Objectives In this paper we present some fixed point theorems for self mappings satisfying generalized $$(\phi , \psi )$$ ( ϕ , ψ ) -weak contraction condition in partially ordered complete b-metric spaces. The results presented over here generalize and extend some existing results in the literature. Finally, we illustrate two examples to support our results. Result We obtained a unique fixed point of a self mapping satisfying certain contraction condition which is involving an auxiliary function. Also, the results are presented for the existence of a common fixed point and a coincidence point for generalized $$(\phi , \psi )$$ ( ϕ , ψ ) -weak contraction mappings in partially ordered complete b-metric space.

Objectives The aim of this paper is to establish some fixed point, coincidence point and, coupled coincidence and coupled common fixed point results for generalized $$(\phi , \psi )$$ ( ϕ , ψ ) -contractive mappings in partially ordered b-metric spaces. Our results generalize, extend and unify most of the fundamental metrical fixed point theorems in the existing literature. Few examples are illustrated to justify our results. Result The existence and uniqueness theorems for a fixed point and coincidence point, coupled coincidence point and coupled common fixed points for two mappings satisfying generalized $$(\phi , \psi )$$ ( ϕ , ψ ) -contractive conditions in complete partially ordered b-metric spaces are proved. These results generalize several comparable results in the existing literature.

In this paper, we establish some boundedness conditions for the multidimensional Hausdorff operator on the homogeneous Hardy-Morrey and on the Besov-Morrey space, and we extend some results in the recent papers by Jia and Wang, and by Mazzucato, respectively. The main tool we implement in the study is the decomposition of the given function spaces in terms of atoms (smooth atoms for Besov-Morrey space) concentrated on dyadic cubes. The atomic decomposition of the classical Hardy space and Besov space is our study model, however, our case is a quite different one. Particularly, we will combine the Calderón reproducing formula with the atomic decomposition when we establish the boundedness of the Hausdorff operator on the Besov-Morrey space. MSC: 42B35; 42B30

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