,b(4.1)andbthecontinuitofthequantities x(t), (t), (t)intheargument displaed,thereexist t 1, t, T < t 1 < t suchthat (4.5) (4.6) x (t 1 ) + (t 1 ) + (t 1 ) = D 9, x (t ) + (t ) + (t ) = D 11 and such that (4.7) D 9 x (t) + (t) + (t) D 11, t 1 t t. But,writing V (t) V (x(t), (t), (t)),since D 9 D 3,(4.7)obviouslimplies[in view of(3.4) that V (t ) < V (t 1 ), and this contradicts the conclusion[from(4.3) and(4.6) V (t ) > V (t 1 ). Hence(4.4) holds. This completes the proof of(3.4), and the theorem now follows. 36

10 References [1 Ademola, T.A., Ogundiran, M.O.,Arawomo, P.O,Adesina, O.A:Boundednessresults for a certain third order nonlinear differential equation. Appl. Math. Comput. 16 (010), [ Afuwape, A. U.: Frequenc-domain criteria for dissipativit of some third order differential equations. An. Stiint. Univ. Al. I. Cua Iasi, n. Ser., Sect. Ia 4(1978), [3 Afuwape, A. U.: An application of the frequenc-domain criteria for dissipativit of a certain third order non-linear differential equation. Analsis 1(1981), [4 Afuwape, A. U., Omeike, M. O.: Further ultimate boundedness of solutions of some sstem of third order nonlinear ordinar differential equations. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 43(004), 7 0. [5 Afuwape, A. U., Omeike, M. O.: Convergence of solutions of certain third order sstems of nonlinear ordinar differential equations. J. Nigerian Math. Soc. 5(006), 1 1. [6 Afuwape, A. U., Omeike, M. O.: Convergence of solutions of certain non-homogeneous third order ordinar differential equations. Kragujevac J. Math. 31(008), [7 Bereketoglu, H., Gori, I.: On the boundedness of the solutions of a third-order nonlinear differential equation. Dnam. Sstems Appl. 6(1997), [8 Chukwu, E. N.: On the boundedness of solutions of third order differential equations. Ann. Mat. Pur. Appl. 104(1975), [9 Eeilo,J.O.C.:Anelementarproofofaboundednesstheoremforacertainthirdorder differential equation. J. Lond. Math. Soc. 38(1963), [10 Eeilo,J.O.C.:Someresultsforthesolutionsofacertainsstemofdifferentialequations. J. Math. Anal. Appl. 6(1963), [11 Eeilo,J.O.C.:Ageneraliationofaboundednesstheoremforacertainthirdorder differential equation. Proc. Cambridge Philos. Soc. 63(1967), [1 Eeilo,J.O.C.:Newpropertiesoftheequation x + ax + bx + h(x)=p(t, x,x, x ) forcertainspecialvaluesoftheincrementarratio 1 h(x+) h(x)}(p.janssons, J. Mawhin, N. Rouche, eds.). Equations differentielles et functionalles non-lineares, Hermann Publishing, Paris, 1973, pp [13 Eeilo, J. O. C.: A generaliation of some boundedness results b Reissig and Tejumola. J. Math. Anal. Appl. 41(1973), [14 Eeilo,J.O.C.:Afurtherresultontheexistenceofperiodicsolutionsoftheequation x+ψ(ẋ)ẍ+ϕ(x)ẋ+v(x,ẋ, ẍ)=p(t)withabound ν.atti.accad.na.linceirend. Cl. Sci. Fis. Mat. Natur. 55(1978), [15 Haddad, W. A., Chellaboina, V. S.: Nonlinear Dnamical Sstems and Control A Lapunov-Based Approach. Princeton Universit Press, Princeton, 008. [16 Hara, T.: On a uniform ultimate boundedness of the solution of certain third order differential equations. J. Math. Anal. Appl. 80(1981), [17 Qian, C.: Asmptotic behavior of third-order nonlinear differential equations. J. Math. Anal. Appl. 84(003), [18 Reisssig, R.: Über die Existen periodischer Lösungen bei einer nichtlinearen Differentialgleichung dritter Ordnung. Math. Nachr. 3(1966), [19 Reisssig, R., Sansone, G., Conti, R.: Nonlinear Differential Equations of Higher Order. Noordhoff, Groningen, [0 Swick, K. E.: Boundedness and stabilit for a nonlinear third order differential equation. Atti. Accad. Na. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 56(1974), [1 Tejumola, H. O.: On the boundedness and periodicit of solutions of certain third-order non-linear differential equations. Ann. Mat. Pura Appl., IV Ser. 83(1969), [ Tunç, Cemil: Boundedness of solutions of a third order nonlinear differential equation. J. Inequal. Pure Appl. Math. 6(005), Article 3, 6 pp.(electronic). 363

11 [3 Yoshiawa, T.: Stabilit Theor b Lapunov s Second Method. Mathematical Societ of Japan, Toko, Authors addresses: Anthon Afuwape, Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No , Medellín AA 16, Colombia, aafuwape@ahoo.co.uk; M. O. Omeike, Department of Mathematics, Universit of Agriculture, Abeokuta, Nigeria, moomeike@ahoo.com. 364