Let G be a tamely ramified reductive p -adic group. We study distinction of a class of irreducible admissible representations of G by the group of fixed points H of an involution of G. The representations corre-spond to G -conjugacy classes of pairs (T; f), where T is a tamely ramified maximal torus of G and f is a quasicharacter of T whose restriction to the maximal pro-p -subgroup satisfies a regularity condition.Under mild restrictions on the residual characteristic of F, we derive necessary conditions for H -distinction of a representation corresponding to (T; f), expressed in terms of properties of T and f relative to the involution.We prove that if an H -distinguished representation arises from a pair (T; f) such that T is stable under the involution and compact modulo (T Ç H) Z (here, Z is the centre of G), then the representation is H -RELATIVELY SUPERCUSPIDAL.

In this paper, we present a study of RELATIVELY connected sublocales. Connected sublocales are RELATIVELY connected, not conversely. We study conditions under which RELATIVELY connected sublocales are connected. The development of this study is subsequently utilized to characterize what we call C-normal frames. We show that normal frames are C-normal but not conversely. Some results concerning J-frames are presented; amongst other things, we prove that regular continuous frames are rim-compact. A rim-compact J-frame is regular continuous. The latter is used to show that the least compactification of a regular continuous J-frame coincides with its Freudenthal compactification. In turn, this contributes to the known conditions under which the least compactification is perfect.

The general RELATIVELY isotropic mean Landsberg metrics contain the general RELATIVELY isotropic Landsberg metrics. A class of Finsler metrics is given, in which the mentioned two concepts are equivalent. In this paper, an interpretation of general RELATIVELY isotropic mean Landsberg metrics is found by using C-conformal transformations. Some necessary conditions for a general RELATIVELY isotropic mean Landsberg metric, as well as generalized Landsberg metric to be a Riemannian metric are also found.

In this paper the general RELATIVELY isotropic L -curvature Finsler metrics are studied. It is shown that on constant RELATIVELY Landsberg spaces, the concepts of weakly Landsbergian, Landsbergian and generalized Landsbergian metrics are equivalent. Some necessary conditions for a RELATIVELY isotropic L -curvature Finsler metric to be a Riemannian metric are also found.

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The class of generalized Berwald metrics contains the class of Berwald metrics as a special case. Let F= αΦ(s), s=β/α, be a generalized Berwald (α, β)-metric on manifold M. We show that F has vanishing S-curvature  S=0 and is of RELATIVELY isotropic Landsberg curvature  L+ cFC=0 if and only if B=0, where c=c(x) is a scalar function on M.

X. Qin and Y. Su proved a strong convergence theorems of modified Ishikawa iteration by CQ method for RELATIVELY nonexpansive mappings in a Banach space [Xiao long Qin, Yongfu Su, Nonlinear Anal. 67 (2007), no. 6, 1958–1965]. The result of this paper extends and improves the result of X. Qin and Y. Su in the two respects: (1). By using the monotone CQ method to modify the CQ method, so that the new method of proof is used. (2). Relax the restriction on T from uniformly continuous to continuous. The result of this paper also extends and improves the recent ones announced by Nakajo, Takahashi, Kim, Martinez-Yanes, Xu and some others.

In this article we survey the existence of best proximity points for a class of non-self mappings which satisfy a particular nonexpansiveness condition. In this way, we improve and extend a main result of Abkar and Gabeleh [A. Abkar, M. Gabeleh, Best proximity points of non-self mappings, Top, 21, (2013), 287-295] which guarantees the existence of best proximity points for nonexpansive non-self mappings in the setting of uniformly convex Banach spaces.We also introduce a new notion, called RELATIVELY asymptotic center, on a nonempty, bounded, closed and convex pair of subsets of a Hadamard metric space and as a result of our main conclusions, we will show that the asymptotic center of any sequence in a nonempty, bounded, closed and convex subset of a Hadamard space is singleton. Moreover, we obtain the other existence results of best proximity points for generalized nonexpansive mappings using the appropriate geometric properties of Hadamard spaces. Finally, we provide some examples to illustrate our main results.