Abstarct: Special Banach lattices appear naturally in many areas of analysis. They provide a wealth of interesting Banach spaces with non-trivial properties. In this thesis, we first define the best ε−simultaneous approximation in complete Banach lattice spaces. Then, we present a method that identifies and characterizes the best ε−simultaneous approximation. We also define ε−suns and ε−simultaneous suns and investigate the properties of these quantities. Finally, we prove the modified M-theorem for convex sets in the ε−simultaneous approximation mode. The notion of S-suns and BS-suns extends the well-known notion of suns in Banach spaces to the case of ff-simultaneous approximation. We introduce the εsimultaneous approximation and determine the best ε-simultaneous approximation baxsed on S-suns and BS-suns as sequences. We also present some necessary and sufficient conditions to study the best approximation of sets that do not correspond to proximal sets. We call these new sets “du-sets”. “D” and “u” refer to the relation that these sets can have with downward and upward. The closed Du-sets are not necessarily convex, upward, downward or star-shaped, but they are still proximal. We will also obtain some useful and practical results. In the following, we study the characterization of simultaneous and ε-simultaneous approximations of Du sets, some results are discussed here. One of the most important results is that we study how to approximate sets that do not correspond to proximal sets of known sets.