Krasnosel'skii-Guo Cone Compression Theorem

• Krasnosel’skii-Guo Cone Compression Theorem is a fixed point theorem that localizes nontrivial solutions in ordered Banach spaces using conical shells.
• It employs component-wise and index-theoretic methods to handle compact operators with singular or discontinuous nonlinearities in both scalar and vector settings.
• The theorem underpins multiple applications, including differential equations, boundary-value problems, and physical models such as Nambu solutions in QCD.

The Krasnosel’skii-Guo Cone Compression Theorem is a central tool in nonlinear analysis for locating fixed points of compact operators in ordered Banach spaces, particularly within cones. Distinguished from general fixed-point theorems by its geometric localization mechanism, this theorem is notable for both its scalar and product-space (vector) versions, enabling coexistence solutions with nontrivial components. It achieves existence results even for operators exhibiting singular or discontinuous nonlinearities, and has significant applications in differential equations, especially periodic and boundary-value problems. The theorem's modern component-wise and discontinuous generalizations utilize fixed-point index theory to establish precise localization and multiplicity.

1. Framework: Cones, Norms, and Conical Shells

Let $(X_i,\|\cdot\|_i)$ ($i=1,\dots,n$) be real Banach spaces. The product space is $X = X_1 \times \cdots \times X_n$ with the supremum norm

$$\|x\| = \max_{1\leq i\leq n}\|x_i\|_i, \quad x = (x_1,\dots,x_n).$$

Each $X_i$ has a closed, convex cone $K_i \subset X_i$ satisfying $\lambda K_i\subset K_i$ for $\lambda \geq 0$ and $K_i \cap (-K_i) = \{0\}$. The product cone is $K = K_1\times\cdots\times K_n \subset X$, inducing the partial order $x \leq y$ iff $y-x \in K$.

Given vectors $r=(r_1, ..., r_n), R=(R_1, ..., R_n)$ with $0 < r_i < R_i$, define the conical shell

$$K_{r,R} = \{x\in K : r_i < \|x_i\|_i < R_i \ \forall i\},$$

with closure

$$\overline{K}_{r,R} = \{x\in K : r_i \leq \|x_i\|_i \leq R_i \ \forall i\}.$$

This geometric structure is foundational for the placement and localization of fixed points via cone compression.

2. The Component-wise Krasnosel’skii-Guo Theorem

The theorem, as established by Fernández–Pardo & Rodríguez‐López (Fernández-Pardo et al., 3 Apr 2025), generalizes scalar cone-compression results to product spaces. For $n=2$ components, it reads:

Let $K_i \subset X_i$ $(i=1,2)$ be cones, $K = K_1 \times K_2$, and $0 < r_i < R_i$. Suppose $$\mathcal T=(\mathcal T_1,\mathcal T_2):\overline{K}_{r,R} \to K$$ is compact and for each $i$ either

• (compressive): $\|\mathcal T_i x\|\geq\|x_i\|$ if $\|x_i\|=r_i$, $\|\mathcal T_i x\|\leq\|x_i\|$ if $\|x_i\|=R_i$,
• or (expansive): $\|\mathcal T_i x\|\leq\|x_i\|$ if $\|x_i\|=r_i$, $\|\mathcal T_i x\|\geq\|x_i\|$ if $\|x_i\|=R_i$.

Then there exists a fixed point $x=(x_1,x_2) \in K$ such that $r_i \leq \|x_i\| \leq R_i$ for both $i$. This localization excludes trivial solutions in any coordinate, guaranteeing coexistence.

Noteworthy remarks:

• Compactness may be replaced by complete continuity.
• No monotonicity in the partial order is required—only norm-type boundary comparisons.
• The result recovers the classical scalar theorem when $n=1$.

3. Fixed Point Index Theory and Proof Structure

The main technical device is the fixed-point index in cones, $i_K(T,U)$, defined when $T:\overline{U}\to K$ is compact and $T$ has no fixed points on $\partial_K U$. Its standard properties include:

• Additivity: If $U=U_1\cup U_2$ disjoint, $i_K(T,U)=i_K(T,U_1)+i_K(T,U_2)$.
• Existence: $i_K(T,U)\neq 0 \implies$ existence of a fixed point in $U$.
• Homotopy invariance and normalization.

One-dimensional index computations:

• If $T x \neq \lambda x$ for $x\in \partial U, \lambda>1$, then $i_K(T,U)=1$.
• If $x-Tx \neq \mu S x$ for some $S>0$ on $\partial U$, then $i_K(T,U)=0$.

