On the existence for solutions of evolutionary equations, there are many works and methods 1–7 . For example, the fixed principle 1, 3–5, 7 and Galerkin approximations 2, 6 . They are very classical methods to prove existence and uniqueness. Generally speaking, there are four solution concepts. That is, weak solution, mild solution, strong solution, and classical solution. We can obtain differe...

In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwellian molecules. We first show that the global existence in time of the mild solution of the viscosity equation ( , ) t v f Q f f f          . We then study the asymptotic behaviour of the mild solution as the coefficients 0    , and an estimate on 0 f f   is derived.

In this article, we study a class of impulsive stochastic neutral partial functional differential equations in a real separable Hilbert space. By using Banach fixed point theorem, we give sufficient conditions for the existence and uniqueness of a mild solution. Also the exponential p-stability of a mild solution and its sample paths are obtained.

We consider Upadhay's three species aquatic food chain model, with the inclusion of spatial spread. This is a well established food chain model for the interaction of three given aquatic species. It exhibits rich dynamical behavior, including chaos. We prove the existence of a global weak solution to the diffusive system, followed by existence of local mild and strong solution.

The paper is concerned with existence of mild solution of evolution equation with Hilfer fractional derivative which generalized the famous Riemann–Liouville fractional derivative. By noncompact measure method, we obtain some sufficient conditions to ensure the existence of mild solution. Our results are new and more general to known results. Nowadays, fractional calculus receives increasing at...

The paper is concerned with the existence of solution of nonlinear second order neutral stochastic differential inclusions with infinite delay in a Hilbert Space. Sufficient conditions for the existence are obtained by using a fixed point theorem for condensing maps. Keywords—Mild solution, Convex multivalued map, Neutral stochastic differential inclusions.

This paper is concerned with the existence and uniqueness of mild solution to functional fractional fuzzy impulsive differential equations. For the existence and uniqueness we use the Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type.