Weak nonlinearities

Abstract

The present chapter is concerned with the study of non-linear equation

$$x(t) = Af[t,x(t)]$$

((19.1))

in various spaces of scalar-valued functions x(t). We will assume that the function f(t, x) : Ω × ℝ¹ → ℝ ¹ has the form

$$f(t,x) = \alpha x + \varphi (t,x) + b(t)$$

((19.2))

where the proper nonlinearity — the function φ(t, x) — is locally bounded in x, i.e.,

$$\mathop{{\sup }}\limits_{{\left| x \right| \leqslant {{x}_{0}}}} \mathop{{\sup }}\limits_{{t \in \Omega }} \left| {\varphi (t,x)} \right| < \infty$$

((19.3))

for any nonnegative x⁰, and, in addition,

$$\mathop {\lim sup {x^{ - 1}}}\limits_{\left| x \right| \to \infty t \in \Omega } \left| {\varphi \left( {t,x} \right)} \right| = 0.$$

((19.4))