In this paper, we obtain some sufficient conditions for the existence of $2\pi$-periodic solutions of some semilinear equations at resonance where the kernel of the linear part has dimension $2n(n\ge 1)$. Our technique essentially bases on the Brouwer degree theory and Mawhin's coincidence degree theory

Shiwang, M.

Wang, Z.

Electronic Journal of Qualitative Theory of Differential Equations

Ma Shiwang

Z. Wang

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Electronic journal of qualitative theory of differential equations

Periodic solutions of semilinear equations at resonance with a 2n-dimensional kernel

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Periodic Solutions of Semilinear Equationsat Resonance with a 2n-Dimensional KernelMa Shiwang1, Wang Zhicheng2Abstract. In this paper, we obtain some sufficient conditions for the existence of2pi-periodic solutions of some semilinear equations at resonance where the kernel ofthe linear part has dimension 2n(n ≥ 1). Our technique is essentially based on theBrouwer degree theory and Mawhin’s coincidence degree theory.Key words: Semilinear equation, resonance, periodic solution, kernel, dimension,Brouwer degree, coincidence degree.1991 AMS Classification: 34K151. IntroductionFor a long time, many authors have payed much attention to the existence prob-lem of periodic solutions for the perturbed systems of ordinary as well as functionaldifferential equations. In recent years, we see an increasing interest in the moredifficult problem “at resonance” in the sense that the associated linear homoge-nous system has a nontrivial periodic solution. In this side, some useful techniques,say the averaging method, have been developed and many significant results havebeen obtained for the existence of periodic solutions to some nonlinear systems offirst order differential equations at resonance that involve a small parameter (see[1,2] and references therein).Much research has also been devoted to the study of existence results for somenonlinear systems whose nonlinearities satisfy so-called Landesman-Lazer condi-tions. Several of these results are mentioned in [3]. However, less is known whenthe linear part has a two-dimensional kernel. Some work has been done by Lazer& Leach [4], Cesari[5], Iannacci & Nkashama[6], Nagle & Sinkala[7,8] and Ma, Wang& Yu[9]. To the best of our knowledge, few authors have considered the case whenthe linear part has dimension greater than two. In this direction, an example withThis project was supported by NNSF of China1). Department of Automatic Control, Huazhong University of Science and Technology2). Dept. of Appl. Math., Hunan University, Changsha 410082, P. R. ChinaTypeset by AMS-TEXEJQTDE, 1999 No. 2, p.1a three-dimensional kernel and a fourth order ordinary differential equation areconsidered in [8] and [10] respectively. In a recent paper[12], the results in [8] havebeen improved and unified by Ma, Wang & Yu.This paper is concerned with the existence of 2pi-periodic solutions for the non-linear system of first order functional differential equations of mixed type(1.1) x˙j(t) = Bjxj(t) + Fj(t, x(t + ·)) + pj(t), j = 1, 2, · · · , nwhere xj(t) ∈ R2, x(t + ·) ∈ BC(R, R2n) is defined by x(t + s) = (x1(t + s), x2(t +s), · · · , xn(t + s)), pj ∈ C(R, R2) is 2pi-periodic, and Fj : R × BC(R, R2n) → R2is continuous, bounded and 2pi-periodic in its first variable t. The constant matrixBj has a pair of purely imaginary eigenvalues ±imj with mj some positive integer.Without loss of generality, we assumeBj =(0 mj−mj 0), j = 1, 2, · · · , n.In this paper, we also need the following hypothesis(F) There exists a permutation k1, k2, · · · , kn consisting of 1, 2, · · · , n and for anypositive integer j with 1 ≤ j ≤ n, there exist τj ∈ R, Hj ∈ BC(R2, R2) with theasymptotic limits Hj(±,±) = limr,s→±∞ Hj(r, s) and Gj : R×BC(R, R2n) → R2,which is continuous, bounded and 2pi-periodic with respect to its first variable t,such that for any t ∈ R and ϕ ∈ BC(R, R2n),Fj(t, ϕ) = Hj(ϕ2kj−1(−τj), ϕ2kj (−τj)) + Gj(t, ϕ).2. Main ResultsIn order to state our main results, we need some notations. For any positiveinteger N , we will denote by | · | the Euclidean norm in RN . We always denote byA the matrixA =(0 1−1 0).Let m and l be some positive integers. If p ∈ C(R, R2) is 2pi-periodic, we set(2.1) p(m) :=12pi∫ 2pi0eAT (ms)p(s)ds.where “T” denotes the transpose and e· denotes the