PERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE p(t)-laplacian EQUATION. R. Ayazoglu (Mashiyev), I. Ekincioglu, G. Alisoy

Transcription

1 Acta Universitatis Apulensis ISSN: No. 47/216 pp doi: /j.aua PERIODIC SOLUTIONS FOR A KIND OF LIÉNARD-TYPE pt)-laplacian EQUATION R. Ayazoglu Mashiyev), I. Ekincioglu, G. Alisoy Abstract. Based on Manásevich-Mawhin s continuation theorem, the existence of periodic solutions for the Liénard-type pt)-laplacian equation with a deviating argument x t) ) pt) 2 x t) + x t) + g t, x t τt))) = et). 21 Mathematics Subject Classification: 35J35; 35J6; 35J7; 35D3 Keywords: pt)-laplacian, periodic solution, Nemytskii operator, deviating argument. 1. Introduction The speciality of this paper is that we discuss the existence of T -periodic solutions for Liénard-type pt)-laplacian equation such an equation that can be derived from many fields, such as fluid mechanics and nonlinear elastic mechanics) of the from x t) ) pt) 2 x t) + x t) + g t, x t τt))) = et), 1.1) where p, τ, e : R R and g : R R R are continuous functions, p, τ, e are T -periodic, g is T -periodic in its first argument and T >. In recent years, the problem of the existence of periodic solutions of Duffing type, Liénard-type and Rayleigh type p-laplacian equations with deviating argument has received much attention. We refer the reader to see, for example, [1, 11, 12, 13]) and the references there in. These papers were studied when p > 1 constant number. However, when p = pt) is a function, i.e. p is not a constant, the situation is very crucial, and therefore, the equation like 1.1) become a very challenging problem. As a consequence, one needs additional and different approaches to deal such problems. 61

2 Motivated by the ideas in [2], we will make some a priori estimates and use topological degree theory to obtain the existence of positive periodic solutions for 1.1). This work is organized as follows. In Section 1, we introduce some necessary and preliminary results. In Section 2, we give the statement of our main result and its proof. Definition 1. We denote C T = x : x C R, R), xt + T ) xt) with the norm x = max t [,T ] xt), and C 1 T = x : x C1 R, R), xt + T ) xt) with the norm x 1 = x + x. Clearly, C T,. ) and C 1 T,. 1) are two Banach spaces. Define Let p C[, T ], R) with 1 < min pt) = t [,T ] p pt) p + = max pt) <. t [,T ] L pt) [, T ], R) = x L 1 [, T ], R) : xt) pt) dt <. The variable exponent space L pt) T [, T ], R) denotes the Banach space of T -periodic functions on [, T ] with values in R under the norm x L pt) := x pt) = inf δ > : xt) pt) δ dt 1. Define W 1,pt) [, T ], R) = x L pt) [, T ], R) : x L pt) [, T ], R) with the norm x W 1,pt) := x 1,pt) = x pt) + x pt). The space W 1,pt) [, T ], R) is denoted by the closure of C 1,pt) [, T ], R) in WT [, T ], R), where W 1,pt) T [, T ], R) := x : x W 1,pt) [, T ], R), xt + T ) xt) 62

3 for all x W 1,pt) [, T ], R). We will use x 1,pt) and x pt) for x W 1,pt) [, T ], R) in the following discussions. Moreover, if 1 < pt) p + < the spaces L pt) [, T ], R), W 1,pt) [, T ], R) and W 1,pt) [, T ], R) are separable and reflexive Banach spaces see [1],[3]). Proposition 1. [1, 3, 5] If we denote ρ px) u) = Ω ux) px) dx, u L px) Ω), then i) u px) < 1= 1; > 1) ρ px) u) < 1= 1; > 1); ii) u px) ρ px) u) u p > 1 u p px) ρ px) u) u p+ px), u px) < 1 u p+ px) px) ; iii) ρ px) u) max u p px), u p+ px) u p px) + u p+ px). Proposition 2. [6] Assume that r L + Ω) and p C + Ω). If u rx) L px) Ω), then we have min u r+ rx)px), u r rx)px) u rx) px) max u r+ rx)px), u r rx)px). Proposition 3. [1, 4] If p and q L 1 Ω) satisfy 1 < px) qx) < p x) = Npx) N px), px) < N; which satisfies for all x Ω, then there is a continuous em- +, px) N, bedding W 1,px) Ω) L qx) Ω) and then there exists a positive constant C such that u qx) C u 1,px). Theorem 1. [4] If G maps L p 1x) Ω) into L p 2x) Ω), then G is continuous and bounded, and there is a constant b and a non-negative function a L p 2x) Ω) such that p 1 x) p 2 x) fx, u) ax) + b u, 1.2) for x Ω and u R. On the other hand, if f satisfies 1.2), then G maps L p 1x) Ω) into L p 2x) Ω), and thus G is continuous and bounded. Remark 1. N f u, υ) L q t) [, T ], R) if and only if 63

