Boundedness and Stability of Solutions of Some Nonlinear Differential Equations of the Third-Order.

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1 Boundedness Sabili of Soluions of Some Nonlinear Differenial Equaions of he Third-Order. A.T. Ademola, M.Sc. * P.O. Arawomo, Ph.D. Deparmen of Mahemaics Saisics, Bowen Universi, Iwo, Nigeria. Deparmen of Mahemaics, Universi of Ibadan, Ibadan, Nigeria. * ademola67@ahoo.com ABSTRACT Sufficien condiions are esablished for he uniform ulimae boundedness of soluions of a hird-order nonlinear differenial equaion (. When p (,, ', '' =, crieria under which all soluions (, is firs second derivaives end o ero as are given., (Kewords: hird-order, differenial equaions, sabili, uniform-bounded, ulimae boundedness INTRODUCTION Nonlinear hird-order differenial equaions have been eensivel sudied wih high degree of generali. In paricular, here have been ineresing works on asmpoic behavior, boundedness, periodici, sabili of soluions for nonlinear differenial equaions of he hird-order. Auhors ha have worked in his direcion include Ademola e. al., [,, 3], Afuwape [], Berekeoğlu Giöri [5], Eeilo [6], Omeike [7], Swick [9], o menion a few. All he above menion works were done b using he Lapunov s second mehod ecep in [] [], where Yoshiawa funcion frequenc domain echnique were used. In his paper, we shall invesigae uniform ulimae boundedness sabili of soluions of he hird-order nonlinear ordinar differenial equaion: + f(,,, + qg ( (, + h (,, = p(,,, ( or is equivalen ssem: =, =, = p(,,, f (,,, qg ( (, h (,, ( where f, g, h, p q are coninuous in heir respecive argumens,, denoe he firs, second hird derivaives of he funcion ( wih respec o. The derivaives: f (,,, / = f (,,,, f (,,, / = f(,,,, f (,,, / = f(,,,, g (, / = g (,, h (,,/ = h (,,, h (,,/ = h (,,, h (,,/ = h (,, dq(/ d = q ( eis are coninuous. Moreover, he eisence uniqueness of soluions of ( will be assumed. In 5, Tunç [] discussed crieria for boundedness of soluions of a hird-order nonlinear differenial equaion: + f(,, + g(, + h (,, = p(,,, (3 In 8, Ademola e. al., [] Omeike [7] esablished condiions for he ulimae boundedness of soluions of a hird-order differenial equaion (3 using a complee Yoshiawa a complee Lapunov funcions, respecivel. However, he problem of sabili boundedness of soluions of hird-order differenial equaions where he nonlinear, The Pacific Journal of Science Technolog 87 hp:// Volume. Number. November 9 (Fall

