We consider the porous medium equation on a compact Riemannian manifold and give a new proof of the contraction of its semigroup in the Wasserstein distance. This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient flow trajectories, and on an Eulerian formulation for the Wasserstein distance using smooth curves....

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possess...

In this paper, we present some new fixed point theorems for both single-valued and multi-valued maps controlled by the contraction conditions with perturbed linear operators in continuous function spaces. Our results can be applied to various integral operators. Some previous results are generalized in this literature. As applications, the existence and uniqueness of solutions of impulsive peri...

The main result is a common fixed point theorem for a pair of multivalued maps on a complete metric space extending a recent result of D − orić and Lazović 2011 for a multivalued map on a metric space satisfying Ćirić-Suzuki-type-generalized contraction. Further, as a special case, we obtain a generalization of an important common fixed point theorem of Ćirić 1974 . Existence of a common soluti...

The focus of this paper is to establish a new concept b-hybrid fuzzy contraction regarding the study fixed-point theorems in setting b-metric spaces. This idea harmonizes and refines several well-known results direction point-valued, multivalued, fuzzy-set-valued maps comparable literature. To attract researchers field, some important are shown be corollaries. Moreover, result presented suffici...

P L Powar and GRK Sahu [14] noticed that the fixed point may not exist for some self maps T defined on a set X. To deal with such cases idea of Serendipity had been introduced by in 2015. Considering lighter concept completeness viz. weak completeness, existence uniqueness established under contraction condition. In this paper we result serendipity PGA [15] Akhilesh Pathak 2018. This be useful ...