Existence of positive solutions of a third order nonlinear differential equation with positive and negative terms

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1 Lo Advances in Difference Eqaions 208) 208:87 hps://doi.org/0.86/s R E S E A R C H Open Access Exisence of posiive solions of a hird order nonlinear differenial eqaion wih posiive and negaive erms Demo Lo * * Correspondence: scldm@63.com School of Applied Mahemaics, Gangdong Universiy of Technology, Gangzho, P.R. China Absrac In his aricle we invesigae he exisence of posiive solions for a hird order nonlinear differenial eqaion wih posiive and negaive erms. The main ool employed here is Kigradze s lemma of classificaion of posiive solions. The asympoic properies of solions are also discssed. Two examples are also given o illsrae or resl. MSC: Primary 34A34; 34K3; secondary 34K30; 34L30 Keywords: Posiive solions; Asympoic properies; Delay argmen; Posiive and negaive erm Inrodcion In 993, Kigradze and Chanria [] inrodced he heory of asympoic properies of solions of nonaonomos ordinary differenial eqaions as a mehod of coninm calcli. Since Kigradze s grondbreaking work, here has been a significan growh in he heory of nonaonomos differenial eqaions wih deviaing argmen covering a variey of differen problems; see [2 4] and he references herein. Le T be a ime scale sch ha 0 T. In his aricle, we are ineresed in he analysis of qaliaive heory of posiive solions of hird order nonlinear differenial eqaions. Moivaed by he papers [, 5] and he references herein, we consider he following dynamic eqaion: b) a)x ) ) ) + p)f x τ) )) q)g x σ ) )) 0, 0..) Throgho his paper we assme ha H ) a), b), p), q), τ), σ ) C[ 0, )) are posiive; H 2 ) f ), h) CR), f )>0, h)>0for 0, g is bonded, f is nondecreasing; H 3 ) f v) f v) f )f v) for v >0,andf ) ; H 4 ) τ), lim τ), lim σ ). We consider he canonical case of.), ha is, H 5 ) 0 as) ds 0 ds. bs) By a solion of Eq..), we can easily ndersand a fncion x) wih derivaives a)x ), b)a)x )) coninos on [T x, ), T x 0, which saisfies Eq..)on[T x, ). The Ahors) 208. This aricle is disribed nder he erms of he Creaive Commons Aribion 4.0 Inernaional License hp://creaivecommons.org/licenses/by/4.0/), which permis nresriced se, disribion, and reprodcion in any medim, provided yo give appropriae credi o he original ahors) and he sorce, provide a link o he Creaive Commons license, and indicae if changes were made.

2 Lo Advances in Difference Eqaions 208) 208:87 Page 2 of 2 We consider only hose solions x) of.) which saisfy sp{ x) : T} >0forall T T x. The research of he higher order ordinary differenial eqaions ODE) see [ 7]) essenially akes advanage of some recapilaion of Kigradze s lemma [, 2]. In he lemma, from he fixed sign of he highes derivaive, we can infer he form of possible nonoscillaory solions. We canno fix he sign of he forh order qasi-derivaive for an limaely posiive solion becase he posiive and negaive erms are inclded in.). So he ahors primarily invesigae he properies of.) in he parial case when eiher p) 0 or q) 0. In wha follows we shall assme ha H 6 ) 0 a) bs) s q) d ds d <. The organizaion of his paper is as follows. In Secion 2, we inrodce some definiions and lemmas and declare some preliminary maerial needed in laer secions. We will sae some facs abo he differenial eqaions wih deviaing argmen as well as Kigradze s lemma of classificaion of posiive solions. For deails on Kigradze s heorem, we refer he reader o []. In Secion 3, we esablish or main resls for posiive solions by applying Kigradze s classificaion of posiive solions heorem. In Secion 4, we presen he asympoic properies of solions. In Secion 5, we give wo examples o illsrae or resls. The resls presened in his paper exend he main resls in [5]. 2 Preliminaries A ime scale is an arbirary nonempy closed sbse of real nmbers. The research of dynamic eqaions on ime scales is an incredibly new area, and he nmber of sdies on his sbjec is rapidly growing. The heory of dynamic eqaions nifies he heories of differenial eqaions and difference eqaions. We sppose ha he reader is familiar wih he basic conceps concerning he calcls on ime scales for dynamic eqaions. Oherwise one can find mos of he maerial needed o read his paper in Kigradze and Chanria s books []. Definiion 2. []) A solion of.) is ermed oscillaory if i has arbirarily large zeros on [T x, ), oherwise i is ermed nonoscillaory. Eq..) is said o be oscillaory if all is solions are oscillaory. Definiion 2.2 []) The differenial eqaion x n) )p)x) 2.) has propery A if every solion of 2.) forn even is oscillaory and for n odd eiher is oscillaory or saisfies he condiion x i) ) 0as + i,2,...,n ). Assme ha.) possesses an evenally posiive solion x)on[t x, ), T x 0.We inrodce he axiliary fncion z)associaedwihx)by z)x)+ av) v bs) s q)g x σ ) )) d ds dv. 2.2) I follows from H 6 ) and he bondedness of h) ha he definiion of fncion z) is correc and z) exiss for all T x.iisseflonoicehaz)>x)>0,z )<x ),