exponential of an operater.For H ∈ C(R2, R2), whenever the asymptotic limitsH(±,±) = limr,s→±∞H(r, s)EJQTDE, 1999 No. 2, p.2exist, we set(2.2)WH :=12pi[(1 −11 1)H(+, +) +(−1 −11 −1)H(+,−)+(−1 1−1 −1)H(−,−) +(1 1−1 1)H(−, +)];(2.3)WH(m, l) :=12pi[∫ pi/20L(m, l)(s)dsH(+, +) +∫ pipi/2L(m, l)(s)dsH(+,−)+∫ 3pi/2piL(m, l)(s)dsH(−,−) +∫ 2pi3pi/2L(m, l)(s)dsH(−, +)],where the matrix value mapping L(m, l) : R → R2×2 is defined by(2.4) L(m, l)(s) =1ll−1∑k=0eAT ( ml(s+2kpi)).It is easy to verify that if m = l, then W H(m, l) = W H . Finally, let X be a normedspace, if G : X → RN is continuous and bounded, we denote by MG the supremumof G, i.e.,(2.5) MG := supx∈X|G(x)|.Theorem 2.1. If, in addition to (F), we assume that for any 1 ≤ j ≤ n,(2.6) |mj −mkj | <12mkjand(2.7) |W Hj (mj , mkj )| >12(MGj + |pj(mj)|) +12(n∑i=1(MGi + |pi(mi)|)2)1/2hold, then Eq.(1.1) has at least one 2pi-periodic solution.The following is a direct corollary of Theorem 2.1.Corollary 2.1. If, in addition to (F), we assume that for any 1 ≤ j ≤ n, (2.6)and(2.8) |W Hj (mj , mkj )| >12MGj +12(n∑i=1M2Gi)1/2+ n∑j=1|pj(mj)|21/2hold, then Eq.(1.1) has at least one 2pi-periodic solution.EJQTDE, 1999 No. 2, p.33. Proof of Main ResultsLet X and Z be real normed spaces with respective norms ‖ · ‖X and ‖ · ‖Z ,and L : domL ⊂ X → Z be a linear Fredholm mapping of index zero. Let P bea continuous projection in X onto ker L, I − Q be a continuous projection in Zonto ImL, and KP : ImL → domL ∩ kerP be the (unique) pseudo-inverse of Lassociated to P in the sense that LKP z = z for all z ∈ ImL and PKP = 0. LetJ : ImQ → ker L be an isomorphism. In addition, we assume that N : X → Z isL-completely continuous and that 〈·, ·〉 is an inner product on ker L.The folowing useful lemma is proved in [11].Lemma 3.1 [11]. Assume that dim kerL ≥ 2 and there exist M > 0, a boundedopen subset Ω0 ⊂ ker L with 0 ∈ Ω0 and ∂Ω0 a connected subset in ker L, such thatthe following conditions hold:(I) ‖KP (I −Q)Nx‖X ≤ M, for all x ∈ X;(II) For any x ∈ X with Px ∈ ∂Ω0 and ‖(I − P )x‖X < M ,(3.1) 〈JQNx, JQNx〉 > 0;(III) There exist a continuous mapping η : Ω¯0 → kerL and a family of contin-uous mappings ηi : kerL → ker L (i = 1, 2, · · · , N) satisfying(3.2) 〈ηi(u), ηi(u)〉 > 0, 〈u, ηi(u)〉 6= 0 for i = 1, 2, · · · , N and u 6= 0,such that for any u ∈ ∂Ω0,(3.3) 〈JQNu− η(u), η1η2 · · · ηN (u)〉 6= 0;(3.4) 〈JQNu, JQNu− η(u)〉 6= 0.Then Lx = Nx has at least one solution x satisfyingPx ∈ Ω¯0 and ‖(I − P )x‖X ≤ M.Let N be a positive integer. Set P(N)2pi = {x ∈ C(R, RN) : x(t + 2pi) = x(t), ∀t ∈R}, ‖x‖ = supt∈R |x(t)| = supt∈[0,2pi] |x(t)|. Then P (N)2pi ⊂ BC(R, RN) is a Banachspace.Let D = diag(B1, B2, · · · , Bn), whereBj =(0 mj−mj 0).EJQTDE, 1999 No. 2, p.4Define the operator L : P(2n)2pi → P (2n)2pi by Lx(t) = x˙(t)−Dx(t),domL = {x ∈ P (2n)2pi : x˙(t) exists and is continuous}.It is not hard to check that L is a Fredholm mapping of index zero. Let J : ker L →kerL be the identical operator and let P = Q : P(2n)2pi → P (2n)2pi be the projectionsdefined by(3.5) Px(t) =12pieDt∫ 2pi0eDT sx(s)ds.Then the (unique) pseudo-inverse of L associated to P , denoted by K : ImL →domL ∩ ker P , is a compact operator with ‖K‖ ≤ 2pi (see [11] for details).Define the operator N : P(2n)2pi → P (2n)2pi byNx(t) = (N1x(t), N2x(t), · · · , Nnx(t)),Njx(t) = Fj(t, x(t + ·)) + pj(t), j = 1, 2, · · · , n.Then N is continuous and takes bounded sets into bounded sets, and hence is L-completely continuous. Moreover, Eq.(1.1) is equivalent to the operater equationLx = Nx.It is easy to see that H : R2n → kerL defined byH(a) = eDta, for a ∈ R2nis an isometric isomorphism. In this paper, we identify a ∈ R2n with its imageH(a) ∈ kerL, i.e., H(a) = a, a ∈ R2n.For the sake of convenience, we also introduce the following notations. Let m, lbe some positive integers and H ∈ C(R2, R2). For any real number ρ ≥ 0, we set(3.6) MH(ρ) :=12pi∫ 2pi0eAT sH((ρ sin s, ρ cos s)T )ds;(3.7) MH(ρ, m, l) :=12lpi∫ 2lpi0eAT ( msl)H((ρ sin s, ρ cos s)T )ds.It is easy to know that for any positive integer m,MH(ρ, m, m) = MH(ρ)In what follows, the following lemmas are needed.EJQTDE, 1999 No. 2, p.5Lemma 3.2 [11]. Let m, l be some positive integers and 0 < r0 < 1. If H ∈C(R2, R2) is bounded and the asymptotic limits H(±,±) = limr,s→±∞ H(r, s) exist,then(3.8) limρ→∞MH(rρ, m, l) = W H(m, l)and(3.9) limρ→∞MH(rρ) = W Huniformly for r in [r0, 1].Lemma 3.3. For any permutation k1, k2, · · · , kn