JUNXIA MENG. 2. Preliminaries. 1/k. x = max x(t), t [0,T ] x (t), x k = x(t) dt) k

Transcription

1 Electronic Journal of Differential Equations, Vol. 29(29), No. 39, pp ISSN: URL: or ftp ejde.math.txstate.edu POSIIVE PERIODIC SOLUIONS FOR LIÉNARD YPE p-laplacian EQUAIONS JUNXIA MENG Abstract. Using topological degree theory, we obtain sufficient conditions for the existence and uniqueness of positive periodic solutions for Liénard type p-laplacian differential equations. 1. Introduction In recent years, the existence of periodic solutions for the Duffing equation, Rayleigh equation and Liénard type equation has received a lot of attention. We refer the reader to [3, 5, 6, 7, 8, 9] and the references cited therein. However, as far as we know, fewer papers discuss the existence and uniqueness of positive periodic solutions for Liénard type p-laplacian differential equation. In this paper we study the existence and uniqueness of positive -periodic solutions of the Liénard type p-laplacian differential equation of the form: (ϕ p (x (t))) + f(x(t))x (t) + g(x(t)) = e(t), (1.1) where p > 1 and ϕ p : R R is given by ϕ p (s) = s p 2 s for s and ϕ p () =, f and g are continuous functions defined on R. e is a continuous periodic function defined on R with period, and >. By using topological degree theory and some analysis skill, we establish some sufficient conditions for the existence and uniqueness of -periodic solutions of (1.1). he results of this paper are new and they complement previously known results. For convenience, let us denote 2. Preliminaries C 1 := {x C 1 (R, R) : x is -periodic}, which is a Banach space endowed with the norm x = max{ x, x }, and ( 1/k. x = max x(t), t [, ] x = max t [, ] x (t), x k = x(t) dt) k 2 Mathematics Subject Classification. 34K15, 34C25. Key words and phrases. p-laplacian; positive periodic solutions; Liénard equation; topological degree. c 29 exas State University - San Marcos. Submitted February 17, 29. Published March 19, 29. Supported by grant 2765 from Scientific Research Fund of Zhejiang Provincial Education Department. 1

2 2 J. MENG EJDE-29/39 For the periodic boundary-value problem (ϕ p (x (t))) = f(t, x, x ), x() = x( ), x () = x ( ) (2.1) where f is a continuous function and periodic in the first variable, we have the following result. Lemma 2.1 ([11]). Let Ω be an open bounded set in C 1, if the following conditions hold (i) For each λ (, 1) the problem (ϕ p (x (t))) = λ f(t, x, x ), x() = x( ), x () = x ( ) has no solution on Ω; (ii) he equation F (a) := 1 f(t, a, ) dt = has no solution on Ω R; (iii) he Brouwer degree of F satisfies deg(f, Ω R, ), hen the periodic boundary value problem (2.1) has at least one periodic solution on Ω. Set Ψ(x) = x We can rewrite (1.1) in the form where q > 1 and 1 p + 1 q = 1. f(u)du, y(t) = ϕ p (x (t)) + Ψ(x(t)). (2.2) x (t) = y(t) Ψ(x(t)) q 1 sign(y(t) Ψ(x(t))), y (t) = g(x(t)) + e(t), Lemma 2.2. Suppose that the following condition holds. (A1) g is a continuously differentiable function defined on R, and g x(x) <. hen (1.1) has at most one -periodic solution. (2.3) Proof. Suppose that x 1 (t) and x 2 (t) are two -periodic solutions of (1.1). hen, from (2.3), we obtain Set it follows from (2.4) that x i(t) = y i (t) Ψ(x i (t)) q 1 sign(y i (t) Ψ(x i (t))), y i(t) = g(x i (t)) + e(t), i = 1, 2. (2.4) v(t) = x 1 (t) x 2 (t), u(t) = y 1 (t) y 2 (t), (2.5) v (t) = y 1 (t) Ψ(x 1 (t)) q 1 sign(y 1 (t) Ψ(x 1 (t))) y 2 (t) Ψ(x 2 (t)) q 1 sign(y 2 (t) Ψ(x 2 (t))), u (t) = [g(x 1 (t)) g(x 2 (t))], (2.6)

