At a news conference with other hostages’ relatives at Sheba Medical Center, where the three former captives are being treated, Ms. Ashkenazi said her sister was “doing well, strong and brave, but she has a long road of recovery ahead of her.”

Ms. Ashkenazi also conveyed a message from her sister: “To continue fighting, never give up, take to the streets, all phases of the deal must be implemented.”

Meirav Leshem Gonen, the mother of one of the released hostages, Romi Gonen, 24, thanked Israeli Prime Minister Benjamin Netanyahu for “bravely and ethically making the responsible decision to initiate this critically important process.”

Ms. Leshem Gonen also mentioned Mr. Trump, “whose courageous support of Israel and reinforcement of human good in the world helped drive and bring about this moment.”

She has become a particularly familiar face in Israel, appearing frequently on television and in public gatherings to speak about her daughter and the plight of more than 200 hostages abducted from Israel in the Hamas-led attack on Oct. 7, 2023.

On Monday, after getting to embrace her daughter again, she teared up when she declared Romi was now a “former hostage.”

Mandy Damari, the mother of the third released hostage, Emily Damari, 28, a British-Israeli national, added a thank you to the British government. She urged Israeli leaders and the public to “keep on fighting for the remaining 94 hostages who need to come home. There are too many other families who are waiting to hug their loved ones or bring them back for a proper burial.”

“Our Emily, we’re proud of you and admire you for your strength,” said Ms. Damari’s brother, Tom Damari. “You are the symbol of courage. You’ve won.”

The Israeli military “is operating in accordance with the cease-fire agreement and will keep to the defensive lines as agreed,” the statement said, adding that its forces remain “prepared in defense and offense.”

Earlier in the day, Wafa reported that four people were wounded in Rafah when Israeli forces opened fire at people in the center and south of the city and that three others were wounded when an Israeli drone fired on people. The agency said that three children were wounded when a “suspicious object from the remnants of the Israeli military” exploded in Rafah.

The Israeli military, in response to a request for comment on those earlier reports of injuries, said in a statement that “several suspects” were identified approaching Israeli soldiers who were operating in Gaza in “accordance with the terms of the cease-fire agreement,” and that “the soldiers responded with warning shots.” It did not say whether anyone had been injured.

Some continued violence is typical after cease-fire agreements, according to experts. In the case of the parallel war between Israel and Hezbollah in Lebanon, in which a truce went into effect in November, both sides have claimed violations but neither has suggested the deal was over.

Breaches can be relatively minor, like going too far in a demilitarized zone, gunfire initiated by low-level commanders or combatants, or accidents.

Most signatories to a truce do not want to be seen as the ones blowing it up, said Dan Byman, director of the Warfare, Irregular Threats, and Terrorism Program at the Center for Strategic and International Studies in Washington. But even a minor or accidental violation can sometimes serve as a pretext for resuming fighting, he said.

“The real question is a political one, which is when does the violation reach the level where you want to end the agreement,” Mr. Byman said.

The deal between Israel and Hamas is fragile, Mr. Byman noted. Not only is it a phased agreement in its first stage, one that has barely begun, but there is also ambiguity about what will happen in subsequent phases, should they be reached, and the deal has strong opponents.

Hard-line elements of Israel’s coalition government have denounced the deal, saying that Hamas must be completely destroyed and that because it leaves Hamas in control of Gaza for now, the agreement threatens Israel’s long-term security.

Hamas has controlled the enclave since 2007, handling matters from health care to law enforcement, and so far, there is no alternative governing authority. Since the cease-fire went into effect, Hamas has shown that it is still a force, with members emerging openly in the enclave.

The Ministry of Health in Gaza on Monday reported more than 120 deaths verified in the past day, including more than 60 missing people who had been missing, along with more than 340 injured. The ministry does not distinguish between civilian and combatant deaths. It said that the death toll in Gaza since the war began on Oct. 7, 2023 had surpassed 47,000.

Civil Defense emergency responders in Gaza said on Monday that after more than 470 days of war, nearly half of its personnel had been killed, injured or detained, while 85 percent of its vehicles and 17 out of 21 facilities had been damaged or destroyed.

A significant element of the first phase of the cease-fire agreement is a series of swaps of Israeli hostages in Gaza for Palestinians jailed by Israel. The International Committee of the Red Cross, which on Sunday facilitated the exchange of three Israeli hostages for 90 Palestinian prisoners on Monday stressed the urgency of both sides’ continued commitment to the agreement.

“More families are waiting anxiously for their loved ones to come home,” Mirjana Spoljaric, president of the I.C.R.C., said in a statement. “We call on all parties to continue to adhere to their commitments to ensure the next operations can take place safely.”

Hiba Yazbek and Rawan Sheikh Ahmad contributed reporting.

The prisoners to be released range from people detained in Gaza on suspicion of what Israel says are offenses related to its national security, to others who have been convicted of involvement in deadly attacks and are serving life prison sentences.

Defense Minister Israel Katz’s office said in a statement that he had directed the military to “take all measures to avoid the repeat of mass Palestinian release celebrations and parades for the releases of terrorists” in the Israeli-occupied West Bank.

“The victory celebrations for terrorists must not provide further encouragement to terrorism,” he added.

With Israel’s wars in Gaza and Lebanon apparently concluded, Mr. Katz said, the West Bank is a top priority, now the primary “arena of friction” where “shootings and terrorism against Israel persist.”

During more than 15 months of war in Gaza, Israeli forces and settlers killed more than 800 Palestinians in the West Bank, according to Palestinian officials, in clashes and military raids on militants. Over the years, Israel has steadily encroached on the West Bank, with increasing settlements, roads, barriers and checkpoints.

On Sunday, Israeli settlers in the West Bank attacked at least three Palestinian villages, according to residents and the Israeli military. Two days earlier, Mr. Katz ordered the immediate release of five settlers detained for their alleged involvement in attacks on Palestinians, including in recent riots.

On Saturday, one man stabbed another at a sidewalk cafe in Tel Aviv — the authorities called it a terrorist attack — and then the assailant was fatally shot.

“Checkpoints are usually less crowded and dangerous at night. However, last night, all checkpoints and closed roads seemed to be meant to prevent and prolong the distance between us and our loved ones,” said Rasha Zughaibi, whose sister-in-law, Nida Zughaibi, a mother of four, was one of those released.

Rasha Zughaibi said that the round trip had taken more than 24 hours, given the delays.

Settlers attacked at least three Palestinian villages in the days leading up to the releases, according to residents and the Israeli military. Israeli security forces and settlers have killed more than 800 Palestinians in the West Bank since Hamas led a deadly attack on Israel on Oct. 7, 2023, according to the U.N. human rights office.

Israel released the 90 detainees early Monday morning, shortly after midnight, hours after three Israeli hostages held by Hamas in Gaza had been freed and taken home, as part of the cease-fire between Israel and Hamas that took effect on Sunday.

Red Cross buses ferried the former inmates shortly after from Ofer prison, an Israeli prison in the West Bank near Jerusalem, to Beitunia in the West Bank, where they were met by a cheering crowd. As the buses arrived, Palestinians set off fireworks to celebrate their release.

Some family members awaiting the detainees held wreaths of red and white flowers — colors from the Palestinian flag — while others carried jackets and warm scarves for those returning. Others in the crowd that came to greet the prisoners climbed onto the roof of one of the buses and waved green flags associated with Hamas, as well as a Palestinian flag.

The authorities in Ramallah, the administrative hub of the occupied West Bank, put 10 former detainees into hotels, along with their families, according to Bakeer Wheidi, chief of staff for Ramallah’s governor.

They included Naheel Masalma, 36, a resident of the South Hebron area, who said she stayed at a hotel with four siblings and relatives who had come to greet her on her release. She said travel overnight would have been a risk because “we’ll have to maneuver through settlers who might attack us and military checkpoints where we could be arrested again at any moment.”

She also said she had been freed without shoes.

Some of the checkpoints as well as temporary closures that had been set up in recent days amid an intensification of violence in the West Bank by settlers had been dismantled on Monday, according to residents.

The release of the prisoners was the first stage in a series of hostage-for-prisoner swaps slated to take place over the course of the phased 42-day truce. The next exchange is set for Saturday, when four Israeli hostages are expected to be exchanged for more Palestinian prisoners.

Under the terms of the cease-fire agreement, 30 more hostages, most of whom Israeli authorities believe are living, are slated to be released in this first phase of the agreement in exchange for more than 1,000 Palestinian prisoners.

Mr. Abu Marzouk’s remarks may also indicate that Hamas feels buoyed by the cease-fire and believes there could be an opportunity to expand the group’s international relations.

“We’re prepared for a dialogue with America and achieving understandings on everything,” Mr. Abu Marzouk, the first leader of Hamas’s political office, said in a phone interview on Sunday.

Beyond the United States, numerous Western countries also consider Hamas to be a terrorist organization, including Britain and Canada. But Hamas has made some efforts to improve relations with Western governments, including by issuing a policy document in 2017 that took more moderate positions than its founding charter. The document called the establishment of a Palestinian state along the pre-1967 borders between Israel and the West Bank and Gaza a “formula of national consensus,” but it also rejected recognition of Israel.

The group has refused to renounce violence and recognize Israel, and after the Hamas-led assault on southern Israel on Oct. 7, 2023, it was forcefully condemned by the United States and European countries. The attack, analysts say, was a reflection of the ascendancy of hard-line leaders in the group advocating for violent conflict over long-term understandings with Israel.

President Trump, however, has previously demonstrated willingness to meet with longtime foes of the United States such as Kim Jong-un, the leader of North Korea.

Mr. Abu Marzouk, a native of Gaza and a former resident of Northern Virginia, is a member of Hamas’s political office, but it is not clear whether he speaks on behalf of all senior Hamas leaders, including Mohammed Sinwar and Izzeldin al-Haddad, hard-line military commanders in Gaza.

He is considered to be one of the more pragmatic voices in Hamas and leads Hamas’s foreign relations office. Other senior Hamas officials have invested more in developing ties with members of the so-called axis of resistance, including Iran, the Lebanese militant group Hezbollah, Iraqi militias, and the Houthis in Yemen.

Mr. Abu Marzouk, 74, said Hamas was also ready to welcome an envoy from the Trump administration in the coastal enclave, despite longstanding American policy to provide Israel with billions of dollars in weapons and defend it at international institutions. Hamas, he added, would even provide such a visitor with protection.

“He can come and see the people and try to understand their feelings and wishes so that the American position can be based on the interests of all the parties, and not only one party,” he said.

On Saturday, NBC News reported that Steve Witkoff, Mr. Trump’s Middle East envoy, was weighing a visit to Gaza to help maintain the cease-fire agreement, citing an anonymous Trump transition official with direct knowledge of the cease-fire process.

The comments by the Hamas leader were in stark contrast to many Hamas statements during the first Trump administration that blasted the relocation of the U.S. Embassy in Israel to Jerusalem, the recognition of Jerusalem as Israel’s capital, cutting off aid to Palestinians, and a peace plan that heavily favored Israeli positions.

After the previous Trump administration presented its plan, Hamas referred to it as “the deal of shame.”

One reason Hamas may want to reach out to the United States is to secure the entry of materials needed to reconstruct Gaza without conditions that could undermine its ability to remain the most dominant Palestinian group in the territory.

Hamas will likely need to offer some compromises if it wants enough aid to rebuild Gaza to flow into the enclave. Until now, Hamas leaders have expressed readiness to give up civilian governance in Gaza, but without dismantling its military wing — a dynamic that analysts have said would be similar to Hezbollah’s role in Lebanon before its last conflict with Israel.

Steven Cheung, Mr. Trump’s incoming White House director of communications, didn’t immediately respond to a request for comment.

Mr. Abu Marzouk also offered high praise for Mr. Trump for his involvement in helping broker the cease-fire agreement between Israel and Hamas and called him a “serious president.”

“If not for President Trump, his insistence on ending the war, and his dispatching of a decisive representative, the deal wouldn’t have happened,” said Mr. Abu Marzouk, referring to Mr. Witkoff.

While the deal calls for an eventual “cessation of military operations and hostilities permanently,” Prime Minister Benjamin Netanyahu of Israel has repeatedly suggested that the Israeli military will resume attacking Hamas after some hostages held by militants are released.

Alongside Qatar and Egypt, members of the Biden administration, including White House official Brett McGurk and C.I.A. director Bill Burns, played critical roles in putting together the cease-fire deal. But Mr. Witkoff helped encourage Israel to approve it by telling Mr. Netanyahu that Mr. Trump wanted it completed, according to officials briefed on the talks who spoke on condition of anonymity to discuss sensitive diplomacy.

Still, Mr. Abu Marzouk insisted Mr. Trump’s role was crucial to getting the agreement across the finish, especially in applying pressure on Mr. Netanyahu.

“Truthfully, Trump gets the credit for ending the war,” he said.

During the war, fewer than 100 trucks per day had been entering the enclave, and deliveries had at times been suspended. Relief agencies accused Israel of overly restricting deliveries with stringent inspections and the closure of border crossings, which Israel denied, and have said that at least 200 trucks per day were required to provide food, medicines, fuel, clean water and other essentials.

With the cease-fire between Israel and Hamas taking effect, aid convoys seemed to enter Gaza without issues, and no attempts at stealing or looting the aid were reported on Sunday or Monday, according to Juliette Touma, a spokeswoman for the main U.N. agency that assists Palestinians. She added that local police officers were present in some parts of Gaza to secure the convoys, while in other areas security was not needed.

Uniformed police officers and armed fighters, rarely seen in the open during the war, were visible in cities and towns across Gaza after the cease-fire took effect. It was an apparent show by Hamas, which had controlled the enclave for years, that it is still in charge and plans to hold onto power.

Videos posted to social media showed convoys driving through Gaza on Sunday, as people gathered calmly on roadsides, refraining from approaching the trucks. It was a stark contrast to the apocalyptic scenes of wartime aid deliveries, when desperate crowds swarmed and climbed onto the trucks in hopes of securing a food package or bag of flour, sometimes resulting in violence.

“What was very noticeable is that none of the trucks that entered yesterday were looted,” said Nebal Farsakh, a spokeswoman for the Palestinian Red Crescent. “And this was the first time in 15 months that this many trucks entered Gaza,” she added.

What was less clear was how efficiently and equitably aid was being distributed after it entered Gaza, and some residents said they have yet to receive or even see the aid. Ms. Touma said that UNRWA staff and aid workers from other agencies were still sorting through the aid that has come in since Sunday before it is eventually delivered to people.

“It’s the second day of the cease-fire and they said that aid and flour have come in but unfortunately we haven’t seen any of it yet,” said Moustafa al-Aloul, a 22-year-old from northern Gaza. “Currently, the markets literally have nothing,” he added.

The Gazan ministry of social development, a part of the Hamas-led administration, said in a statement on Monday that it had “made all preparations to receive aid” and will provide all the permits needed for aid organizations to receive and distribute the aid. “Work will be done in a coordinated manner between all partners to ensure the fair distribution of aid to all citizens,” the ministry said, adding that aid will be provided to families inside and outside shelter centers and tent encampments.

Israeli officials have accused Hamas of hoarding essential supplies to serve its own members and to wield control over the population, and there have been reports of profiteers seizing aid and then selling it on the black market. Aid officials say the solution is to end the scarcity.

Ms. Touma said the convoys entering Gaza have included some trucks carrying commercial goods for sale, which have rarely reached Gaza during the war.

“There were several goods that people take for granted that were very much missing from the market,” Ms. Touma said in an interview on Monday. “So it’s very good that commercial supplies have also come in because you can’t turn the two million people in Gaza into a nation that is solely dependent on aid,” she added.

The World Food Program said that on Sunday it delivered ready-to-eat meals and bags of wheat flour. It aims to send at least 150 truckloads of food into Gaza every day, along with other supplies, as well as restocking bakeries and providing nutritional supplements for children facing malnutrition.

Ameera Harouda contributed reporting from Qatar, Matthew Mpoke Bigg from London and Vivian Yee from Cairo.

The military said that it had detained two suspects and handed them over to the Israeli police. The police did not respond to a request to confirm the episodes or the detentions.

