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This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u ″ (t) + A u (t) + b (t) u ′ (t) = 0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b (t) ∼ (1 + t) − β (− 1 < β < 1). In the case b (t) ≡ 1 , it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v ′ (t) + A v (t) = F n (t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b (t) v ′ (t) + A v (t) = 0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n -th order profile u ˜ n is given via v ′ (t) + m n (t) A v (t) = F n (t) which coefficient m n (time-scale) differs each other. [ABSTRACT FROM AUTHOR]

In this paper the half-linear differential equation with one-dimensional p -Laplacian (p = α + 1), (| u ′ | α sgn u ′) ′ = α (λ α + 1 + b (t)) | u | α sgn u , t ≥ t 0 , is considered, where α and λ are positive constants. It is proved that if the function b (t) is absolutely integrable on [ t 0 , ∞) , then the above equation has two nonoscillatory solutions u + (t) and u − (t) such that u ± (t) ∼ c e ± λ t and u ± ′ (t) ∼ ± c λ e ± λ t (t → ∞), and moreover, any nontrivial solution u (t) of the above equation satisfies either u (t) ∼ c e λ t , u ′ (t) ∼ c λ e λ t (t → ∞) or u (t) ∼ c e − λ t , u ′ (t) ∼ − c λ e − λ t (t → ∞). Here, c is a nonzero constant. A weaker result is also shown under the weaker condition that b (t) is conditionally integrable on [ t 0 , ∞). [ABSTRACT FROM AUTHOR]

This paper considers compound operators on a general vector space. Explicit representations for these operators are obtained in various contexts with emphasis on the compound differential equations corresponding to a given linear differential equation. Applications to estimation of eigenvalues of self-adjoint boundary value problems and of the codimension of the stable manifold of a linear differential equation in a Banach space are given. [ABSTRACT FROM AUTHOR]