Valves / Petrol Pump Problem

Hi

Please help. Recently purchased an X5 4.8IS on low mileage. Ive been driving the car for almost 3 months now. Whenever i start the car, there is a delay in starting as if the starter is giving in or perhaps the problem with the battery. I checked the following with the mechanics: battery, alternator, starter and all are in good working order.

Could it be that the injectors are not getting enough gas before starting, or does it mean there is a problem with my valves? Or does it mean my petrol pump is gving in? Your assistance is appreciated.

My 03 x5 was stopping intermediary but re starting after a few minutes but now it turns over but will not start could it be fuel filter or fuel pump