A Boundedness Theorem for a Certain Third Order Nonlinear Differential Equation

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1 Journal of Mathematics Statistics 4 ): 88-93, 008 ISSN Science Publications A Bouneness Theorem for a Certain Thir Orer Nonlinear Differential Equation AT Aemola, R Kehine OM Ogunlaran Department of Mathematics Statistics, Bowen University, Iwo, Nigeria Abstract: In this stuy, we establish conitions for ultimate bouneness of solutions for a certain thir orer nonlinear ifferential equation using a complete Yoshizawa function The result inclues extens the earlier results in the literature Keywors: Bouneness of solutions, nonlinear thir orer equation, complete Yoshizawa functions, incomplete Lyapunov functions only on the arguments isplaye explicitly, the INTRODUCTION ots, as usual, ente ifferentiation with respect to the inepenent variable The partial erivatives It is well known from relevant literature that there have been eep thorough stuies on the qualitative behaviour of solutions of thir orer nonlinear ifferential equations in recent years Mean while, many articles have been evote to the investigation of bouneness of solutions for various forms of thir orer nonlinear ifferential equations; see for instance Yoshizawa [0] Reissiq et al, [14] which are backgroun survey books respectively, Aemola et al, [1],Afuwape [], Anres [3], Bereketolu Györi [4], Chukwu [5] [ 6, 7, 8, 9, 10,, Ezeilo 11], Ezeilo Tejumola [1], Hara [14], Swick [16, 17], Tejumola [18], Tunç [19] These works were one with the ai of complete Yoshizawa incomplete Lyapunov functions Our observation in relevant literature reveals that works on the ultimate bouneness of solutions for the thir orer nonlinear ifferential equation 1) using a complete Yoshizawa function are scarce The purpose of this paper is to stuy the ultimate bouneness of solutions of the thir orer nonlinear non-autonomous orinary ifferential equation On setting equation 1) is equivalent to the system of ifferential equations In which It is suppose that the functions epen " Corresponing Author: AT Aemola, Department of Mathematics Statistics, Bowen University, Iwo, Nigeria 88 # $ " $ " $ " " exist are continuous Here we shall use a complete Yoshizawa function as our basic tool to achieve the esire result RESULT AND DISCUSSION Theorem: In aition to the basic assumptions on, suppose there are positive constants %&' ) * such that the following conitions are satisfie % + +,, for all - + /for all, &+ " #,, 0 1 for all %,,314,, + ),,- 5 +' for all 6 ' / 9 %,, for all - 0 " 0 " : 0 " 0;<%==-,, + *,,,, for all 0 > Then there exists a constant? whose magnitue epens on %& ' ) %/@*as well as the

2 J Math & Stat, 4 ): 88-93, 008 functions %/@ such that every solution of the system ) ultimately satisfies,, +?,, +?,, +? for all 0 > For the rest of this article,? ),?,?, the D s st for positive constants Their ientities are preserve throughout this paper The proof of the theorem epens on the funamental properties of the continuously ifferentiable function A A efine as: A A ) A B A ) D E F GG) I H /K,, 0,, A J /K,, +,,> L M Observe that A ) is a further generalization of the incomplete Lyapunov function A in [16], while A is the same as a complete signum function in [5] The require properties of A is containe in the following lemmas Lemma 1 Subject to the conitions