Relaxed One-Sided Lipschitz (ROSL)

• ROSL is a condition for set-valued mappings that generalizes monotonicity and classical Lipschitz continuity by allowing flexible selection requirements.
• It guarantees the existence, stability, and numerical convergence of solutions, particularly when the constant l is negative, through explicit ball localization.
• ROSL underpins advanced techniques in nonlinear analysis, variational inequalities, and partial differential inclusions, offering practical frameworks for robust computations.

A relaxed one-sided Lipschitz (ROSL) condition is a structural property for set-valued mappings that subsumes monotonicity and classical one-sided Lipschitz requirements, while permitting greater flexibility for operators used in nonlinear analysis, variational inequalities, and differential inclusions. A mapping is called ROSL with constant $l\in\mathbb{R}$ if, for every pair of points and every selection in one image, a suitable selection in the other image can be found so that the inner product of image and domain differences is bounded by $l$ times the squared norm of the domain difference. This property, particularly with $l<0$, guarantees powerful solvability, stability, and numerical convergence results for a broad range of nonlinear and set-valued problems.

1. Formal Definition and Core Properties

Let $F : X \rightrightarrows X^*$ be a set-valued mapping with nonempty, closed, bounded, convex values, where $X$ is a real Hilbert space equipped with inner product $(\cdot,\cdot)_X$ and norm $\|\cdot\|_X$. The mapping $F$ is called $l$-relaxed one-sided Lipschitz (or $l$-ROSL) if

$l$0

If $l$1, the mapping displays a one-sided contraction property with respect to the underlying duality pairing. In finite dimensions and when $l$2 takes values directly in $l$3, the same inequality applies with the inner product replacing duality (Rieger et al., 2015, Mordukhovich et al., 2014, Eberhard et al., 2019).

The ROSL condition relaxes the classical (two-sided) Lipschitz continuity, which requires uniform comparability of mapped points for all pairs: $l$4 and the more restrictive one-sided Lipschitz (OSL) condition for single-valued $l$5: $l$6 The ROSL form allows arbitrary $l$7, only requires pair-existence of selections, and does not enforce bounds on pointwise norms or Hausdorff distances (Mordukhovich et al., 2014).

Maximal monotone operators are $l$8-ROSL; classical dissipativity and monotonicity are included as special cases. For single-valued $l$9, $l<0$0 is ROSL if the symmetric part of its Jacobian is bounded above by $l<0$1 (Mordukhovich et al., 2014).

2. Solvability and Localization Results

The existence and localization of solutions for inclusions $l<0$2 under the ROSL hypothesis are governed by explicit ball estimates. Consider $l<0$3 {nonempty compact convex subsets of $l<0$4}, upper semicontinuous and $l<0$5-ROSL with $l<0$6. For any $l<0$7 and $l<0$8, one can define

$l<0$9

Then, there exists at least one $F : X \rightrightarrows X^*$0 with $F : X \rightrightarrows X^*$1 and $F : X \rightrightarrows X^*$2 (Beyn et al., 2013, Rieger et al., 2015).

This is a strictly localized version of metric regularity: solutions lie within a ball whose boundary passes through $F : X \rightrightarrows X^*$3. If $F : X \rightrightarrows X^*$4 is also $F : X \rightrightarrows X^*$5-Lipschitz, solutions cannot be arbitrarily close to $F : X \rightrightarrows X^*$6; $F : X \rightrightarrows X^*$7 (Beyn et al., 2013, Eberhard et al., 2019).

For Hilbert spaces, analogous theorems state that under compact upper hemicontinuity and boundedness, every $F : X \rightrightarrows X^*$8 sufficiently close to a nominal value admits a solution $F : X \rightrightarrows X^*$9 localized as above (Rieger et al., 2015). In Gelfand triple settings, weighted inner products are constructed to preserve the ROSL structure for composite mappings, enabling solution existence for broad classes of PDE inclusions (Rieger et al., 2015, Beyn et al., 2017).

