Actually, in the process of proofs, we can find passivity is a higher abstraction level of stability.
From the process of proof in Theorem 1, we can get the following result.
Finally, using [4, Lemma 21] and repeating the process of proof in Lemma 2.8, one can show that this solution is actually in (C_{1-gamma}^{gamma}(J,E)).
Proof The process of proof is divided into four parts.
In the process of the proof of Corollary 2.3, we also call the set a generalized comparison type precompact-inducing subadmissible subset of.
In the process of the proof of Theorem 2.1, we call the set a Meir-Keeler type precompact-inducing subadmissible subset of.
Actually, we can find that passivity is a higher abstraction level of stability in the process of the proof, and the corresponding Lyapunov function (3.2) can be regarded as the storage function.
For Theorem 1.4, it is easy to see that it has the same shape as the classical Mazur-Ulam theorem, but it is not trivial since we must make full use of the relations between random normed modules and classical normed spaces in the process of the proof.
In the process of the proof of Theorem 2.4, it is worth pointing out that, since the space variant x and the time variant t appearing in (varphi x, t)) are inseparable, we cannot directly use the method of Lin [7].
Since the system case is different from the scalar case, we will come across some new difficulties, and more computing skills are needed in the process of our proofs.
