A stationary point theorem characterizing metric completeness
We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle.
Common Endpoints for Suzuki Mappings in Uniformly Convex Hyperbolic Spaces
The existence of endpoints for Suzuki mappings in uniformly convex hyperbolic spaces is proved. A common endpoint theorem for a commuting pair of single-valued and multivalued Suzuki mappings is also established.
Phase diagram and magnetic properties of the diluted fcc system NipMg1-pO.
Starting from Ni and Mg nitrates, about 20 samples of ${\mathrm{Ni}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O samples (with 0.06\ensuremath{\le}p\ensuremath{\le}0.86) were prepared and x-ray-diffraction studies showed the samples to have the NaCl structure with the lattice constant fitting the expression a(p)=4.2115-0.0340p \AA{}. Temperature-dependent magnetic-susceptibility (\ensuremath{\chi}) studies of the samples were carried out between 1.8 and 600 K using a superconducting-quantum-interference-device magnetometer and N\'eel temperatures ${\mathit{T}}_{\mathit{N}}$'s were determined from the peak in \ensuremath{\partial}(\ensuremath{\chi}T)/\ensuremath{\partial}T. The variation of t=${\mathit{T}}_{\mathit{N}}$(p)/${\mathit{T}}_{\mathit{N}}$(1) vs p is compared with that of ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O. For both systems, the variations for pg0.31 are found to fit the predicted values for a simple-cubic Heisenberg antiferromagnet and a theoretical basis for this anomalous result is advanced. The experimental percolation threshold ${\mathit{p}}_{\mathit{c}}$=0.15\ifmmode\pm\else\textpm\fi{}0.01 and for ${\mathit{p}}_{\mathit{c}}$p0.31, \ensuremath{\chi} below ${\mathit{T}}_{\mathit{N}}$ shows irreversible behavior for the zero-field-cooled and field-cooled cases, suggestive of spin-glass-like behavior also observed in other diluted fcc antiferromagnets, such as ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O and ${\mathrm{Eu}}_{\mathit{p}}$${\mathrm{Sr}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$Te. It is suggested that the differences in the t-vs-p variations for p0.31 in ${\mathrm{Ni}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O, ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O, and ${\mathrm{Eu}}_{\mathit{p}}$${\mathrm{Sr}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$Te may be related to the differences in the ratio of the next-nearest-neighbor to nearest-neighbor exchange constants in these systems.