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Fixed points and stability in nonlinear equations with variable delays. (English) Zbl 1214.34061
The following nonlinear delay differential equation is considered $$x'(t)=-a(t) f(x(t-r_1(t)))+b(t) g(x(t-r_2(t)))$$ (with special attention to $f(y)=y$), where $r_1, r_2:[0,\infty)\to [0,\infty)$, $a,b:[0,\infty)\to\Bbb R$, $f,g:\Bbb R\to\Bbb R$ are continuous functions. The main assumptions are that $r_1$ is differentiable and the functions $t\to t-r_i(t)$, $i=1,2$, are strictly increasing and tending to $\infty$ as $t\to\infty$. On the basis of the contraction mapping principle, new results are obtained for the boundedness and the stability of the solutions, which improve some recently obtained results of the same type. It is pointed out that, investigating stability, the fixed point method is some alternative of the Lyapunov direct method. The main difference is that while the Lyapunov method usually requires pointwise conditions, the fixed point method, needs average conditions.

##### MSC:
 34K20 Stability theory of functional-differential equations 47N20 Applications of operator theory to differential and integral equations
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##### References:
 [1] [2]