Existence of Two Positive Periodic Solutions for a Neutral Multi-Delay Logarithmic Population Model with a Periodic Harvesting Rate
نویسندگان
چکیده
and Applied Analysis 3 The following lemma 11, page 213 will play a key role in this paper. Lemma 2.2. Let L be a Fredholm mapping of index zero and let N : Ω → Z be L-k-set contractive on Ω, k < 1. Suppose i Lx/ λNx for every x ∈ domL ∩ ∂Ω and every λ ∈ 0, 1 ; ii QNx/ 0 for every x ∈ ∂Ω ∩ KerL; iii Brouwer degree degB JQN, Ω ∩ KerL, 0 / 0. Then Lx Nx has at least one solution in domL ∩Ω. 3. Main Result LetC0 T denote the linear space of real valued continuous T -periodic functions on R. The linear space C0 T is a Banach space with the usual norm for x ∈ C0 T given by |x|0 maxt∈R|x t |. Let C1 T denote the linear space of T -periodic functions with the first-order continuous derivative. C1 T is a Banach space with norm |x|1 max{|x|0, |x′|0}. Let X C1 T and Z C 0 T and let L : X → Z be given by Lx dx/dt. Since |Lx|0 |x′|0 ≤ |x|1, we see that L is a bounded with bound 1 linear map. Under the transformation of y t e t , 1.1 can be rewritten as x′ t a t − n ∑ j 1 bj t x ( t − σj t ) − m ∑ i 1 ci t ( 1 − τ ′ i t ) x′ t − τi t − h t ex t . 3.1 Next define a nonlinear map N : X → Z by N x t a t − n ∑ j 1 bj t x ( t − σj t ) − m ∑ i 1 ci t ( 1 − τ ′ i t ) x′ t − τi t − h t ex t . 3.2 Now, if Lx Nx for some x ∈ X, then the problem 3.1 has a T -periodic solution x t . In the following, we denote g 1 T ∫T 0 g t dt, g min t∈ 0,T g t , g max t∈ 0,T g t , 3.3 where g t is a continuous nonnegative T -periodic solution. From now on, we always assume that H1 a t , bj t ∈ C R, 0, ∞ , σj t , ci t , τi t ∈ C1 R, 0, ∞ , for all j ∈ {1, 2, . . . , n}, for all i ∈ {1, 2, . . . , m}. H2 σ ′ j t < 1, τ ′ i t < 1, for all t ∈ R, and Γ t n ∑ j 1 bj ( μj t ) 1 − σ ′ j ( μj t ) − m ∑ i 1 c′ i ( γi t ) 1 − τ ′ i ( γi t ) > 0, 3.4 4 Abstract and Applied Analysis where μj t is the inverse function of t−σj t , γi t is the inverse function of t−τi t , for all j ∈ {1, 2, . . . , n}, for all i ∈ {1, 2, . . . , m}. H3 Let 1 ln h Γu < a Γl ≤ h l Γu < 1. 3.5
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تاریخ انتشار 2014