Existence of periodic solutions for a class of

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1 Chang and Qiao Bondary Vale Problems 213, 213:96 R E S E A R C H Open Access Existence of periodic soltions for a class of p-laplacian eqations Xiaojn Chang 1,2* and Y Qiao 3 * Correspondence: changxj1982@hotmail.com 1 School of Mathematics and Statistics, Northeast Normal University, Changchn, Jilin 1324, P.R. China 2 College of Mathematics, Jilin University, Changchn, Jilin 1312, P.R. China Fll list of athor information is available at the end of the article Abstract This paper is devoted to the existence of periodic soltions for the one-dimensional p-laplacian eqation φ p )) = ft, ), where φ p )= p 2 1 < p <+ ), f C[, 2π] R, R). By sing some asymptotic interaction of the ratios ft,) and p ft,s) ds p 2 p with the Fčík spectrm of φ p )) related to periodic bondary condition, we establish a new existence theorem of periodic soltions for the one-dimensional p-laplacian eqation. Keywords: periodic soltions; p-laplacian;fčík spectrm; Leray-Schader degree; Borsk theorem 1 Introdction and main reslts In this paper, we are concerned with the existence of soltions for the following periodic bondary vale problem: φ p )) = f t, ), ) = 2π), ) = 2π), 1.1) where φ p ) = p 2 1 < p <+ ), f C[, 2π] R, R). A soltion of problem 1.1) means that is C 1 and φ p ) is absoltely continos sch that 1.1) issatisfiedfora.e. t [, 2π]. Existence and mltiplicity of soltions of the periodic problems driven by the p- Laplacian have been obtained in the literatre by many people see [1 5]). Many solvability conditions for problem 1.1) were established by sing the asymptotic interaction at infinity of the ratio f x,) with the Fčík spectrm for φ p 2 p )) nder periodic bondary condition see e.g.,[2, 4, 6 9]). In [6], Del Pino, Manásevich and Mrúa firstly defined the Fčík spectrm for φ p )) nder periodic bondary vale condition as the set p consisting of all the pairs λ +, λ ) R 2 sch that the eqation φ p )) = λ+ + ) p 1 λ ) p Chang and Qiao; licensee Springer. This is an Open Access article distribted nder the terms of the Creative Commons Attribtion License which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited.

2 Changand Qiao Bondary Vale Problems 213, 213:96 Page 2 of 11 admits at least one nontrivial 2π-periodic soltion see [1]for p =2).Let π p =2p 1) 1 p 1 By [6], it follows that 1 t p ) 1 p dt =2p 1) 1 p π p sin π p ). { 1 p = λ +, λ ) R 2 : π p p + 1 ) λ+ p = 2π }. λ k, k Z+ Then they applied the Strm s comparison theorem and Leray-Schader degree theory to prove that problem 1.1) is solvable if the followingrelations hold: p 1 lim inf + q 1 lim inf f t, ) lim sp p 2 + f t, ) lim sp p 2 f t, ) p 2 p 2, f t, ) p 2 q 2, niformly for a.e. t [, 2π]withp 1, q 1, p 2, q 2 > satisfying 2π k +1)π p < 1 p p2 + 1 p q2 1 p p1 + 1 p q1 < 2π kπ p, k Z +. Clearly, in this case, we have [p 1, q 1 ] [p 2, q 2 ]) p =, which is sally called that the nonlinearity f is nonresonant with respect to the Fčík spectrm p.in[11], Anane and Dakkak obtained a similar reslt by sing the property of nodal set for eigenfnctions. If f f t,) is resonant with respect to p, i.e.,thereexistsλ +, λ ) p sch that lim + = λ p 2 +, f t,) lim = λ p 2 niformly for a.e. t [, 2π], together with the Landesman-Lazer type condition, Jiang [9] obtained the existence of soltions of 1.1) by applying the variational methods and symplectic transformations. In these works, either f is resonant or nonresonant with respect to p, the solvability of problem 1.1) was assred by assming that the ratio f t,) stays at infinity in the pointwise sense asymptotically between two p 2 consective crves of p.notethat f t, s) 2Ft, s) 2Ft, s) lim inf lim inf lim sp lim sp s ± s s ± s 2 s ± s 2 s ± f t, s), s we can see that the conditions on the ratio 2Ft,s) are more general than that on the ratio s 2. Recently, Li andli [2] stdied the nondissipative p-laplacianeqation f t,s) s φ p )) = c φp )) + g) pt), 1.2) where c > is a constant. Define G)= gs) ds. They proved that 1.2)issolvablender the following assmptions: 1) There exist b, d 1, d 2 >sch that d 1 g) d p 2 2 for all b; pg) pg) 2) lim + p = λ +, lim p = λ with λ +, λ )/ p. Here, the potential fnction G is nonresonant with respect to p and the ratio g) is not p 2 reqired to stay at infinity in the pointwise sense asymptotically between two consective branches of p and it may even cross at infinity mltiple Fčík spectrm crves.

