Please consider the following MWE. I have two problems here.
The space between (i) and (ii) are OK but the space between 1st and 2nd line; 2nd and 3rd line; 6th and 7th line and 7th and 8th line are more. I need no space there.
The last line crosses the right margin. I was suggested to use \allowbreak for this type problem. But it does not work especially here. Don't know why. (See my previous problem.)
\documentclass[11pt]{article}
\usepackage{amsmath, enumerate}
\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{mathpazo}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\begin{document}
\begin{theorem}
\noindent Let bla bla . Then
\begin{enumerate}[(a)]
\item there exists bla bla such that
\begin{enumerate}[(i)]
\item $\varpi >0$, there exist in $\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$ that it describes the long time $\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$
\item there exist in $\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$ that it describes the long time $\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$.
\end{enumerate}
\item there exist bla bla such that
\begin{enumerate}[(i)]
\item there exist in $\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$ that it describes the long time $\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$
\item there exist in $\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$ that it describes the long time $\left\{\left(\theta, r+\rho\right): \allowbreak 0\leq \frac{2\pi}{3}-\theta\delta<\epsilon_4, \,\, 0<r+\rho<r_4\right\}$
\end{enumerate}
\end{enumerate}
\end{theorem}
\end{document}
The second point happens because TeX wants to break the line at the \rho symbol, not where you put \allowbreak. Give it a newline (\\) right before the last math inline and it should break there instead.
I couldn't see what you meant for the first problem---I see no problems with the spacing, so I don't know what you want me to fix.
As I said, the second problem is because the line actually needs to break before the \allowbreak you have added (i.e., the \allowbreak command isn't helping in this particular spot). Options are (a) include a manual break, as I suggested; (b) edit the text so the line breaks in a different spot; or (c) change the margins or font size so the line breaks in a different spot.
There is no other way around the second problem. If TeX can't figure out how to break the line, it gives up and you have to help it out.
Regarding my 1st problem, I want to say that, I want to minimize the spacing between the lines. In the attached pdf file, it may be seen that there is an unequal spacing between the lines and I need no extra spacing between them.
Maybe you want to switch from enumerate to enumitem which let's you easily specify the itemsep and parsep for each list. The synatx for the labelling is a bit different, though:
\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{lmodern}
\usepackage[T1]{fontenc}
\usepackage{mathpazo}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\usepackage{enumitem}
\begin{document}
\begin{theorem}
\noindent Let bla bla . Then
\begin{enumerate}[label=(\alph*),noitemsep]
\item there exists bla bla such that
\begin{enumerate}[label=(\roman*),noitemsep]
\item $\varpi >0$, there exist in $\left( \alpha\phi-\delta \right)
\allowbreak
\left( N+\gamma \epsilon\right)$ that it describes the long time
\newline
$\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$
\item there exist in $\left( \alpha\phi-\delta \right)
\allowbreak
\left( N+\gamma \epsilon\right)$ that it describes the long time
\newline
$\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$.
\end{enumerate}
\item there exist bla bla such that
\begin{enumerate}[label=(\roman*),noitemsep]
\item there exist in $\left( \alpha\phi-\delta \right)
\allowbreak
\left( N+\gamma \epsilon\right)$ that it describes the long time
\newline
$\left( \alpha\phi-\delta \right) \allowbreak\left( N+\gamma \epsilon\right)$
\item there exist in $\left( \alpha\phi-\delta \right)
\allowbreak
\left( N+\gamma \epsilon\right)$ that it describes the long time
\newline
$\left\{\left(\theta, r+\rho\right): \allowbreak 0\leq
\frac{2\pi}{3}-\theta\delta<\epsilon_4, \,\, 0<r+\rho<r_4\right\}$
\end{enumerate}
\end{enumerate}
\end{theorem}
\end{document}