3 EJDE-29/39 POSIIVE PERIODIC SOLUIONS 3 Now, we prove that u(t) for all t R. Contrarily, in view of u C 2 [, ] and u(t + ) = u(t) for all t R, we obtain max u(t) >. t R hen, there must exist t R (for convenience, we can choose t (, )) such that u(t ) = max t [, ] which, together with g (x) <, implies that u(t) = max u(t) >, t R u (t ) = [g(x 1 (t )) g(x 2 (t ))] =, x 1 (t ) = x 2 (t ), u (t ) = ( (g(x 1 (t)) g(x 2 (t)))) t=t = [g x(x 1 (t ))x 1(t ) g x(x 2 (t ))x 2(t )]. hen u (t ) = g x(x 1 (t ))[x 1(t ) x 2(t )] = g x(x 1 (t ))[ y 1 (t ) Ψ(x 1 (t )) q 1 sign(y 1 (t ) Ψ(x 1 (t ))) y 2 (t ) Ψ(x 2 (t )) q 1 sign(y 2 (t ) Ψ(x 2 (t )))] = g x(x 1 (t ))[ y 1 (t ) Ψ(x 1 (t )) q 1 sign(y 1 (t ) Ψ(x 1 (t ))) y 2 (t ) Ψ(x 1 (t )) q 1 sign(y 2 (t ) Ψ(x 1 (t )))]. (2.7) (2.8) In view of and g x(x 1 (t )) >, u(t ) = y 1 (t ) y 2 (t ) >, (2.9) It follows from (2.8) that y 1 (t ) Ψ(x 1 (t )) q 1 sign(y 1 (t ) Ψ(x 1 (t ))) y 2 (t ) Ψ(x 1 (t )) q 1 sign(y 2 (t ) Ψ(x 1 (t ))) >. u (t ) = g x(x 1 (t ))[ y 1 (t ) Ψ(x 1 (t )) q 1 sign(y 1 (t ) Ψ(x 1 (t ))) y 2 (t ) Ψ(x 1 (t )) q 1 sign(y 2 (t ) Ψ(x 1 (t )))] >, (2.1) which contradicts the second equation of (2.7). his contradiction implies that u(t) = y 1 (t) y 2 (t) for all t R. By using a similar argument, we can also show that y 2 (t) y 1 (t) for all t R. herefore, we obtain y 2 (t) y 1 (t) for all t R. hen, from (2.6), we get g(x 1 (t)) g(x 2 (t)) for all t R, again from g x(x) <, which implies that x 2 (t) x 1 (t) for all t R. Hence, (1.1) has at most one -periodic solution. he proof is complete.

4 4 J. MENG EJDE-29/39 3. Main Results Using Lemmas 2.1 and 2.2, we obtain our main results: heorem 3.1. Let (A1) hold. Suppose that there exists a positive constant d such that (A2) g(x) e(t) < for x > d and t R, g(x) e(t) > for x and t R. hen (1.1) has a unique positive -periodic solution. Proof. Consider the homotopic equation of (1.1) as follows: (ϕ p (x (t))) + λf(x(t))x (t) + λg(x(t)) = λe(t), λ (, 1) (3.1) By Lemma 2.2, and (A1), it is easy to see that (1.1) has at most one positive - periodic solution. hus, to prove heorem 3.1, it suffices to show that (1.1) has at least one -periodic solution. o do this, we shall apply Lemma 2.1. Firstly, we will claim that the set of all possible -periodic solutions of (3.1) is bounded. Let x(t) C 1 be an arbitrary solution of (3.1) with period. By integrating two sides of (3.1) over [, ], and noticing that x () = x ( ), we have (g(x(t)) e(t)) dt =. (3.2) As x() = x( ), there exists t [, ] such that x (t ) =, while ϕ p () = we see ϕ p (x (t)) = (ϕ p (x (s))) ds t (3.3) λ f(x(t)) x (t) dt + λ g(x(t)) dt + λ e(t) dt, where t [t, t + ]. From (3.2), there exists a ξ [, ] such that g(x( ξ)) e( ξ) =. In view of (A2), we obtain x( ξ) d. hen, we have and x(t) = x( ξ) + x(t) = x(t ) = x( ξ) ξ x (s)ds d + ξ t Combining the above two inequalities, we obtain Denote x = max x(t) = max x(t) t [, ] t [ ξ, ξ+ ] max t [ ξ, ξ+ ] d ξ x (s) ds, t [ ξ, ξ + ], ξ x (s)ds d + x (s) ds, t [ ξ, ξ + ]. t {d ( x (s) ds + ξ x (s) ds. E 1 = {t : t [, ], x(t) > d}, ξ t x (s) ds)} E 2 = {t : t [, ], x(t) d}. (3.4)

5 EJDE-29/39 POSIIVE PERIODIC SOLUIONS 5 Since x(t) is -periodic, multiplying x(t) and (3.1) and then integrating it from to, in view of (A2), we get x (t) p dt = (ϕ p (x (t))) x(t)dt = λ [g(x(t)) e(t)]x(t)dt + λ E 1 [g(x(t)) e(t)]x(t)dt E 2 D x, max{ g(x(t)) e(t) : t R, x(t) d} x(t) dt (3.5) where D = max{ g(x) e(t) : x d, t R}. For x(t) C(R, R) with x(t + ) = x(t), and < r s, by using Hölder inequality, we obtain ( 1 this implies that ) 1/r ( 1 x(t) r dt ( ( 1 = hen, in view of (3.4), (3.5) and (3.6), we can get ( ( x(t) r ) s/r dt) r/s ( 1/s, x(t) dt) s ) 1/r 1dt) s r s x r s r rs x s, for < r s. (3.6) x (t) dt) p p 1 x (t) p p = p 1 x (t) p dt p 1 D x p D(d x (s) ds). (3.7) Since p > 1, the above inequality allows as to choose a positive constant M 1 such that In view of (3.3), we have x (s) ds M 1, x d x (s) ds M 1. x p 1 = max { ϕ p(x (t)) } t [, ] { t [t,t + ] = max t (ϕ p (x (s))) ds } f(x(t)) x (t) dt + g(x(t)) dt + e(t) dt [max{ f(x) : x M 1 }]M 1 + [max{ g(x) : x M 1 } + e ]. (3.8) hus, we can get some positive constant M 2 > M such that for all t R, x (t) M 2. Set Ω = {x C 1 : x M 2 + 1}, then we know that (3.1) has no solution on Ω as λ (, 1) and when x(t) Ω R, x(t) = M or

