Existence and Nonexistence of Positive Solutions for High-Order Fractional Differential Equation Boundary Value Problems at Resonance

We investigate positive solution for high-order fractional differential equations with resonant boundary value conditions. By using the spectral theory and the fixed point index theory, both the nonexistence and existence results of the resonant boundary value problems are given.

A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.

Isolated singularities of positive solutions for Choquard equations in sublinear case

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent [Formula: see text]. Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation.

Existence of positive periodic solutions in shifts δ± for a nonlinear first order functional dynamic equation on time scales

Let T ? R be a periodic time scale in shifts ?? with period P ? [t0,?)T. In this paper we consider the nonlinear functional dynamic equation of the form x?(t) = a(t)x(t)- ?b(t) f (x(h(t))), t ? T. By using the Krasnoselski?, Avery-Henderson and Leggett-Williams fixed point theorems, we present different sufficient conditions for the nonexistence and existence of at least one, two or three positive periodic solutions in shifts ?? of the above problem on time scales. We extend and unify periodic differential, difference, h-difference and q-difference equations and more by a new periodicity concept on time scales.

On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities

We consider the class of polynomial differential systems of the form , where Pn and Qn are homogeneous polynomials of degree n. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence