Jerusalem (CNN) -- An uptick of violence along the Israel-Gaza border marred expressed optimism for success in a fresh round of Middle East peace talks that continued Wednesday in Jerusalem.
"I'm well aware of the obstacles that stand in the way of peace," Clinton said. "I know that this long history of conflict and distrust hangs over everything. But I see the future that can deliver on the aspirations of Israelis and Palestinians."
Her words came as an Israeli airstrike killed a Palestinian man in southern Gaza, Palestinian security sources said. The Israeli military confirmed the strike on a tunnel in Rafah.
The Israel Defense Forces said Palestinians fired nine rockets and mortar shells Wednesday that landed in Israeli areas just east of the border with Gaza. It said 14 such attacks have occurred since Sunday.
Two of the nine mortars fired contained white phosphorous, Israeli police told CNN. Israel itself has been accused of using the incendiary material in its Gaza offensive that began in late 2008.
"Over the past few days, we have witnessed an increase in attempts by terrorist organizations to attack using rockets in the area surrounding the Gaza Strip, in light of the progress of political negotiations," Israeli Brig. Gen. Eyal Eisenberg said Tuesday.
"The Hamas organization controls the area and we hold it accountable for everything that takes place there," he said.
The peace talks trudged forward, though few details emerged Wednesday of Clinton's trilateral talks in Jerusalem with Palestinian Authority President Mahmoud Abbas and Israeli Prime Minister Benjamin Netanyahu.
"I have sat with these two men individually and together ... and they have begun to grapple issues that can only be resolved through face to face negotiations," Clinton said. "I believe they are serious about reaching an agreement that results in two states living side by side in peace and security."
Tensions are running high over the possibility of new Israeli settlements in the West Bank. Palestinians have said construction would torpedo the talks, but Israel has said some construction is likely.
The latest round of talks started in Egypt on Tuesday, when Abbas and Netanyahu began "a serious discussion on core issues," said U.S. special envoy to the region George Mitchell, who is attending the negotiations.
"They have agreed to begin first on working to achieve a framework agreement for permanent status," Mitchell said Tuesday. "That work is now well under way."
Netanyahu is under pressure from the Palestinians and the Obama administration to extend a 10-month moratorium on building Israeli settlements in the disputed West Bank territory. That moratorium is set to expire September 26.
Mitchell said he wouldn't divulge many details about the talks, noting the importance of confidentiality and sensitivity. But he said "our vision is for a two-state solution."
"That includes a Jewish, democratic state of Israel living side by side in peace and security with a viable, independent, sovereign and contiguous state of Palestine," he said. "But of course, this is one of many sensitive issues the parties need to resolve themselves, and that's the point of negotiations."
The peace talks are aimed at resolving all core issues within 12 months in a process that kicked off recently during a meeting of Abbas, Netanyahu and Clinton in Washington.
Mitchell reiterated the Obama administration's position that Israel should extend the moratorium on settlement construction in the West Bank.
Other issues include the future of Palestinian refugees, Israeli security and the status of Jerusalem.
After the negotiations in Jerusalem on Wednesday, Clinton is scheduled to meet with King Abdullah in Jordan while Mitchell will travel to Syria and Lebanon to update senior officials in those two countries.
CNN's Ben Wedeman, Jason Hanna and Laurie Ure contributed to this report.