Positive periodic solutions for singular fourth-order differential equations with a deviating argument

Abstract

In this paper, we study the singular fourth-order differential equation with a deviating argument:By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.

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

“…In this section, we present an example to demonstrate the main results. Remark 4.2 From the above example, we see that the degree μ associated with the restoring force term ϕ(t)x μ is μ = 3 2 , which is different from the corresponding ones of μ ∈ (0, 1] in [23][24][25]. Furthermore, since the degree μ in (1.1) is required μ ∈ (0, 1], even if c = 0, the results of the present paper are different from Theorem 1.1.…”

“…(1.1). However, the study of periodic solutions for delay functional differential equation with a singularity is relatively infrequent [23][24][25]. For example, Wang in [23] studied the problem of periodic solutions for the singular delay Liénard equation of repulsive type…”

“…has been investigated in [24] and [25], where ϕ(t) and e(t) are T-periodic with ϕ, e ∈ L 1 [0, T], γ and μ are positive constants. We notice that the degree associated with the term ϕ(t)x μ is required μ ∈ (0, 1).…”