2 specificall he resoring, erms depend on he independen variable muliple of he funcions of are scare. Moivaion for his sud comes from he works of Ademola e. al, [, 3], Omeike [7] Swick [9]. The purpose of his paper, herefore, is o invesigae crieria under which all soluions (, is firs second derivaive, when p (,,, =, end o ero as. Sufficien condiions were also obained for uniform ulimae boundedness of soluions of a hird-order differenial Equaion (. Here, he Lapunov second mehod is used o achieve he desired resuls. Our resuls do no onl generalie, o hird-order equaion, he resuls in [, 3, 9] bu also include eend he resul in [7]. Some eising resuls on hird-order nonlinear differenial equaions, which have been discussed in [8], are also generalied. MAIN RESULTS In he case p (,, ', '' =, Equaion ( becomes: =, =, = f(,,, qg ( (, h (,, ( wih he following resul. THEOREM. In addiion o he basic assumpions on he funcions f, g, h, p q, suppose ha here are posiive consans aa,, bb,, cq,, α, βδ, μ such ha for all, he following condiions are saisfied: (i h (,, =, δ h (,,/ ; b g(, / b (ii g (, =, for all, ; (iii a f a (,,, for all,, ; (iv μ q (, q'( ; (v f (,,,, f (,,,, g (,, h (,, c for all, ; h h (,,, (vi (,,, f for all,,. (,,, Then ever soluion ( (, (, ( of ( is uniform-bounded saisfies: (, (, ( as. (5 REMARK. Observe ha he hpoheses: a f (,,,, b g(, /, δ h (,,/, h (,, c μ q ( of Theorem impl he eisence of arbirar posiive consans α β saisfing: c α a bμ < < (6a β min{ bμ;( abμ c η; ( a α η} (6b where, η = [ + a+ δ μ [ g(, / b] ] η = [ + δ [ f(,,, a] ] for all,,. REMARK 3. (i Noe ha f (,,, f (, qg ( (, g( h (,, h(, ssem (. reduces o ha invesigaed b Ademola e. al, in [3]. (ii Also, whenever f (,,, f (,,, g(, g( h (,, rh ( ( ssem (. specialies o ha sudied b Swick in [9]. (iii Furhermore, he hpoheses on ( are considerabl weaker han hose in [3] [9]. The Pacific Journal of Science Technolog 88 hp:// Volume. Number. November 9 (Fall

3 Hence, our resul generalies he resuls in [3] [9]. The proofs of our resuls depend on some cerain fundamenal properies of a coninuousl differeniable Lapunov funcion V = V(,,, defined b: V = ( α + a h( ξ,, dξ + q( g(, τ dτ + h (,, + + β + ( α + a τ f (,, τ, dτ + ( α + a + β + bβq( + aβ (7 whereα β are defined in (6. Namel, his funcion is ime derivaive saisf some fundamenal inequaliies which are discussed in he following lemmas. LEMMA. Subjec o he hpoheses of Theorem, V (,,, = here are posiive consans D = D ( a, b, c, α, βδμ,, D = D( abca,,,, b, q, α, βδ, such ha: (i (ii D + + V D ( + + ; ( (, (, (, ( V (, (, (, ( + +. Furhermore, for an soluion ( (, (, ( of ( d (iii V V(, (, (, ( d D + +. as PROOF. I is clear ha V (,,, =. Since h (,, = b q(, we observe ha he funcion V defined in (7 can be rearranged as follows: ( V = [( α + a bq( hξ ( ξ,,] h( ξ,, dξ bq( + q ( [ g(, τ/ τ b] τdτ + β + β[ bq ( β] τ[( α a f(,, τ, ( α a ] dτ ( α ( β + a+ + [ h(,, + bq( ]. bq( (8 Now, since μ q (, h (,, c h (,,/ δ, if follows ha, [( abq ( h(,,] h(,, d bq( α + ξ ξ ξ ξ [( αbμ c + ( abμ c] δb μ. (9a Also, g(, / b, implies ha, (9b q ( [ g (, τ/ τ b] τdτ. Furhermore, from he inequaliies in condiion (iii of Theorem, we obain: [ τ ( α + a f(,, τ, ( α + a ] dτ α( a α. (9c Combining esimaes (9a - (9c wih (8, we obain: V {[( αbμ c + ( abμ c] δb μ + ( α+ + β[ bμ β]} + [ α( a α + β] ( ( + β+ a+ + b μ δ+ bμ. The Pacific Journal of Science Technolog 89 hp:// Volume. Number. November 9 (Fall

