Caspian Journal of Mathematical Sciences
Volume:5 Issue: 1, Winter and Spring 2016

Abstract. In the present paper a generalized Lotka-Volterra food chain system has been studied and also its dynamic behavior such as locally asymptotic stability has been analyzed in case of existing interspecies competition. Furthermore, it has been shown that the said system is permanent (in the sense of boundedness and uniformly persistent). Finally, it is proved that the nontrivial equilibrium point of the above system is locally asymptotically stable.

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R^3.

Let (X, d) be a compact metric space and let K be a nonempty compact subset of X. Let α ∈ (0, 1] and let Lip(X, K, dα ) denote the Banach algebra of all continuous complex-valued functions f on X for which p(K,dα)(f) = sup{ |f(x)−f(y)| dα(x,y) : x, y ∈ K, x 6= y} < ∞ when equipped with the algebra norm ||f||Lip(X,K,dα) = ||f||X +p(K,dα)(f), where ||f||X = sup{|f(x)| : x ∈ X}. We denote by lip(X, K, dα ) the closed subalgebra of Lip(X, K, dα ) consisting of all f ∈ Lip(X, K, dα ) for which |f(x)−f(y)| dα(x,y) → 0 as d(x, y) → 0 with x, y ∈ K. In this paper we show that every proper closed ideal of (lip(X, K, dα ), k · kLip(X,K,dα)) is the intersection of all maximal ideals containing it. We also prove that every continuous point derivation of lip(X, K, dα ) is zero. Next we show that lip(X, K, dα ) is weakly amenable if α ∈ (0, 1 2 ). We also prove that lip(T, K, d 1 2 ) is weakly amenable where T = {z ∈ C : |z| = 1}, d is the Euclidean metric on T and K is a nonempty compact set in (T, d).

We prove the existence of solutions for the neutral periodic integro-differential equation with infinite delay x 0 (t) = G(t, x(t), x(t − τ (t))) + d dtQ(t, x(t − τ (t))) + Z t −∞ Xn j=1 gj (t, s)  f(x(s))ds, x(t + T) = x(t). A Krasnoselskii and Banach’s fixed point theorems are employed in establishing our results.

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