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We obtain asymptotic representations for one class of solutions of a second-order difference equation with power nonlinearity. Statement of the Problem and Main ResultsWe consider the second-order difference equationand p n is positive for n ∈ N. For this equation, we investigate the problem of the existence and asymptotic representation of P (λ)-solutions defined as follows: Definition 1.1. A solution (y n ) +∞ n=1 of Eq. (1.1) is called a P (λ)-solution if the following relations are true: lim n→+∞ y n = y 0 , y 0 = ⎧ ⎨ ⎩ either 0 or ±∞, lim n→+∞ nΔ 2 y n Δy n = λ. (1.2)Equation (1.1) is a discrete analog of the well-known differential equation of the Emden-Fowler type, whose asymptotic properties are fairly well studied in [1][2][3][4][5][6]. In the present paper, we attempt to generalize the method of investigation presented in [6] to an analogous class of difference equations, complementing the results of [7,8], where, for an equation of the form (1.1), conditions for the existence of solutions asymptotically equivalent to cn and solutions belonging to the classes of functions c 0 and l 2 were obtained.We formulate the obtained results in the form of theorems. Theorem 1.1. Let +∞ m=1 mp m = +∞.

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We obtain asymptotic representations for one class of solutions of a second-order differential equation with nonlinearity close to an exponential. Statement of the Problem and Main ResultsConsider the second-order differential equation(1.2) and the functions L k : (0, +∞) → R \ {0} and F k : I → R are defined by the recurrence relations L r (z) = ln |L r−1 (z)| , L 1 (z) = ln z, z ∈ (0, +∞), F r (y) = L r−1 (ϕ(y)) (F r−1 (y) − 1) , F 1 (y) = ϕ (y)ϕ(y) ϕ 2 (y) , r = 2, . . . , k.(1.3)In the present paper, we investigate the asymptotic behavior of solutions of Eq. (1.1) defined in the left neighborhood of the point ω and such that lim t→ω y(t) = y 0 .The asymptotic behavior of solutions of Eq. (1.1) that belong to the class of so-called P ω (λ)-functions, where the function ϕ(y) satisfies the limit relation lim y→y 0 F 1 (y) = γ, γ ∈ R\{0},