In this paper we define a generalized Cesàro sequence space and consider it equipped with the Luxemburg norm under which it is a Banach space, and we show that the space posses property (H) and property (G), and it is rotund, where is a bounded sequence of positive real numbers with for all .
In this paper, we define the direct sum of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that has the H-property if and only if each has the H-property, and has the Schur property if and only if each has the Schur property. Moreover, we also show that is rotund if and only if each is rotund.
In this paper, we introduce the new concept of proximal mapping, namely proximal weak contractions and proximal Berinde nonexpansive mappings. We prove the existence of best proximity points for proximal weak contractions in metric spaces, and for proximal Berinde nonexpansive mappings on starshape sets in Banach spaces. Examples supporting our main results are also given. Our main results extend and generalize some of well-known best proximity point theorems of proximal nonexpansive mappings in...
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