Jerusalem (CNN) President Donald Trump met with both Israeli Prime Minister Benjamin Netanyahu and Palestinian Authority President Mahmoud Abbas during the second leg of his first foreign trip as president, pressing forward with his bid to advance the peace process.

Trump doesn't expect to emerge from his two-day visit to Israel and the West Bank with a deal in hand, but he is the latest president to arrive here with the lofty goal in mind.

"During my travels I have seen many hopeful signs that lead me to believe that we can truly achieve a more hopeful future for people of this religion and for people of all faiths and all beliefs and frankly all over the world," Trump said, delivering remarks Monday evening alongside Netanyahu.

The President and first lady wave from Air Force One after landing in Riyadh.

The President and first lady wave from Air Force One after landing in Riyadh.

Trump is welcomed by King Salman after arriving at the King Khalid International Airport in Riyadh.

Trump is welcomed by King Salman after arriving at the King Khalid International Airport in Riyadh.

Soldiers on horseback carry the US and Saudi flags as they escort Trump to the Saudi Royal Court in Riyadh.

Soldiers on horseback carry the US and Saudi flags as they escort Trump to the Saudi Royal Court in Riyadh.

The first lady chats with Saudi Crown Prince Muhammad bin Nayef at the medal ceremony on May 20.

The first lady chats with Saudi Crown Prince Muhammad bin Nayef at the medal ceremony on May 20.

King Salman presents Trump with a gilded necklace and medal, the country's highest honor. The distinction also was bestowed upon Presidents Barack Obama and George W. Bush.

King Salman presents Trump with a gilded necklace and medal, the country's highest honor. The distinction also was bestowed upon Presidents Barack Obama and George W. Bush.

The Trumps look at a display of modern art at the Saudi Royal Court in Riyadh.

The Trumps look at a display of modern art at the Saudi Royal Court in Riyadh.

Trump and King Salman take part in a signing ceremony at the Saudi Royal Court in Riyadh on May 20. The two leaders oversaw the signing of a defense deal worth nearly $110 billion.

Trump sits with members of his staff and Cabinet before a meeting with Saudi King Salman on May 20.

Trump sits with members of his staff and Cabinet before a meeting with Saudi King Salman on May 20.

President Trump and Egyptian President Abdel Fattah el-Sisi share a laugh during a meeting on May 21. El-Sisi complimented Trump on his "unique personality that is capable of doing the impossible." Trump exchanged pleasantries back, praising el-Sisi's shoes.

First lady Melania Trump chats with children during a visit to the American International School in Riyadh on May 21.

First lady Melania Trump chats with children during a visit to the American International School in Riyadh on May 21.

Trump meets with other heads of state in Riyadh on May 21.

Trump meets with other heads of state in Riyadh on May 21.

Saudi King Salman shakes hands with Trump on May 21. Trump is the first US president to start his first foreign trip in the Middle East.

Trump sits at the summit, which included leaders from 55 Muslim-majority countries. He urged them to do more to eradicate terrorist groups that claim the mantle of Islam. "We can only overcome this evil if the forces of good are united and strong and if everyone in this room does their fair share and fulfills their part of the burden," Trump said. "Muslim-majority countries must take the lead in stamping out radicalization."

Trump sits at the summit, which included leaders from 55 Muslim-majority countries. He urged them to do more to eradicate terrorist groups that claim the mantle of Islam. "We can only overcome this evil if the forces of good are united and strong and if everyone in this room does their fair share and fulfills their part of the burden," Trump said. "Muslim-majority countries must take the lead in stamping out radicalization."

Trump speaks in Riyadh during the Arab Islamic American Summit on Sunday, May 21. Trump looked to make it clear that the United States is not at war with Islam. "This is not a battle between different faiths, different sects or different civilizations," he said. "This is a battle between barbaric criminals who seek to obliterate human life, and decent people of all religions who seek to protect it. This is a battle between good and evil."

While in Riyadh, Saudi Arabia, Trump attends the inauguration ceremony for the Global Center for Combating Extremist Ideology. Joining him here are Saudi King Salman bin Abdulaziz Al Saud, center, and Egyptian President Abdel Fattah el-Sisi, left.

While in Riyadh, Saudi Arabia, Trump attends the inauguration ceremony for the Global Center for Combating Extremist Ideology. Joining him here are Saudi King Salman bin Abdulaziz Al Saud, center, and Egyptian President Abdel Fattah el-Sisi, left.

On the way to Tel Aviv, US Secretary of State Rex Tillerson speaks with reporters aboard Air Force One.

On the way to Tel Aviv, US Secretary of State Rex Tillerson speaks with reporters aboard Air Force One.

Trump is welcomed by Netanyahu upon arriving in Tel Aviv on May 22. Trump started his trip with two days in Saudi Arabia.

