The Existence Of Solutions Of Nonlinear Periodic Boundary Value Problem

In this paper, we consider the existence of solutions of the following periodicboundary value problem ??u′(′(0t))=+um( u(t) = f (t,u(t),u′(t)) 2 (I) ?u 2π),u′(0) = u′(2π) When m∈(0, )and f:I ×R+ ×R → R+ is continuous,where I =[0,2π ],we 1 2obtainthefollowing result, applyingthe theorem from the reference[5]. Theorem 1 If f (t,u,v) satisfies one of the following conditions ?lim f (t, u , v) | t ∈ [0,2π ], v ≤ ku} > λ1 , ?k > 0 (1) ??? inf inf{ u → 0 + u ???limu f (t, u, v) sup sup{ | t ∈ [0,2π ], v ≤ ku} < λ1 , ?k >0 → ∞ u ?lim f (t, u , v) | t ∈ [0,2π ], v ≤ ku } < λ1, ?k >0 (2) ??? sup sup{ u → 0+ u ???limu f (t, u , v ) inf inf{ | t ∈ [0,2π ], v ≤ ku} > λ1, ?k > 0 → ∞ uthe problem (I) has at least one positive solution, where λ1 represents the firsteigenvalue of relevant linear problem. When m = ,and f:I × R+ → R+is continuous,we prove the following theorem, 1 2using the fixed point theorem in cone. Theorem 2 If f (t,u) satisfies one of the following conditions f (t,u)= m1 < λ1 = , lim inf 1 f (t,u)= M1 > λ1 = ; 1 (1) lim sup u→0+ u 4 u→∞ u 4 f (t,u) 1 f (t,u) 1 (2) lim inf = M1 > λ1 = , lim sup = m1 < λ1 = u→0+ u 4 u→∞ u 4the problem (I) has at least one positive solution, where λ1 represents the firsteigenvalue of relevant linear problem. When m is a nonzero positive integer,and f:I × R → R is continuous,weobtain the existence of the solution of problem (I),using Mawhin alternative theorem.It extends the result of the reference [14].