3 Lo Advances in Difference Eqaions 208) 208:87 Page 3 of 2 and b) a)z ) ) ) p)f x τ) )) < ) Theorem 2.3 Assme ha all he solions of z )+p)z) 0 2.4) are nonoscillaory. Then every posiive solion z) of.) saisfieseiher z) N 0 a)z )<0, b) a)z ) ) >0, b) a)z ) ) ) <0, or z) N 2 a)z )>0, b) a)z ) ) >0, b) a)z ) ) ) <0. Corollary 2.4 []) Assme ha lim inf 2 p)< 2 3 3, 2.5) hen he se N of all posiive solions of.) has he following decomposiion: N N 0 N 2. In his aricle, o ensre qaliaive heory of he sdied eqaion, some easily verifiable condiions ms be esablished. To prediges or noaion, we denoe ha B) bs) ds and s A) d ds. as) b) Definiion 2.5 []) A solion x) of.) defined on an inerval [,+ ) [ 0,+ ) is said o be a Kneser solion if ) i x i) )x) 0 for 0 i 0,,...,n ). 3 Exisence of posiive solions In his secion we shall invesigae he exisence of posiive solions for Eq..). The main resl is in he following heorem.

4 Lo Advances in Difference Eqaions 208) 208:87 Page 4 of 2 Theorem 3. Le, for all large enogh, ps)f A τs) )) ds. 3.) Assme ha { τ) lim sp ps)f A τs) )) Bs) ds Bτ)) + ps)f A τs) )) ds + f B τ) )) ) } Aτs)) ps)f ds Bτs)) τ) > lim sp 0 f ). 3.2) Then he posiive solion class N 2. Proof Assme on he conrary ha.) possesses an evenally posiive solion z) N 3.Usinghefachab)a)z )) is decreasing, we have a)z ) bs) as)z s) ) bs) ds b) a)z ) ) bs) ds b) a)z ) ) B). 3.3) In view of 3.3), we can see ha a)z ) ) 0; conseqenly, a)z ) is decreasing. Then B) B) x) x s) ds z s) ds a)z ) B) Bs) as) ds a)z ) A). B) as)z s) Bs) Bs) as) ds Seing he las esimae ino 2.3), we see ha y)a)z ) is a posiive solion of he differenial ineqaliy b)y ) ) Aτ)) + p)f Bτ)) y τ) ) ) ) Wha is more, y) B) is decreasing and b)y )>0. On he oher hand, an inegraion of 3.4)from o and hen from o yields y) b) b) + b) Aτs)) ps)f Bτs)) y τs) ) ) ds d Aτs)) ps)f Bτs)) y τs) ) ) ds d Aτs)) ps)f Bτs)) y τs) ) ) ds d

5 Lo Advances in Difference Eqaions 208) 208:87 Page 5 of 2 Aτs)) ps)f Bτs)) y τs) ) ) Bs) ds Aτs)) + B) ps)f Bτs)) y τs) ) ) ds. Having replaced by τ)inhelasineqaliy,weobain y τ) ) τ) Aτs)) ps)f Bτs)) y τs) ) ) Bs) ds + B τ) ) Aτs)) ps)f τ) Bτs)) y τs) ) ) ds + B τ) ) Aτs)) ps)f Bτs)) y τs) ) ) ds. Employing H 3 )andhefachay) is increasing and y) is decreasing, we have B) y τ) ) f ) yτ)) τ) ps)f A τs) )) Bs) ds Bτ)) ) + B τ) ) f yτ)) Bτ)) + B τ) ) f y τ) )) ps)f Therefore, seing yτ)) Bτ)),wege f ) τ) Bτ)) ps)f A τs) )) ds τ) ps)f A τs) )) Bs) ds + + f B τ) )) Aτs)) ps)f Bτs)) ) Aτs)) ds. 3.5) Bτs)) τ) ps)f A τs) )) ds ) ds. 3.6) Condiion 3.)garaneesha y) y) 0as. Indeed, if we admi B) B) l >0,hen l and seing he las ineqaliy ino 3.4), we obain y) B) b)y ) ) + p)f l)f A τ) )) b)y ) ) + p)f la τ) )) 0. An inegraion from o yields b )y ) f l) ps)f A τs) )) ds, which conradics condiion 3.). Now, we can ake lim sp on boh sides of 3.6), one ges a conradicion o 3.2). Obviosly, we have he following easily verifiable crierion for some special cases of.). Corollary 3.2 Le, for all large enogh, ps)a τs) ) ds. 3.7)