3 Changand Qiao Bondary Vale Problems 213, 213:96 Page 3 of 11 In this paper, we want to obtain the solvability of problem 1.1) bysingtheasymptotic interaction at infinity of both the ratios f x,) pft,) and p 2 p with the Fčík spectrm for φ p )) nder periodic bondary condition. Here, Ft, )= f t, s) ds.thegoalisto obtain the existence of soltions of 1.1) by reqiring neither the ratio f x,) stays at infinity in the pointwise sense asymptotically between two consective branches of p nor p 2 pft,) the limits lim ± p exist. We shall prove that problem 1.1) admits a soltion nder the assmptions that the nonlinearity f has at most p 1)-linear growth at infinity and the ratio f t,) has a p 2 L1 limit as ±,whiletheratio pft,) p stays at infinity in the pointwise sense asymptotically between two consective branches of p. Or reslt will complement the reslts in the literatre on the solvability of problem 1.1) involvingthe Fčík spectrm. For related works on resonant problems involving the Fčík spectrm, we also refer the interested readers to see [12 19] and the references therein. Or main reslt for problem 1.1) now reads as follows. Theorem 1.1 Assme that f C[,2π] R) and the following conditions hold: i) There exist constants C 1, M >sch that f t, ) C 1 1+ p 1 ), a.e. t [, 2π], M; 1.3) ii) There exists η ± L, 2π) sch that 2π f t, ) p 2 η± t) dt as ± ; 1.4) iii) There exist constants p 1, p 2, q 1, q 2 >sch that pft, ) pft, ) p 1 lim inf lim sp p + p + p 2, 1.5) pft, ) pft, ) q 1 lim inf lim sp q p p 2 1.6) hold niformly for a.e. t [, 2π] with [p1, p 2 ] [q 1, q 2 ] ) p =. 1.7) Then problem 1.1) admits a soltion. Remark If f t, )=at) p 2 + bt) p 2 + e)+ht), where a, b, h C[, 2π] with p 1 at) p 2, q 1 bt) q 2 and p 1, p 2, q 1, q 2 >satisfy1.7), e is continos on R and e) lim + =,thenlim p 2 + pft,) pft,) p = at)andlim p = bt). By Theorem 1.1, it follows that problem 1.1) admits a soltion. It is easily seen that the reslt of [16] cannot be applied to this case. Note that one can also obtain the solvability of 1.1)inthiscaseby the reslt of [6], while in Theorem 1.1 we do not reqire the pointwise limit at infinity of the ratio f t,) as in [6]. p 2 For convenience, we introdce some notations and definitions. L p, 2π) 1<p < ) denotes the sal Sobolev space with inner prodct, p and norm p,respectively.