consisting of 1, 2, · · · , n thereexists a family of continuous mappings ηi : R2n → R2n (i = 1, 2, · · · , N1) with(3.10) 〈ηi(u), ηi(u)〉 > 0, 〈u, ηi(u)〉 > 0, for u ∈ R2n\{0},such that for any aj ∈ R2 (j = 1, 2, · · · , n),(3.11) η1η2 · · ·ηN1(a1, a2, · · · , an) = (ak1 , ak2 , · · · , akn)holds.Proof. It suffices to show that there exists a family of continuous mappings ζi :R2n → R2n (i = 1, 2, · · · , n1) with(3.12) 〈ζi(u), ζi(u)〉 > 0, 〈u, ζi(u)〉 > 0, for u ∈ R2n\{0},such that for any aj ∈ R2 (j = 1, 2, · · · , n) and 1 ≤ j1 < j2 ≤ n,(3.13) ζ1ζ2 · · · ζn1(a1, · · · , aj1, · · · , aj2 , · · · , an) = (a1, · · · , aj2 , · · · , aj1, · · · , an),Define ζi : R2n → R2n (i = 1, 2, · · · , 6) byζi(u) = (ζ(1)i , ζ(2)i , · · · , ζ(n)i ),ζ(k)i =uk, k 6= j1, j2√22uj1 +√22uj2 , k = j1−√22uj1 +√22uj2 , k = j2, i = 1, 2EJQTDE, 1999 No. 2, p.6ζ(k)i =uk, k 6= j1((√22,−√22)uTj1 , (√22,√22)uTj1), k = j1, i = 3, 4, 5, 6where u = (u1, u2, · · · , un) ∈ R2n, uk ∈ R2, k = 1, 2, · · · , n.It is easy to see that ζi (i = 1, 2, · · · , 6) is continuous, moreover, (3.12) and(3.13) hold with n1 = 6. This completes the proof.We are now in a position to prove our main result.Proof of Theorem 2.1. Let M = 4pi[(∑nj=1 M2Fj)1/2 + (∑nj=1 ‖pj‖2)1/2], thenfor any x ∈ P (2n)2pi , ‖K(I −Q)Nx‖ ≤ M , and hence the condition (I) of Lemma 3.1holds.Let ρ > 0, takeΩ0 = {u ∈ ker L : u = (r1ρa1, r2ρa2, · · · , rnρan), as ∈ ∂B1(0) ⊂ R2,0 ≤ rs < 1, s = 1, 2, · · · , n}.Then Ω0 is a bounded open set in ker L and∂Ω0 =n⋃j=1{u ∈ ker L : u = (r1ρa1, r2ρa2, · · · , rnρan), as ∈ ∂B1(0) ⊂ R2,0 ≤ rs ≤ 1, s = 1, 2, · · · , n, rj = 1}For x ∈ P (2n)2pi with xj(t) = rjρeBjtaj + x¯j(t), aj ∈ ∂B1(0) = {a ∈ R2 : |a| =1} ⊂ R2; x¯ = (x¯1, x¯2, · · · , x¯n) ∈ ImL, x¯j(t) ∈ R2, ‖x¯j‖ ≤ M, j = 1, 2, · · · , n, it isnot hard to verify that(3.14) JQNx = ((JQNx)1, (JQNx)2, · · · , (JQNx)n),(3.15) (JQNx)j = eBTj τj Yj(ρ, akj , rkj ) + Xj(x) + pj(mj),where(3.16) Xj(x) =12pi∫ 2pi0eBTj sGj(s, x(s + ·))ds,(3.17) Yj(ρ, akj , rkj ) =12pi∫ 2pi0eBTj sHj(rkj ρeBkj sakj + x¯kj (s + τ))ds.EJQTDE, 1999 No. 2, p.7By using the fact that ‖x¯kj‖ ≤ M and a similar argument used in the proof ofLemma 3.2, it is not hard to show that(3.18) limρ→∞|Yj(ρ, akj , 1)| = |W Hj (mj , mkj )|uniformly for akj in ∂B1(0) ⊂ R2 and ‖x¯kj‖ ≤ M .It follows from (2.7) and (3.16) that(3.19) |W Hj (mj , mkj )| > MGj + |pj(mj)| ≥ |Xj(x)|+ |pj(mj)|If rj0 = 1 for some j0, then (3.14), (3.15), (3.18) and (3.19) imply that for ρsufficiently large,JQN(x) 6= 0.Thus, we have proved that for ρ sufficiently large,JQN(x) 6= 0for any x ∈ P (2n)2pi with Px ∈ ∂Ω0 and ‖(I − P )x‖ ≤ M , that is, the condition (II)of Lemma 3.1 also holds.Define the mapping η : Ω¯0 → kerL byη(u) = (η(1)(u), η(2)(u), · · · , η(n)(u)),η(j)(u) =12pi∫ 2pi0eBTj sGj(s, r1ρeB1(s+·)a1, · · · , rnρeBn(s+·)an)ds + pj(mj),u = (r1ρa1, · · · , rnρan), aj ∈ ∂B1(0) ⊂ R2, 0 ≤ rj ≤ 1, j = 1, 2, · · · , n.Then it is easy to see that η is continuous.Let βj (−pi < βj ≤ pi) be defined bysin βj =WHj1 (mj , mkj )|WHj (mj , mkj )|, cos βj =WHj2 (mj , mkj )|WHj (mj , mkj )|hereWHj (mj , mkj ) = (WHj1 (mj , mkj ), WHj2 (mj , mkj )).Let N2 be a positive integer satisfying∣∣∣∣τj − βj/mjN2∣∣∣∣ < pi2mj , j = 1, 2, · · · , n.EJQTDE, 1999 No. 2, p.8Define ηi : ker L → ker L (i = 1, 2, · · · , N2) byηi(u) = (η(1)i (u), η(2)i (u), · · · , η(n)i (u)),η(j)i (u) = eBTj γj uj , γj =τj − βj/mjN2, j = 1, 2, · · · , n,where u = (u1, u2, · · · , un), uj ∈ R2 (j = 1, 2, · · · , n). Then it is clear that ηi (i =1, 2, · · · , N2) is continuous and (3.2) holds.By Lemma 3.3, we can also define ηi : kerL → ker L (i = N2 +1, · · · , N2 +N1),which are continuous and satisfy (3.2), such thatηN2+1ηN2+2 · · · ηN2+N1(a1, a2, · · · , an) = (ak1 , ak2 , · · · , akn)holds for any aj ∈ R2 (j = 1, 2, · · · , n).In the sequel, we assume that u = (r1ρa1, r2ρa2, · · · , rnρan) ∈ ∂Ω0, aj ∈∂B1(0) ⊂ R2, 0 ≤ rj ≤ 1 j = 1, 2, · · · , n. Clearly, we may assume, withoutloss of generality, that rkj0 = 1 for some j0.Let αj = αj(aj) (−pi < αj ≤ pi) defined by sin αj = a(1)j , cosαj = a(2)j , whereaj = (a(1)j , a(2)j ).Therefore, we have(3.20) η¯(u) := η1η2 · · · ηN2+N1(u) = (η¯1(u), η¯2(u), · · · , η¯n(u)),(3.21)η¯j(u) = rkj ρeBTj (τj−βj/mj)akj = ρrkj eBTj τj eAαkj WHj (mj , mkj )/|W Hj (mj , mkj )|(3.22) ξ¯(u) := JQNu− η(u) = (ξ¯1(u), ξ¯2(u), · · · , ξ¯n(u)),(3.23)ξ¯j(u) =12pieBTj τj∫ 2pi0eBTj sHj(rkj ρeBkj sakj )ds= eBTj τj eA(mjαkj /mkj )MHj (rkj ρ, mj, mkj ).It follows from (3.20)-(3.23) that(3.24)〈ξ¯(u), η¯(u)〉= ρn∑j=1rkj|WHj (mj , mkj )|〈WHj (mj , mkj ), eA$(j)αkj MHj (rkj ρ, mj, mkj )〉.EJQTDE, 1999 No. 2, p.9where $(j) =mj−mkjmkj.Since |αkj | ≤ pi and |mj −mkj | < 12mkj , we have∣∣$(j)αkj ∣∣ ≤ |$(j)pi| < pi2 , j = 1, 2, · · · , n.Hence,〈WHj (mj , mkj ), eA$(j)αkj WHj (mj , mkj )〉 = |W Hj (mj , mkj )|2 cos($(j)αkj)(3.25) ≥ |W Hj (mj , mkj )|2 cos ($(j)pi) > 0,For j 6= j0, we setI0j =[0, |W Hj0 (mj0 , mkj0 )| cos ($(j0)pi)/(4nMHj )),I1j =[|WHj0 (mj0 , mkj0 )| cos ($(j0)pi)/(4nMHj ), 1] ,then by (3.24), and noting that rkj0 = 1, we have(3.26) 〈ξ¯(u), η¯(u)〉 = ρ [Z0 + Z1 + Z2]where,(3.27)Z0 =1|WHj0 (mj0 , mkj0 )|〈WHj0 (mj0 , mkj0 ), eA($(j0)αkj0)MHj0 (ρ, mj0 , mkj0 )〉(3.28)Z1 =∑j 6=j0,rkj∈I0jrkj|WHj (mj , mkj )|〈WHj (mj , mkj ), eA($(j)αkj )MHj (rkj ρ, mj , mkj )〉(3.29)Z2 =∑j 6=j0,rkj∈I1jrkj|WHj (mj , mkj )|〈WHj (mj , mkj ), eA($(j)αkj )MHj (rkj ρ, mj , mkj )〉Since MHj (rkj ρ, mj , mkj ) → W Hj (mj , mkj ) (ρ → ∞) uniformly for rkj in I1j byLemma 3.2, we have(3.30) Z2 > 0,for large ρ.EJQTDE, 1999 No. 2, p.10By the Schwartz inequility, we also find(3.31) |Z1| ≤∑j 6=j0,rkj∈I0jrkj MHj ≤ |W Hj0 (mj0 , mkj0 )| cos ($(j0)pi)/4.Therefore, it follows from (3.26)-(3.31) that for ρ suffciently large,〈ξ¯(u), η¯(u)〉 > 0.Thus, (3.3) holds for any u ∈ ∂Ω0.On the other hand, it is not hard to show that〈JQNu, JQNu− η(u)〉=n∑j=1|MHj (rkj ρ, mj , mkj )|2+n∑j=1〈MHj (rkj ρ, mj, mkj ), eAT (mjαkj /mkj )eBjτj Xj(u)〉+n∑j=1〈MHj (rkj ρ, mj, mkj ), eAT (mjαkj /mkj )eBjτj pj(mj)〉,whereXj(u) =12pi∫ 2pi0eBTj sGj(s, r1ρeB1(s+·)a1, · · · , rnρeBn(s+·)an)ds.By using the Schwartz inequality, it follows that〈JQNu, JQNu− η(u)〉 ≥n∑j=1|MHj (rkj ρ, mj, mkj )|2−n∑j=1|MHj (rkj ρ, mj, mkj )|[MGj + |pj(mj)|]=n∑j=1[|MHj (rkj ρ, mj, mkj )| −12(MGj + |pj(mj)|)]2− 14n∑j=1(MGj + |pj(mj)|)2.Since rkj0 = 1 for some j0, and MHj0 (ρ, mj0, mkj0 ) → W Hj0 (mj0 , mkj0 ) (ρ →∞),it follows from (2.7) that for ρ sufficiently large,〈JQNu, JQNu− η(u)〉 > 0.EJQTDE, 1999 No. 2, p.11Thus, (3.4) also holds for any u ∈ ∂Ω0.By virtue of Lemma 3.1, Eq.(2.1) has at least one 2pi-periodic solution and theproof is complete.Finally, we give an example to illustrate our main results.Example Consider the system(3.32)x′1 = x2 + x3/(1 + x23) + arctan x1 + p1(t)x′2 = −x1 + 3 arctanx4 +12arctanx2 + p2(t)x′3 = x4 + x5e−x25 +√2 sin x3 + p3(t)x′4 = −x3 −√6 arctan x6 +√2 cos x3 + p4(t)x′5 = x6 + arctan x5 − 2 arctanx1 + p5(t)x′6 = −x5 +12arctan x6 − 2 arctanx2 + p6(t)where pj(j = 1, 2, · · · , 6) are continuous, 2pi-periodic functions. By Corollary 2.1,it is easy to check that Eq.(3.32) has at least one 2pi−periodic solution provided√|c1|2 + |c2|2 + |c3|2 <√6−√22− 14√2√5pi2 + 32,whereck =12pi∫ 2pi0(cos s − sin ssin s cos s)(p2k−1(s)p2k(s))ds, k = 1, 2, 3.References1. J. K. Hale, Ordnary Differential Equations, Wiley Interscience, New York, 1969.2. R. K. Nagle, Nonlinear boundary value problems for ordinary differential equations with asmall parameter, SIAM J. Math. Analysis 9 (1978), 719-729.3. S. Fucik, Solvability of Nonlinear Equations and Boundary Value Problems, D. Reidel Pub-lishing, Dordrecht, Holland, 1980.4. A. C. Lazer & D. E. Leach, Bounded perturbations of forced harmonic oscillations at reso-nance, Ann. Mat. Pura. Appl., 82 (1969), 49-68.5. L. Cesari, Non-linear problems across a point of resonance for non-self-adjoint systems, Ac-demic Press (1978), 43-67.6. R. Iannacci & M. N. Nkashma, Unbounded perturbations of forced second order ordinarydifferential equations at resonance, J. Diff. Eqns. 69 (1978), 289-309.7. R. K. Nagle & Z. Sinkala, Semilinear equations at resonance where the kernel has dimensiontwo, in Differential Equations:Stability and Control (Edited by S.Elaydi) Marcel Dekker, NewYork (1991).EJQTDE, 1999 No. 2, p.128. R. K. Nagle & Z. Sinkala, Existence of 2pi-periodic solutions for nonlinear systems of first-order ordinary differential equations at resonance, Nonlinear Analysis (TMA) 25 (1995),1-16.9. S. W. Ma, Z. C. Wang & J. S. Yu, Coincidence degree and periodic solutions of Duffingequations, Nonlinear Analysis (TMA) 34 (1998), 443-460.10. J. D. Schuur, Perturbation at resonance for a fourth order ordinary differential equation, J.Math. Anal. Appl. 65 (1978), 20-25.11. S. W. Ma, Z. C. Wang & J. S. Yu, An abstract existence theorem at resonance and itsapplications, J. Diff. Eqns., 145 (1998), 274-294.EJQTDE, 1999 No. 2, p.13