6 Lo Advances in Difference Eqaions 208) 208:87 Page 6 of 2 { τ) lim sp ps)a τs) ) Bs) ds + ps)a τs) ) ds Bτ)) τ) + B τ) ) ps) Aτs)) } Bτs)) ds. 3.8) Then he posiive solion class N 2 of hird order rinomial differenial eqaion b) a)x ) ) ) + p)x τ) ) q)g x σ ) )) 0 3.9) is empy. Theorem 3.3 Le, for all large enogh, b) ps)f ) Aτs)) ds d. 3.0) Bτs)) Assme ha { ) τ) lim sp f Bτ)) + B τ) ) f Bτ)) + B τ) ) ps)f > lim sp ps)f A τs) )) Bs) ds ) ps)f A τs) )) ds τ) Aτs)) Bτs)) ) } ds f ). 3.) Then he posiive solion class N 2 of.) is empy. Proof Assme ha x) isaposiive solion of.).proceedingexaclyasinheproofof Theorem 3., we verify ha he associaed fncion z) belongs o he siaion of z) N 2. If z) N 2,heny) a)z ) saisfies3.5). We claim ha condiion 3.0) implies ha y) as.really,ifno,heny) L as. An inegraion of 3.4) from o yields Aτs)) b)y ) ps)f Bτs)) y τs) ) ) ds. Inegraing once more, we ge y) b) f y τ ) )) b) Aτs)) ps)f Bτs)) y τs) ) ) ds d ps)f ) Aτs)) ds d. Bτs)) This is in conradicion o 3.0), andwe concldeha y) as.

7 Lo Advances in Difference Eqaions 208) 208:87 Page 7 of 2 Ths, we can se ha yτ)), and we obain ) τ) f ) f ps)f A τs) )) Bs) ds Bτ)) ) + B τ) ) f Bτ)) + B τ) ) ps)f ps)f A τs) )) ds τ) Aτs)) Bτs)) ) ds. 3.2) Now, we can ake lim sp on boh sides of 3.2), one ges a conradicion o 3.). The proof is complee now. Theorem 3.4 Le, for all large enogh, av) v bs) s p) d ds dv. 3.3) Then he posiive solion class N 0. Proof Assme on he conrary ha.) possesses an evenally posiive solion z) N 0. Le s denoe ha Z ) z) andz) >0.Deoz) >0andz ) <0,hen zs) ds z) z)andz ) z) 0, we obain Z)Z )+ zs) ds z)+z ) z) z). Using he fac ha f )>0andf is nondecreasing, we have f xτ))) ε >0.Aninegraion of 2.3)from o yields b) a)z ) ) ps)f x τs) )) ds ε ps) ds. Inegraing from o and conseqenly from o,weobain z ) ε av) v bs) s p) d ds dv. We ge a conradicion o 3.3), and heproofis complee. Theorem 3.5 Le 2.5) hold. Assme ha all he condiions of Theorems 3. and 3.4 hold. Then Eq..) has no posiive solions. Theorem 3.6 Le 2.5) hold. Assme ha all he condiions of Theorems 3.3 and 3.4 hold. Then Eq..) has no posiive solions. Theorem 3.7 Le 2.5) hold. Assme ha all he condiions of Corollary 3.2 and Theorem 3.4 hold. Then Eq.3.9) has no posiive solions.

