Periodic solutions for second order differential equations with indefinite singularities

Abstract

In this paper, the problem of periodic solutions is studied for second order differential equations with indefinite singularities $$\begin{array}{} \displaystyle x''(t)+ f(x(t))x'(t)+\varphi(t)x^m(t)-\frac{\alpha(t)}{x^\mu(t)}+\frac{\beta(t)}{x^y (t)}=0, \end{array}$$ where f ∈ C((0, +∞), ℝ) may have a singularity at the origin, the signs of φ and α are allowed to change, m is a non-negative constant, μ and y are positive constants. The approach is based on a continuation theorem of Manásevich and Mawhin with techniques of a priori estimates.

Correctness conditions for high-order differential equations with unbounded coefficients

“…is also closed and invertible, and the inverse operator P -1 m is defined on the whole space L 2 . So, it follows that, for eachf ∈ L 2 ,ŷ = P -1 mf ∈ D(P m ) andŷ is a solution of equation (17). By (19), we deduce that…”

“…In other words, the cases of some degenerate equations are covered. Note that the criteria for the existence of positive periodic solutions for differential equations with indefinite singularity and pseudo almost periodic solutions of an iterative functional differential equations, respectively, were found in [17] and [18].…”

New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

“…So, it would be interesting to extend the results of this paper to the fractional delay differential equations. Moreover, it is interesting to study the periodicity behavior of solutions of the studied equation as an extension of the works [19,23].…”

“…Some of these works that are concerned with improving the oscillation criteria of DDEs are [3][4][5][6]. In addition, many improved methods and interesting results can be found in studies [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], which study the oscillatory behavior of the NDDEs of different order.…”

Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type

“…After the pioneering paper [6] came out, many scholars put their attention to the periodic problems of second order singular differential equations without friction term (see [7][8][9][10][11][12][13] and the references therein). Beginning with the paper of Habets-Sanchez in [14], the interest in the problem of periodic solution for second order singular differential equations with friction term has increased [15][16][17][18][19][20][21]. Hakl, Torres, and Zamora considered the periodic problem for the singular equation of attractive type…”