[Paper Review] Non-trivial solutions of local and non-local Neumann boundary-value problems

Gennaro Infante, Paolamaria Pietramala+1 more|arXiv (Cornell University)|Apr 4, 2014

Differential Equations and Numerical Methods54 references41 citations

TL;DR

This paper establishes new existence, non-existence, localization, and multiplicity results for nontrivial solutions of local and nonlocal Neumann boundary-value problems via topological methods, particularly fixed point index theory applied to perturbed Hammerstein integral equations. The key contribution is a general framework that handles sign-changing solutions and nonlocal conditions through spectral radius comparisons and improved Green's function estimates, validated through three detailed examples including a thermostat model and a nonlinear eigenvalue problem.

ABSTRACT

We prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary value problems with local and nonlocal boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

Motivation & Objective

• To develop a general topological framework for analyzing nontrivial solutions in second-order differential equations with Neumann-type boundary conditions.
• To address the challenge of non-existence of Green's functions when the spectral parameter λ = 0 is an eigenvalue, by employing a shift technique to construct solvable related problems.
• To extend existing results on positive solutions to include solutions that change sign, broadening applicability to models like thermostats with feedback control.
• To provide sharp localization and non-existence criteria via spectral radius comparisons and kernel sign analysis.
• To validate the theory through explicit examples, including a nonlocal thermostat model and a nonlinear eigenvalue problem from Bonanno and Pizzimenti (2013).

Proposed method

• Formulate the BVP as a perturbed Hammerstein integral equation involving Stieltjes integrals for nonlocal boundary conditions.
• Apply fixed point index theory on a cone K defined by lower bounds on u(t) over [a,b] and nonnegativity of functional evaluations α[u], β[u].
• Establish existence via conditions (I0ρ) and (I1ρ) on the nonlinearity f, ensuring index 0 and index 1 on small and large balls.
• Use spectral radius comparisons with associated linear operators to derive growth restrictions on f, leveraging the Krein-Rutman theorem.
• Introduce multiple linear operators to handle the sign-changing nature of the Green's function, improving upon prior estimates.
• Verify conditions via numerical checks on constants like c, M, fρ,ρ/c, and integrals of KA(s), KB(s) over [a,b] in examples.

Experimental results

Research questions

• RQ1 Under what conditions does a second-order differential equation with Neumann boundary conditions admit at least one nontrivial solution that may change sign?
• RQ2 How can fixed point index theory be adapted to handle nonlocal boundary conditions expressed via Stieltjes integrals when the standard Green's function does not exist?
• RQ3 What role does the spectral radius of associated linear operators play in determining the existence or non-existence of nontrivial solutions?
• RQ4 Can sharp localization and non-existence results be derived for nonlinearities that are not necessarily positive?
• RQ5 How do improved estimates of Green's function kernels enhance the applicability and precision of existence criteria?

Key findings

• For the nonlocal BVP (6.3) with ω ∈ (π/2, π), a nontrivial solution exists in the cone K for all ω in this interval, as verified by checking (I0ρ) and (I1ρ) conditions numerically.
• In Example 6.3, the BVP −u′′ + u = λte^u with u′(0) = u′(1) = 0 admits at least one positive solution for λ ∈ (0, (e+1)/e²) ≈ (0, 0.546), improving the range from (0, 2e⁻²) ≈ (0, 0.271) in Bonanno and Pizzimenti (2013).
• A non-existence result is proven: no nontrivial solution exists in the cone K when λ > (e+1)/(e(e−1)) ≈ 0.797, which is sharper than previous bounds.
• The method yields a localization: for λ = 1/4, the solution satisfies 0.064 ≤ u(t) ≤ 0.16 for all t ∈ [0,1], obtained via index theory and interval analysis.
• The paper improves kernel estimates in Section 5, particularly for the Green’s function associated with the operator −u′′ + ω²u, providing tighter bounds than in earlier works.
• The framework successfully handles sign-changing nonlinearities, as demonstrated in two examples where solutions change sign, extending beyond the scope of positive solution theory.