Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian

Journal of Inequalities and Applications, Nov 2012

By means of the generalized Mawhin’s continuation theorem, we present some sufficient conditions which guarantee the existence of at least one T-periodic solution for a generalized Liénard neutral functional differential system with p-Laplacian. MSC: 34B15, 34L30.

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Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian

Journal of Inequalities and Applications p-Laplacian Qing Yang 0 1 2 Bo Du 0 1 2 0 Jiangsu 223300 , P.R. China 1 Huaiyin Normal University , Huaiyin 2 Department of Mathematics By means of the generalized Mawhin's continuation theorem, we present some sufficient conditions which guarantee the existence of at least one T -periodic solution for a generalized Liénard neutral functional differential system with p-Laplacian. n periodic solutions; neutral equations; generalized Mawhin's continuation 1 Introduction tial system: ϕp(x) = |x|p–x = A : CT → CT , continuous bounded inverse A– satisfying A–f (t) = if |c| < , ∀f ∈ CT , j≥ c–jf (t + jτ ), if |c| > , ∀f ∈ CT . j= ji= c(t – (i – )τ )f (t – jτ ), c < , ∀f ∈ CT , ∞ j∞= ji+= c(t+iτ) f (t + jτ + τ ), σ > , ∀f ∈ CT . ⎧  A–f (t) dt ≤ ⎨⎩ σ––c TT||ff((tt))||ddtt,, cσ><,,∀∀ff ∈∈CCTT., 2 Main lemmas |x| = max x(t) , ≤t≤T x(t) = U is a complex such that UBU– = Eλ = diag(J, J, . . . , Jn) is a Jordan’s normal matrix, where A : CT → CT , Furthermore, we suppose that γ (t) ∈ C(R, R) with γ (t) < , ∀t ∈ R. It is obvious that the function t – γ (t) has a unique inverse denoted by μ(t). Lemma . ([]) Suppose that the matrix U and the operator A are defined by (.) and (.), respectively, and for all i = , , . . . , l, |λi| = . Then A has its inverse A– : CT → CT with the following properties: () A– ≤ |U–||U|σ, σ = li= jn=i jk= |–λi|k . () For all f ∈ CT , T |[A–f ](s)|p ds ≤ |U–|p|U|pσ T |f (s)|p ds, p ∈ [, +∞), where p ∈ [, +∞) and q >  is a constant with /p + /q = . R () A–f ∈ CT , [A–f ] (t) = [A–f ](t), for all f ∈ CT , t ∈ . Definition . ([]) Let X and Z be two Banach spaces with norms · X , · Z, respectively. A continuous operator M : X ∩ dom M → Z M : X ∩ dom M → Z Gl(a) = Gl(a) = . Lemma . ([]) The function Gl has the following properties: () For any fixed l ∈ Z, there must be a unique a˜ = a˜ (l) such that the equation 3 Main results A : Z → Z, R (Ax)(t) = x(t) – Bx(t – τ ), t ∈ , M : dom M ∩ X → Z, (Mx)(t) = ϕp (Ax) then (Nλx)(t) = λF. By (.)-(.), Eq. (.) is equivalent to the operator equation Nx = Mx, where N = N . Then we have Im M = z ∈ Z : P : X → Ker M, Q : Z → Z/ Im M, " λ F r, x(r) – (QF)(r) dr ds , t ∈ [, T ], " λ F r, x(r) – (QF)(r) dr ds, t ∈ [, T ], Hence, R(x, λ) is equicontinuous on ¯ × [, ]. By using the Arzelà-Ascoli theorem, we have R(x, λ) is completely continuous on ¯ × [, ]. Secondly, we show that Nλ is M-compact in four steps, i.e., the conditions of Definition . are all satisfied. Step . By Q = Q, we have Q(I – Q)Nλ( ¯ ) = θ , so (I – Q)Nλ( ¯ ) ⊂ Ker Q = Im M, here θ is an n-dimension zero vector. On the other hand, ∀z ∈ Im M. Clearly, Qz = θ , so z = z – Qz = (I – Q)z, then z ∈ (I – Q)Z. So, we have (I – Q)Nλ( ¯ ) ⊂ Im M ⊂ (I – Q)Z. Step . We show that QNλx = θ , λ ∈ (, ) ⇔ QNx = θ , ∀x ∈ . Because QNλx = T λF dr = θ , we get T F dr = θ , i.e., QNx = θ . The inverse is true. Step . When λ = , from the above proof, we have ax = θ . So, we get R(·, ) = θ . ∀x ∈ λ = {x ∈ ¯ : Mx = Nλx}, we have (ϕp[(Ax) ]) = λF and QF = θ . In this case, when ax = ϕp[(Ax) ()], we have " λ F r, x(r) – (QF)(r) dr ds ϕp (Ax) (r)  s (Ax) (s) ds  t  t  t = A– = A– = A– (Ax) (s) ds = (I – P)x (t).  