Periodic solutions of semilinear equations at resonance with a $2n$-dimensional kernel

Shiwang Ma

Zhicheng Wang

University of Szeged

Periodic Solutions of Semilinear Equationsat Resonance with a 2n-Dimensional KernelMa Shiwang1, Wang Zhicheng2Abstract. In this paper, we obtain some sufficient conditions for the existence of2π-periodic solutions of some semilinear equations at resonance where the kernel ofthe linear part has dimension 2n(n ≥ 1). Our technique is essentially based on theBrouwer degree theory and Mawhin’s coincidence degree theory.Key words: Semilinear equation, resonance, periodic solution, kernel, dimension,Brouwer degree, coincidence degree.1991 AMS Classification: 34K151. IntroductionFor a long time, many authors have payed much attention to the existence prob-lem of periodic solutions for the perturbed systems of ordinary as well as functionaldifferential equations. In recent years, we see an increasing interest in the moredifficult problem “at resonance” in the sense that the associated linear homoge-nous system has a nontrivial periodic solution. In this side, some useful techniques,say the averaging method, have been developed and many significant results havebeen obtained for the existence of periodic solutions to some nonlinear systems offirst order differential equations at resonance that involve a small parameter (see[1,2] and references therein).Much research has also been devoted to the study of existence results for somenonlinear systems whose nonlinearities satisfy so-called Landesman-Lazer condi-tions. Several of these results are mentioned in [3]. However, less is known whenthe linear part has a two-dimensional kernel. Some work has been done by Lazer& Leach [4], Cesari[5], Iannacci & Nkashama[6], Nagle & Sinkala[7,8] and Ma, Wang& Yu[9]. To the best of our knowledge, few authors have considered the case whenthe linear part has dimension greater than two. In this direction, an example withThis project was supported by NNSF of China1). Department of Automatic Control, Huazhong University of Science and Technology2). Dept. of Appl. Math., Hunan University, Changsha 410082, P. R. ChinaTypeset by AMS-TEXEJQTDE, 1999 No. 2, p.1a three-dimensional kernel and a fourth order ordinary differential equation areconsidered in [8] and [10] respectively. In a recent paper[12], the results in [8] havebeen improved and unified by Ma, Wang & Yu.This paper is concerned with the existence of 2π-periodic solutions for the non-linear system of first order functional differential equations of mixed type(1.1) ẋj(t) = Bjxj(t) + Fj(t, x(t + ·)) + pj(t), j = 1, 2, · · · , nwhere xj(t) ∈ R2, x(t + ·) ∈ BC(R, R2n) is defined by x(t + s) = (x1(t + s), x2(t +s), · · · , xn(t + s)), pj ∈ C(R, R2) is 2π-periodic, and Fj : R × BC(R, R2n) → R2is continuous, bounded and 2π-periodic in its first variable t. The constant matrixBj has a pair of purely imaginary eigenvalues ±imj with mj some positive integer.Without loss of generality, we assumeBj =(0 mj−mj 0), j = 1, 2, · · · , n.In this paper, we also need the following hypothesis(F) There exists a permutation k1, k2, · · · , kn consisting of 1, 2, · · · , n and for anypositive integer j with 1 ≤ j ≤ n, there exist τj ∈ R, Hj ∈ BC(R2, R2) with theasymptotic limits Hj(±,±) = limr,s→±∞ Hj(r, s) and Gj : R×BC(R, R2n) → R2,which is continuous, bounded and 2π-periodic with respect to its first variable t,such that for any t ∈ R and ϕ ∈ BC(R, R2n),Fj(t, ϕ) = Hj(ϕ2kj−1(−τj), ϕ2kj (−τj)) + Gj(t, ϕ).2. Main ResultsIn order to state our main results, we need some notations. For any positiveinteger N , we will denote by | · | the Euclidean norm in RN . We always denote byA the matrixA =(0 1−1 0).Let m and l be some positive integers. If p ∈ C(R, R2) is 2π-periodic, we set(2.1) p(m) :=12π∫ 2π0eAT (ms)p(s)ds.where “T” denotes the transpose and e· denotes the exponential of an operater.For H ∈ C(R2, R2), whenever the asymptotic limitsH(±,±) = limr,s→±∞H(r, s)EJQTDE, 1999 No. 2, p.2exist, we set(2.2)WH :=12π[(1 −11 1)H(+, +) +(−1 −11 −1)H(+,−)+(−1 1−1 −1)H(−,−) +(1 1−1 1)H(−, +)];(2.3)WH(m, l) :=12π[∫ π/20L(m, l)(s)dsH(+, +) +∫ ππ/2L(m, l)(s)dsH(+,−)+∫ 3π/2πL(m, l)(s)dsH(−,−) +∫ 2π3π/2L(m, l)(s)dsH(−, +)],where the matrix value mapping L(m, l) : R → R2×2 is defined by(2.4) L(m, l)(s) =1ll−1∑k=0eAT ( ml(s+2kπ)).It is easy to verify that if m = l, then W H(m, l) = W H . Finally, let X be a normedspace, if G : X → RN is continuous and bounded, we denote by MG the supremumof G, i.e.,(2.5) MG := supx∈X|G(x)|.Theorem 2.1. If, in addition to (F), we assume that for any 1 ≤ j ≤ n,(2.6) |mj − mkj | <12mkjand(2.7) |W Hj (mj , mkj )| >12(MGj + |pj(mj)|) +12(n∑i=1(MGi + |pi(mi)|)2)1/2hold, then Eq.