[Paper Review] Fixed points of Hammerstein-type equations on general cones

Rubén Figueroa, F. Adrián F. Tojo|arXiv (Cornell University)|Nov 8, 2016

Nonlinear Differential Equations Analysis34 references3 citations

TL;DR

This paper establishes new existence and multiplicity results for fixed points of Hammerstein-type integral equations in general cones by combining continuous functionals with fixed point index theory. The key contribution is a novel characterization of cones via functionals satisfying subadditivity and positivity conditions, enabling unified analysis of boundary value problems with concave, nonnegative, or other structured solutions.

ABSTRACT

We obtain new results on the existence and multiplicity of fixed points of Hammerstein equations in very general cones. In order to achieve this, we combine a new formulation of cones in terms of continuous functionals with fixed point index theory. Many examples and an application to boundary value problems are also included.

Motivation & Objective

• To develop a unified framework for analyzing fixed points of Hammerstein-type equations in general cones.
• To generalize Krasnosel’skiï´ski-type theorems by reformulating cones in terms of continuous subadditive functionals.
• To provide conditions ensuring existence and multiplicity of fixed points in cones arising from boundary value problems.
• To apply the abstract results to concrete examples, particularly concave solutions of differential equations with integral constraints.

Proposed method

• Characterize cones in normed spaces using continuous subadditive functionals satisfying specific positivity and non-degeneracy conditions.
• Define a cone $ K_\alpha = \{ u \in N : \alpha(u) \geq 0 \} $ for $ \alpha \in \mathcal{A} $, where $ \mathcal{A} $ consists of functionals with subadditivity and homogeneity properties.
• Use the fixed point index theory on these functional-defined cones to derive existence and multiplicity theorems.
• Establish conditions (C1)–(C8) on the kernel and nonlinearity to ensure the operator satisfies the required properties for index computation.
• Apply the abstract results to a second-order boundary value problem with a nonlinear term $ f(t,u) = 4/(|u|+4) $, using $ \|u\|_2 $, $ \|u\|_1 $, and concavity constraints.
• Verify conditions (I_ρ₁⁰), (I_ρ₂¹), and (S₁) to confirm the existence of a solution in a specified annular region of the cone.

Experimental results

Research questions

• RQ1 How can cones in general normed spaces be systematically characterized using continuous subadditive functionals?
• RQ2 What conditions on the kernel and nonlinearity of a Hammerstein equation ensure the existence of fixed points in a given cone?
• RQ3 Can the fixed point index theory be extended to abstract cones defined via functionals, beyond classical nonnegative or concave function cones?
• RQ4 Under what conditions does a Hammerstein equation admit multiple fixed points in a cone, particularly when constrained by $ L^1 $, $ L^2 $, or maximum norm bounds?
• RQ5 Can the abstract framework be applied to prove the existence of concave, nonnegative solutions to boundary value problems with integral constraints?

Key findings

• The paper proves that every cone in a normed space can be represented as $ K_\alpha = \{ u \in N : \alpha(u) \geq 0 \} $ for some $ \alpha \in \mathcal{A} $, where $ \mathcal{A} $ is a class of continuous, subadditive, and non-degenerate functionals.
• A new existence and multiplicity result (Theorem 3.6) is established for Hammerstein equations in general cones, under conditions (C1)–(C8) and index-type inequalities.
• For the specific problem $ -u''(t) = 4/(|u(t)|+4) $, $ u(0)=u(1)=0 $, a concave, nonnegative solution exists with $ \int_0^1 u(s) ds \geq 1/20 $ and $ \|u\|_2 \leq 1/2 $, satisfying $ f_{\rho_1} \int \psi_\gamma g \, ds > 1 $ and $ f^{\rho_2} \int \psi_\beta g \, ds < 1 $.
• The functional $ \alpha(u) = \min\{ \text{concavity term}, u(0), -u(0), u(1), -u(1) \} $ characterizes the cone of continuous concave functions vanishing at endpoints.
• The results are applied to show that $ \|u\|_1 \geq \frac{1}{2} \|u\|_2 $ for functions in the cone, enabling the construction of bounds $ b(\rho) = 2\rho $ and $ c(\rho) = \rho $.
• The example confirms that $ \rho_1 = 1/20 $, $ \rho_2 = 1/2 $ satisfy $ \rho_2 > b(\rho_1) $, and the index conditions $ (I_{\rho_1}^0) $ and $ (I_{\rho_2}^1) $ hold, ensuring a solution in the annular region $ K_\alpha^{\beta,\rho_2} \setminus K_\alpha^{\gamma,\rho_1} $.