Characterization of the Best Approximation and Establishment of the Best Proximity Point Theorems in Lorentz Spaces

Since the monotonicity of the best approximant is crucial to establish partial ordering methods, in this paper, we, respectively, characterize the best approximants in Banach function spaces and Lorentz spaces ${\Gamma }_{p,w}$, in which we especially focus on the monotonicity characterizations. We first study monotonicity characterizations of the metric projection operator onto sublattices in general Banach function spaces by the property $H_g$. The sufficient and necessary conditions for monotonicity of the metric projection onto cones and sublattices are then, respectively, established in ${\Gamma }_{p,w}$. The Lorentz spaces ${\Gamma }_{p,w}$ are also shown to be reflexive under the condition $R{B}_p$, which is the basis for the existence of the best approximant. As applications, by establishing the partial ordering methods based on the obtained monotonicity characterizations, the solvability and approximation theorems for best proximity points are deduced without imposing any contractive and compact conditions in ${\Gamma }_{p,w}$. Our results extend and improve many previous results in the field of the approximation and partial ordering theory.