Hello,
looking at screenshot, you are referring my blog Let’s code CRUDQ and Function Import operations in OData service! but please understand that when you are using search function, it actually hits get_entityset method of DPC_EXT class. I recently found out similar discussion. please refer Filtering OData Model data at client side (SAP UI5 program) Go through it and implement the code. Put external breakpoint in get_entityset method and debug the code to understand more.
Regards,
Chandra