Honenu, a legal organization that has represented settlers, on Monday issued a statement on behalf of one of its lawyers, Daniel Shimshilashvili, saying that two people accused by the police of setting fire to the homes of Palestinians in the West Bank had been released after a court hearing. It did not name the two.

The violence against Palestinians in the West Bank comes as many far-right Israelis have been protesting the cease-fire agreement between Hamas and Israel, hoping to disrupt implementation of the deal.

On Sunday, WhatsApp groups administered by Israeli settler groups were full of angry comments about the release of Palestinian prisoners to the West Bank and calls to block their entry with organized protests. Many users said that the released prisoners, some convicted of mass-casualty attacks on Israelis, posed a threat to citizens of Israel.

In Sinjil, a village south of Nablus, dozens of men, some carrying slingshots, rampaged through the village throwing stones and setting houses on fire, according to residents and videos verified by the Times. One clip showed vehicles burning in the village.

“People screamed as their homes were burning,” said Ayed Jafry, 45, a resident of Sinjil. He said that several people were injured, including an 86-year-old man.

On Sunday morning, a WhatsApp group for settlers posted a video of an Israeli man outside a shop with shattered glass, in Funduq, a Palestinian town.

“The next stage is destroying it,” he declared.

In a video taken from a CCTV camera in Turmus Aya, at least 20 masked men can be seen throwing stones and entering the village, as Israeli police cars appear in the distance. Villagers said the officers stayed clear of the violence and did not try to stop it.

Lafi Adeeb, the mayor of Turmus Aya, said that the police didn’t enter the village but that the military eventually dispersed the riot using tear gas. “I was shocked,” he said, because “this is the first time the army stopped the settlers here.”

Palestinian villagers in Turmus Aya and Sinjil, some of whom are U.S. citizens, alerted the American embassy in Israel about the attacks, and a senior U.S. diplomat called the office of Defense Minister Israel Katz to inform him, according to a U.S. official who spoke on condition of anonymity to discuss sensitive diplomatic information.

Nader Ibrahim and Fatima AbdulKarim contributed reporting.

And on Monday morning, Ms. Damari posted on Instagram thanking God, her family and friends, and saying, “I have returned to life” and was “the happiest person in the world.”

Ms. Damari was freed Sunday along with Romi Gonen, 24, and Doron Steinbrecher, 31, all of them kidnapped during the Hamas-led assault on southern Israel on Oct. 7, 2023, in which an estimated 1,200 people were killed.

Video released by the Israeli military showed the three hostages being reunited with their families in emotional scenes, and Israeli news anchors openly expressed their relief as the first images emerged from Gaza and they were seen entering a Red Cross vehicle unaided.

But while the early scenes captured on video and in photographs were ones of elation, much is unknown about the women’s conditions.

The Israeli health ministry and Sheba Medical Center, where the three women are staying in a closed wing with family, have provided little information about the conditions of the women, saying in statements that their primary commitment was to safeguard the privacy of the returnees while they receive medical and psychological care.

In brief statements on Sunday night, two doctors at Sheba hospital suggested that the women were not in immediate need of emergency treatment.

“I’m happy to report that they are in stable condition,” said Prof. Itai Pessach, adding, “That allows us, and them, to focus on what is the most important thing for now — reuniting with their families.”

Sheba Medical Center has been the first stop for dozens of captives who were seized in the October assault and later freed, including many of those released in an exchange of hostages for Palestinian prisoners in November 2023, as well as the four hostages rescued in an audacious and deadly Israeli military raid in June 2024.

Professor Pessach, who has led the Sheba medical team for returning hostages, cautioned in an interview in June that first impressions can be deceptive.

“The thing I definitely know is to expect the unexpected,” he said then, after receiving the four hostages who were rescued. “After eight months,” he said, “we had a notion that they’d be much more broken, maybe look differently than they were.”

They had lost less weight than had been expected, he said. But then, he said, the results from medical tests start coming in, along with initial psychological evaluations, and “you start to grasp what they’ve been through.”

All four had come back suffering from severe malnutrition, Dr. Pessach said, adding that the lack of sunlight, abuse and psychological stress they had endured would have long-term implications for their health.

