Stability in Semilinear Problems

Abstract

In the paper, we derive results concerning a continuous dependence of solutions on the right-hand side for a semilinear operator equation Lu={g(u), by assuming that L : D(L)/H Ä H (H&a Hilbert space) is self-adjont, with a closed range, and g: H Ä R is continuous convex on H and Ga^teaux differentiable on D(L). Using these results, we obtain theorems on the continuous dependence of solutions on functional parameters for a semilinear problem of the second order u +au= D u F(t, u, |), t # [0, ?] a.e., with the Dirichlet boundary conditions u(0)= u(?)=0, where a 1, and | is a functional parameter. Academic Press

Stability of solutions for an abstract Dirichlet problem

“…Reasoning similarly to the last part of the proof of the main theorem of [4] we may prove that [2]), G * denotes the Fenchel-Young dual of the convex functional G (see [1]) and…”

“…In case G has quadratic growth a problem similar to ours has been considered in [4], for L not necessarily positive definite. However, for a superquadratic nonlinearity the method from [4] does not work since in this case both the action and the dual action functionals are unbounded. We believe that the variational method from [2] may contribute to this research when combined with some stability results from [4].…”

“…However, for a superquadratic nonlinearity the method from [4] does not work since in this case both the action and the dual action functionals are unbounded. We believe that the variational method from [2] may contribute to this research when combined with some stability results from [4]. Now we specify the assumptions under which we shall consider system (1.1).…”

On the existence and the stability of solutions for higher-order semilinear Dirichlet problems

“…Stability for abstract problems satisfying quadratic growth conditions is considered in [8]. This approach is based on pioneering works [19], [20], where the question of stability in case of non unique solution is properly stated.…”

“…The dual method was first applied to proving the continuous dependance on parameters in [16], where a certain type of differential equation with a nonlinearity being separated in the state variable and a parameter. Later, using the ideas from [8] and [16] for the stability of solution, the problem similar to ours and with the additional assumption that L is positive definite, has been considered in [6]. Our approach being somehow different allows us to prove the stability of the solutions and contrary to [6] we do need to use the spectral theory.…”