Coincidence points results for families of four relatively nonexpansive mappings on nonconvex domains in $q$-normed spaces have been obtained in the present work. As applications, best approximation results have been given. These results extend and generalize previously known results to a more general class of non commuting relatively nonexpansive mappings in a~space which is not necessarily locally convex.

The aim of this paper is to give some properties for\ the $\mathcal{L}_{p,q}^{\lambda,s}$ spaces, especially concerning embeddings and equivalent norms based of maximal functions and local means.

The convergence of families of linear polynomial operators in the scale of the $L_p$-spaces with $0<p<+\infty$ is studied. The convergence conditions are formulated in terms of the Fourier transform of the generator of the kernel. The results are applied to methods generated by classical kernels.

Fix a prime $l$ and let $M$ be an integer such that $l\not|M$. Let $f\in S_2(\Gamma_1(M l^2))$ be a newform which is supercuspidal at $l$ of a fixed type related to the nebentypus and special at a finite set of primes. Let $\mathbf{T}^\psi$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\mathbf{T}^\psi$ acts on a module $\mathcal M^\psi_f$ coming from the cohomology of a Shimura curve. It follows from the Taylor-Wiles criterion and a recent Savitt's theorem, that $\mathbf{T}^\psi$ is the universal deformation ring of a global Galois deformation problem associated to $\orho_f$. Moreover $\mathcal M^\psi_f$ is free of rank 2 over $\mathbf{T}^\psi$. If $f$ occurs at minimal level, we prove a result about congruences of ideals and we obtain a raising the level result. The extension of these results to the non minimal case is still an open problem.

We give a nontrivial estimate of a certain sum involving the Jacobi symbol in $\mathbb{Z}[i]$ which is a generalization of Heath-Brown's character sum estimate.