Two weeks ago, Facebook (FB) shares — offered in May at $38 a share — slipped as low as $25.52, a mere two bits away from fulfilling the price targets of $25 a share that some of the more bearish analysts had put on the stock.

Since then, Facebook has staged an impressive rebound. The stock has risen as much as 25% from its June 6 low to $31.91 as of Tuesday’s close (The shares fell 1% to $31.60 on Wednesday). Facebook’s stock is still 17% down from its offering price, but there are reasons for Facebook bulls — yes, there are still some out there — to believe it can in time regain that ground again, and then some.

Over the past few weeks, Facebook has rolled out a number of initiatives intended as first steps toward addressing its weakness in areas that concerned investors in the wake of its IPO: in particular, a way to monetize mobile and the need to win a bigger slice of the ad budgets of big consumer brands.

Facebook’s revenue in the first quarter of 2012 grew by 45%, below the 88% growth rate for all of 2011; its net profit in the quarter fell 9% to $205 million. Concern about slowing revenue growth mounted as the IPO approached, as General Motors (GM) and other companies expressed doubt about the effectiveness of Facebook ads. But in recent days, two other big brands — Ford (F) and Coca-Cola (KO) — have stepped forward with endorsements.

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And in an early but encouraging sign for mobile revenue, data from a number of Facebook’s API ad partners showed that Facebook users accessing its social network through mobile devices were significantly more likely to click on sponsored stories than they were on Facebook’s web site. “Mobile is potentially going to be the big revenue driver that Facebook needs,” an executive at TBG Digital, a Facebook partner, told Techcruch.

Facebook, meanwhile, has been busy making deals and announcing new features, none of which are revolutionary in themselves. But the slow, steady stream of developments are starting to add up to what could be a strong strategy for Facebook to capture a larger share of its users attention, particularly on mobile devices. And that would mean more opportunities for serving ads.

Among some of the recent developments that could strengthen Facebook’s role on the mobile web: The company confirmed it’s buying Face.com, whose facial-recognition technology aids in automatically tagging photos. Along with the Instagram acquisition in April and the possibility of another deal with browser maker Opera, Facebook is building a foothold on the mobile web.

Other recent moves could increase ad revenue but raise further privacy concerns: new mobile ads targeting users ads based on their location and a real-time bidding ad system based on user visits to third-party sites.

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Most of these initiatives are so new it’s hard to say how much they will help Facebook keep growing. But what’s clear is that Facebook isn’t sitting still while rivals like Google (GOOG) strengthen their hands in online ads and the mobile web. The urgency that Google has shown in fighting to overcome its weakness in social media is being answered by Facebook as it tries to increase its mobile revenue in the wake of its IPO.

There is another, more technical reason for Facebook rally over the past two weeks. The Nasdaq trading malfunction that marred Facebook’s IPO left some investors and brokerages with more shares than they wanted. According to CNBC, UBS was left with millions of Facebook shares when it repeated buy orders multiple times after not receiving trading confirmations after Facebook debuted on Nasdaq.

If others were also left holding unwanted Facebook shares, they could have added to the selling pressure during the first few grim weeks after Facebook’s IPO. With Facebook’s price slumping day after day, few buyers were willing to step in and pick up the shares. Once that selling pressure passed, it became easier for Facebook to rise on positive news.

But all this raises the question that has been vexing Facebook’s stock even before it went public: How much is it really worth? $25 a share? $45 a share? Or somewhere in between? Facebook is still trading at 70 times its 2011 earnings, a valuation that’s hard to justify even with the encouraging news Facebook has seen in recent weeks.

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Facebook’s shares could rise even further. Underwriters backing the IPO will start publishing research reports on Facebook, offering arguments for a more bullish view. They will need to overcome a lot of remaining skepticism among investors if Facebook is to surpass its offering price.

It’s more likely that Facebook won’t trade in the $40 range until it posts a couple of quarters of strong revenue and profit growth. But compared with only a few weeks ago, the idea of Facebook winning over investors is looking a little more possible.