Two half brothers and their respective mothers are alive. But the father of the two half brothers is expired.

Is it possible to find out from the blood samples of the respective persons that the biological father of two half brothers are one and the same?

This question is a medical one and should be moved to a more apt forum.

People with a lot of emotions and opinions can't help you here. :)

Yes you can. You can get it through DNA Relationship Testing, which is similar to DNA Paternity Testing but involves testing close members of the family to confirm various types of relationships. Tests can be performed between siblings, uncle/aunt and niece/nephew and grandparents.

From e-articles here is something that can be done if the above options aren't available:
Case 1: If the person has just died and it is still possible to obtain biological material from the body (usually this period is not more than one week from time of death), it is recommended that the Client tries to obtain (where possible) hair samples with root as well as fingernail cuttings. A technically qualified laboratory should be able to extract DNA from these samples and use the DNA to perform the paternity comparison.
Case 2: If the body has already been buried, samples may be obtained in an indirect manner - for example through a toothbrush, comb (might contain useful hairs), dentures or recently smoked cigarettes. These samples are all likely to contain DNA material that can be used to perform the test. However, success in obtaining DNA from such samples depend on a number of factors most importantly the condition of the sample and how much DNA it contains (e.g. a fully smoked cigarette vs a relatively unused one.)
Case 3: In cases where the body has been buried for a number of years, and the samples available are skeletal remains, it is recommend that a bone fragment from the shaft of the femur and/or the humerus weighing approximately 2 grams and/or two teeth per individual are obtained.