4 i) a L q t) [, T ], R),at) and there are constants c 1, c 2 such that ft, u, υ) at) + c 1 u qt) q t) + c 2 υ st) q t) for u L qt) [, T ], R) and υ L st) [, T ], R). ii) if i) is satisfied, N f continuous and bounded from L qt) [, T ], R) L st) [, T ], R) into L q t) [, T ], R). Lemma 2. [2] Assume that Ω is an open bounded set in CT 1 conditions hold. such that the following 1) For each λ, 1) the problem Φp x ) ) = λ ft, x, x ), x) = xt ), x ) = x T ) 1.3) has no solution on Ω, where ft, x, x ) is a continuous function and T -periodic in the first variable. 2) The equation has no solution on Ω R N. 3) The Brouwer degree F a) = 1 T ft, a, )dt =. deg F, Ω R N, ). Then the perodic boundary value problem 1.3) has at least one T -periodic solution in Ω. 2. Main result and its proof We denote ft, x, x ) := et) x t) g t, x t τt))). Assume that f and g satisfies the following conditions: 64

5 F1) f : R R R R, a L q t) R, R), at) and s, q : R R are continuous functions with s t), q t) < pt) < p t) and there are constants c 1, c 2 such that ft, u, υ) at) + c 1 u qt) q t) + c 2 υ st) q t). F2) There exist a constant M > for any D M such that g t, D) > et) or for all t R. g t, D) < et) F3) There exist a constant L > such that for all t, x 1, x 2 R. gt, x 1 ) gt, x 2 ) L x 1 x 2 2 Consequently, we obtain the auxiliary result. For simplicity of natation, we write X = W 1,pt) [, T ], R), X = W 1,p t) [, T ], R), Y qt),st) = L qt) [, T ], R) L st) [, T ], R), Yqt) [, T ], R). Let us recall some results borrowed from [3, 7, 9] about p t)-laplacian and Nemytskii operator N f. Firstly, since s t), q t) < p t) < p t) for all t [, T ], X is compactly embedded in Y qt),st) see [3, 7, 9]). Denote by i the compact injection of X in Y qt),st) and by i : Yqt) X, i ϕ = ϕ i for all ϕ Yqt), its adjoint. Since the Caratheodory function f satisfies CAR), the Nemytskii operator N f generated by f, N f x, x ) t) = ft, x t), x t)), is well defined from Y qt),st) into Yqt), continuous, and bounded by Remark 1). To prove that Eq. 1.1) has a solution is sufficient to prove that the equation Φpt) x ) ) = x t) ) pt) 2 x t) = i N f i) x, x ) 2.1) has a solution in X. If x X satisfies 2.1) then, for all ϕ X, one has Φpt) x ) ), ϕ X,X = i N f i) x, x ), ϕ X,X = N f ix, ix ) ), iϕ Y t),st),y qt) 65.

6 Since Φ pt) is a homeomorphism of X onto X, 2.1) may be equivalently written as x = Φ pt) ) 1 [ i N f i) x, x ) ]. 2.2) Thus, compact operator Λ = Φ pt) ) 1 i N f i) : X X has a fixed point. According to the classical result of Leray, Schauder and Schaefer, a sufficient condition for Λ to have a fixed point is that a constant R > exists such that S = x X : x = λλx, x ) for some λ [, 1] B, R). Since, for λ =, the only solution of equation x = λλx, x ) is x =, it is enough to show that there exists a constant R > such that, any x X which satisfies for some λ, 1], belongs to B, R). So Then, we obtain x = λ Φ pt) ) 1 [ i N f i) x, x ) ] 2.3) ) ) x x Φ pt), λ λ X,X = = 1 λ p x t) λ pt) dt x t) pt) dt 1 λ p ρ pt) x t) ). x t) px) dt λ p 1 i N f i) x, x ), x X,X i Y qt) X N f ix, ix ) Y qt) x X. 2.4) Consider the homotopic equation of Eq. 1.1) as follows: Φpt) x t)) ) = λet) λx t) λg t, x t τt))), λ, 1). 2.5 λ ) The main result of the present paper is: Theorem 3. If the assumptions F 1), F 2), F 3) hold and > 2, then the Eq.1.1) admits at least one positive periodic solution. 66