Trump is welcomed by Netanyahu upon arriving in Tel Aviv on May 22. Trump started his trip with two days in Saudi Arabia.

First lady Melania Trump, in white, visits the Western Wall. At far left is Ivanka Trump.

First lady Melania Trump, in white, visits the Western Wall. At far left is Ivanka Trump.

Trump stands in the Western Wall plaza. To his left, in black, is Shmuel Rabinowitz, the rabbi of the Western Wall.

Trump stands in the Western Wall plaza. To his left, in black, is Shmuel Rabinowitz, the rabbi of the Western Wall.

Trump touches the Western Wall, Judaism's holiest prayer site, while in Jerusalem on May 22. Trump became the first sitting US president to visit the wall.

Trump talks to reporters as he meets with Israeli Prime Minister Benjamin Netanyahu on May 22. Trump sought to rebut claims that he damaged Israeli intelligence capabilities by revealing highly classified information to Russian operatives earlier this month. "Just so you understand, I never mentioned the word or the name Israel," Trump told reporters as he began the second leg of his first foreign tour.

Melania Trump and Israeli first lady Sara Netanyahu speak to children during their visit to the Hadassah hospital in Jerusalem on May 22.

Melania Trump and Israeli first lady Sara Netanyahu speak to children during their visit to the Hadassah hospital in Jerusalem on May 22.

Israeli and American activists hold signs Monday, May 22, during an anti-Trump protest next to the US embassy in Tel Aviv, Israel.

Israeli and American activists hold signs Monday, May 22, during an anti-Trump protest next to the US embassy in Tel Aviv, Israel.

Trump meets with Mahmoud Abbas, the President of the Palestinian Authority, on May 23. Trump met with Israeli leaders the day before and said he believes both sides "are ready to reach for peace."

With the help of US Marines, Trump and his wife lay a wreath at Yad Vashem.

With the help of US Marines, Trump and his wife lay a wreath at Yad Vashem.

People take pictures of the message Trump wrote at Yad Vashem, the Holocaust memorial in Jerusalem, on May 23.

People take pictures of the message Trump wrote at Yad Vashem, the Holocaust memorial in Jerusalem, on May 23.

President Trump and Israeli Prime Minister Benjamin Netanyahu wave at the Israel Museum in Jerusalem on Tuesday, May 23. Trump gave a speech there, reaffirming his country's commitment to Israel while also holding up Judaism's historical ties to the United States.

Melania Trump arrives at the Vatican on May 24. With Vatican protocol in mind, she wore a black veil and long-sleeved black dress draped down to her calf. Ivanka Trump wore a similar outfit with a larger veil.

Trump speaks to reporters in Rome during a meeting with Italian Prime Minister Paolo Gentiloni, right, on May 24.

Trump speaks to reporters in Rome during a meeting with Italian Prime Minister Paolo Gentiloni, right, on May 24.

Trump and his wife look at the ceilings of the Sistine Chapel.

Trump and his wife look at the ceilings of the Sistine Chapel.

Trump and the Pope exchange gifts. Trump presented the Pope with a first-edition set of Martin Luther King's writings. The Pope gave Trump an olive-tree medal that the Pope said symbolizes peace.

Trump and the Pope exchange gifts. Trump presented the Pope with a first-edition set of Martin Luther King's writings. The Pope gave Trump an olive-tree medal that the Pope said symbolizes peace.

Pope Francis stands with Trump and his family during a private audience at the Vatican on May 24. Joining the President, from left, are Trump's son-in-law, White House senior adviser Jared Kushner; Trump's daughter and adviser Ivanka Trump; and first lady Melania Trump.

Protesters in Brussels demonstrate with effigies of Trump and Michel on May 24.

Protesters in Brussels demonstrate with effigies of Trump and Michel on May 24.

Trump stands with Belgian Prime Minister Charles Michel while the national anthem is played during Trump's arrival in Belgium on May 24.

Trump stands with Belgian Prime Minister Charles Michel while the national anthem is played during Trump's arrival in Belgium on May 24.

Trump speaks with King Philippe of Belgium as Queen Mathilde and Melania Trump chat during a reception at the Royal Palace in Brussels on Wednesday, May 24.

Trump speaks with King Philippe of Belgium as Queen Mathilde and Melania Trump chat during a reception at the Royal Palace in Brussels on Wednesday, May 24.

Trump, third from right, attends a meeting with leaders at the European Council.

Trump, third from right, attends a meeting with leaders at the European Council.

Trump walks with European Council President Donald Tusk, center, and European Commission President Jean-Claude Juncker, right, after they met at the European Council in Brussels on May 25.

Trump walks with European Council President Donald Tusk, center, and European Commission President Jean-Claude Juncker, right, after they met at the European Council in Brussels on May 25.

