Motivation: I'm trying to understand the proof of Theorem 3.1 in the this paper. Though in the following I'm raising kind of abstract question.

Consider non-linear Schrodinger equation (NLS): $$i\partial_tu +\Delta u = |u|^{2}u, \quad u(t_0, x)= \varphi(x).$$

Suppose $X$ (space of functions on $\mathbb R^d$) is Banach space and assume that $$\|u\|_{L_I^{\infty}X} \lesssim \|\varphi\|_{X} + |I|\|u\|^3_{L_{I}^{\infty}X}$$ here $I$ is small time interval

Question: (1) How to use the standard continuity argument to say that: there exists a solution $u$ to NLS in $I \times \mathbb R^d$ such that $$\|u\|_{L^{\infty}_{I} X} \leq C \|\varphi \|_{X}$$ for sufficiently small $I$? (2) What is standard continuity argument?

Motivation: I'm trying to understand the proof of Theorem 3.1 in Antonelli, Saut, and Sparber - Well-Posedness and averaging of NLS with time-periodic dispersion management. Though in the following I'm raising kind of abstract question.

Consider non-linear Schrödinger equation (NLS): $$i\partial_tu +\Delta u = |u|^{2}u, \quad u(t_0, x)= \varphi(x).$$

Suppose $X$ (space of functions on $\mathbb R^d$) is Banach space and assume that $$\|u\|_{L_I^{\infty}X} \lesssim \|\varphi\|_{X} + |I|\|u\|^3_{L_{I}^{\infty}X}$$ here $I$ is small time interval.

Questions: (1) How to use the standard continuity argument to say that: there exists a solution $u$ to NLS in $I \times \mathbb R^d$ such that $$\|u\|_{L^{\infty}_{I} X} \leq C \|\varphi \|_{X}$$ for sufficiently small $I$? (2) What is standard continuity argument?