4 From esimaes (6a (6b, we haveαbμ c >, abμ c >, a α > bμ β >. I follows ha he V defined in (7 is posiive definie. Hence, here eiss a posiive such ha: consan δ = δ ( abc,,, α,, δ, μ β V δ ( + +. ( I is clear from ( ha: V (,,, as + +. ( Le us observe ha q ( implies, q ( q( = q since h (,, = hen h (,, c implies h (,, c. These ogeher wih g (, / b, f (,,, a Schwar inequali Equaion (7 becomes: V ( α + a c + b q + c( + aa α a + ( α ( + ( + + β + bβq + aβ( + + β( +. Rearranging he erms, here eiss a posiive consan δ = δ( abca,,,, b, q, α, β such ha: V δ ( + +. ( To deal wih hpohesis (iii of Lemma, le ( (, (, ( be an soluion of ( consider he funcion V = V(, (, (, (. B an elemenar calculaion using ( (7, we have: V = W + W W W + a + β (. 3 β βq ([ g (, / b ] β[ f(,,, a ] (3 Where, W = q g d + b ( (, τ τ β, W = ( α + a τ f(,, τ, dτ + τ f(,, τ, dτ + q ( g(, τ dτ, h (,, W3 = β + f + a g (, + ( α + aq ( h (,,, h (,, h (,, W = ( α + a h (,, h (,, + f(,,, f(,,, + ( α + a [ (,,, ( α + ] B hpohesis (iv q'( for all. If q'( = hen W =. For hose s such ha q'( <, we have, W = q g d + b since, ( (, τ τ β g (, τ dτ + bβ b β + f or all. Thus, on combining he wo cases, we have: W for all,. In view of condiion (v of Theorem, since α are posiive consans q ( μ >, we have W. Moreover, h (,,/ δ, g(, / b, h (,, c, f (,,, a q ( μ, we have W3 βδ + [( α + a bμ c] + [ a α] a. The Pacific Journal of Science Technolog 9 hp:// Volume. Number. November 9 (Fall

5 Also, from hpohesis (vi of Theorem, we have he following inequaliies: W h (,, h (,, = ( α + a = + ( α ah (, θ,, θ, a α are posiive consans, buw = when =. Hence, W for all. Similarl, when, we have : W = h (,, h (,, = bu W = when =. h(,, θ, θ, Hence, W for all. Finall, when, we have: f(,,, f(,,, W3 = ( α + a = + af bu W = when =. Thus ( α (,, θ θ3, 3 3 for all W,,. W, W W3 On combining esimaes, we obain W for all,,. W, W, W 3 W On gahering he esimaes wih (3 complee he squares o ge: V (. βδ ( αbμ c ( a α g (, abμ c β + a + δ μ b ( [ (,,, ] a α β δ f + a βδ g (, + δ μ b βδ + δ [ f(,,, a]. Since β δ are posiive consans, i follows ha g (, + δ μ b δ [ (,,, ] + f a for all,,. Hence, b (6a (6b, here eiss a posiive consan δ ( abc,,,,,, = δ α β δ μ such ha: δ ( + + (. V. ( This complees he proof of Lemma. PROOF OF THEOREM. From hpoheses (i - (iii of Lemma i follows ha he soluion ( (, (, ( of ( is uniform-bounded (see [] p Moreover, from Lemma, V D ( + +. Now, le W( X D ( + + a posiive definie funcion wih respec o a closed se Ω {(,, =, =, = } V (, X W( X. From he coninui of h (,, q (, he fac ha he funcions f (,,, g(, are bounded above, i follows ha he funcion FX (, defined as: FX (, = f (,,, qg ( (, h (,, is bounded. Since g(, = = h(,,, he onl se conained in Ω is he origin. Then b Theorem. p. 6 6 in [], (5 follows. This complees he proof of Theorem. The Pacific Journal of Science Technolog 9 hp:// Volume. Number. November 9 (Fall

