Existence Of Periodic Solutions Of 2n-Order Delay Differential Equations

In this paper,we consider the existence of solution of 2n-order periodic boundary value problem for n > 1.u⁽²ⁿ⁾(t) + D(t, u(t - r), u(t),..., u^(2n-2)(t))u^(2n-1) (t)+g(t, u(t - r), u(t),..., u^(2n-2)(t)) = h(t),t ∈ [0, T], (1)where h ∈ L¹(0,T),D ∈ C([0,T] × R²ⁿ, R),and g : [0,T] × R²ⁿ → R be a Caratheodory function, T-periodic in the first variable.Habets and Sanchez [1] has studied the existence of solution of second-order periodic boundary value problem,which can be written as follow:Mawhin in [2] first gave the upper and lower solution method for the following Duffing equationwith boundary conditionw(0) = u(T), and u'(0) = u'{T),under the continuity of /. Nkashama generalized this method to the Caratheodory case in [3] for the first-order differential equation. In [1], Habets et al. obtained some similar results to the Caratheodory case for Lienard equation, which is more general than the Duffing equation. But their results are only applicable to the case k > 0. In [4], Nieto et al. extended these results in a way. Recently, Wang in [5] extended these results under a Caratheodory case for k € M\{0}, the upper and lower solutions which he used may no longer be periodic.In 1943, Leray and Schauder introduced some " nonlinear alternative " theorems for compact maps. These theorems have enhanced greatly the theory of ordinary differential equations and are still used widely today. There are two major approaches to modern nonliner alternative theory, the first uses degree theory and the second uses a theory based on essential maps. Recently, there are many investigations extended the related topics about second-order differential equations to high-order differential equations. For these investigations, we refer to [6]-[9] and references therein. In [10] Fu-Hsiang Wong et al. generalized the above-mentioned results with respect to high-order periodic boundary value problems. They established some sufficient conditions for theexistence of solutions of following high-order periodic boundary value problem for n > 2.u(n\t) + D{t, u(t),..., u{n-2\t))u{n^x\t) + g(t, u(t),..., u{n²)(t))= h(t),te[0,T\,??(0)=0, i = 0,l,2,...,n-3,where h G Ll{Q,T),D G C([O,T] x Rn^l, R),and g : [0,T] x Rn^l -> R be a Caratheodory function,T-periodic in the first variable.In this article, we attempt to generalized these results to the 2n-order delay differential eqation. We will establish some sufficient conditions for the existence of solutions of periodic bounderay value problem(l).For the abbreviation of our discussion, we need the following definitions.(Ai) a and (3 continuous on [0,T], such thata(i)(t) < P{l)(t), i = 0,l,...,2n-2, ï¿¡G[0,T],and a(t) < 0 < /3(t), t e [-r, 0].(A2) E := {(t,x-i,xo,xu...,x2n-i)\t G [0,T],a(t - r) < X-X < (3(t -r) n>(^(t} < r < B^(t) ?-(l 1 On — 9 To , <= R\(A3) /:ï¿¡"—> i? is a Caratheodory function on ï¿¡", i.e., a function with the following properties :(i) For each x, G i?, i = —1, 0,1,..., 2n—1, the function /(?, x_i, a;o,Xi,... ,x2n-i), with domain {t G [0, T]|(i,a;_i, x0, xi,... ,X2n-i) G ï¿¡} is measurable.(ii) The function /(?, a{- - r), ?(?), ?(1)(-)? ? ? ?, ?(2""2)(-), V) and /(?, /?(? -r), f3(-), P^(-),..., ^2n²\-)- y) are measurable for each fixed y G R.(iii) For almost every t G [0,T],the function f(t,X-i,x0, â–  â–  â–  ,x2n-i) with domain{(x_!,x0,..., x2n-i)\oi{t - r) < x_i < 0{t-r),a{i)(t) !,a;o,...,a;2n-2,0) G ï¿¡", |/(t,a:₁, xo,xl7..., a:2n-2, 0)| < h(t)holds.We also need the assumptions as follows.(Co) There are two real numbers ai,a2, and a\ < 0,ai +a2 > 0, such thatg(t,x-i,xo,...,x2n-2) > h(t), \ix2n-2 < au and t e [0. T], g(t, x-i,xo,..., x2n-2) < h(t), if ai < x2n^2 < a2, and t G [0, T]. (Ci) For (t, x_i, x0,..., x2n-2,2/i), (ï¿¡, a;_i, x0,..., :r2n-2, t/2) G -B, there exists ate I/H0^) such that\f{t,X-i,X0, . . . ,X2n-2,yi) - f(t,X-UX0,...,X2n-2,y2)\ < L{t)\yx - y2\.(C2) a,/5 G W1'71-1^,^ satisfy(i) a^(t) > f(t, a(t - r), a(t),a.e. in [0, T],a(0) Ht,t3(t-r),P(t),0l1Ht),...,fi2n-V(t)) a.e. in [0,T],(C3) There exists a m ï¿¡ Ll(Q,T), such that for any with \y\ < 1, the inequalitymax{|/(t, a(t-r), a(t),..., a^^(t), y)\,\f(t, (3(t-r), /3(t),..., ^n²\t)., h(t),L(t)}l,te[O,T],(2)*(*) =r), *(<),..., xVn 6(*)),(4)where F(t,x(t - r),x{t),..., x^-^(t)) is defined as follows f(t, a(t - r), a(t),..., a(2nF:=- r), x(t),..., x(2"-2)(i), x^/(t, /3(t - r),W ?,z = O, ...,2n-2and 0 < A < 1.In order to obtain our main results, we need the following useful lemma.LEMMA 1 : ( See [11] Corollary IV.7.) Let X, Z be real vector spaces , L : domL C X —> Z a linear Fredholm mapping of index zero , Q C X an open bounded subset , N : Q -^ Z a L—compact mapping. If kerL = {0},0 € VL andLx - XNx ^ 0for every (x, A) G {domL n dfi) x (0,1). Then, equationhas at least one solution in domL DIt follows from the above-mentioned definitions and assumptions, we finished the proof of the following theorem.THEOREM 1: Boundary value problem (2) has at least one solutionx(t) which satisfies<*'(*)