Police in Texas City, 48 miles (77 kilometers) southeast of Houston on the Gulf Coast, went to Pointe Ann Apartments on Thursday evening on a welfare check and discovered the children along with a woman with a gunshot wound to the head.
One of the children was an infant, another was 2 years old and the third was 5 years old, police said in a statement.
The woman was listed in stable condition at a Galveston hospital. Her relationship to the children was not immediately clear.
Later Thursday, Houston police received a call from a man who identified himself as the person being sought and asking to surrender. He was apprehended peacefully in Houston, said Houston police officer Ben Starr. He declined to identify the man, who was sought as a person of interest.
Texas City police Lt. Kenneth Brown told the Houston Chronicle that the wounded woman is unable to speak to investigators. He described her injuries as severe.
"It's a very intense scene over there," he told the newspaper. "Everybody involved is either deceased or injured."
Investigators have not indicated a motive in the case.
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