This paper deals with, in the framework of absolute stability, boundary stabilization for a nonlinear axially moving beam under boundary velocity feedback controls. The nonlinear boundary control that satisfies a slope-sector condition covering many types of nonlinear control schemes is a negative feedback of the transverse velocity at the right eyelet of the moving beam. Under the nonlinear control scheme, the well-posedness of the nonlinear partial differential equation which depends continuously on the initial value is investigated by means of the Faedo-Galerkin approximation and priori estimates. By exploiting the integral-type multiplier method, the exponential stability of the closed-loop system is established, where a novel energy like function is constructed. The numerical simulation examples using the finite element method are presented to illustrate the effectiveness of the established criterion of the controller.

A new fractional function space X^{k(⋅),α(⋅)}(Ω) with variable exponents k, α and its relaxed properties are established in this paper. Under this configuration, the comparison principle of the fractional α(⋅)-Laplace operator and a priori estimate of weak solutions for a variable-order fractional α(⋅)-Laplacian equation are obtained. Then, the application of a fixed point theorem ensures the existence of weak solutions for a nonlinear elliptic equation with the variable-order fractional α(⋅)-Laplacian.

In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem { x ˙ + A ( t , x ) + B x = f ( t ) a.e. t ∈ R , x ( t + T ) = − x ( t ) , where A ( t , x ) is a nonlinear map and B is a bounded linear operator from R N to R N . Sufficient conditions for the existence of the solution set are presented. Also, we consider the nonlinear evolution problems with a perturbation term which is multivalued. We show that, for this problem, the solution set is nonempty and weakly compact in W 1 , 2 ( I , R N ) for the case of convex valued perturbation and prove the existence theorems of anti-periodic solutions for the nonconvex case. All illustrative examples are provided.