Periodic Solutions of Multispecies Mutualism System with Infinite Delays

and Applied Analysis 3 The projectors are defined by P : X → X and Q : Z → Z by


Introduction
Recently, there are many papers considering the existence of periodic solutions for competitive Lotka-Volterra system based on Mawhin's coincidence degree theory (see [1][2][3][4]).But there are few papers considering the periodicity of mutualism system; for example, one can refer to [5][6][7].However, the references mentioned above only considered two-dimensional mutualism system.To the best knowledge of the authors, there is no paper considering the existence of periodic solutions for -species mutualism system.It should be noted that the method used in [5][6][7] is difficult to be extended to the -dimensional system.So, we employ the method used in [2][3][4].However, the problem considered in this paper is completely different from those mentioned above.On the other hand, the above-mentioned works considered the models with constant discrete delays or without delays.In practice, there will be a distribution of transmission delays.In this case, the transmission of species is no longer instantaneous and cannot be modelled with discrete delays.A more appropriate way is to incorporate distributed delays.Therefore, the studies of the model with distributed delays have more important significance than the ones of the model with discrete delays.Thus, in this paper, we considered the following mutualism system with distributed delays: where   ,    ( 0 ) =  0  ,  0  > 0,  = 1, 2, . . .. (2)
Then  =  has at least one solution in Ω ∩ Dom .
Proof.Note that every solution () = ( 1 (),  2 (), ...  ())  of system (1) with the initial value condition is positive.Make the change of variables Then system (1) is the same as Obviously, if system (8) has at least one -periodic solution, then system (1) has at least one -periodic solution.
To prove Theorem 4, we should find an appropriate open set Ω satisfying Lemma 1.We divide the proof into three steps.
Step 1.We verify that (i) of Lemma 1 is satisfied.For any () ∈ , by periodicity, it is easy to check that And define  : Dom  ⊂  →  and  :  →  as follows: where The projectors are defined by  :  →  and  :  →  by It is easy to follow that  is a Fredholm mapping of index zero.Furthermore, the generalized inverse (to )   : Im  → Dom  ∩ Ker  exists, which is given by Then  :  →  and   ( − ) :  →  are defined by where Using similar arguments to Step 1 in [2], it is easy to show that (  (−))(Ω) is relatively compact in the space (, ‖⋅‖ 1 ).