8 Lo Advances in Difference Eqaions 208) 208:87 Page 8 of 2 4 Asympoic properies In his secion some asympoic properies of posiive solions of.) are reaed. Firsly, we need o ransform.). Le s denoe ha and D 0 ) p)f xτ))) q)gxσ ))), a)b) D ) a )b )+a )b), a)b) D 2 ) a)b )+2a )b), a)b) F, x), x ), x ) ) D 2 )x )+D )x )+D 0 ) ). Obviosly, Eq..) will be ransformed as follows: x )+D 2 )x )+D )x )+D 0 )0. 4.) Below we esablish he exisence crieria for Kneser solions of 4.)andsdyheasympoics of hese solions. Theorem 4. Le F,0,0,0) q)g0) p)f 0) a)b) for, m {0, }, r 0, + ) and 0 4.2) ) m+3 F, x, x 2, x 3 ) 0 4.3) for,0 ) m+i x i r i, i,2,3.then Eq.4.) has a coninm of Kneser solions saisfying he condiions ) m+i x i) ) 0 4.4) for, i 0,,2. Proof Se F )max { F, x, x 2, x 3 ) :0 ) m+i x i r i}, i,2,3, and choose δ 0, + )sosmallha 2 +) 2 +δ F τ) dτ τ. 4.5)

9 Lo Advances in Difference Eqaions 208) 208:87 Page 9 of 2 To prove he heorem, i sffices o show ha if r 0 δ +δ )2 r 2, hen for any c 0 [0, r 0 ] Eq. 4.) ogeher wih he condiion x ) ) m c ) Consider he fncions 0, ) m+i x <0, χ i, x) x, 0 ) m+i x r i, r i, ) m+i x > r i and F, x, x 2, x 3 )F, χ, x ), χ 2, x 2 ), χ 3, x 3 )). Obviosly, F, x, x 2, x 3 )F, x, x 2, x 3 ) 4.7) for,0 ) m+i x i r i, i,2,3.and F, x, x 2, x 3 ) F ) 4.8) for,x, x 2, x 3 ) R 3. On he oher hand, 4.2)and4.3)yield F, x, x 2, x 3 ) 0 4.9) for, ) m+i x i 0, i,2,3.and ) m+3 F, x, x 2, x 3 ) 0 4.0) for, ) m+i x i 0, i,2,3. By Lemma0. of [], for any posiive ineger k, he differenial eqaion x ) F, x), x ), x ) ) has a solion x k ) saisfying he bondary condiions x k ) ) m c 0, x i k + k)0, i, 2. 4.) Assming ha ) m+2 x k + k)<0,from4.9)weobain x k ) 2 x k + k) k) 2 for + k, ) m x k )<0. Bhisisimpossiblebecase ) m x k )c 0 0. So ) m+2 x k + k) )

10 Lo Advances in Difference Eqaions 208) 208:87 Page 0 of 2 According o 4.0), 4.), and 4.2), ) m+i x i) k ) 0 4.3) for + k, i 0,,2,3, 2 i0 x i) k ) i! for + k,and ) i x ) r0 4.4) x i) k + δ) i!r0 δ i, i 0,, ) Applying 4.5), 4.7), and 4.5), we ge x i) k ) 2 ji x j) k + δ) j i)! + δ ) j i + 2 i)! 3!r 0 δ i + +δ +δ τ ) 4 i F τ) dτ F τ) dτ [ ) + δ i 3!r δ) i ] δ 2 +) r i 2 r i 4.6) for + δ, i 0,,2.Onheoherhand,4.4)implies x i) k ) r i for + δ + k, i 0,,2.Therefore, x i) k ) r i 4.7) for + k, i 0,,2. I follows from 4.7), 4.3), and 4.7), ha for any posiive ineger k he fncion x k )is asolionofeq.4.)onheinerval[, + k]. By Lemma 0.2 of [], {x k )} + k conains asbseqence{x kl )} + l sch ha {x k l ) i) } + l, i 0,,2, converge niformly on every finie sbinerval of [,+ ), and x)lim l + {x kl )} for is a solion of Eq. 4.). In view of 4.)and4.3), x) saisfies condiions 4.4)and4.6). The proof of Theorem 4. is complee. 5 Examples Example 5. Consider he hird order rinomial differenial eqaion 3 2 x ) ) ) p + xλ) q arcan x σ ) )) 0 5.) 3 3 6

