Global Solvability of Hammerstein Equations with Applications to BVP Involving Fractional Laplacian

Abstract

Some sufficient conditions for the nonlinear integral operator of the Hammerstein type to be a diffeomorphism defined on a certain Sobolev space are formulated. The main result assures the invertibility of the Hammerstein operator and in consequence the global solvability of the nonlinear Hammerstein equations. The applications of the result to nonlinear Dirichlet BVP involving the fractional Laplacian and to some specific Hammerstein equation are presented.

Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian

“…The global solvability of some related problem under different conditions guaranteeing the integral operator to be a global diffeomorphism was considered in [9].…”

Sensitivity of a nonlinear ordinary BVP with fractional Dirichlet-Laplace operator

“…. Equality (5) and the fact that isolated points of the spectrum of a self-adjoint operator are the eigenvalues imply that…”

“…This property can be used in optimal control for system (1). Similar method but based on a global diffeomorphism theorem ( [13]) and applied to a nonlinear integral Hammerstein equation is presented in [5] with an application to the problem λ(−∆) σ 2 x(t) + h(t, x(t)) = (−∆) σ 2 z(t), t ∈ (0, 1), with the exterior Dirichlet boundary condition…”

Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

“…Some continuous dependence results for homogenous Dirichlet boundary problem involving the fractional Laplacian one can find in [12] where coercive case is examined by the direct method of calculus of variations. Differentiable continuous dependence on parameters, or in other words robustness result are presented in [13] where the application of theorem on diffeomorphism leads to the stability result for the problem involving one-dimensional fractional Laplacian with zero boundary condition. In the present paper we obtain the existence and the continuous dependence results for the exterior boundary value problem involving the equation with the fractional Laplacian by adopting the approach presented in [11] were superlinear elliptic boundary value with the nonhomogeneous Dirichlet boundary condition was examined.…”