Trump shakes hands with Macron in Brussels, Belgium, on Thursday, May 25. They were attending a NATO summit as the alliance officially opened a new $1 billion headquarters.

Melania Trump arrives at the City Hall in Catania, Italy, on May 26. She was wearing a $51,500 Dolce & Gabbana jacket as she met with other spouses of G-7 leaders.

Trump and other leaders pose for a group photo at the G-7 summit on May 26. From left are European Council President Donald Tusk, Canadian Prime Minister Justin Trudeau, German Chancellor Angela Merkel, Trump, Italian Prime Minister Paolo Gentiloni, French President Emmanuel Macron, Japanese Prime Minister Shinzo Abe, British Prime Minister Theresa May and European Commission President Jean-Claude Juncker.

Trump and other leaders pose for a group photo at the G-7 summit on May 26. From left are European Council President Donald Tusk, Canadian Prime Minister Justin Trudeau, German Chancellor Angela Merkel, Trump, Italian Prime Minister Paolo Gentiloni, French President Emmanuel Macron, Japanese Prime Minister Shinzo Abe, British Prime Minister Theresa May and European Commission President Jean-Claude Juncker.

President Donald Trump and his wife, Melania, arrive for a concert of the La Scala Philharmonic Orchestra while in Taormina, Italy, on Friday, May 26. The Trumps are in Italy for a two-day G-7 summit.

President Donald Trump and his wife, Melania, arrive for a concert of the La Scala Philharmonic Orchestra while in Taormina, Italy, on Friday, May 26. The Trumps are in Italy for a two-day G-7 summit.

Leaders of the G-7 and some African nations pose for a photo on May 27, on the second day of the G-7 summit in Taormina, Italy.

Leaders of the G-7 and some African nations pose for a photo on May 27, on the second day of the G-7 summit in Taormina, Italy.

President Trump and first lady Melania Trump arrive on May 27, to address US military personnel and families at Naval Air Station Sigonella.

President Trump and first lady Melania Trump arrive on May 27, to address US military personnel and families at Naval Air Station Sigonella.

President Trump addresses US troops and their families on May 27, at the Sigonella Naval Air Station.

President Trump addresses US troops and their families on May 27, at the Sigonella Naval Air Station.

President Trump greets people on May 27, after speaking to US troops at Naval Air Station Sigonella.

President Trump greets people on May 27, after speaking to US troops at Naval Air Station Sigonella.

President Donald Trump and first lady Melania Trump board Air Force One on Saturday, May, 27, 2017, at Naval Air Station Sigonella in Italy. They were headed back to the United States after a nine-day trip to the Middle East and Europe.

President Donald Trump and first lady Melania Trump board Air Force One on Saturday, May, 27, 2017, at Naval Air Station Sigonella in Italy. They were headed back to the United States after a nine-day trip to the Middle East and Europe.

Standing alongside Abbas on Tuesday in the West Bank, Trump said he was "truly hopeful" that his administration could broker a peace agreement and called his meetings with Arab leaders in the region "deeply productive."

But even as Trump points to a behind the scenes rapprochement between Israel and Arab countries in the Middle East and with his advisers insisting the new US administration is approaching the conflict differently from those that preceded it, the political conditions between Israelis and Palestinians appear largely unchanged.

The current Israeli government led by Netanyahu is one of the most conservative in the country's history, and Abbas' hold on power is as tenuous as ever -- and mistrust reigns between the two.

But Trump, on his second stop of a three-country tour highlighting the world's three largest monotheistic religions, is continuing to ramp up expectations, expressing optimism that a peace deal can finally be brokered to end the nearly 70-year-old conflict and even suggesting a resolution may not be so difficult to obtain after all.

"It is something that I think is, frankly, maybe, not as difficult as people have thought over the years," Trump said last month during a meeting with Abbas.

He offered even more evidence to back up his optimism on Monday, signaling that his visit to Saudi Arabia brought him confidence that Arab leaders would begin to throw more weight toward reaching a peace deal to resolve the Israeli-Palestinian conflict, pointing in particular to growing unity in the face of Iranian actions in the region.

"There's a great feeling for peace throughout the world," Trump said. "I've seen such a different feeling toward Israel from countries that, as you know, were not feeling so well about Israel."

He added: "I was deeply encouraged by my conversations with Muslim world leaders in Saudi Arabia, including King Salman, who I spoke to at great length. King Salman feels very strongly, and I can tell you would love to see peace between Israel and the Palestinians."

Former Middle East peace envoy George Mitchell said that Gulf countries like Saudi Arabia "have increasingly awakened to the reality that their real challenge comes not from Israel, but rather from Iran.."

"The prospects of their uniting to do something to try to reduce their differences with Israel ... creates an opportunity," Mitchell said Monday on CNN.

Still, Mitchell stressed that the challenges in Israel and the Palestinian territories remain just as stiff as ever as Trump joins the long list of US presidents who have tried to broker Middle East peace.