(1.1) has at least one 2π-periodic solution.The following is a direct corollary of Theorem 2.1.Corollary 2.1. If, in addition to (F), we assume that for any 1 ≤ j ≤ n, (2.6)and(2.8) |W Hj (mj , mkj )| >12MGj +12(n∑i=1M2Gi)1/2+n∑j=1|pj(mj)|21/2hold, then Eq.(1.1) has at least one 2π-periodic solution.EJQTDE, 1999 No. 2, p.33. Proof of Main ResultsLet X and Z be real normed spaces with respective norms ‖ · ‖X and ‖ · ‖Z ,and L : domL ⊂ X → Z be a linear Fredholm mapping of index zero. Let P bea continuous projection in X onto ker L, I − Q be a continuous projection in Zonto ImL, and KP : ImL → domL ∩ kerP be the (unique) pseudo-inverse of Lassociated to P in the sense that LKP z = z for all z ∈ ImL and PKP = 0. LetJ : ImQ → ker L be an isomorphism. In addition, we assume that N : X → Z isL-completely continuous and that 〈·, ·〉 is an inner product on ker L.The folowing useful lemma is proved in [11].Lemma 3.1 [11]. Assume that dim kerL ≥ 2 and there exist M > 0, a boundedopen subset Ω0 ⊂ ker L with 0 ∈ Ω0 and ∂Ω0 a connected subset in ker L, such thatthe following conditions hold:(I) ‖KP (I − Q)Nx‖X ≤ M, for all x ∈ X;(II) For any x ∈ X with Px ∈ ∂Ω0 and ‖(I − P )x‖X < M ,(3.1) 〈JQNx, JQNx〉 > 0;(III) There exist a continuous mapping η : Ω̄0 → kerL and a family of contin-uous mappings ηi : kerL → ker L (i = 1, 2, · · · , N) satisfying(3.2) 〈ηi(u), ηi(u)〉 > 0, 〈u, ηi(u)〉 6= 0 for i = 1, 2, · · · , N and u 6= 0,such that for any u ∈ ∂Ω0,(3.3) 〈JQNu − η(u), η1η2 · · · ηN (u)〉 6= 0;(3.4) 〈JQNu, JQNu− η(u)〉 6= 0.Then Lx = Nx has at least one solution x satisfyingPx ∈ Ω̄0 and ‖(I − P )x‖X ≤ M.Let N be a positive integer. Set P(N)2π = {x ∈ C(R, RN) : x(t + 2π) = x(t), ∀t ∈R}, ‖x‖ = supt∈R |x(t)| = supt∈[0,2π] |x(t)|. Then P(N)2π ⊂ BC(R, RN) is a Banachspace.Let D = diag(B1, B2, · · · , Bn), whereBj =(0 mj−mj 0).EJQTDE, 1999 No. 2, p.4Define the operator L : P(2n)2π → P(2n)2π by Lx(t) = ẋ(t) − Dx(t),domL = {x ∈ P (2n)2π : ẋ(t) exists and is continuous}.It is not hard to check that L is a Fredholm mapping of index zero. Let J : ker L →kerL be the identical operator and let P = Q : P(2n)2π → P(2n)2π be the projectionsdefined by(3.5) Px(t) =12πeDt∫ 2π0eDT sx(s)ds.Then the (unique) pseudo-inverse of L associated to P , denoted by K : ImL →domL ∩ ker P , is a compact operator with ‖K‖ ≤ 2π (see [11] for details).Define the operator N : P(2n)2π → P(2n)2π byNx(t) = (N1x(t), N2x(t), · · · , Nnx(t)),Njx(t) = Fj(t, x(t + ·)) + pj(t), j = 1, 2, · · · , n.Then N is continuous and takes bounded sets into bounded sets, and hence is L-completely continuous. Moreover, Eq.(1.1) is equivalent to the operater equationLx = Nx.It is easy to see that H : R2n → kerL defined byH(a) = eDta, for a ∈ R2nis an isometric isomorphism. In this paper, we identify a ∈ R2n with its imageH(a) ∈ kerL, i.e., H(a) = a, a ∈ R2n.For the sake of convenience, we also introduce the following notations. Let m, lbe some positive integers and H ∈ C(R2, R2). For any real number ρ ≥ 0, we set(3.6) MH(ρ) :=12π∫ 2π0eAT sH((ρ sin s, ρ cos s)T )ds;(3.7) MH(ρ, m, l) :=12lπ∫ 2lπ0eAT ( msl)H((ρ sin s, ρ cos s)T )ds.It is easy to know that for any positive integer m,MH(ρ, m, m) = MH(ρ)In what follows, the following lemmas are needed.EJQTDE, 1999 No. 2, p.5Lemma 3.2 [11]. Let m, l be some positive integers and 0 < r0 < 1. If H ∈C(R2, R2) is bounded and the asymptotic limits H(±,±) = limr,s→±∞ H(r, s) exist,then(3.8) limρ→∞MH(rρ, m, l) = W H(m, l)and(3.9) limρ→∞MH(rρ) = W Huniformly for r in [r0, 1].Lemma 3.3. For any permutation k1, k2, · · · , kn consisting of 1, 2, · · · , n thereexists a family of continuous mappings ηi : R2n → R2n (i = 1, 2, · · · , N1) with(3.10) 〈ηi(u), ηi(u)〉 > 0, 〈u, ηi(u)〉 > 0, for u ∈ R2n\{0},such that for any aj ∈ R2 (j = 1, 2, · · · , n),(3.11) η1η2 · · ·ηN1(a1, a2, · · · , an) = (ak1 , ak2 , · · · , akn)holds.Proof. It suffices to show that there exists a family of continuous mappings ζi :R2n → R2n (i = 1, 2, · · · , n1) with(3.12) 〈ζi(u), ζi(u)〉 > 0, 〈u, ζi(u)〉 > 0, for u ∈ R2n\{0},such that for