Dependence on parameters for nonlinear equations—Abstract principles and applications

Abstract

We provide parameter‐dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)2. Potential equations and the convergence of their Euler action functionals are also investigated. Applications towards the dependence on parameters for both potential and nonpotenial nonlinear Dirichlet boundary problems are given.

On variational approach to fourth‐order problems with unbounded weight

“…Note that we do not need uniqueness here. In case the solution was unique, we could apply the abstract results from Bełdzinski et al [13] to reach the assertion, and this is why we do not pursue this further. Instead, we modify the method indicated in Ledzewicz et al [14] which works also in case when the solutions are not unique.…”

On the Solvability of the Discrete s(x,·)-Laplacian Problems on Simple, Connected, Undirected, Weighted, and Finite Graphs

“…We can consider the following family of problems for with assumptions (independent on the parameter) leading to the existence and uniqueness of solutions employed in Section 3 . By using the continuity of the solution operator in the uniqueness case as follows from the approach suggested in [ 16 ], we may obtain the Hadamard well-posedness, which says, roughly speaking, that small deviations from the unique solution return to such solution in the limit. Nevertheless, in case the solution is nonunique, we suggest some continuous dependence on parameters result as well.…”

Continuous dependence of fixed points on parameters, remetrization theorems, and an application to the initial value problem

“…Extensions of Theorem 2 for mappings satisfying a nonlinear contractive condition were obtained, inter alia, by Dugundji and Granas, 8 Jachymski and Jó źwik, 9 and Jachymski. 10 Very recently, another generalization of Theorem 2 was established in Bełdzi ński et al 11 : Theorem 3 (Bełdzi ński-Galewski-Kossowski). Let (X, d) be complete and F ∶ Λ × X → X be continuous in the first variable.…”

“…As an application, Bełdziński et al 11 used Theorem 3 to obtain a result on the continuous dependence on parameters of a family of Dirichlet problems for ordinary differential equations of the second order.…”