6 THEOREM 5. Suppose ha aa,, bb,, cq,, α, βδμ,, are posiive consans P are such ha: (i hpoheses (i - (vi of Theorem hold; p (,,, P <. (ii Then he soluion ( (, (, ( of ( is uniform ulimael bounded. REMARK 6. If f (,,, f (,, q (, hen he ssem ( reduces o ha invesigaed b Omeike in [7]. Moreover, he condiion required here on f (,,, o impl ha ever soluion ( (, (, ( of ( o be uniform ulimael bounded is weaker here han ha used b Omeike in [7] for he nonlinear hirdorder differenial equaion (3, since here i was required ha f (,, > a. LEMMA 7. Subjec o he assumpions of Theorem 5, here eiss a posiive consan D = D ( a, b, c, α, βδμ,, such ha along an soluion ( (, (, ( of ( V D( + +. PROOF. Along a soluion ( (, (, ( of (, we have: V (. = V (. + [ β + ( α + a + ] p(,,,. From esimae (, hpohesis (ii of Theorem 5 he Schwar inequali, we obain: V δ ( δ ( + + / (. 3 (5 / whereδ3 = 3 P ma{ β; α + a;}. Choose / ( + + δ = δ δ3, inequali (5 becomes V δ ( + +, (. 5 where δ5 = δ. This complees he proof of Lemma 7. PROOF OF THEOREM 5. From condiions (i (ii of Lemma, Lemma 7 Theorem. in [] p, i follows ha he soluion ( (, (, ( of ( is uniform ulimael bounded. REFERENCES. Ademola, A.T Arawomo, P.O. 8. On he Sabili Ulimae Boundedness of Soluions for Cerain Third-Order Differenial Equaions. J. Mah. Sa. (: -8.. Ademola, A.T, Kehinde, R., Ogunlaran, M.O. 8. A Boundedness Theorem for a Cerain Third-Order Nonlinear Differenial Equaions. J. Mah. Sa. (: Ademola, A.T., Ogundiran, M.O., Arawomo, P.O., Adesina, O.A. 8. Sabili Resuls for he Soluions of a Cerain Third-Order Nonlinear Differenial Equaion. Mah. Sci. Res. J.(6: -3.. Afuwape, A.U. 6. Remarks on Barbashin- Eeilo Problem on Third-Order Nonlinear Differenial Equaions. J. Mah. Anal. Appl. 37: Berekeoğlu, H. Göri, I On he Boundedness of Soluions of a Third-Order Nonlinear Differenial Equaion. Dnam. Ssems Appl. 6(: Eeilo, J.O.C On he Sabili of Soluions of Some Third-Order Differenial Equaions. J. London Mah. Soc. 3: Omeike, M.O. 8. New Resul in he Ulimae Boundedness of Soluions of a Third-Order Nonlinear Ordinar Differenial Equaion. J. Inequal. Pure Appl. Mah. 9 (:Ar. 5, Reissig, R., Sansone,G., Coni, R. 97. Nonlinear Differenial Equaions of Higher Order. Noordhoff Inernaional Publishing: Leeden, The Neherls. 9. Swick, K.E On he Boundedness Sabili of Soluions for Some Non-Auonomous Differenial Equaions of he Third-Order. J. London Mah. Soc. : Tunç, C. 5. Boundedness of Soluions of a Third-Order Nonlinear Differenial Equaion. J. Inequal. Pure Appl. Mah. 6 (:-6.. Yoshiawa, T Sabili Theor b Liapunov s Second Mehod. The Mahemaical Socie of Japan. The Pacific Journal of Science Technolog 9 hp:// Volume. Number. November 9 (Fall

7 ABOUT THE AUTHORS Adeleke Timoh Ademola, is a Lecurer in he Deparmen of Mahemaics Saisics a Bowen Universi, Iwo, Nigeria. His area of research is in differenial equaions applicaions. Dr. P.O. Arawomo is a member of he Facul of he Deparmen of Mahemaics, Universi of Ibadan, Ibadan, Nigeria. Dr. Arawomo s research ineress are in differenial equaions applicaions. SUGGESTED CITATION Ademola, A.T. P.O. Arawomo. 9. Boundedness Sabili of Soluions of Some Nonlinear Differenial Equaions of he Third- Order. Pacific Journal of Science Technolog. (: Pacific Journal of Science Technolog The Pacific Journal of Science Technolog 93 hp:// Volume. Number. November 9 (Fall