11 Lo Advances in Difference Eqaions 208) 208:87 Page of 2 wih p >0,q >0,λ 0, ). Now f x)x is nondecreasing and hx)arcanx)isbonded. We can easily calclae ha ps)f A τs) )) ds. ps)a τs) ) ds p s 3 6 λs v d dv ds v 2 3 Then condiions 3.)and3.7)holdre.Deo av) v 2, bs) v v s 3 s s p) d ds dv p 3 6 d ds dv condiion 3.3) holds re. Therefore, he posiive solion class N 0. Simple compaion shows ha A) , B) and 3.2), 3.0)akeheform pλ ln λ ) + 27 ) > 7 and b) 6p 7. 3 ps)f 3 ) Aτs)) ds d Bτs)) 9 7 s 7 6 p s 3 6 s s 2 3 ds d ) ds d Therefore, he posiive solion class N 2.WecanconcldehaEq.5.)hasnoposiive solions. Orreslsarealsoapplicableforhecasewhenτ). Example 5.2 Consider he hird order differenial eqaion 3 2 x ) ) ) p + x) q arcan x σ ) )) 0 5.2) 3 3 6

12 Lo Advances in Difference Eqaions 208) 208:87 Page 2 of 2 wih p >0,q > 0. We can easily calclae ha ps)f A τs) )) ds. ps)a τs) ) ds p s 3 6 s v 2 v 3 d dv ds The following seps are he same as in Example 5., hs we omi hem. Acknowledgemens This work was sppored in par by he Naional Naral Science Fondaion of China nder Gran No , in par by he Naral Science Fondaion of Gangdong Province nder Gran No. 204A , and in par by he Projecs of Science and Technology of Gangzho nder Gran No The ahors are graefl o he referee for carefl reading of he paper and for his or her sefl commens which helped hem o improve he paper. Compeing ineress The ahors declare ha hey have no compeing ineress. Ahors conribions All he ahors conribed eqally and significanly in wriing his paper. All he ahors read and approved he final manscrip. Pblisher s Noe Springer Nare remains neral wih regard o jrisdicional claims in pblished maps and insiional affiliaions. Received: 28 November 207 Acceped: 7 Febrary 208 References. Kigradze, I.T., Chanria, T.A.: Asympoic Properies of Solions of Nonaonomos Ordinary Differenial Eqaions. Klwer Academic, Dordrech 993) 2. Kigradze, I.T.: On he oscillaory characer of solions of he eqaion dm d m + a) n sign 0.Ma.Sb.65, ) in Rssian) 3. Agarwal, R.P., Grace, S.R., O Regan, D.: Oscillaion Theory for Difference and Fncional Differenial Eqaions. Klwer Academic, Dordrech 2000) 4. Kiamra, Y.: Characerizaion of oscillaion of forh order fncional differenial eqaions wih deviaing argmens. Ann. Ma. Pra Appl. 24), ) 5. Li, T., Baclíková, B., Džrina, J., Zhang, C.: Oscillaion of forh-order neral differenial eqaions wih p-laplacian like operaors. Bond. Vale Probl. 204,Aricle ID ) 6. Ryn, M., Haiyan, W.: On he exisence of posiive solions of forh-order ordinary differenial eqaions. Appl. Anal. 59 4), ) 7. Ladde, G.S., Lakshmikanham, V., Zhang, B.G.: Oscillaion Theory of Differenial Eqaions wih Deviaing Argmens. Marcel Dekker, New York 987) 8. Elsgols, L.E., Norkin, S.B.: Inrodcion o he Theory and Applicaion of Differenial Eqaions wih Deviaing Argmens. Elsevier, Amserdam 973) 9. Baršek, M., Došlá, Z.: Asympoic problems for forh-order nonlinear differenial eqaions. Bond. Vale Probl. 203, Aricle ID ) 0. Ho, C., Cheng, S.S.: Asympoic dichoomy in a class of forh-order nonlinear delay differenial eqaions wih damping. Absr. Appl. Anal. 2009,Aricle ID ). Agarwal, R., Grace, S.R., Manojlovic, J.V.: Oscillaion crieria for cerain forh order nonlinear fncional differenial eqaions. Mah. Comp. Model. 44), ) 2. Liang, H.: Asympoic behavior of solions o higher order nonlinear delay differenial eqaions. Elecron. J. Differ. Eq. 204, Aricle ID ) 3. Swanson, C.H.A.: Comparison and Oscillaion Theory of Linear Differenial Eqaions. Elsevier, Amserdam 968) 4. Elias, U.: Oscillaion Theory of Two-Term Differenial Eqaions. Mahemaics and Is Applicaions, vol Springer, Dordrech 203) 5. Džrina, J., Baclíková, B.: Propery A of differenial eqaions wih posiive and negaive erm. Elecron. J. Qal. Theory Differ. Eq. 207, AricleID27 207)