“As wonderful as it is to see Emily’s resilience, these are still early days,” Mandy Damari, Emily’s mother, acknowledged in a statement on Monday, in which she noted that Emily was “doing much better than any of us could ever have anticipated.”

In a recent television interview, Yamit Ashkenazi, Ms. Steinbrecher’s sister, said she was expecting to receive “a different Doron.” Ms. Ashkenazi also worried about telling her sister that so many of her friends were killed in the Hamas attack.

But in the email dated Sunday and sent by a Houthi-linked group that communicates with the shipping industry, the Houthis warned that if the United States or Britain directly attacked Yemen, they would resume their assaults on vessels associated with these countries.

A military spokesman for the Houthis, Yahya Saree, said that another statement was planned for later Monday, which was expected to elaborate on the decision.

Since November 2023, the Houthis have been launching missiles at Israel and targeting commercial vessels they believe are headed for Israel, in what they say is a show of solidarity with Hamas, their Iran-backed ally in Gaza.

The attacks have severely disrupted global shipping trade through the Red Sea, a key route for traffic between Europe, Asia and the Middle East. Many shipping companies have rerouted their cargo around the Cape of Good Hope, located at the southern tip of Africa. This alternative path has added approximately 4,000 miles and 10 additional days to shipping journeys, requiring significantly more fuel.

Images and video released by the Israeli government showed the former captives, all young women, walking under their own power and embracing relatives, but no details have been made public yet about their condition or what they endured.

Hostages who have been released previously have described a wide range of experiences that have taken physical and psychological tolls on them. Here is what some have said:

• Andrey Kozlov was one of four captives rescued in an Israeli military operation in June. He was held for eight months along with two other hostages, and was moved to six locations in the first two months — sometimes with only a pail for a toilet and scarce food — while shackled at the wrists and ankles. For the next six months, Mr. Kozlov, 27, and the two others were kept in the apartment of a Hamas operative. There, they were unshackled and ate regularly, Mr. Kozlov said, but they were psychologically abused and threatened. Mr. Kozlov, a Russian Israeli, said he focused on surviving by reciting mantras to himself in Russian, such as “You are alive; every day a gift.”

• Amit Soussana, a lawyer abducted from her home in Kibbutz Kfar Azza, was the first Israeli hostage to speak publicly about being sexually assaulted during captivity. She said she was held alone in a child’s bedroom, chained by her left ankle. Sometimes a guard would enter, sit beside her on the bed and touch her, and weeks into her captivity, she was forced at gunpoint to engage with him sexually, she said. Ms. Soussana, then 40, was later moved to another location, where she said different captors beat her. She was released as part of the first cease-fire agreement in November 2023.

• Chen Goldstein-Almog was abducted along with three of her children from Kibbutz Kfar Aza, while her husband and eldest daughter were killed in the Hamas-led attack of Oct. 7, 2023. Ms. Goldstein-Almog, who was 48 at the time, later said that she and the children were treated “respectfully” and were not physically harmed. Even so, she said, “We were in daily danger. It was fear at a level we didn’t know existed.”

• Ofelia Adit Roitman, who was 77 when she was kidnapped from Kibbutz Nir Oz, described her captivity in a video statement shortly after she was released in the previous cease-fire. “I was very scared the first two weeks,” Ms. Roitman said. “I thought I was crazy because I was alone. There was barely any light. There was barely any food.”

• Mia Schem was shot and taken hostage after fleeing the Nova music festival. Released as part of the 2023 cease-fire, she described her experience to Israeli news media, saying she underwent surgery for her gunshot wound in her first days in Gaza, but received no painkillers afterward and had to replace her own bandages. Ms. Schem, then 21, was held by a family that sometimes withheld food from her for entire days, even as they ate, she said.

• Margalit Moses, who was 78 when she was taken, said she spoke to her captors in Arabic, serving as an interpreter for a group of hostages in poor health who were held together. She was released as part of the last cease-fire and invited her guards for coffee “when peace comes” as she left. After she was freed, she realized she had actually been relatively fortunate. “Now that I’ve heard the horror stories of those who came back,” she said, “I understand I was kept in conditions others did not have.”