Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument

Advances in Difference Equations, Feb 2016

In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument: ( φ p ( x ′ ( t ) ) ) ′ + f ( x ( t ) ) x ′ ( t ) + g ( t , x ( t − σ ) ) = e ( t ) . By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established. MSC: 34C25, 34K13, 34K40.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.advancesindifferenceequations.com/content/pdf/s13662-015-0721-2.pdf

Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument

Xin and Cheng Advances in Difference Equations Positive periodic solution of p-Laplacian Yun Xin 0 1 3 Zhibo Cheng 0 1 2 0 University , Jiaozuo, 454000 , China 1 Technology , Henan Polytechnic 2 School of Mathematics and Information Science, Henan Polytechnic University , Jiaozuo, 454000 , China 3 College of Computer Science and Technology, Henan Polytechnic University , Jiaozuo, 454000 , China In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument: By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established. positive solution; p-Laplacian; Liénard equation; singularity; deviating - available at the end of the article argument 1 Introduction with singularity and deviating argument: ϕp x (t) + f x(t) x (t) + g t, x(t – σ ) = e(t), (.) solutions of the Liénard equation with a singularity and a deviating argument, x (t) + f x(t) x (t) + g t, x(t – σ ) = , where σ is a constant. When g has a strong singularity at x =  and satisfies a new small force condition at x = ∞, the author proved that the given equation has at least one positive T -periodic solution. However, the Liénard type differential equation (.), in which there is a p-Laplacian Liénard type differential equation, has not attracted much attention in the literature. There are not so many existence results for (.) even as regards the p-Laplacian Liénard type differential equation with singularity and deviating argument. In this paper, we try to fill this gap and establish the existence of a positive periodic solution of (.) using coincidence degree theory. Our new results generalize in several aspects some recent results contained in [, ]. 2 Preparation Let X and Y be real Banach spaces and L : D(L) ⊂ X → Y be a Fredholm operator with index zero, here D(L) denotes the domain of L. This means that Im L is closed in Y and dim Ker L = dim(Y / Im L) < +∞. Consider supplementary subspaces X, Y of X, Y , respectively, such that X = Ker L ⊕ X, Y = Im L ⊕ Y. Let P : X → Ker L and Q : Y → Y denote the natural projections. Clearly, Ker L ∩ (D(L) ∩ X) = {} and so the restriction LP := L|D(L)∩X is invertible. Let K denote the inverse of LP. Let be an open bounded subset of X with D(L) ∩ = ∅. A map N : → Y is said to be L-compact in if QN ( ) is bounded and the operator K (I – Q)N : → X is compact. Lemma . (Gaines and Mawhin []) Suppose that X and Y are two Banach spaces, and L : D(L) ⊂ X → Y is a Fredholm operator with index zero. Let ⊂ X be an open bounded set and N : → Y be L-compact on . Assume that the following conditions hold: () Lx = λNx, ∀x ∈ ∂ ∩ D(L), λ ∈ (, ); () Nx ∈/ Im L, ∀x ∈ ∂ ∩ Ker L; () deg{JQN , ∩ Ker L, } = , where J : Im Q → Ker L is an isomorphism. Then the equation Lx = Nx has a solution in ∩ D(L). For the sake of convenience, throughout this paper we will adopt the following notation: |u|p =  T |u|∞ = max u(t) , t∈[,T] |u| = min u(t) , t∈[,T]  T |u|p dt  p T h(t) dt. Lemma . ([]) If ω ∈ C(R, R) and ω() = ω(T ) = , then ω(t) p dt ≤ T πp p  T ω (t) p dt, where  ≤ p < ∞, πp =  (p–)/p (– psd–ps )/p = pπs(ipn–(π)/p/)p .  T  T x(t) p dt x(t) p dt  p  p T ≤ πp T T   = ≤ T ≤ πp = πTp  T T  T  x (t) p dt  + dT p .  p  p T   p  p ω(t) + x(t) p dt ω(t) p dt  p + x(t) p dt  p ω (t) p dt x (t) p dt  + dT p  + dT p . Lemma . If x ∈ C(R, R) with x(t + T ) = x(t), and t ∈ [, T ] such that |x(t)| < d, then In order to apply the topological degree theorem to study the existence of a positive periodic solution for (.), we rewrite (.) in the form ⎧⎨ x(t) = ϕq(x(t)), ⎩ x(t) = –f (x(t))x(t) – g(t, x(t – σ )) + e(t), where p + q = . Clearly, if x(t) = (x(t), x(t)) is an T -periodic solution to (.), then x(t) must be an T -periodic solution to (.). Thus, the problem of finding an T -periodic solution for (.) reduces to finding one for (.). Now, set X = Y = {x = (x(t), x(t)) ∈ C(R, R) : x(t + T ) ≡ x(t)} with the norm x = max{|x|∞, |x|∞}. Clearly, X and Y are both Banach spaces. Meanwhile, define L : D(L) = x ∈ C R, R : x(t + T ) = x(t), t ∈ R ⊂ X → Y by (Lx)(t) = x(t) x(t) and N : X → Y by (Nx)(t) =  T T Ker L ∼= R, Im L = y ∈ Y : y(s) y(s) ds =   ψ (t) = lim sup g(t, x) x→+∞ xp– g(t, x) ≤ ψ (t) + ε x + gε(t), exists uniformly a.e. t ∈ [, T ], i.e., for any ε >  there is gε ∈ L(, T ) such that for all x >  and a.e. t ∈ [, T ]. Moreover, ψ ∈ C(R, R) and ψ (t + T ) = ψ (t). For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel: (H) (Balance condition) There exist constants  < D < D such that if x is a positive continuous T -periodic function satisfying g t, x(t) dt = , then  T D ≤ x(τ ) ≤ D, for some τ ∈ [, T ]. (.) (.) (.) (H) (Degree condition) g¯(x) <  for all (...truncated)


This is a preview of a remote PDF: http://www.advancesindifferenceequations.com/content/pdf/s13662-015-0721-2.pdf

Yun Xin, Zhibo Cheng. Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument, Advances in Difference Equations, 2016, pp. 41, 2016, DOI: 10.1186/s13662-015-0721-2