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In this paper, using the Fourier series expansion and fixed point methods, we investigate the existence and uniqueness of Besicovitch almost periodic solutions for a class of second order differential equations involving reflection of the argument. Lipschitz nonlinear case is considered.

We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: (|f′|p−2f′)′+βrf′+αf+(fq)′=0 satisfying a specific decay rate: lim⁡r→∞rα/βf(r)=0 with α:=(p−1)/(pq−2p+2) and β:=(q−p+1)/(pq−2p+2). Here p>2 and q>p−1. Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term ut=(|ux|p−2ux)x+(uq)x defined on the half line [0,+∞).

We generalize the comparison result 2007 on Hamilton-Jacobi equations to nonlinear parabolic equations, then by using Perron's method to study the existence and uniqueness of time almost periodic viscosity solutions of nonlinear parabolic equations under usual hypotheses.

In this paper, we consider the boundedness of solutions for a class of impact oscillators with time dependent polynomial potentials, {ẍ + x2n+1 ...pi(t)xi = 0, for x(t) > 0, x(t) ≥ 0, ẋ(t0+) = -ẋ (t0-), if x(t0) = 0, where n ∈ ℕ+, pi(t + 1) = pi(t) and pi(t) ∈ C5 (ℝ/ℤ). [ABSTRACT FROM AUTHOR]