7 First, let s give some lemmas and proofs that are necessary to prove the Theorem 3. We will prove the following lemma based on Shiping Lu and Weigao Ge s article [8]. Lemma 4. Let > 2 and α T be a constant, s CT 1 with s α. Then for x CT 1, we have x t s t)) x t) 2 dt 4α2 x t) pt) dt + 2α2 T p + 2). Proof. By using the Lemma 1 given in [8], and the Young s inequality, we obtain the inequality as follows: x t s t)) x t) 2 dt 2α 2 4α2 x t) 2 dt x t) pt) dt + 2α2 T p + 2). 2.6) This completes the proof of Lemma 4. Next, we will show that the xt) must be uniform bounded in CT 1 for any λ [, 1]. In order to make this we use F 1), 2.4), Propositions 1, 2 and 3, we have x t) pt) dt i Y qt) X N f ix, ix )) Y x X qt) qt) st) k 1 at) + c q t) 1 ix + c2 ix q t) x X Y qt) k 1 at) Y + c 1 qt) ix qt) q t) k 1 at) Y + c 1 k 1 max qt) Y + c 2 ix qt) ix q+ 1 Y qt) +c 2 k 1 ix s+ 1, ix s 1 x Y qt) Y qt) X st) q t) Y qt), ix q 1 Y qt) ) x X x X 67

8 ) k 1 M 1 + c 1 k 1 k 2 C 1 max x q+ 1 X, x q 1 X x X +c 2 k 1 k 3 C 2 max x s+ 1 X, x s 1 X x X = k 1 M 1 x X + c 1 k 1 k 2 C 1 max x q+ X, x q X + c 2 k 1 k 3 C 2 max x s+ X, x s X, 2.7) i s+ 1 Y qt) where k 1 = i Y qt) X, k 2 = max i q+ 1 Y qt), i q 1 Y qt), k 3 = max and M 1 = a Y. Since s +,q + < in particular, if x X satisfies 2.3) for some qt) λ, 1], we derive from 2.7) the set S. Consequently, for x X we obtain, i s 1 Y qt) x t) pt) dt K. 2.8) Let xt) be a T -periodic solution of Eq.2.5 λ ). Then, integrating Eq.2.5 λ ) from to T, and noticing that x ) = x T ) =, we have This implies that there exists ξ [, T ] such that g t, x t τt))) et) dt =. 2.9) gξ, xξ τξ)) eξ) =. Thus, taking this together with 2.9), we have xξ τξ)) < M. Since xt) is a T -periodic function, then there is an integer k and a constant ξ [, T ] such that ξ τξ) = kt + ξ. According to the Young s inequality, we have x x ξ ) + x t) dt M + 1 x t) pt) dt + p+ 1 T + M. If we take into account the inequality of 2.8), then we get x t) dt x K + p+ 1 T + M := K ) 68

9 By the boundary condition x) = xt ), it is easy to see that there exists t [, T ], such that x t ) =. Integrating 2.5 λ ) from t i to t, and from F 3), Lemma 2 and Young s inequality, we obtain t x t) pt) 2 x t)) dt = t t et) x t) g t, x t τt))) ) dt t t t x t) pt) 2 x t) x t) dt + g t, x t τt))) dt + t t 1 x t) pt) dt + p+ 1 T + 1 g t, x t τt))) g t, x) dt + x t) pt) dt + p+ 1 +T g d + T e 1 x t) pt) dt + p+ 1 t = x t) pt) 1 t et) dt g t, x) dt + et) dt T + L x t τt)) xt) 2 dt T + 4α2 L x t) pt) dt + 2α2 T L p + 2) +T g d + T e 1 + 4α 2 L ) K + e + g d + 2α2 L p + 2) + p + ) 1 T := K 2, 2.11) where g d = max gs, x) : s [, T ], x [ M, M]. So from 2.11), we obtain x 1 p t) max K 1 1 p 2, K := K 3. That is, x K 3. Combining this inequality with 2.1), we obtain x 1 K 1 + K 3 := K 4. Thus, we obtain that xt) is uniform bounded in CT 1 for any λ [, 1]. Let Ω = x W 1,pt) [, T ], R) : x) = xt ), x ) = x T ) and let us define the set Ω 1 = x Ω R : F x) =, F x) = 69 g t, x t τt))) et) dt.