of the theorem, there is a constant? such that A G? N for all A 9 % 9> Proof From 4), it is clear that A ) can be rearrange in the form A ) & O) PQ& & & S C &S@ E O) C S@> T We now observe, view of hypothesis i), that the integral O) C S@ 0 ) % 9) Next, Chukwu s estimate in [5] shows quite clearly that UPVWX 8XSYX 0 for,[, 0 1 because the integr satisfies VWX 8XS\V]X 0 for,x, 0 1 > But if,[, + 1 we have UC VWX 8XSYX 0 1 ^),_W,`a 4 for all W [ Finally, in view hypothesis iii) 8 O) C Q8U_ b c R_c> Yc 0 8 O) 8 S Ue C _c Yc> On combining 9), 10), 11) 1), 8) becomes f ) 0 8 O) 8Ue C _c Yc ) U7 g [ ) Uh[ `8[_W g g S `_ g 1 i^),_w,`a 4 jb Now, since U 8 are positive constants it follows that U 8 8[_W S 0 for all W [> Also by hypothesis iv), for sufficiently large,w,,_w, 6 _ W > On gathering these estimates into 13), we obtain f ) 0 U U7 [ U Uh[ ) & O) 8U78e^)1 SP _c Yc `U8k ^) _ E1 4 14) Since 8 is a positive constant, it follows by 5) that 8: e G now assume that ^) is so small that 8Ue G 8e^)1 I there exist positive constants m ) m m such that f ) 0 m ) C _c Ycm [ ) Uh g [ m M Finally, as in [5], [10] [11], 6) yiels f 0,h,>N On gathering the estimates 16) 17) in 3), we get f 0 m ) C _c Ycm [ ) Uh[,h, g m T In view of hypothesis iv) of the theorem the fact that : is a positive constant Lemma 1 is immeiate Lemma Let xt), yt), zt)) be any solution of the ifferential system ) the function f n fo be efine by fo n fpowo [o ho q> Then the limit fr o n stu v9 w v exist there are positive constants m { m such that fr o + m { IfW [ h 0 m l xyv yv zyv SOxy y zy S 89

3 J Math & Stat, 4 ): 88-93, 008 Proof Let W[h n Wo [o ho be any solution of ) Also in [5] [10] [13], the limit fr o corresponing to W[h clearly exists An elementary calculation from ) 3) will infact yiel the following equation fr o f)f} Where f)o [C UV WX X~ WX SYX [Uh _W[h _W S [h~w[h ~W[ S VW[ U_ [ W [ U~W[h Sh [Uh ow[h [\V]ht~,h, 0,W, ~W[h hvw[ ƒ _W[h S\V]W fo ow[h \V]W L t~,h, +,W, ƒ Observe that the first component in 0), in view of hypothesis v) of the theorem, becomes [C : V WX X~ WX SYX + ) for all W [>The next two components in 0), by mean value theorem assumption vi) of the theorem, are non-negative That is for all W[h [Uh _W[h _W S [ _ W ) [ Uh _ z W[ h [h_ z W[ h U[h_ W ) [ 0 3) + + t [h~w[h ~W[ S h [~ z W[ 0 4) + + > On gathering estimates ), 3) 4), 0) becomes f) + iˆ U_ W j[ U~W[h Sh [Uh ow[h > 5) From 1) 5), we have the following estimates for 19) fr o + [Uh ow[h [\V]h 6) if,h, 0,W, fr o + ~W[h hvw[ _W[h S\V]W \V]W[: h ow[h 7) if,h, +,W, n iˆ Let U_ W j[ U~W[h Sh ) [\V]h[: h ow[h 8) if,h, 0,W, \V]W[Uh ow[h if,h, +,W, ) are group of terms in 6) 7) respectively Using the inequality,[,,h, + [ h hypothesis vii) of the theorem, we get ) + m Š,[,,h, m ^[ h 9) + Œm,[,,h, m ^[ h 30) m u7wu m Š u7wm Œ m ŒS m m ^> Let Ž G be sufficiently small such that Now set U G 7 Ž>B W[h n ) ) UŽ ~W[h Sh 3) An Ž~W[h h ~W[h h\v]wsm,h, ^m h 33) is containe partly in 7) partly in 30) In view of hypothesis i) of the theorem, ) + UŽ 7Sh 34) Since : Ž)7 G by 31), + ŽU^m h Um,h, 35) provie ^ is chosen so small that Ž7 G ^m An estimate for will be calculate for the two cases when z is arbitrarily large when z is boune When z is arbitrarily large, say z 0 m 6 m 6 then + provie Ž G ^m 7 O) On gathering the estimates ) when, z, 0 m for all x, y, we have + UŽ 7Sh 36) when z D9, Ž 6 ^m 7 O) for all W[, we obtain from 33) that Ž~W[h ^m Sh +[~W[h +m ],h, + ŽU^m Sh ^,_W[h,m S,h, + ^,_W[h,m S,h,BN 90