4 Changand Qiao Bondary Vale Problems 213, 213:96 Page 4 of 11 C m [, 2π] m N) denotes the space of m-times continos differential real fnctions with norm x C m = max xt) + max ẋt) + + max x m) t). t [,2π] t [,2π] t [,2π] 2 Proof of the main reslt Denote by deg the Leray-Schader degree. To prove Theorem 1.1, we need the following reslts. Lemma 2.1 [2] Let be a bonded open region in a real Banach space X. Assme that K : R is completely continos and p / I K) ). Then the eqation I K)x)=p has a soltion in if degi K,, p). Lemma 2.2 Borsk Theorem [2]) Assme that X is a real Banach space. Let be a symmetric bonded open region with θ. Assme that K : R is completely continos and odd with θ / I K) ). Then degi K,, θ) is odd. Proof of Theorem 1.1 Take λ +, λ ) [p 1, p 2 ] [q 1, q 2 ]. Consider the following homotopy problem: φ p )) =1 μ)λ + + ) p 1 λ ) p 1 )+μft, ) f μ t, ), 2.1) ) = 2π), ) = 2π), where μ [, 1]. By 1.3) and the reglarity argments, it follows that C 1 [, 2π], and frthermore there exists a, b R + sch that, if is a soltion of problem 2.1), then C 1 a + b. 2.2) In what follows, we shall prove that there exists C > independent of μ [, 1] sch that C for all possible soltion t)of2.1). Assme by contradiction that there exist a seqence of nmber {μ n } [, 1] and corresponding soltions { n } of 2.1)schthat n + as n ) Set z n = n n.obviosly, z n =1.Define f t, n ), α n t)= n p 2 n t)>m, n, n t) M and f t, n ), β n t)= n p 2 n t)< M, n, n t) M.

5 Changand Qiao Bondary Vale Problems 213, 213:96 Page 5 of 11 By 1.3), there exists M >schthat α n t), β n t) M, a.e.t [, 2π]. Then there exist α, β L, 2π)schthat α n t) α t), β n t) β t) inl, 2π). 2.4) In addition, sing 1.3) and the reglarity argments, there exists M 1 >schthat,for each n, wehave z n C 1 M 1, and ths there exists z C 1 [, 2π] sch that, passing to a sbseqence if possible, z n z in C 1 [, 2π]. 2.5) Clearly, z =1.Inviewof{μ n } [, 1], there exists μ [, 1] sch that, passing to a sbseqence if possible, μ n μ as n ) Note that for μ =,problem2.1) has only the trivial soltion, it follows that μ, 1]. Denote ᾱt)=1 μ )λ + + μ α t), βt)=1 μ )λ + μ β t). It is easily seen that z is a nontrivial soltion of the following problem: φ p z )) = ᾱt)z + )p 1 βt)z )p 1, z ) = z 2π), z ) = z 2π). 2.7) We now distingish three cases: i) z changes sign in [, 2π]; ii) z t), t [, 2π]; iii) z t), t [, 2π]. In the following, it will be shown that each case leads to a contradiction. Case i). Let I + = { t [, 2π]:z t)> }, I = { t [, 2π]:z t)< }, I = { t [, 2π]:z t)= }. Then, as n +,weget n t) +, t I +, n t), t I. In addition, as shown in [11], we have I =.Define ηt), t I +, η + t)= α t), t I,

6 Changand Qiao Bondary Vale Problems 213, 213:96 Page 6 of 11 and β t), t I +, η t)= ηt), t I. By 1.4)and2.4), it follows that α t) η + t), β t) η t), a.e. t [, 2π]. Ths, z satisfies φ p z )) = αt)z + )p 1 βt)z )p 1, z ) = z 2π), z ) = z 2π). 2.8) Here, αt)=1 μ )λ + + μ η + t), βt)=1 μ )λ + μ η t). Now we prove that there exist n Z + and < κ 1 <1<κ 2 sch that κ 1 max n min n κ 2, n n. 2.9) In fact, if not, we assme, by contradiction, that there exists a sbseqence of { n },westill denote it as { n } with max n and min n,schthat max n min n or max n min n +. Combing with 2.5), z =1andthefactthatz changes sign, we obtain max / n min n n n A contradiction. Hence, 2.9)holds. For any t, μ) [, 2π] [, 1], define ) max z min z >. f 1 t, s, μ)=f μ t, s) αt)s p 1, s R +, f 2 t, r, μ)=f μ t, r) βt) r p 2 r, r R and F 1 t, s, μ)= s f 1 t, τ, μ) dτ, r F 2 t, r, μ)= f 2 t, τ, μ) dτ. Denote s n = max n, r n = min n.thenby2.9) it follows that s n + and r n. Taking t n sch that n t n )=s n, tn is the nearest point satisfying t n < t n and n tn )=. Since tn, t n [, 2π], there exist t, t [, 2π]schthat t n t, t n t as n )