10 We will prove that for any D Ω 1, it holds D M. For any D Ω 1, by F D) =, we have We can confirm that D M. Otherwise, by F 2) or g t, D) et) dt =. 2.12) gt, D) et) > 2.13) gt, D) et) < 2.14) which implies that F D) < or F D) >. That is a condradiction to 2.12). Finally we will prove that the condition 3) of Lemma 2 is also satisfied. For any x Ω R, from F2) we know that if D > M, such that gt, D) et) > or gt, D) et) <. In the following, we assume 2.13). As for the case of 2.14), the proof is similar, so we omit it. Let us define Ω 2 = x Ω R : λd ζ) + 1 λ)f D) =, λ [, 1], where ζ R with < ζ < M. Our aim is to show that Ω 2 is bounded by M. Argue by contrary; there exists ζ Ω 2 with D > M. Moreover, we have λd ζ) + 1 λ)f D) >, which contradicts the definition of Ω 2. That is for any D Ω 2, we always have D M. From the definition of F, it is easy to see that F : R R is completely continuous. Let h λ D) = λd ζ) + 1 λ)f D), and define Ω = x Ω : x 1 < K 4 + M. Then clearly h λ Ω R) for any λ [, 1]. By virtue of the invariance property of homotopy, we obtain deg F, Ω R, ) = deg h, Ω R, ) = deg h 1, Ω R, ) = deg [, T ], Ω R, ζ) = 1. Hitherto, we have proved that the conditions 1)-3) in Lemma 2 are all satisfied. Therefore, by Lemma 2, the Eq. 1.1) admits a solution in Ω. This completes the proof of Theorem 3. 7

11 References [1] L. Diening, P. Harjuletho, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag 211), Berlin. [2] R. Manasevich, J. Mawhin, Periodic solutions for nonlinear systems with p- Laplacian-like operators, J. Differential Equations ), [3] X.L. Fan, Dun Zhao, On the Spaces L px) Ω) and W 1,px) Ω), Journal of Mathematical Analysis and Applications ), [4] X. L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces W k,px) Ω), J. Math. Anal. Appl ), [5] O. Kovăčik, J. Răkosnik, On spaces L px) and W k,px), Czechoslovak Math. J ) 1991), [6] B. Cekic, R. Ayazoglu Mashiyev), Existence and localization results for px) -Laplacian via topological methods, Fixed Point Theory and Applications, V.21, Article ID [7] G. Dinca, A fixed point method for the p.)-laplacian, C. R. Acad.Sci.,Paris I ), [8] S. Lu, W. Ge, Periodic solutions for a kind of Liénard equation with a deviating argument, J. Math. Anal. Appl ), [9] R. Ayazoglu Mashiyev), M. Avci, Positive periodic solutions of nonlinear differential equations system with nonstandard growth, Applied Mathematics Letters ), 5-9. [1] M. L. Tang, X. G. Liu, Periodic solutions for a kind of Duffing type p-laplacian equation, Nonlin. Anal., 71 29), [11] W. S. Cheung, J. L. Ren, On the existence of periodic solutions for p-laplacian generalized Liénard equation, Nonlin. Anal., 6 25), [12] B. Xiao, B. W. Liu, Periodic solutions for Rayleigh-type p-laplacian equation with a deviating argument, Nonlin. Anal. RWA,1 29), [13] S. P. Lu, Z. J. Gui, On the existence of periodic solutions to p-laplacian Rayleigh differential equation with a delay, J. Math.Anal. Appl., ), Rabil Ayazoglu Mashiyev) Faculty of Education, Bayburt University, Bayburt, Turkey a rayazoglu@bayburt.edu.tr 71

12 Ismail Ekincioglu Faculty of Science and Arts, Dumlupınar University, Kütahya, Turkey b ismail.ekincioglu@dpu.edu.tr Gülizar Alisoy Faculty of Science and Arts, Namık Kemal University, Tekirdağ, Turkey c galisoy@nku.edu.tr 72