Air strikes hit around Gaza's derelict international airport near the southern city of Rafah and in nearby farming areas, said the territory's Hamas-run interior ministry.
Security sources in Gaza said the jets fired two missiles at the airport but said no one was hurt in the strikes.
Israel's army said its warplanes had hit five sites operated by Hamas.
"The (Israel armed forces) will continue to operate in order to protect the civilians of Israel from all Hamas terrorist threats above & beneath ground," spokesman Peter Lerner said on Twitter.
Israeli tanks shelled the Palestinian enclave at least twice, saying it was in response to mortar fire across the border, while the army designated an Israeli border town a closed military zone.
A military official said up to five mortar shells in total were fired at Israeli territory, while forces were fired upon near the small border community of Nahal Oz.
"In accordance with security assessments, a closed military zone has been established in the area of Nahal Oz," an army statement said.
Earlier the Israeli army twice confirmed its tanks had fired on targets in Gaza, saying both times it was in response to mortar rounds fired from inside the strip.
Palestinian security sources reported tank fire in Al-Tufah in eastern Gaza.
There were no reported casualties and no claims of responsibility for the mortar fire.
Hamas and Islamic Jihad, another militant Palestinian group, released separate statements warning Israel against any escalation along the border.
Hamas' armed wing, the Ezzedine al-Qassem Brigades, said it was ready to respond to Israel's strikes.
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