4 we have use the hypothesis i) the fact that [ U^m ] z 0 for all z if U G ^m δ is chosen so small A combination of 33) 37) when 9 for x, y yiels + UŽ 7Sh m ^,_W[h,m S 38) Next, we let W4 be group of terms containe partly in 7) partly in 30), efine as { n ˆ J Math & Stat, 4 ): 88-93, 008 z D U_ W[h [ VW[ \V]W m,[,^m [ ŒB} Again, the terms in W4 will be estimate for two cases: When y is arbitrarily large, say, y D10 on applying hypotheses ii) iii) of the theorem, we obtain { + 8Ue ^m S[ 8m,[,Œ + 8Ue ^m S[ m )),[,S D11 max[ b + D8, M ] Choosing δ so small that b αc > δ D 40) 6 then { + m ) [ m )),[, +41) provie y D10 D [ b α c) δ D ] > y D In the case 10, we obtain { + 8Ue ^m S[ m m ),, Œ + m ) u7w,,3 4,VW[,S + m ),_W,SD D max[1, D D + M ], since D y 1 0 for all y, provie that 40) hols Utilizing the estimates 36) 41) in 7), we obtain fr o + _W[W \V]WUŽ 7Sh 43) for z D9, y D10 when x z, provie that ^ is chosen so small so that 8Ue G ^m On gathering the estimates 38) 4) in 7), we have f + _W[W \V]W UŽ 7Sh m Q^,_W[h, m ) ^),_W,SR 44) 91 for z D9, y D10 when x z Next, we fin a suitable similar upper boun for 6) when z x To see this, let n iˆ U_ W[h j[ ^m [ m Š,[, W5 the group of terms is partly containe in 6) 9) Now + 8Ue ^m S[ m Š,[, +? ) m Š,[, 45) provie y D14, say y D In the case 14, then + 8Ue ^m S[ m Š m ){ + m ) 46) since D1 y 0, for all y D15 D7D 14 Finally, let W6 be the group of terms partly containe in 6) 9), efine as n U~W[h Sh m Š,h,^m h + m ) h m Š,h, + m 47) provie δ is chosen so small that αa 1 > δ D6 Summing 45) 47) for, z x we obtain f + m ) [ m Š,[,m + 48) provie y D14 D 1 > 0 But in the event that y D14 the inequality in 6) together with 46) when z x yiels f + m ) h m Š,h,m ) + 49) provie z D17 Proof of the Theorem: As in [5], let D18 max[ D17, D14, D10, D9 ] We now estimate 6) 7) when: y + z D 18 x D 18, that is when y + x is large in comparison with x ; x is large in comparison with, say x D18 but y + z D18 The final estimate will follow on combining i) ii) y + z In the first case we estimate V in 43), 48) 49) When z x 43) is use, we obtain

5 J Math & Stat, 4 ): 88-93, 008 f + _W[h \V]WUŽ 7Sh + 50) 50) provie z D18 since h x, y, z)sgn x + as x, then h x, y, z)sgn x = h x, y, z) D for all x, y, z When 48) 49) are use, z x, we have f + provie [ h 0 m ) 51) From 50) 51) when z x, we conclue that f + whenever [ h 0 m ) 5) In the secon case we consier f in 43) when x z, we have f + _W[h \V]WUŽ 7Sh m ^,_W[h,? Sm ) ^),_W,S + ^m,_w[h,^)m ),_W,Sm ) since h x, y, z)sgn x = h x, y, z) when x is sufficiently large, [ α ) a 1] z 0 for all z provie α ) a > 1 D19 D8 D9 + D13 δ 0 D If δ 0 is chosen so small that 1> 9 by hypothesis iv) of the theorem, we obtain V 1 provie x D0 53) but y z are small It is now clear from 5) 53) that f + whenever W [ h 0 m m ) This completes the proof of the Theorem Remark 1: Whenever n n the hypotheses conclusion of theorem coincie with those of Chukwu [5], except hypotheses i) iii) of the theorem which are consierable weaker than those in [5] Remark : If n n n then 1) reuces to the case stuie by Ezeilo [10] Remark 3: Suppose n % G % is constant, n n n then the system ) specializes to that investigate by Ezeilo Tejumola in [13] Moreover, the hypotheses conclusion