Israeli Prime Minister Ariel Sharon says he could meet soon with newly elected Palestinian President Mahmoud Abbas, who has been invited to the White House for talks with President Bush. Such meetings would be seen as important steps towards re-starting peace talks, but there are many challenges each leader must face in the weeks ahead.

Prime Minister Sharon told his newly formed cabinet that he would be calling Mr. Abbas to discuss a possible meeting.

Many of the ministers favor resuming talks with the Palestinians and will be pleased with the news. It appears likely such a meeting will take place soon, but whether much can come it remains a question mark.

Commentator Akiva Eldar of the Ha'aretz newspaper says that is because the new Israeli government might not last that long.

"It is not very promising for the new government. I am not even sure if they will last until the end of the countdown for the disengagement," he said. "I believe that before the evacuation of the settlements, we are going to see early elections."

Mr. Sharon has had to cobble together a coalition of diverse ideological views after an earlier government fell apart over his plan to withdraw Jewish settlements from the Gaza Strip and parts of the West Bank. Much of his own Likud Party is against the plan and in open revolt, while other nationalist and religious parties also oppose it.

So, Mr. Sharon is relying on leftist parties to see him through. Many experts wonder how long this will last.

Despite his landslide victory on Sunday, Mahmoud Abbas has problems of his own from Palestinian militants. They have answered his repeated calls for an end to attacks on Israel, by firing rockets and mortars into Jewish settlements in Gaza and into Israel proper.

Akiva Eldar says the challenge is clear. "The message is [if] you do not deliver your way, we will deliver our way," he said.

Palestinian analyst Nader Sa'id of Birzeit University says reigning in the militants is not a simple task. "Getting rid of rockets is not just a decision by a president who does not really have real authority on the ground," he said. "The authority on the ground and the control here is by Israel, not by the Palestinians."

Professor Sa'id says Sunday's election was a confirmation of the Palestinians' desire to move ahead with peace efforts. But, he says Mr. Abbas will have to deliver results.

"I think improvements and major tangible achievements in the peace process, real changes on the ground will help him very much," he said.

The question in the coming weeks will be how much progress can be made with a Palestinian president with little room to maneuver and a need to deliver change and an Israeli prime minister whose own ability to move forward could be hampered by the fragility of his coalition government.