7 Changand Qiao Bondary Vale Problems 213, 213:96 Page 7 of 11 By 2.5), we obtain z t )=,z t) =max t [,2π] z t). Note that n +, wehave n t) +, t t, t). Hence, together with μ n μ and 1.4), there exist sbseqences of { n } and {μ n }, we still denote them as { n } and {μ n },schthat,fora.e.t [, 2π], f 1 t, n τ), μ n ) n τ)) p 1, a.e. τ t, t ). Using 1.3), for a.e. t [, 2π], { f 1 t, n τ),μ n ) n τ)) p 1 } is niformly bonded with respect to τ t, t), we obtain by the Lebesge dominated convergence theorem that t t f 1 t, n τ), μ n ) n τ)) p 1 dτ, niformly for a.e. t [, 2π]. Ths, tn f 1 t, n τ), μ n ) n τ)) p 1 dτ t = f 1 t, n τ), μ n ) t n τ)) p 1 dτ + f 1 t, n τ), μ n ) tn n τ)) p 1 dτ + f 1 t, n τ), μ n ) n τ)) p 1 dt t n t n t, niformly for a.e. t [, 2π]. 2.11) t By 1.4)and2.2), we get p F 1 t, s n, μ n ) = ) ) ) p F 1 t, n t n ), μ n p F 1 t, n t n, μn tn ) = p f 1 t, n τ), μ n n τ) dτ p = p C tn tn tn tn tn tn t n ) f 1 t, n τ), μ n dτ n f 1 t, n τ), μ n ) n τ)) p 1 z n τ) ) p 1 dτ n p 1 n f 1 t, n τ), μ n ) n τ)) p 1 dτsp n. In view of 2.11), we obtain that p F 1 t, s n, μ n ) 2.12) s p n holds niformly for a.e. t [, 2π]. Similarly, p F 2 t, r n, μ n ) r n p 2.13) holds niformly for a.e. t [, 2π].

8 Changand Qiao Bondary Vale Problems 213, 213:96 Page 8 of 11 On the other hand, for {s n }, {r n } satisfying 2.12)-2.13), denoting pft, s n ) ξ 1 t)=lim inf, n + s n p pft, r n ) ξ 2 t)=lim inf, n + r n p we obtain by 1.5)-1.6)that p 1 ξ 1 t) q 1, p 2 ξ 2 t) q 2, a.e.t [, 2π]. 2.14) Using μ n μ,wehave p F 1 t, s n, μ n ) lim inf n + s n p [ = lim inf n + 1 μ n )λ + + μ n pft, s n ) s n p αt) = μ ξ1 t) η + t) ), a.e.t [, 2π], 2.15) p F 2 t, r n, μ n ) lim inf n + r n p [ ] pft, r n ) = lim inf 1 μ n )λ + μ n βt) n + r n p ] = μ ξ2 t) η t) ), a.e.t [, 2π]. 2.16) We claim that there exists sbinterval I 1 [, 2π]with I 1 >schthat ξ 1 t) η + t), t I 1, 2.17) or sbinterval I 2 [, 2π]with I 2 >schthat ξ 2 t) η t), t I ) Indeed, if not, we assme that η + t)=ξ 1 t), η t)=ξ 2 t), a.e. t [, 2π]. Together with the choosing of λ +, λ and 2.14), we get p 1 αt) q 1, p 2 βt) q 2, a.e.t [, 2π]. Then by 1.7), it follows that z. A contradiction. Combining 2.12)-2.13)with2.15)- 2.18), we obtain a contradiction. Case ii). In this case, we have s n + and {r n } is niformly bonded. Using similar argments as in Case i), by 1.4)and2.4) it follows that α t) η + t), t [, 2π]. Taking f + = f 1, F + = F 1,a.e.t [, 2π]. We can see that there exists sbseqence

