A Pogorelov estimate and a Liouville-type theorem to parabolic k-Hessian equations

We consider Pogorelov estimates and Liouville-type theorems to parabolic [Formula: see text]-Hessian equations of the form [Formula: see text] in [Formula: see text]. We derive that any [Formula: see text]-convex-monotone solution to [Formula: see text] when [Formula: see text] satisfies a quadratic growth and [Formula: see text] must be a linear function of [Formula: see text] plus a quadratic polynomial of [Formula: see text].

Monotone Solution of Cauchy Type Weighted Nonlocal Fractional Differential Equation

In this paper we will consider a nonlinear fractional di fferential equation withweighted initial and nonlocal conditions and will obtain monotone solution by thesequence of successive approximations starting at a lower solution converges monotonicallyto the solution of the related cauchy type weighted nonlocal fractionaldi fferential equation under some suitable conditions.

On the Behaviour of the Solutions of a Second-Order Difference Equation

We study the difference equationxn+1=α−xn/xn−1,n∈ℕ0, whereα∈ℝand wherex−1andx0are so chosen that the corresponding solution(xn)of the equation is defined for everyn∈ℕ. We prove that whenα=3the equilibriumx¯=2of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao. For the caseα=1, we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the caseα=0.