Existence of periodic solution for fourth-order generalized neutral Emphasis Type="Italic"p/Emphasis-Laplacian differential equation with attractive and repulsive singularities

摘要

In this paper, we investigate the existence of a positive periodic solution for the following fourth-order p-Laplacian generalized neutral differential equation with attractive and repulsive singularities: $$igl(arphi_{p}igl(u(t)-c(t)uigl(t-delta(t)igr) igr)''igr)''+f igl(u(t)igr)u'(t)+gigl(t,u(t)igr)=k(t), $$ where g has a singularity at the origin. The novelty of the present article is that we show that attractive and repulsive singularities enable the achievement of a new existence criterion of a positive periodic solution through an application of coincidence degree theory. Recent results in the literature are generalized and significantly improved.KeywordsPositive periodic solution??p-Laplacian??Fourth-order??Attractive and repulsive singular??Generalized neutral operator??MSC34C25??34K13??34K40??1 IntroductionIn this paper, we consider the existence of a positive periodic solution for the following fourth-order p-Laplacian generalized neutral differential equation with singularity: $$ { } igl(arphi_{p}igl(u(t)-c(t)uigl(t-delta(t) igr)igr)''igr)''+f igl(u(t)igr)u'(t)+gigl(t,u(t)igr)=k(t), $$ (1.1) where (pgeq2), (arphi_{p}(u)=|u|^{p-2}u) for (ueq0) and (arphi_{p}(0)=0); (f:mathbb{R}omathbb{R}) is a continuous function, (|c(t)|eq1), for all (tin[0,T]), (c, deltain C^{2}(mathbb{R},mathbb{R})) and c, ?′ are T-periodic functions for some (T 0), (delta'(t)1) for all (tin[0,T]); (k:mathbb{R}ightarrowmathbb{R}) is continuous periodic functions with (k(t+T)equiv k(t)) and (int^{T}_{0}k(t),dt=0); (g(t,u)=g_{0}(u)+g_{1}(t,u)), (g_{1}:mathbb{R}imes(0,+infty)omathbb {R}) is an (L^{2})-Carath??odory function and (g_{1}(t,cdot)=g_{1}(t+T,cdot)); (g_{0}:(0,+infty)o mathbb{R}) is a continuous function. g can come with a singularity at the origin, i.e., $$ lim_{uightarrow0^{+}} g(t,u)=+infty quadBigl(mbox{or } lim _{uightarrow0^{+}} g(t,u)=-inftyBigr),quadmbox{uniformly in } t. $$ It is said that (1.1) is of repulsive type (resp. attractive type) if (gightarrow+infty) (resp. (gightarrow-infty)) as (uightarrow0^{+}).In recent years, the study of periodic solutions for neutral differential equations has attracted the attention of many researchers; see [2, 3, 4, 5, 6, 7, 8, 9, 14, 16, 17, 18, 20, 21] and the references cited therein. For related books, we refer the reader to [1, 12]. Most work concentrated on the neutral operator ((A_{1}u)(t):=u(t)-cu(t-delta)) (see [6, 7, 14, 16, 21]) or the neutral operator with variable parameter ((A_{2}u)(t):=u(t)-c(t)u(t-delta)) (see [3, 8]) or the neutral operator with variable delay ((A_{3}u)(t):=u(t)-cu(t-delta(t))) (see [4, 5]). However, the study of a neutral operator with linear autonomous difference operator ((Au)(t):=u(t)-c(t)u(t-delta(t))) is relatively rare.