A Note on Existence and Convergence of Best Proximity Points for Pointwise Cyclic Contractions

We introduce a notion of pointwise cyclic contraction T satisfying TA ? B and TB ? A to obtain the existence of a point x ∈ A, such that d(x, Tx) = dist(A, B), known as a best proximity point for such a map. We also prove that for any x ∈ A, the Picard iteration {T²ⁿx} converges to a best proximity point.     

Synthesis of a palladium complex bearing 2‐phenylbenzothiazole and its application to Suzuki–Miyaura coupling reaction

The palladacycle complex [L^sPdOAc]₂ bearing 2‐phenyl benzothiazole was synthesized and characterized by NMR and X‐ray crystallography. [L^sPdOAc]₂ was used as a catalyst in the Suzuki–Miyaura cross coupling reaction of 4‐bromotoluene with phenylboronic acid, which resulted in a conversion of >90% with 5 mol% of the Pd complex within 10 min at 60°C.     

Extensions of Edelstein's Theorem on Contractive Mappings

The aim of this article is to provide extensions of Edelstein's theorem for a class of contractive mappings, namely cyclic contractive mappings. We prove the existence and convergence of best proximity points of a cyclic contractive map. We also discuss continuity properties of cyclic contractive maps. Finally, we give a characterization of such maps in the setting of a Hilbert space.     

Internal resonance in the higher-order modes of a MEMS beam: experiments and global analysis

This work investigates the dynamics of a microbeam-based MEMS device in the neighborhood of a 2:1 internal resonance between the third and fifth vibration modes. The saturation of the third mode and the concurrent activation of the fifth are observed. The main features are analyzed extensively, both experimentally and theoretically. We experimentally observe that the complexity induced by the 2:1 internal resonance covers a wide driving frequency range. Constantly comparing with the experimental data, the response is examined from a global perspective, by analyzing the attractor-basins scenario. This analysis is conducted both in the third-mode and in fifth-mode planes. We show several metamorphoses occurring as proceeding from the principal resonance to the 2:1 internal resonance, up to the final disappearance of the resonant and non-resonant attractors. The shape and wideness of all the basins are examined. Although they are progressively eroded, an appreciable region is detected where the compact cores of the attractors involved in the 2:1 internal resonance remain substantial, which allows effectively operating them under realistic conditions. The dynamical integrity of each resonant branch is discussed, especially as approaching the bifurcation points where the system becomes more vulnerable to the dynamic pull-in instability.