President Bush said Wednesday that 60 years of Israel's existence is cause for optimism for democratic change throughout the Middle East. "What happened here is possible everywhere," Bush said, opening a trip divided between ceremonial duties and a new push for Israeli-Palestinian peace.

The president, trying to hold together peace talks in his waning months in office, said modern Israel gives him a strong example to preach optimism to other nations in the region.

"I suspect if you looked back 60 years ago and tried to guess where Israel would be at that time, it would be hard to be able to project such a prosperous, hopeful land," Bush said during a meeting with Israeli President Shimon Peres. "No question, people would have said, 'We'd be surrounded by hostile forces."'

Yet troubling realities offset Bush's rosy message.

A weakened Israeli Prime Minister Ehud Olmert fended off corruption allegations. New bursts of violence erupted in the Gaza Strip. And an Israeli Cabinet minister claimed he's won approval to expand settlement activity in the West Bank, a development that could undermine peace talks with Palestinians.

Bush has expressed confidence, though more tempered lately, that an Israeli-Palestinian peace agreement would be struck before his term ends. But he and his aides are holding out little hope for a major breakthrough during this five-day trip to Israel, Saudi Arabia and Egypt.

Secretary of State Condoleezza Rice said ahead of the trip that reaching a deal to end one of the world's longest-running and most difficult disputes within the next eight months "might be improbable, but it's not impossible."

Meanwhile, the violence continued.

A rocket fired from Gaza exploded in a shopping center in the southern Israeli city of Ashkelon, wounding at least three people and leaving at least two trapped under the rubble, officials said. Two other people were killed in recent days in attacks by Palestinian militants on Israeli communities outside Gaza, even as an Egyptian mediator met with Israeli officials to try to work out a truce between Israel and Gaza's Hamas rulers.

"What happened today is entirely intolerable and unacceptable," Olmert said. "The government of Israel is committed to stopping it, and we will take the necessary steps so that this will stop."

The White House also condemned the attack. "It's clear Hamas isn't interested in peace or helping the people of Gaza lead better lives," Bush spokesman Gordon Johndroe said. "Political goals will never be achieved by launching rockets from Gaza onto innocent women and children."

In a speech Thursday before the Knesset, the Israeli parliament, Bush plans to discuss his vision for the country "on its 120th anniversary" -- a vision that includes peaceful coexistence with a Palestinian state, Johndroe said.

Bush will say the celebration of Israel's founding is a time to look forward, as well as back.

"The United States and Israel share a belief that all people have the right to live in peace, that democracy is the best way to ensure human rights, that religious liberty is fundamental to civilized society and that using violence to achieve political objectives is always wrong," Johndroe said in a preview of the address' themes.

While focusing primarily on celebrating Israel's birthday, Bush also will use the speech to acknowledge -- briefly -- that Palestinians view the anniversary much differently, said a senior administration official, speaking on condition of anonymity. Israel's establishment resulted in the displacement of hundreds of thousands of Palestinians, so they call the milestone "al-naqba" or the catastrophe.

Bush planned to state his concern for the daily lives of Palestinians and the Israeli responsibility for helping to improve them, the official said. Olmert, in Bush's view, is a partner who agrees with this. The president also was to meet Thursday with international Mideast envoy Tony Blair for an update on progress for improving Palestinian civic institutions and economic conditions. He spoke about the issue in his meetings Wednesday with both Olmert and Peres, the official said.

Peres, meeting with Bush in the trellis-covered sandstone portion of his gardens, backed Bush's hope for an accord, saying Israelis want to work with Palestinians.

"We would like to see the Palestinians living together," he said. "They have suffered a great deal of their life. The separation is a tragedy for them and for the rest of us."

Israel has imposed a closure on the West Bank and Gaza Strip during Bush's visit, though, preventing Palestinians from entering the country. Normally, tens of thousands of Palestinians are permitted into Israel each day for work, health care and family visits.

And just hours before Bush arrived, Eli Yishai, a right-wing minister in Olmert's Cabinet, said the prime minister had agreed to the construction of hundreds of homes in a West Bank settlement. Olmert spokesman Mark Regev disputed the claim, saying no decision had been made. In current peace talks, the Palestinians demand that Israel stop building in areas they both want for a future state, and Israel's failure to do so -- despite pressure from the Bush administration -- has increased Palestinian disappointment and frustration.

Also Wednesday, Israeli military raids on the Gaza Strip killed two Palestinian civilians and three militants, Palestinian medical officials said.

Israel frequently raids Gaza to try to stop militants from firing rockets and mortars at Israeli border communities. But the attacks occur almost daily, and Olmert threatened a larger Israeli military incursion into Gaza after two hours of meetings with Bush at his official residence.

In the Gaza Strip, Hamas called the Bush visit a "bad omen." "No greetings to you, Bush, on our holy land," said Hamas strongman Mahmoud Zahar. "Your people will punish you one day."

Upon Bush's arrival at the airport at Tel Aviv, he hugged Olmert, the subject of a new criminal investigation that could push him from office. Earlier, broadcasters' microphones had picked up Olmert's assurances to Hadley: "Holding on, holding on, don't worry."

Olmert has rejected charges that he accepted illegal campaign contributions and possibly bribes. But he has also pledged to step down if he is indicted.

At night, Bush spoke at a conference in Jerusalem celebrating the Jewish state's birthday.

A huge cheer erupted as Bush and his wife, Laura, entered the large theater at the convention center, their images visible on giant screens for those too far back to see the floor. Then came a gasp when the moderator opened the evening's program by delivering news of the Ashkelon attack.

The evening amounted to a glowing tribute to the American-Israeli relationship, and Bush was clearly moved by the praise he received.

"You are an unusual person. You are an unusual leader. And you are an unusual friend of the people of Israel," Olmert told the U.S. president.

Bush's eyes teared up as he tried to hold back his emotion. He received an ovation from the audience, and soon got a hug from Olmert.

Bush said there will be a time when every child of the Middle East can live in peace and freedom.

When that day comes, Bush told Israeli leaders, "the United States of America will be at your side."