Since the number of these single men has recently increased, a new phrase 弁当男子 (弁当 means lunch boxes and 男子 means men) has appeared to describe them.

Under this circumstance, restaurants whose sales depend on businessmen during the lunch break suffer from their poor performance.

They are also forced to be engaged in a price war.

Especially, a price battle between three popular chain restaurants to expand their market share has recently caught public attention.

When I heard the news, I wondered why these three restaurants did that since everybody knows their attempts will end up fueling the dangerous cycle of falling prices and wages that we encounter.

Under these　 circumstances　, restaurants whose sales depend on businessmen during the lunch break suffer from their poor performance.　*It's not incorrect, but I changed it to 'these circumstances because the circumstances are: people keeping tight wallets, prolonged economic stagnation, men making their own lunches, and not going out for lunch. Since you listed more than one thing, I put the word in the plural form.

Especially, a price battle among　 three popular chain restaurants to expand their shares has recently caught public attention.　* rule of thumb: Usually, between shows a difference of up to two things, or between things that are specifically distinct (with actual names), and among shows a relationship of 3 or more things, or things that are not specific in nature (like groups of people etc.) Many native speakers get this wrong, so don't worry too much about actually getting it right lol. (http://grammar.quickanddirtytips.com/between-versus-among.aspx)

These three are well known 　for their　beef-on-rice bowls (specifically, a bowl of rice topped with boiled beef and onions　that is known as　牛丼 in Japanese.）　英語の　「〜is known as」＝〜と言います。　料理は牛丼と言いますだから。

I hope that our economy will get out　 this　 deflationary spiral soon.　* I put 'this' because it is a specific deflation spiral you were talking about in the entry, so this just makes it a little more clear. Using 'the deflation spiral' kind of makes it sound like it is an actual term used for the economic recession instead of your specific word choice for the situation

In Japan, people keep a tight hold on their wallets due to the prolonged economic stagnation. (men don't carry purses ^^)

Especially, A price battle between three popular chain restaurants to expand their market share has caught the public's attention lately.

Surprisingly, all three of them started selling the normal size of the dish at less than three US dollars ( 250yen to 270yen, 1US dollar = 90 yen) several days ago, although the price is expected to get back to around four dollars in the week when the campaign ends.

I hope that our economy will get out of this deflationary spiral soon.

In Japan, people are keeping a tight hold on their purse stings due to the prolonged economic stagnation. It is okay to say "purse strings" here because it's a well known phrase. In the media, they talk about the government purse strings, but obviously the government doesn't carry a purse either :) Also, I think Japanese men do carry purses (man-bags).

According to the news, many businessmen/women bring lunch boxes with them to their offices in order to save money. (sounds more natural in plural)

Under these circumstances, restaurants whose sales depend on businessmen during the lunch break suffer from their poor performance. It sounds like "their" is referring to the businessmen not the restaurant.

These three are well known as beef-on-rice bowls (specifically, a bowl of rice topped with boiled beef and onions, and 牛丼 in Japanese. In Australia, gyu-don is well known and called by that name :)