* S&P 500, Nasdaq end higher
* Dollar breaks into positive territory for year
* Oil falls after recent strong gains
* Apple results in focus after reports of weak iPhone demand (Updates with closing U.S. market levels)
By Caroline Valetkevitch
NEW YORK, May 1 (Reuters) - The S&P 500 and Nasdaq rose on Tuesday, helped by optimism over U.S. trade negotiations, while the dollar broke into positive territory for the year.
Mexican Economy Minister Ildefonso Guajardo said Mexico will respond to the latest U.S. proposals to rework automotive sector rules under a revised North American Free Trade Agreement (NAFTA) when ministers meet next week.
Also, U.S. President Donald Trump's chief trade negotiator, Robert Lighthizer, said he hoped to further open China's economy to U.S. companies.
The comments helped reverse early losses in stocks in the S&P 500, though the Dow ended lower.
"There is a certain level of fear that has seeped into this market," said Tim Ghriskey, chief investment strategist at Inverness Counsel in New York. "But there are positive outlooks out there on trade."
Apple shares rose about 4 percent after the closing bell after the company beat revenue and profit expectations in its March quarter. The shares ended the regular session up 2.3 percent.
Technology sector results so far – at least from the likes of Amazon, Alphabet, Microsoft, Samsung and SAP – have broadly beaten forecasts for the first quarter, and overall aggregate U.S. earnings growth is tracking seven-year highs of almost 25 percent.
The recent rise in oil prices fueled expectations the Federal Reserve could flag more interest rate hikes at its policy meeting this week.
The dollar attracted attention as it turned positive for 2018 just ahead of a statement Wednesday from the Fed. While U.S. central bank is not expected to raise interest rates, investors will be on the lookout for any clues on the outlook for rates or the economy.
A divergence between growth and the rate outlook versus those of other countries prompted investors to push the currency higher.
The dollar index rose 0.68 percent, with the euro down 0.72 percent to $1.199.
"We're pretty much back to where we were at the beginning of the year, so a lot of the dollar weakness has been pretty much wiped out," said Sireen Harajli, foreign exchange strategist at Mizuho in New York.
The Dow Jones Industrial Average fell 64.1 points, or 0.27 percent, to 24,099.05, the S&P 500 gained 6.75 points, or 0.25 percent, to 2,654.8 and the Nasdaq Composite added 64.44 points, or 0.91 percent, to 7,130.70.
MSCI's gauge of stocks across the globe shed 0.15 percent.
May Day holidays across Asia and Europe meant trading was thinner than usual.
For Europe's stocks followers, only London's FTSE and Denmark's bourse were open.
U.S. Treasury yields rose, with prices pressured ahead of a quarterly refunding announcement. The U.S. Treasury is scheduled to announce its findings on a refunding survey on Wednesday, with analysts projecting an increase in auction sizes, or new issuance at different points on the yield curve.
Benchmark 10-year notes last fell 9/32 in price to yield 2.9681 percent, from 2.936 percent late on Monday.
Brent oil prices eased off four-month highs of just over $75 a barrel set on Monday on worries that Trump may pull out of the 2015 Iran nuclear deal and thus bring back sanctions on its oil output.
The White House had said on Monday that information provided by Israel on Iran's nuclear program had provided "new and compelling details."
U.S. crude fell $1.32 to settle at $67.25 a barrel, while Brent dropped $1.56 to $73.13.
Gold slid to a four-month low as the dollar strengthened. Spot gold was down 0.7 percent at $1,305.72, off an earlier low of $1,301.51, its weakest since Dec. 29.
Additional report by Karen Brettell in New York, Marc Jones and Jan Harvey in London and Hideyuki Sano in, Tokyo Editing by Hugh Lawson, Chizu Nomiyama, Susan Thomas and Jonathan Oatis