6 6 J. MENG EJDE-29/39 x(t) = M 2 1, from (A 2 ), we can see that 1 1 { g(m 2 + 1) + e(t)} dt = 1 { g( M 2 1) + e(t)} dt = 1 so condition (ii) is also satisfied. Set H(x, µ) = µx (1 µ) 1 and when x Ω R, µ [, 1] we have xh(x, µ) = µx 2 (1 µ)x 1 hus H(x, µ) is a homotopic transformation and deg{f, Ω R, } = deg{ 1 {g(m 2 + 1) e(t)} dt >, {g( M 2 1) e(t)} dt <, {g(x) e(t)} dt, {g(x) e(t)} dt >. {g(x) e(t)} dt, Ω R, } = deg{x, Ω R, }. so condition (iii) is satisfied. In view of the previous Lemma 2.1, there exists at least one solution with period. Suppose that x(t) is the -periodic solution of (1.1). Let t be the global minimum point of x(t) on [, ]. hen x ( t) = and we claim that (ϕ p (x ( t))) = ( x ( t) p 2 x ( t)). (3.9) Assume, by way of contradiction, that (3.9) does not hold. hen (ϕ p (x ( t))) = ( x ( t) p 2 x ( t)) <, and there exists ε > such that (ϕ p (x (t))) = ( x (t) p 2 x (t)) < for t ( t ε, t+ ε). herefore, ϕ p (x (t)) = x (t) p 2 x (t) is strictly decreasing for t ( t ε, t + ε), which implies that x (t) is strictly decreasing for t ( t ε, t + ε). his contradicts the definition of t. hus, (3.9) is true. From (1.1) and (3.9), we have In view of (A2), (3.1) implies x( t) >. hus, x(t) g(x( t)) e( t). (3.1) min x(t) = x( t) >, for all t R, t [, ] which implies that (1.1) has at least one positive solution with period. completes the proof. 4. An Example As an application, let us consider the following equation his (ϕ p x (t)) + e x(t) x (t) (x 9 (t) + x(t) 12) = cos 2 t, (4.1) where p = 5. We can easily check the conditions (A1) and (A2) hold. By heorem 3.1, equation (4.1) has a unique positive 2π-periodic solution. Since the periodic solution of p-laplacian equation (4.1) is positive, one can easily see that the results of this paper are essentially new.

7 EJDE-29/39 POSIIVE PERIODIC SOLUIONS 7 References [1] Xiankai Huang, Zigui Xiang; On the existence of 2π periodic solution for delay Duffing equation x (t) + g(t, x(t τ)) = p(t), Chinese Science Bullitin, 39(3), (1994). [2] Yongkun Li; Periodic solutions of the Liénard equation with deviating arguments, J. Math. Research and Exposition, 18(4), (1998). (in Chinese). [3] G. Q. Wang and S. S. Cheng; A priori bounds for periodic solutions of a delay Rayleigh equation, Applied Math, Lett., 12, (1999). [4] Genqiang Wang and Jurang Yan; On existence of periodic solutions of the Rayleigh equation of retarded type, Internat. J. Math.& Math. Sci. 23(1), (2). [5] Shiping Lu and Weigao Ge; Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument, Nonlinear Anal. MA, 56, , (24). [6] Shiping Lu and Zhanjie Gui; On the existence of periodic solutions to p-laplacian Rayleigh differential equation with a delay, Journal of Mathematical Analysis and Applications, 325, , (27). [7] Minggang Zong and Hongzhen Liang; Periodic solutions for Rayleigh type p-laplacian equation with deviating arguments, Applied Mathematics Letters, 26, 43-47, (27). [8] Bingwen Liu; Periodic solutions for Liénard type p-laplacian equation with a deviating argument, Journal of Computational and Applied Mathematics, In Press, doi:1.116/ j.cam [9] Fuxing Zhang and Ya Li, Existence and uniqueness of periodic solutions for a kind of duffing type p-laplacian equation, Nonlinear Analysis: Real World Applications, In Press, Available online 14 February 27. [1] R. E. Gaines and J. Mawhin; Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568, Springer-Verlag, Berlin, New York, (1977). [11] R. Manásevich, J. Mawhin; Periodic solutions for nonlinear systems with p-laplacian-like operators, J. Differential Equations, 145, (1998). Junxia Meng College of Mathematics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 3141, China address: mengjunxia1968@yahoo.com.cn