I'm assuming you don't really have JSON (a string) in resp (remember, JSON is a textual notation for data exchange; if you're dealing with JavaScript source code, and not dealing with a string, you're not dealing with JSON), but that it's already been parsed into an array of objects (perhaps jQuery did it for you, if resp is the response of an ajax call). (If resp is a string containing JSON, parse it with JSON.parse and then continue with the rest of the answer.)
inArray does an === (type and value) check. Your array contains numbers. Your object's ext property values are strings.
Either change your array to contain strings, or perhaps convert the ext property values to numbers as you compare:
if(jQuery.inArray(+obj.ext,arr) == -1){
// ---------------^
Also note that you'd normally use the return value of jQuery.map for something; otherwise, you'd use jQuery.each.
Example (keeping jQuery.map in case you are using the result):
var resp = [{"ext":"20177","name":"SIP\/20177-000001e8","state":"Ringing"},{"ext":"20122","name":"SIP\/20122-000001e7","state":"Ringing"}];
var arr = [20122, 20133, 20144, 20155, 20177];
jQuery.map(resp, function(obj) {
if(jQuery.inArray(+obj.ext,arr) == -1){
// -------------^
console.log("Not found: " + obj.ext);
} else {
console.log("Found: " + obj.ext);
}
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
Side note: If you're using an even vaguely modern browser, you can use built-in array features rather than jQuery.map, jQuery.each, and jQuery.inArray: They're the map, forEach, and indexOf methods on arrays themselves. They're all polyfillable/shimmable on older browsers as well.
For example:
var resp = [{"ext":"20177","name":"SIP\/20177-000001e8","state":"Ringing"},{"ext":"20122","name":"SIP\/20122-000001e7","state":"Ringing"}];
var arr = [20122, 20133, 20144, 20155, 20177];
resp.forEach(function(obj) {
if(arr.indexOf(+obj.ext) == -1){
console.log("Not found: " + obj.ext);
} else {
console.log("Found: " + obj.ext);
}
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>