Narrow Results

Consider in a Hilbert space $H$ the Cauchy problem $(P_0)$: $u^{\prime }(t)+Au(t)+Bu(t)\ni f(t),t\in [0,T];u(0)=u_0$, and associate with it the second-order problem $(P_{𝜀})$: $-𝜀u^{\prime \prime }(t)+u^{\prime }(t)+Au(t)+Bu(t)\ni f(t),t\in [0,T];u(0)=u_0,u^{\prime }(T)=0$, where $A:D(A)\subset H\to H$ is a (possibly set-valued) maximal monotone operator, $B:H\to H$ is a Lipschitz operator, and $𝜀$ is a positive small parameter. Note that $(P_{𝜀})$ is an elliptic-like regularization of $(P_0)$ in the sense suggested by Lions in his book on singular perturbations. We prove that the solution $u_{𝜀}$ of $(P_{𝜀})$ approximates the solution $u$ of $(P_0)$: $\parallel u_{𝜀}-u{\parallel }_{C([0,T];H)}={𝒪}({𝜀}^{1/4})$. Applications to the nonlinear heat equation as well as to the nonlinear telegraph system and the nonlinear wave equation are presented.

In this paper, we study a nonlocal Cahn–Hilliard equation (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn–Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in $L^1$ of the Cahn–Hilliard Equation. We also show that the Cahn–Hilliard equation is the gradient flow of the Ginzburg–Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behavior of the solutions.

The aim of this paper is to study the existence and uniqueness of solutions for a boundary value problem associated with a fractional nonlinear differential equation with higher order posed on the half-line. An appropriate continuous embedding for suitable Banach spaces are proved and the Minty–Browder theorem for monotone operators is used in the proof of existence of solutions for a boundary value problem of fractional order posed on the half-line.

In this paper we consider optimal control problem for a class of infinite dimensional uncertain systems. Necessary conditions of optimality is presented under the assumption that the principal operator is monotone in a reflexive Banach space. A computational algorithm is also given.

A variant of the monotone operator method is applied to the study of solvability of nonlinear singular integral equations of Hammerstein type in Orlicz spaces.