Mostafa Fazly

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In this Note, sharp sufficient conditions for the existence of periodic solutions of a nonautonomous discrete time semi-ratiodependent predator–prey system with functional responses are derived. In our results this system with any monotone functional response bounded by polynomials in R+, always has at least one ω-periodic solution. In particular, this(More)
We prove that the following pointwise inequality holds −∆u ≥ √ 2 (p + 1)− cn |x| a 2 u p+1 2 + 2 n− 4 |∇u|2 u in R where cn := 8 n(n−4) , for positive bounded solutions of the fourth order Hénon equation that is ∆u = |x|u in R where a ≥ 0 and p > 1. Motivated by the Moser iteration argument in the regularity theory, we develop an iteration argument to prove(More)
In this paper we explore the existence of periodic solutions of a nonautonomous semi-ratio-dependent predator-prey dynamical system with functional responses on time scales. To illustrate the utility of this work, we should mention that, in our results this system with a large class of monotone functional responses, always has at least one periodic(More)
This paper deals with the question of existence of periodic solutions of nonautonomous predator–prey dynamical systems with Beddington–DeAngelis functional response. We explore the periodicity of this system on time scales. New sufficient conditions are derived for the existence of periodic solutions. These conditions extend previous results presented in(More)
We consider the following elliptic system ∆u = ∇H(u) in R , where u : R → R and H ∈ C(R), and prove, under various conditions on the nonlinearity H that, at least in low dimensions, a solution u = (ui) m i=1 is necessarily one-dimensional whenever each one of its components ui is monotone in one direction. Just like in the proofs of the classical De(More)
We consider the problem of non-existence of solutions for the following Hénon-Lane-Emden system  −∆u = |x|v in R , −∆v = |x|u in R , when pq > 1, p, q, a, b ≥ 0, and (p, q) are under the critical hyperbola, i.e. N+a p+1 + N+b q+1 > N − 2. We show that there is no positive bounded solution in dimension N = 3, extending a result established recently by(More)
We classify finite Morse index solutions of the following nonlocal Lane-Emden equation (−∆)u = |u|p−1u R for 1 < s < 2 via a novel monotonicity formula. For local cases s = 1 and s = 2 this classification is provided by Farina in [10] and Davila, Dupaigne, Wang and Wei in [8], respectively. Moreover, for the nonlocal case 0 < s < 1 finite Morse index(More)
Assume that Nm(x) denotes the density of the population at a point x at the beginning of the reproductive season in the mth year. We study the following impulsive reaction-diffusion model for any m ∈ Z+  u (m) t = div(A∇u(m) − au(m)) + f(u(m)) for (x, t) ∈ Ω× (0, 1] u(m)(x, 0) = g(Nm(x)) for x ∈ Ω Nm+1(x) := u(m)(x, 1) for x ∈ Ω for functions f, g, a(More)
We establish Liouville-type theorems for stable and finite Morse index weak solutions of −∆pu = f(x)F (u) in R . For a general non-linearity F ∈ C(R) and f(x) = |x|, we prove such theorems in dimensions N ≤ 4(p+α) p−1 +p, for bounded radial stable solutions. Then, we give some point-wise estimates for not necessarily bounded solutions. Also, similar(More)
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