s  s " λ F r, x(r) – (QF)(r) dr ds ϕp (Ax) (r) Step . ∀x ∈ ¯ , we have = #ϕp#! = #ϕp# (Ax)() + AA– " " $$ λ F r, x(r) – (QF)(r) dr ds " λ F r, x(r) – (QF)(r) dr ds (H) xigi(xi) > , ∀xi ∈ R, |xi| > D, for each i = , , . . . , n, (H) |gi(u) – gi(u)| ≤ l|u – u|, u, u ∈ R, for each i = , , . . . , n, (H) |fi(xi)| ≥ σ , xi ∈ R, for each i = , , . . . , n. σ > √l max γ (t) t∈[,T] σ > √l max γ (t) , t∈[,T] for  < q <  or U– |U|σ√nTfR <  for q = , ϕp (Ax) Integrating both sides of (.) over [, T ], we have g x t – γ (t) dt = θ , xi(ηi) ≤ D, for each i = , , . . . , n. |xi| ≤ D + xi(s) ds, for each i = , , . . . , n. By (.), we have x (s) ds |x| ≤ By (.), we have By assumption (H), we have ϕp (Ax) dt = fi xi(t) xi(t) e(t) dt . and integrating them over [, T ], we have ϕp (Ax) x (t) f x(t) x (t) dt From (.) and (.), we have fi xi(t) fi xi(t) xi(t) e(t) dt. Then by (.), x (s) ds ≤ R. By (.), we have |x| ≤ n(D + R) := R. e(t) dt ϕp (Ax) ≤ fR x (t) + gR + |e|, y  ≤ fR x (t) + gR + |e| by (.), we get y(t) ≤ nT y  ≤ nTfR x (t) + nTgR + nT |e| (Ax) (t) ≤ nTfR x (t) + nTgR + nT |e| By (.) and Lemma ., we have x (t) = A–Ax (t) ≤ U– |U|σ (Ax) (t) nTfR x (t) + nTgR + nT |e| nTfR x (t) + nTgR + nT |e| nTfR x (t) nTfR |x (t)| pendent on p only, such that ( + x)p <  + ( + p)x, ∀x ∈ , h(p) . √ Now, we consider ( nTfR |x (t)| + nTgR + nT |e|)q–. In the formal case, we get x (t) < Case .. If nTfR x (t) + nTgR + nT |e| nT |e| nTfR h nTfR x (t) nTfR x (t) := M. nTfR x (t) nTfR x (t) nT |e|) $ nTgR + √ nT |e| ( nTfR )q– x (t) x (t) ≤ U– |U|σ(√nTfR )q– x (t) √ nTgR + T |e| ( nTfR )q– x (t) nTfR < , we know that there exists a constant M >  such From (.) and (.), we have x (t) ≤ M. x (t) ≤ M. When  < q < , there must be a constant M >  such that Hence, from (.), (.), (.) and (.), we have ∩ Ker M, we H(x, μ) = aμ – ( – μ)  T  = aμ + ( – μ)g(a), –g(a) + e(t) dt then we have deg{JQN , ∩ Ker M, } = deg H(·, ), = deg H(·, ), = deg{I, ∩ Ker M, } = . ∩ Ker M,  ∩ Ker M,  Applying Lemma ., we complete the proof. Remark Assumption (H) guarantees that condition (A) of Lemma . is satisfied. Furthermore, using assumptions (H)-(H), we can easily estimate prior boud of the solution to Eq. (.). As an application, we consider the following example: x(t) = xx((tt)) ∈ R, g(x) = B = e(t) = (sin t, cos t) , τ = γ = π , p = ., T = π , f (x) = ( + sin x,  + cos x). Obviously, λ =  = ±, λ = – = ±, e(t) dt = for|x| > D > , xg(x) = for |x| > D > , gi(u) – gi(u) ≤ |u – u|, u, u ∈ R, for each i = , , f(x) = | + sin x| ≥ , f(x) = | + cos x| ≥ , U = U– = such that UBU– = Competing interests The authors declare that they have no competing interests. Authors’ contributions The first author QY gave an example for verifying the paper’s results. The corresponding author BD gave the proof for all the theorems. QY and BD read and approved the final manuscript. Acknowledgements This work was supported by NSF of Jiangsu education office (11KJB110002), Postdoctoral Fundation of Jiangsu (1102096C), Postdoctoral Fundation of China (2012M511296) and Jiangsu province fund (BK2011407). 1. 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Qing Yang, Bo Du. Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian, Journal of Inequalities and Applications, 2012, 270,