### New Stability Conditions for the Delayed Liénard Nonlinear Equation via Fixed Point Technique

#### Abstract

The class of second order nonlinear neutral differential equations having the form

x+f(t,x,x)x+b(t)g(x(s-τ(s)))=0, for t≥t₀,

is studied by means of contraction mappings. We give some new conditions ensuring that the zero solution is asymptotically stable. Our results are strong and do not require conditions that have been indipensable in previous investigations such as, ((g(x))/x) ≥β>0 and lim((g(x))/x) exists as x→0, the delay τ(t) is differentiable, the map t↦t-τ(t) is strictly increasing (see <cite>pi1</cite>) or the function b:ℝ⁺→ℝ⁺ is bounded and there exists a function c:ℝ⁺→ℝ⁺ such that f(t,x,y)≤F(x,y)c(t) for all t≥0 and x,y∈ℝ.

The results obtained improve those of D.Pi <cite>pi1</cite> and are very significant because, from practical point of view, it is hard to control the factors of nuclear reactors that ensure the delay is smoothly changed if we do not allow dependance between the functions and during all the time.

x+f(t,x,x)x+b(t)g(x(s-τ(s)))=0, for t≥t₀,

is studied by means of contraction mappings. We give some new conditions ensuring that the zero solution is asymptotically stable. Our results are strong and do not require conditions that have been indipensable in previous investigations such as, ((g(x))/x) ≥β>0 and lim((g(x))/x) exists as x→0, the delay τ(t) is differentiable, the map t↦t-τ(t) is strictly increasing (see <cite>pi1</cite>) or the function b:ℝ⁺→ℝ⁺ is bounded and there exists a function c:ℝ⁺→ℝ⁺ such that f(t,x,y)≤F(x,y)c(t) for all t≥0 and x,y∈ℝ.

The results obtained improve those of D.Pi <cite>pi1</cite> and are very significant because, from practical point of view, it is hard to control the factors of nuclear reactors that ensure the delay is smoothly changed if we do not allow dependance between the functions and during all the time.

#### Keywords

Fixed points, Liénard equation, variable delay, stability, asymptotic stability.

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