Key two-dimensional refinement: Combining compressive and expansive boundary conditions in separate components (Proposition 2.3) yields vanishing index on mixed-boundary regions.

The proof uses Lipschitz retractions to extend maps, then indexes are computed on partitioned subdomains of $K$ to leverage the additivity property. The uniqueness of the nonzero index on $K_{r,R}$ yields the existence of a fixed point.

4. Discontinuous and Multivalued Operator Generalizations

The work of Figueroa–López Pousot–Rodríguez‐López (Figueroa et al., 2017) extends the framework to discontinuous operators via the closed-convex envelope $$\widehat{T}(x) = \cap_{\epsilon>0} \operatorname{co} T(B_\epsilon(x)\cap K)$$ and the assumption $\{x\}\cap\widehat{T}(x)\subset\{T(x)\}$.

Under suitable cone-compression and cone-expansion at radii $r_1<r_2<R$, the discontinuous variant yields at least one fixed point $x\in K$ with $r_1<\|x\|<r_2$. The proof invokes multivalued fixed-point index theorems and upper semicontinuity of the envelope.

This approach enables findings even for boundary-value problems with jump discontinuities or where nonlinearities are discontinuous along admissible curves.

5. Generalizations to Multiplicity and Alternative Domains

Rodríguez‐López (Rodríguez-López, 2022) details the further generalization to vector versions (Precup extension) and alternative functional domains. For cones $K_i$ in Banach spaces $X_i$, the theorem holds on intersections of domains defined by continuous, concave functionals or general retract pairs $(U_i, V_i)$.

Multiple annular shells may be nested, yielding multiplicity: if compression-expansion is satisfied on distinct shells $K_{r^{(j)},R^{(j)}}$, the index is nonzero on each, providing multiple fixed points.

This methodology informs the existence and multiplicity of positive solutions for Hammerstein systems and radially symmetric $(p_1, p_2)$-Laplacian equations.

6. Applications: Periodic Boundary Problems and Nonlinear PDEs

The theorem localizes solutions for systems such as

$$x'' + a_1(t)x = f_1(t,x,y), \quad y'' + a_2(t)y = f_2(t,x,y)$$

under $T$-periodicity and suitable nonlinearities $f_i$. The Hammerstein operator representation

$$\mathcal T_i(x,y)(t) = \int_0^T G_i(t,s) f_i(s, x(s), y(s))\, ds$$

with cone

$$K_i = \{u\in C[0,T]: \min_t u(t) \geq c_i\|u\|_\infty \}$$

frequently arises. Example 4.5 (sublinear-superlinear coupling) demonstrates the existence of coexistence solutions localized via norm bounds.

Hybrid situations combining compressive and expansive behavior, as well as singularities, are handled by the component-wise formulation (see (Fernández-Pardo et al., 3 Apr 2025), Theorem 4.9 & Remark 4.11).

7. Physical Example: Nambu Solutions in QCD

Recent work (Roberts, 16 Jan 2026) applies the Krasnosel’skii-Guo theorem to establish existence of positive, continuously decreasing Nambu solutions for the rainbow-ladder gap equation of QCD. Here, the cone comprises nonnegative, decreasing functions in a Banach space $E$, with a norm constructed to control asymptotic behavior.

The nonlinear operator involves the kernel

$$K(k,p) = \frac{kZ^2(k)r(p)}{4\pi^4 r(k)}\int_{S^3}\mathcal G(q^2(k,p)),$$

and a compressive form of the theorem bounds the resulting mass function $B(p)$ between two radii, ensuring existence and localization even for asymptotically perturbative kernels.

A hybrid Krasnosel’skii–Schauder strategy is invoked for coupled systems, distinguishing cone compression for the mass-function equation from a Schauder approach for the $Z$-equation.

8. Key Impact and Novelities

The component-wise, index-theoretic formulation of the Krasnosel’skii-Guo theorem represents a substantial advance over scalar and earlier vectorial versions, particularly in its:

• Guarantee of coexistence (nontrivial components) without monotonicity prerequisites.
• Accommodation of hybrid, singular, or discontinuous nonlinearities.
• Utility in generating multiple solutions by nested index computations.
• Transparent geometric and analytic localization by conical shells in product spaces.

This framework underpins modern approaches to nonlinear systems, boundary-value problems, and models in mathematical physics, notably by providing sharp control over solution norm and component-wise behavior within cones. The fixed-point index machinery is pivotal for both existence and multiplicity, as well as for structural generalizations to broader functional classes.