any aj ∈ R2 (j = 1, 2, · · · , n) and 1 ≤ j1 < j2 ≤ n,(3.13) ζ1ζ2 · · · ζn1(a1, · · · , aj1, · · · , aj2 , · · · , an) = (a1, · · · , aj2 , · · · , aj1, · · · , an),Define ζi : R2n → R2n (i = 1, 2, · · · , 6) byζi(u) = (ζ(1)i , ζ(2)i , · · · , ζ(n)i ),ζ(k)i =uk, k 6= j1, j2√22uj1 +√22uj2 , k = j1−√22uj1 +√22uj2 , k = j2, i = 1, 2EJQTDE, 1999 No. 2, p.6ζ(k)i =uk, k 6= j1((√22,−√22)uTj1 , (√22,√22)uTj1), k = j1, i = 3, 4, 5, 6where u = (u1, u2, · · · , un) ∈ R2n, uk ∈ R2, k = 1, 2, · · · , n.It is easy to see that ζi (i = 1, 2, · · · , 6) is continuous, moreover, (3.12) and(3.13) hold with n1 = 6. This completes the proof.We are now in a position to prove our main result.Proof of Theorem 2.1. Let M = 4π[(∑nj=1 M2Fj)1/2 + (∑nj=1 ‖pj‖2)1/2], thenfor any x ∈ P (2n)2π , ‖K(I − Q)Nx‖ ≤ M , and hence the condition (I) of Lemma 3.1holds.Let ρ > 0, takeΩ0 = {u ∈ ker L : u = (r1ρa1, r2ρa2, · · · , rnρan), as ∈ ∂B1(0) ⊂ R2,0 ≤ rs < 1, s = 1, 2, · · · , n}.Then Ω0 is a bounded open set in ker L and∂Ω0 =n⋃j=1{u ∈ ker L : u = (r1ρa1, r2ρa2, · · · , rnρan), as ∈ ∂B1(0) ⊂ R2,0 ≤ rs ≤ 1, s = 1, 2, · · · , n, rj = 1}For x ∈ P (2n)2π with xj(t) = rjρeBjtaj + x̄j(t), aj ∈ ∂B1(0) = {a ∈ R2 : |a| =1} ⊂ R2; x̄ = (x̄1, x̄2, · · · , x̄n) ∈ ImL, x̄j(t) ∈ R2, ‖x̄j‖ ≤ M, j = 1, 2, · · · , n, it isnot hard to verify that(3.14) JQNx = ((JQNx)1, (JQNx)2, · · · , (JQNx)n),(3.15) (JQNx)j = eBTj τj Yj(ρ, akj , rkj ) + Xj(x) + pj(mj),where(3.16) Xj(x) =12π∫ 2π0eBTj sGj(s, x(s + ·))ds,(3.17) Yj(ρ, akj , rkj ) =12π∫ 2π0eBTj sHj(rkj ρeBkj sakj + x̄kj (s + τ))ds.EJQTDE, 1999 No. 2, p.7By using the fact that ‖x̄kj‖ ≤ M and a similar argument used in the proof ofLemma 3.2, it is not hard to show that(3.18) limρ→∞|Yj(ρ, akj , 1)| = |W Hj (mj , mkj )|uniformly for akj in ∂B1(0) ⊂ R2 and ‖x̄kj‖ ≤ M .It follows from (2.7) and (3.16) that(3.19) |W Hj (mj , mkj )| > MGj + |pj(mj)| ≥ |Xj(x)| + |pj(mj)|If rj0 = 1 for some j0, then (3.14), (3.15), (3.18) and (3.19) imply that for ρsufficiently large,JQN(x) 6= 0.Thus, we have proved that for ρ sufficiently large,JQN(x) 6= 0for any x ∈ P (2n)2π with Px ∈ ∂Ω0 and ‖(I − P )x‖ ≤ M , that is, the condition (II)of Lemma 3.1 also holds.Define the mapping η : Ω̄0 → kerL byη(u) = (η(1)(u), η(2)(u), · · · , η(n)(u)),η(j)(u) =12π∫ 2π0eBTj sGj(s, r1ρeB1(s+·)a1, · · · , rnρeBn(s+·)an)ds + pj(mj),u = (r1ρa1, · · · , rnρan), aj ∈ ∂B1(0) ⊂ R2, 0 ≤ rj ≤ 1, j = 1, 2, · · · , n.Then it is easy to see that η is continuous.Let βj (−π < βj ≤ π) be defined bysin βj =WHj1 (mj , mkj )|WHj (mj , mkj )|, cos βj =WHj2 (mj , mkj )|WHj (mj , mkj )|hereWHj (mj , mkj ) = (WHj1 (mj , mkj ), WHj2 (mj , mkj )).Let N2 be a positive integer satisfying∣∣∣∣τj − βj/mjN2∣∣∣∣<π2mj, j = 1, 2, · · · , n.EJQTDE, 1999 No. 2, p.8Define ηi : ker L → ker L (i = 1, 2, · · · , N2) byηi(u) = (η(1)i (u), η(2)i (u), · · · , η(n)i (u)),η(j)i (u) = eBTj γj uj , γj =τj − βj/mjN2, j = 1, 2, · · · , n,where u = (u1, u2, · · · , un), uj ∈ R2 (j = 1, 2, · · · , n). Then it is clear that ηi (i =1, 2, · · · , N2) is continuous and (3.2) holds.By Lemma 3.3, we can also define ηi : kerL → ker L (i = N2 +1, · · · , N2 +N1),which are continuous and satisfy (3.2), such thatηN2+1ηN2+2 · · · ηN2+N1(a1, a2, · · · , an) = (ak1 , ak2 , · · · , akn)holds for any aj ∈ R2 (j = 1, 2, · · · , n).In the sequel, we assume that u = (r1ρa1, r2ρa2, · · · , rnρan) ∈ ∂Ω0, aj ∈∂B1(0) ⊂ R2, 0 ≤ rj ≤ 1 j = 1, 2, · · · , n. Clearly, we may assume, withoutloss of generality, that rkj0 = 1 for some j0.Let αj = αj(aj) (−π < αj ≤ π) defined by sin αj = a(1)j , cosαj = a(2)j , whereaj = (a(1)j , a(2)j ).Therefore, we have(3.20) η̄(u) := η1η2 · · · ηN2+N1(u) = (η̄1(u), η̄2(u), · · · , η̄n(u)),(3.21)η̄j(u) = rkj ρeBTj (τj−βj/mj)akj = ρrkj eBTj τj eAαkj WHj (mj , mkj )/|W Hj (mj , mkj )|(3.22) ξ̄(u) := JQNu − η(u) = (ξ̄1(u), ξ̄2(u), · · · , ξ̄n(u)),(3.23)ξ̄j(u) =12πeBTj τj∫ 2π0eBTj sHj(rkj ρeBkj sakj )ds= eBTj τj eA(mjαkj /mkj )MHj (rkj ρ, mj, mkj ).It follows from (3.20)-(3.23) that(3.24)〈ξ̄(u), η̄(u)〉= ρn∑j=1rkj|WHj (mj , mkj )|〈WHj (mj , mkj ), eA$(j)αkj MHj (rkj ρ, mj, mkj )〉.EJQTDE, 1999 No. 2, p.9where $(j) =mj−mkjmkj.Since |αkj | ≤ π and |mj − mkj | < 12mkj , we have∣∣$(j)αkj∣∣ ≤ |$(j)π| < π2, j = 1, 2, · · · , n.Hence,〈WHj (mj , mkj ), eA$(j)αkj WHj (mj , mkj )〉 = |W Hj (mj , mkj )|2 cos($(j)αkj)(3.25) ≥ |W Hj (mj , mkj )|2 cos ($(j)π) > 0,For j 6= j0, we setI0j =[0, |W Hj0 (mj0 , mkj0 )| cos ($(j0)π)/(4nMHj )),I1j =[|WHj0 (mj0 , mkj0 )| cos ($(j0)π)/(4nMHj ), 1],then by (3.24), and noting that rkj0 = 1, we have(3.26) 〈ξ̄(u), η̄(u)〉 = ρ [Z0 + Z1 + Z2]where,(3.27)Z0 =1|WHj0 (mj0 , mkj0 )|〈WHj0 (mj0 , mkj0 ), eA($(j0)αkj0)MHj0 (ρ, mj0 , mkj0 )〉(3.28)Z1 =∑j 6=j0,rkj ∈I0jrkj|WHj (mj , mkj )|〈WHj (mj , mkj ), eA($(j)αkj )MHj (rkj ρ, mj , mkj )〉(3.29)Z2 =∑j 6=j0,rkj ∈I1jrkj|WHj (mj , mkj )|〈WHj (mj , mkj ), eA($(j)αkj )MHj (rkj ρ, mj , mkj )〉Since MHj (rkj ρ, mj , mkj ) → W Hj (mj , mkj ) (ρ → ∞) uniformly for rkj in I1j byLemma 3.2, we have(3.30) Z2 > 0,for large ρ.EJQTDE, 1999 No. 2, p.10By the Schwartz inequility, we also find(3.31) |Z1| ≤∑j 6=j0,rkj∈I0jrkj MHj ≤ |W Hj0 (mj0 , mkj0 )| cos ($(j0)π)/4.Therefore, it follows from (3.26)-(3.31) that for ρ suffciently large,〈ξ̄(u), η̄(u)〉 > 0.Thus, (3.3) holds for any u ∈ ∂Ω0.On the other hand, it is not hard to show that〈JQNu, JQNu− η(u)〉=n∑j=1|MHj (rkj ρ, mj , mkj )|2+n∑j=1〈MHj (rkj ρ, mj, mkj ), eAT (mjαkj /mkj )eBjτj Xj(u)〉+n∑j=1〈MHj (rkj ρ, mj, mkj ), eAT (mjαkj /mkj )eBjτj pj(mj)〉,whereXj(u) =12π∫ 2π0eBTj sGj(s, r1ρeB1(s+·)a1, · · · , rnρeBn(s+·)an)ds.By using the Schwartz inequality, it follows that〈JQNu, JQNu − η(u)〉 ≥n∑j=1|MHj (rkj ρ, mj, mkj )|2−n∑j=1|MHj (rkj ρ, mj, mkj )|[MGj + |pj(mj)|]=n∑j=1[|MHj (rkj ρ, mj, mkj )| −12(MGj + |pj(mj)|)]2− 14n∑j=1(MGj + |pj(mj)|)2.Since rkj0 = 1 for some j0, and MHj0 (ρ, mj0, mkj0 ) → WHj0 (mj0 , mkj0 ) (ρ → ∞),it follows from (2.7) that for ρ sufficiently large,〈JQNu, JQNu− η(u)〉 > 0.EJQTDE, 1999 No. 2, p.11Thus, (3.4) also holds for any u ∈ ∂Ω0.By virtue of Lemma 3.1, Eq.(2.1) has at least one 2π-periodic solution and theproof is complete.Finally, we give an example to illustrate our main results.Example Consider the system(3.32)x′1 = x2 + x3/(1 + x23) + arctan x1 + p1(t)x′2 = −x1 + 3 arctanx4 +12arctanx2 + p2(t)x′3 = x4 + x5e−x25 +√2 sin x3 + p3(t)x′4 = −x3 −√6 arctan x6 +√2 cos x3 + p4(t)x′5 = x6 + arctan x5 − 2 arctanx1 + p5(t)x′6 = −x5 +12arctan x6 − 2 arctanx2 + p6(t)where pj(j = 1, 2, · · · , 6) are continuous, 2π-periodic functions. By Corollary 2.1,it is easy to check that Eq.(3.32) has at least one 2π−periodic solution provided√|c1|2 + |c2|2 + |c3|2 <√6 −√22− 14√2√5π2 + 32,whereck =12π∫ 2π0(cos s − sin ssin s cos s)(p2k−1(s)p2k(s))ds, k = 1, 2, 3.References1. J. K. Hale, Ordnary Differential Equations, Wiley Interscience, New York, 1969.2. R. K. Nagle, Nonlinear boundary value problems for ordinary differential equations with asmall parameter, SIAM J. Math. Analysis 9 (1978), 719-729.3. S. Fucik, Solvability of Nonlinear Equations and Boundary Value Problems, D. Reidel Pub-lishing, Dordrecht, Holland, 1980.4. A. C. Lazer & D. E. Leach, Bounded perturbations of forced harmonic oscillations at reso-nance, Ann. Mat. Pura. Appl., 82 (1969), 49-68.5. L. Cesari, Non-linear problems across a point of resonance for non-self-adjoint systems, Ac-demic Press (1978), 43-67.6. R. Iannacci & M. N. Nkashma, Unbounded perturbations of forced second order ordinarydifferential equations at resonance, J. Diff. Eqns. 69 (1978), 289-309.7. R. K. Nagle & Z. Sinkala, Semilinear equations at resonance where the kernel has dimensiontwo, in Differential Equations:Stability and Control (Edited by S.Elaydi) Marcel Dekker, NewYork (1991).EJQTDE, 1999 No. 2, p.128. R. K. Nagle & Z. Sinkala, Existence of 2π-periodic solutions for nonlinear systems of first-order ordinary differential equations at resonance, Nonlinear Analysis (TMA) 25 (1995),1-16.9. S. W. Ma, Z. C. Wang & J. S. Yu, Coincidence degree and periodic solutions of Duffingequations, Nonlinear Analysis (TMA) 34 (1998), 443-460.10. J. D. Schuur, Perturbation at resonance for a fourth order ordinary differential equation, J.Math. Anal. Appl. 65 (1978), 20-25.11. S. W. Ma, Z. C. Wang & J. S. Yu, An abstract existence theorem at resonance and itsapplications, J. Diff. Eqns., 145 (1998), 274-294.EJQTDE, 1999 No. 2, p.13

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Periodic solutions of semilinear equations at resonance with a $2n$ -dimensional kernel

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Periodic solutions of semilinear equations at resonance with a 2n2n2n-dimensional kernel

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Periodic solutions of semilinear equations at resonance with a $2n$ -dimensional kernel