Magento – Magento 2 (Rest api): Get all products based on update_at criteria

Try this one instead, i think you have to encode the date, you have problem in the endpoint

/V1/products?searchCriteria[filter_groups][0][filters][0][field]=updated_at&searchCriteria[filter_groups][0][filters][0][value]=2018-08-24%2008%3A35%3A37&searchCriteria[filter_groups][0][filters][0][condition_type]=eq

First get admin token,

$adminUrl='http:Yourlurl/rest/V1/integration/admin/token';
$ch = curl_init();
$data = array("username" => "admin_username", "password" => "admin_password");
$dataString = json_encode($data);
$ch = curl_init($adminUrl);
curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "POST");
curl_setopt($ch, CURLOPT_POSTFIELDS, $dataString);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_HTTPHEADER, array(
    'Content-Type: application/json',
    'Content-Length: ' . strlen($dataString))
);
$token = curl_exec($ch);
$token = json_decode($token);
print_r($token);
curl_close($ch);

You can make use of below code, If you have many products increase the page_size.

<?php
$productUrl='http:yourlUrl/rest/V1/products?searchCriteria[page_size]=20';

$ch = curl_init($productUrl);
curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "GET");
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_HTTPHEADER, array(
    "Content-Type: application/json",
    "Authorization: Bearer $token"
    )
);
$productList = curl_exec($ch);
$err      = curl_error($ch);
$products = json_decode($productList);
echo '<pre>';print_r($products );
curl_close($ch);