[Paper Review] Dependence on parameters for discrete second order boundary value problems

Marek Galewski|arXiv (Cornell University)|Dec 28, 2009

Nonlinear Differential Equations Analysis10 references4 citations

TL;DR

This paper investigates the continuous dependence of solutions on functional parameters in discrete second-order boundary value problems using a variational approach. By establishing coercivity of the action functional, it proves that convergent parameter sequences induce convergent solution sequences, ensuring stability and existence of positive solutions, with applications to the discrete Emden-Fowler equation.

ABSTRACT

We investigate the dependence on parameters for second order difference equations with two point boundary value conditions by using a variational method in case when the corresponding Euler action functional is coercive. Some applications for discrete Emden-Fowler equation are also given.

Motivation & Objective

• To analyze the continuous dependence of solutions on functional parameters in discrete second-order boundary value problems.
• To establish stability of solutions under convergence of parameter sequences using variational methods.
• To extend results on existence and regularity of positive solutions to discrete Emden-Fowler-type equations.
• To provide a variational framework for parameter dependence that complements topological and lower-upper solution methods.
• To prove that minimizers of coercive action functionals converge when parameters converge, ensuring structural stability.

Proposed method

• Formulates the discrete second-order boundary value problem as a minimization of a Gâteaux-differentiable, coercive action functional on a finite-dimensional Hilbert space.
• Uses the variational method to prove existence of at least one minimizer (solution) for each fixed parameter function $ u \in L_M $.
• Applies coercivity and continuity of the functional to ensure boundedness of solution sets uniformly over $ u \in L_M $.
• Employs compactness arguments and weak convergence in $ \mathbb{R}^T $ to extract convergent subsequences of solutions corresponding to convergent parameter sequences.
• Relies on the equivalence between the Euler-Lagrange equation and the minimization of the action functional to link solutions to critical points.
• Applies the direct method in the calculus of variations to prove existence and convergence of solutions under parameter convergence.

Experimental results

Research questions

• RQ1 Under what conditions does the solution of a discrete second-order boundary value problem depend continuously on a functional parameter?
• RQ2 How can the variational method be used to establish stability of solutions under parameter convergence?
• RQ3 What conditions ensure the existence of positive solutions in discrete Emden-Fowler-type equations via variational techniques?
• RQ4 Can coercivity of the action functional guarantee convergence of solution sequences when parameter sequences converge?
• RQ5 How do assumptions on the nonlinearity $ f(k,x,u) $, such as sign control or growth conditions, affect solution stability?

Key findings

• For any convergent sequence of parameters $ \{u_n\} \to \overline{u} $ in $ L_M $, there exists a subsequence of solutions $ \{x_{n_i}\} $ that converges to a solution $ \overline{x} $ of the problem with parameter $ \overline{u} $.
• The solution set $ V_u $ is non-empty and uniformly bounded for all $ u \in L_M $, ensuring compactness and stability under parameter variation.
• The action functional $ J_u $ is coercive and continuous on $ E $, guaranteeing the existence of at least one minimizer satisfying the discrete boundary value problem.
• When $ M + Q $ is positive definite and $ f $ satisfies subcritical growth conditions, the functional remains coercive, ensuring existence of nontrivial solutions.
• The variational approach allows systematic analysis of parameter dependence, unlike topological methods, and provides a framework for stability proofs.
• Examples show that even when standard assumptions like A2 (sign control) fail, alternative conditions (e.g., A4, A6) can still ensure existence and stability of solutions.