of the theorem coincie with those in [13], Theorem 1 CONCLUSION Obtaining ultimate bouneness of solutions for the nonlinear non-autonomous thir orer ifferential equation 1) using a complete Lyapunov function or the integral omain has remaine ifficult because it is computationally intensive I n this paper, conitions obtaine using a complete Yoshizawa functions accoring to Chukwu [5] ) are exact, see for instance [5], [10], [13] ACKNOWLEDGEMENT The authors woul like to thank the anonymous referee for his valuable suggestions, Bowen University, Nigeria Liverpool Hope University, Unite Kingom for the staff exchange programme REFERENCES 1 Aemola, AT, Oguniran, MO, Arawomo, PO Aesina, OA; 008 Stability results for the soultions of a certain thir orer nonlinear ifferential equation Mathematical Sciences Research Journal, Vol 1, no 6, Afuwape, AU; 006 Remarks on Barbashin- Ezeilo problem on thir orer nonlinear ifferential equations J Math Anal Appl 317: Anres, J; 1986 Bouneness results for solutions of the equation % without the hypothesis / 0 for,, G š>atti Acca Naz Lincei Ren Cl Sci Fis Mat Natur, 8) 80: Bereketolu, H Giyöri, I;1997 On the bouneness of solutions of a thir orer nonlinear ifferential equations Dynam Systems Appl, ) 6: Chukwu, EN; 1975 On the bouneness of solutions of thir orer ifferential equations Ann Mat Pura Appl 4) 104: Ezeilo, JOC; 1961 A note on a bouneness theorem for some thir orer ifferential equations J Lonon Math Soc 36: Ezeilo, JOC; 1963 An elementary proof of a bouneness theorem for a certain thir orer ifferential equation J Lonon Math Soc 38: Ezeilo, JOC; 1963 Further results for the solutions of a thir orer ifferential equation Proc Camb Phil Soc 59:

6 J Math & Stat, 4 ): 88-93, Ezeilo, JOC; 1963 A bouneness theorem for a certain thir orer ifferential equation Proc Lonon Math Soc 3) 13: Ezeilo, JOC; 1970 A generalization of a theorem of Reissign for a certain thir orer ifferential equation Ann Mat Pura Appl 4) 87: Ezeilo, JOC; 1971 A generalization of a bouneness theorem for the equation Atti Acca Naz Lincei Ren Cl Sci Fis Mat Natur 13) 50: Ezeilo, JOC; 1973 A generalization of bouneness results by Reissig Tejumola J Math Anal Appl 41: Ezeilo, JOC Tejumola HO; 1973 Bouneness theorems for certain thir orer ifferential equations Atti Acca Naz Lincei Ren Cl Sci Fis Mat Natur, 55: Hara, T; 1981 On the uniform ultimate bouneness of solutions of certain thir orer ifferential equations J Math Anal Appl, 80: Reissig, R, Sansone, G Conti, R; 1974 Nonlinear ifferential equations of higher orer Noorhoff International Publishing, Leyen 16 Swick, K; 1969 On the bouneness stability of solutions of some non-autonomous ifferential equations of thir orer J Lonon Math Soc 44: Swick, KE; 1974 Bouneness stability for a nonlinear thir orer ifferential equation Atti Acca Naz Lincei Ren Cl Sci Fis Mat Natur,0) 56: Tejumola, HO; 1970 A note on the bouneness of solutions of some nonlinear ifferential equations of the thir orer Ghana Journal of Science ) 11: Tunç, C; 005 Bouneness of solutions of a thir orer nonlinear ifferential equation J Inequal Pure Appl Math; 1) Art 3, 6: Yoshizawa, T; Stability theory existence of perioic solutions almost perioic solutions, Springer-Verlag, New York Heielberg Berlin 1975) 1 Yoshizawa, T; Stability theory by Liapunov s secon metho Mathematical Society of Japan, Tokyo,