In this article, the contraction mapping principle and Liapunov’s method are used to study qualitative properties of nonlinear Volterra equations of the form x(t) = a(t)− ∫ t 0 C(t, s)g(s, x(s)) ds, t ≥ 0. In particular, the existence of bounded solutions and solutions with various L properties are studied under suitable conditions on the functions involved with this equation.

The almost automorphic solution is a generalization of the almost periodic solution. In this paper, the almost automorphic solutions of Cohen-Grossberg neural networks with delays are considered. Using the semi-discretization method and the contraction mapping principle, some sufficient conditions are obtained to ensure the existence and the uniqueness of almost automorphic solutions to Cohen-G...

1 Local Existence of Evolution Equations 2 1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The contraction mapping principle . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Local Existence for Evolution Partial Differential Equations . . . . . . . . . . . . 5 1.4 Local Existence for Nonlinear Heat and Hartree Equations . . . . . . . ...

It is proved under mild regularity assumptions on the data that the Navier-Stokes equations in bounded and unbounded noncylindrical regions admit a unique local-in-time strong solution. The result is based on maximal regularity estimates for the in spatial regions with a moving boundary obtained in [16] and the contraction mapping principle.

We discuss existence, and uniqueness of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Caputo sense and the initial condition is a fuzzy number. To this aim, contraction mapping principle and the fixed point theorem are used and finally an example is given to more illustration of obtained results.

In this paper, we present the existence and stability results for a nonlinear coupled system of sequential fractional differential equations supplemented with new kind boundary conditions. Existence uniqueness are established by using Schaefer’s fixed point theorem Banach’s contraction mapping principle. We examine solutions involved in Hyers–Ulam type. A few examples presented to illustrate ma...

We derive the relative entropy between two Markov transition rate matrices from sample path considerations. This relative entropy is interpreted as a \level 2.5" large deviations action functional. That is, the level two large deviations action functional for empirical distributions of continuous-time Markov chains can be derived from the relative entropy using the contraction mapping principle...

The aim of this paper is to prove the existence and uniqueness of mild and classical solutions of the non-local Cauchy problem for a semilinear integrodifferential equation with deviating argument. The results are established by using the method of semigroups and the contraction mapping principle. The paper generalizes certain results of Lin and Liu.