On some basic properties of cones with proofs independent of Zorn’s Lemma

Without the use of Zorn’s Lemma there are the proofs for such two basic properties of cones given in normed spaces that a regular minihedral cone is strongly minihedral and that an increasing self-mapping on order interval induced by regular cone has minimal and maximal fixed points which have been proved in some references by virtue of Zorn’s lemma. We also show that a strongly minihedral generating cone is minihedral. At last, two cones in continuous function space are discussed as examples.

Cauchy and Poisson Integral of the Convolutor in Beurling Ultradistributions ofLp-Growth

LetCbe a regular cone inℝand letTC=ℝ+iC⊂ℂbe a tubular radial domain. LetUbe the convolutor in Beurling ultradistributions ofLp-growth corresponding toTC. We define the Cauchy and Poisson integral ofUand show that the Cauchy integral of Uis analytic inTCand satisfies a growth property. We represent Uas the boundary value of a finite sum of suitable analytic functions in tubes by means of the Cauchy integral representation ofU. Also we show that the Poisson integral ofUcorresponding toTCattainsUas boundary value in the distributional sense.

Common Fixed Point Theorems of the Asymptotic Sequences in Ordered Cone Metric Spaces

We introduce the notions of the asymptoticSMK-sequence with respect to the stronger Meir-Keeler cone-type mappingξ:int(P)∪{θ}→[0,1)and the asymptoticWMK-sequence with respect to the weaker Meir-Keeler cone-type mappingϕ:int(P)∪{θ}→int(P)∪{θ}and prove some common fixed point theorems for these two asymptotic sequences in cone metric spaces with regular coneP. Our results generalize some recent results.