Fixed point theorems for asymptotically contractive mappings

In this short paper, we prove fixed point theorems for nonexpansive mappings whose domains are unbounded subsets of Banach spaces. These theorems are generalizations of Penot's result in [Proc. Amer. Math. Soc., 131 (2003), 2371--2377].Comment: 7 page

An example for a one-parameter nonexpansive semigroup

We give one example for a one-parameter nonexpansive semigroup.Comment: 9 page

An easily verifiable proof of the Brouwer fixed point theorem

We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.Comment: 4 page

Polymer confinement in undulated membrane boxes and tubes

We consider quantum particle or Gaussian polymer confinement between two surfaces and in cylinders with sinusoidal undulations. In terms of the variational method, we show that the quantum mechanical wave equations have lower ground state energy in these geometries under long wavelength undulations, where bulges are formed and waves are localized in the bulges. It turns out correspondingly that Gaussian polymer chains in undulated boxes or tubes acquire higher entropy than in exactly flat or straight ones. These phenomena are explained by the uncertainty principle for quantum particles, and by a "polymer confinement rule" for Gaussian polymers. If membrane boxes or tubes are flexible, polymer-induced undulation instability is suggested. We find that the wavelength of undulations at the threshold of instability for a membrane box is almost twice the distance between two walls of the box. Surprisingly we find that the instability for tubes begins with a shorter wavelength compared to the "Rayleigh" area-minimizing instability.Comment: 6 pages, 2 figures, submitted to Phys. Rev.

Some Comments on Edelstein’s Fixed Point Theorems in ν-Generalized Metric Spaces

We study deeply two fixed point theorems in n-generalized metric spaces. The two theorems are generalizations of the famous, Edelstein’s fixed point theorem in compact metric spaces

Mizoguchi-Takahashi\u27s fixed point theorem is a real generalization of Nadler\u27s

We give an example which says that Mizoguchi-Takahashi’s fixed pointtheorem for set-valued mappings is a real generalization of Nadler’s. The example isa counterexample to a recent result in Eldred, Anuradha and Veeramani [J. Math.Anal. Appl. (2007), doi:10.1016/j.jmaa.2007.01.087]. We also give a very simple proofof Mizoguchi-Takahashi’s theorem

Numbers on Diameter in n-generalized Metric Spaces

We study some numbers on diameter in n-generalized metric spaces

On the relation between the weak Palais-Smale condition and coercivity given by Zhong

In this paper, we discuss Zhong’s result of that the weak Palais-Smalecondition implies coercivity under some assumption in [Nonlinear Anal., 29 (1997),1421–1431]. We also give a simple proof of Zhong’s result. Further we generalize theresult in Caklovic, Li and Willem [Differential Integral Equations, 3 (1990), 799–800]

Fixed point theorems and convergence theorems for some generalized nonexpansive mappings

We introduce some condition on mappings. The condition is weaker than nonexpansiveness and stronger than quasinonexpansiveness. We present fixed point theorems and convergence theorems for mappings satisfying the condition