9 Changand Qiao Bondary Vale Problems 213, 213:96 Page 9 of 11 of {s n },whichisstilldenotedby{s n },schthat p F + t, s n, μ n ) 2.19) s p n holds niformly for a.e. t [, 2π]. On the other hand, for {s n } satisfying 2.19), denoting ξ + pft, s n ) t)=lim inf, n + s n p we obtain by 1.5)that p 1 ξ + t) q 1, a.e.t [, 2π]. 2.2) Using μ n μ,wehave p F + t, s n, μ n ) lim inf n + s n p [ = lim inf n + 1 μ n )λ + + μ n pfs n ) s n p αt) ] = μ ξ + t) η + t) ), a.e.t [, 2π]. 2.21) We shall show that there exists sbinterval I + [, 2π]with I + sch that ξ + t) η + t), t I ) Infact,ifnot,weassmethatη + t)=ξ + t), a.e. t [, 2π]. By the choosing of λ + and 2.2), we get p 1 αt) q 1,a.e.t [, 2π]. Ths, z is a nontrivial soltion of the following problem: φ p z )) = αt)z p 1, z ) = z 2π), z ) = z 2π). 2.23) Taking 1 as test fnction in problem 2.23), we get 2π = αt)z p 1 dt. 2.24) By αt) p 1 >fora.e.t [, 2π], it follows that z t) =fora.e.t [, 2π], which is contrary to that z =1.Hence,2.22) holds. Clearly, 2.21)-2.22) contradict 2.19). Case iii). In this case, r n and {s n } is niformly bonded. Similar argments as in Case ii) imply a contradiction. In a word, 2.3) cannot hold, and hence by 2.2) thereexistsc > independent of μ [, 1] sch that, if is a soltion of problem 2.1), then C 1 C. 2.25)

10 Changand Qiao Bondary Vale Problems 213, 213:96 Page 1 of 11 Note that, for each h L, 2π), the problem φ p )) + φ p )=ht), ) = 2π), ) = 2π) 2.26) has a niqe soltion G p h) C 1 [, 2π]. Clearly, the operator G p seen as an operator from C[, 2π] intoc 1 [, 2π] is completely continos. Define ψ : C 1 [, 2π] C[, 2π] by ψ)t) =f t, t)). Then solving problem 1.1) is eqivalent to finding soltions in C 1 [, 2π]oftheeqation G p ψ) ) =. Let α, β) [p 1, q 1 ] [p 1, q 2 ]. Define the operator T α,β : C 1 [, 2π] C 1 [, 2π]byT α,β )= G p φ p )+α + ) p 1 β ) p 1 ). Denote B R = { C 1 [, 2π] : C 1 < R, R R}. Clearly, degi T α,β, B R, ) is well defined for all R >.Owingtoλ + λ >,thereisacontinos crve ατ), βτ), τ [, 1], whose image is in R 2 \ p and λ, λ) R \ p sch that α), β)) = λ +, λ ), α1), β1)) = λ, λ). From the invariance property of Leray-Schader degree nder compact homotopies, it follows that the degree degi T ατ),βτ), B R,)isconstant for τ [, 1]. Obviosly, the operator T λ,λ is odd. By the Borsk s theorem, it follows that degi T λ,λ, B R,) forallr >.Ths, degi T λ+,λ, B R,), R >. Consider the following homotopy: Hμ, )=G p φp )+1 μ) λ + + ) p 1 λ ) p 1) + μψ) ), for μ, ) [, 1] C 1 [, 2π]. By 2.25), we can see that there exists R >schthat Hμ, ), μ [, 1], B R. From the invariance property of Leray-Schader degree, it follows that deg I H1, ), B R, ) = deg I H, ), B R, ) = degi T λ+,λ, B R,). Hence, problem 1.1)has a soltion. The proofis complete. Competing interests The athors declare that they have no competing interests. Athors contribtions All athors read and approved the final manscript. Athor details 1 School of Mathematics and Statistics, Northeast Normal University, Changchn, Jilin 1324, P.R. China. 2 College of Mathematics, Jilin University, Changchn, Jilin 1312, P.R. China. 3 College of Mathematics and Information Science, Shaanxi Normal University, Xi an, 7162, P.R. China.

