Abstract
Let f [0, 7r] 
Abstract
This problem is at resonance since the associated linear boundary value problem "(t) = (t), 0 < t < , '(0) = '() = 0, has A = 0 as an eigenvalue. Asymptotic conditions on the nonlinearity .f(t, z(t)) are offered to give existence of solutions for the nonlinear systems. The methods apply to the corresponding system of Lienard-type periodic boundary value problems.
Abstract
Key words and phrases: Second-order system of Neumann boundary value problems, resonance at infinitely many eigenvalues, absence of L-resonance, asymptotic resonance conditions, Fredholm operator
Introduction
Let tt Ar denote the N-dimensional Euclidean space. For z = (Zl,Z2, -..,:gN) t RN, andy = (Yt,Y,...,YN) e RN, let [zl-z-t-z + i'i""+ Zv, and < z,y >= zyl + zy + + zyYy denote the Euclidean norm of z and the inner product of z This paper is devoted to the study of systems of Neumann boundary value problems -z"(t) + f(t,z(t))= e(t), 0 < t < r, '(0) = 0, and z"(t) + f(t,z(t))= e(t), 0 < t < r, ' (o) = o.
Introduction
(1. (ii) If (t,x)l <_ (1.5) for a.e. t [0, 7r] and all z R N with Iz l _< r. We give asymptotic conditions on the behavior of zi-lfi(t, z), i = 1, 2,..., N, at the first two eigenvalues 0 and I of the linear problem z"(t) + Az(t)= 0, 0 < t < 7r, (1.7) x'(0) = x'(r)-0, (1.8) for the problem (1.3)-(1.4).
Introduction
Our methods can be adapted and similar results obtained for Lienard's system of equations 4- x"(t) + [-gradF(x(t))] + f(t,x(t)) = e(t), 0 < t < r, (1.9) z(0)-x(2rr)-z'(0)-x'(2rr)-0, (I.i0) where F tL N -+ tt is in C(I:tN, P), f'[0, 2rr] x 1 N -+ pN satisfies Caratheodory's conditions, and e [0,2r] --+ pN is Lebesgue integrable. The problem (1.9)-(1.10) was studied by Ianacci and Nkashama in [3] , where they give sufficient non-resonance conditions for the existence of a solution. We provide in this paper sufficient resonance conditions for the existence of a solution for the problems (1.1)-(1.2) and (1.3)-(1.4) and accordingly for (1.9)-(1.10)in line with our remark above.
Introduction
Our results and methods are inspired by the results of Gupta and Mawhin [2] for the problem (1.9)-(1.10) when N = 1. We present in Section 2 notations and definitions that we need in this paper. In Section 3 we present some lemmas that are extensions to systems of corresponding lemmas in [3] . We present in Section z"(t)+ f(t,z(t))= e(t), 0 < t < 7r, (4.15) has at least one solution.
Introduction
Proof. Define Since (Lx), Px,) = (Lx:,x),), and given (4.6) We have thus shown that the set of solutions of (4.28) is bounded in X independently of (0, 1).
Introduction
Hence the theorem follows. z"(t) + g(t, z(t)) + h(t, z(t)) = E(t).
Introduction
(4.42)
Introduction
We can next apply Theorem IV.4 of [5] to the boundary value problem posed by(4.42) and (4.36) . It suffices to show that the set of solutions of the family of equations z"(t) + (1 )T(t)Iz(t) + g(t,z(t)) + ,h(t,x(t)) = ,E(t), We note that g = (gi, g2, ...,gN) in (5.6) is such that gi(t, z) = /i(t, z)zi. If we write "(t, z) = ('l(t, z), ..., ZN(t, z)), then g(t, x) = 5/(t, z)TIx.
Introduction
We see that 0 < (1-,)F'(t)+ ,zi(t,z(t)) N r'(t) for i = 1, 2, ...,N in view of (4.40) 

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