<!DOCTYPE HTML>
<html>
<head> </head>
<body>
<input id="text" type="text" value="арбуз">
<input onclick="hasChar('а')" type="button" value="Search">
<script>
var text = document.getElementById('text');
function hasChar(char) {
value = text.value;
for(var i=0; i<value.length; i++) {
if(value.charAt(i) === char) {
alert(true);
}
}
}
</script>
</body>
</html>