11 Changand Qiao Bondary Vale Problems 213, 213:96 Page 11 of 11 Acknowledgements The first athor sincerely thanks Professor Yong Li and Doctor Yixian Gao for their many sefl sggestions and the both athors thank Professor Zhi-Qiang Wang for many helpfl discssions and his invitation to Chern Institte of Mathematics. The first athor is partially spported by the NSFC Grant ), NSFJP Grant ) and FSIIP of Jilin University ). The second athor is partially sp- ported by NSFC Grant ). Received: 26 September 212 Accepted: 5 April 213 Pblished: 19 April 213 References 1. Aizicovici, S, Papageorgio, NS, Staic, V: Nonlinear resonant periodic problems with concave terms. J. Math. Anal. Appl. 375, ) 2. Li, W, Li, Y: Existence of periodic soltions for p-laplacian eqation nder the frame of Fčík spectrm. Acta Math. Sin. Engl. Ser. 27, ) 3. Manásevich, R, Mawhin, J: Periodic soltions for nonlinear systems with p-laplacian-like operators. J. Differ. Eq. 145, ) 4. Reichel, W, Walter, W: Strm-Lioville type problems for the p-laplacian nder asymptotic nonresonance conditions. J. Differ. Eq. 156, ) 5. Yang, X, Kim, Y, Lo, K: Periodic soltions for a generalized p-laplacian eqation. Appl. Math. Lett. 25, ) 6. Del Pino, M, Manásevich, R, Mrúa, A: Existence and mltiplicity of soltions with prescribed period for a second order qasilinear ODE. Nonlinear Anal. 18, ) 7. Fabry, C, Fayyad, D: Periodic soltions of second order differential eqations with a p-laplacian and asymmetric nonlinearities. Rend. Ist. Mat. Univ. Trieste 24, ) 8. Fabry, C, Manásevich, R: Eqations witha p-laplacian and an asymmetric nonlinear term. Discrete Contin. Dyn. Syst. 7, ) 9. Jiang, M: A Landesman-Lazer type theorem for periodic soltions of the resonant asymmetric p-laplacian eqation. Acta Math. Sin. 21, ) 1. Drábek, P: Solvability and Bifrcations of Nonlinear Eqations. Pitman Research Notes in Mathematics, vol ) 11. Anane, A, Dakkak, A: Nonexistence of nontrivial soltions for an asymmetric problem with weights. Proyecciones 19, ) 12. Boccardo, L, Drábek, P, Giachetti, D, Kćera, M: Generalization of Fredholm alternative for nonlinear differential operators. Nonlinear Anal. 1, ) 13. Drábek, P, Invernizzi, S: On the periodic bondary vale problem for forced Dffing eqation with jmping nonlinearity. Nonlinear Anal. 1, ) 14. Fonda, A: On the existence of periodic soltions for scalar second order differential eqations when only the asymptotic behavior of the potential is known. Proc. Am. Math. Soc. 119, ) 15. Habets, P, Omari, P, Zanolin, F: Nonresonance conditions on the potential with respect to the Fčík spectrm for the periodic bondary vale problem. Rocky Mt. J. Math. 25, ) 16. Li, W, Li, Y: Existence of 2π-periodic soltions for the non-dissipative Dffing eqation nder asymptotic behaviors of potential fnction. Z. Angew. Math. Phys. 57, ) 17. Omari, P, Zanolin, F: Nonresonance conditions on the potential for a second-order periodic bondary vale problem. Proc.Am.Math.Soc.117, ) 18. Zhang, M: Nonresonance conditions for asymptotically positively homogeneos differential systems: the Fčík spectrm and its generalization. J. Differ. Eq. 145, ) 19. Zhang, M: The rotation nmber approach to the periodic Fčík spectrm. J. Differ. Eq. 185, ) 2. Deimling, K: Nonlinear Fnctional Analysis. Springer, New York 1985) doi:1.1186/ Cite this article as: Chang and Qiao: Existence of periodic soltions for a class of p-laplacian eqations. Bondary Vale Problems :96.