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Jumpstart to talks?

That's not to say that Netanyahu and Abbas, both wary of being pegged as the unwilling party, aren't projecting an eagerness to jumpstart peace talks with a new president.

"For the first time in my lifetime I see a real hope for change," Netanyahu said Monday with Trump at his side. "The Arab leaders who you met yesterday could help change the atmosphere ... and could help change the conditions for a realistic peace."

Abbas on Tuesday promised Trump that Palestinians are committed to peace and said his meetings with Trump have given Palestinians and Arab countries in the region "so much hope and optimism of the possibility to make true a long-awaited dream and ambition, and that is a lasting peace."

Robert Danin, a former US official with decades of experience in Israeli-Palestinian negotiations, said that while both leaders are eager not to get on Trump's "bad side," the dynamics of reaching a deal remain largely unchanged.

"Behind the smiles and the warm feelings that will be generated on this visit, there is a lot of apprehension," said Danin, the Council on Foreign Relations' senior fellow for Middle East studies. "The reality is that in Israel and in the Palestinian territories and, frankly, here in Washington, very few people see a real change in the dynamics that would lead to a breakthrough towards peace in the fundamentals that impeded the previous presidents."

Anthony Cordesman, a Middle East expert at the Center for Strategic and International Studies, echoed that assessment. He said the most Trump can expect from the visit is for Netanyahu to discuss "preserving at least the shell of an effort to keep the two-state solution going and limit settlements."

But both Netanyahu and Abbas face domestic political challenges.

"The fact is, Israeli domestic politics really don't lend themselves to reaching that particular conclusion," Cordesman said. "It doesn't seem to be one of the prime minister's priorities. You have a divided, weak Palestinian movement, which is not clearly in a position to make concessions that would move this process further any more than Israel is."

Trump administration officials have dismissed experts expressing skepticism at Trump's ability to move the peace process forward as grounded in the same conventional wisdom that has failed to secure a deal in the past and insist the administration is taking a different approach.

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Different approach

In some ways, Trump is approaching the issue differently. Unlike past US presidents who have drawn clear lines for both sides to follow in order to achieve a deal, Trump has offered little -- if not inconsistent -- public guidance for the Israeli and Palestinian leaders.

Instead, Trump has pitched himself as a "mediator, an arbitrator or a facilitator" to help both sides to reach a lasting peace settlement and has resisted stating must-have aspects of such a deal.

Trump shocked longtime observers of the Israeli-Palestinian peace process earlier this year when he declined to endorse a two-state solution to the conflict, breaking with stated US policy under his past two predecessors, Presidents George W. Bush and Barack Obama.

"I'm looking at two-state and one-state, and I like the one that both parties like," Trump said during a joint news conference with Netanyahu at the White House in February. "I'm very happy with the one that both parties like. I can live with either one."

Trump did, however, urge Netanyahu during his visit to Washington to "hold back on settlements for a little bit," though he did not offer any more specifics to back up the request -- including whether it applied to Israeli construction in East Jerusalem.

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Campaign rhetoric

And while Trump has diverged in some ways from past precedent in his approach to the conflict, he is also shirking some of his hardline pro-Israeli rhetoric from the campaign trail in favor of more traditional US policy.

Despite pledging during the campaign to recognize Jerusalem as Israel's capital and move the US embassy to the city, which both Israelis and Palestinians claim as their own, Trump last week backed off tentative plans to announce the bold move during Trump's visit to the city amid a wave of warnings from US foreign policy officials and Arab diplomats.

Administration officials told CNN last week that Trump would not announce the embassy move during the trip, but did not rule out a future decision on the topic.

Where Trump's approach to the conflict has markedly differed from that of past presidents is in how he has staffed the team of top officials leading the peace effort, picking from a crop of individuals from the private sector with no prior government experience -- just like himself.

Overseeing the push is Trump's son-in-law Jared Kushner, a fellow real estate mogul who now serves as the President's senior adviser.

Trump's Secretary of State Rex Tillerson is the former CEO of Exxon Mobil and the President's top envoy for the peace negotiations. Jason Greenblatt was until last year the chief legal officer at the Trump Organization. Trump's ambassador to Israel David Friedman is a well-respected bankruptcy lawyer who worked with Trump.

Greenblatt has sought input from a range of experts and former US officials involved in the peace process, but administration officials insist the officials leading the peace process will bring fresh thinking.

"As the President of America, he feels like it's one of the things that he has to try to do," a senior White House official said. We'll approach it, I think, with a lot of humility. We've looked at what other people have done. But I think that our process -- right now, we have a very good idea what that will be. We've worked with Secretary Tillerson, and Jason Greenblatt's been working with us on this -- we're working very closely and quietly and -- and we'll see."