3. Properties of Inverse Mappings and Explicit Preimage Formulas

If $X$0 is $X$1-ROSL, upper semicontinuous, and bounded, then the inverse $X$2 has closed, nonempty, weakly sequentially closed images and is Lipschitz with constant $X$3 for $X$4: $X$5 Moreover, $X$6 is $X$7-ROSL (Rieger et al., 2015).

An explicit formula for the entire preimage of any $X$8 is available. Given $X$9 $(\cdot,\cdot)_X$0-ROSL with $(\cdot,\cdot)_X$1,

$(\cdot,\cdot)_X$2

where $(\cdot,\cdot)_X$3 is a closed ball of radius $(\cdot,\cdot)_X$4 about $(\cdot,\cdot)_X$5 (Eberhard et al., 2019). For $(\cdot,\cdot)_X$6 convex and compact, the union is restricted to outward-facing extreme points, reducing computational complexity.

These representations yield precise localization for solution sets and facilitate the analysis of stability, regularity, and sensitivity in variational analysis and optimization (Eberhard et al., 2019).

4. Numerical Algorithms and Convergence Guarantees

ROSL properties underpin robust iterative schemes for algebraic inclusions. For $(\cdot,\cdot)_X$7 compact convex sets, $(\cdot,\cdot)_X$8-ROSL and $(\cdot,\cdot)_X$9-Lipschitz with $\|\cdot\|_X$0, the iterative method

$\|\cdot\|_X$1

guarantees geometric convergence to a solution, with sharp residual bounds: $\|\cdot\|_X$2 where $\|\cdot\|_X$3 (Beyn et al., 2013). In one dimension, the rate can be precisely determined (e.g., $\|\cdot\|_X$4 for $\|\cdot\|_X$5).

For operator inclusions in Hilbert spaces, a more general algorithm incorporating error sequences is

$\|\cdot\|_X$6

with contraction factor $\|\cdot\|_X$7 (Rieger et al., 2015). The residuals decay as $\|\cdot\|_X$8 and the iterates remain localized as prescribed by the ball estimates.

For differential inclusions subject to ROSL right-hand sides, the implicit Euler scheme

$\|\cdot\|_X$9

possesses unique solvability for $F$0 (for $F$1). Piecewise linear interpolants $F$2 converge strongly to continuous-time solutions in $F$3 and (if additional convexity/Lipschitz conditions hold) in $F$4 (Mordukhovich et al., 2014).

5. Applications in Partial Differential Inclusions and Control

ROSL hypotheses facilitate broad existence and convergence results for set-valued PDEs, especially in control, elliptic, and parabolic problems. In the Gelfand triple or abstract evolution settings, inclusions of the form

$F$5

with $F$6 maximal monotone or elliptic and $F$7 $F$8-ROSL ensure nonempty sets of weak solutions and strong convergence of Galerkin approximations. Discrete solution sets $F$9 (approximating the infinite-dimensional $l$0) converge in the Hausdorff metric on $l$1 spaces; see (Beyn et al., 2017). Uniform energy estimates and Filippov-type arguments remain available even for nonmonotone, set-valued nonlinearities, substantially expanding the class of tractable differential inclusions.

Systems of elliptic PDE inclusions with set-valued, ROSL nonlinearities (e.g., $l$2) are covered, provided certain spectral gap and Lipschitz conditions are met (Rieger et al., 2015).

The property is instrumental in discrete-time and optimal control settings (generalized Bolza problems), where strong convergence of optimizers and necessary conditions can be derived for the discrete and continuous problems via the implicit Euler scheme (Mordukhovich et al., 2014).

6. Variational Analysis, Sensitivity, and Advanced Modeling

The ROSL condition enables fine-grained sensitivity and stability analysis in nonsmooth optimization, variational inequalities, and multivalued dynamics. The explicit ball localization and preimage formulas allow characterization not only of the existence but of the geometry and structure of solution sets, which is crucial in parametric optimization, nonsmooth analysis, and in the development of set-inversion numerics (Eberhard et al., 2019).

The framework also supports systematic derivation of discrete optimality conditions and mesh-refinement strategies, by providing dual (adjoint) multipliers and error-resilient convergence for nonconvex or nonmonotone perturbations.

In summary, the relaxed one-sided Lipschitz property is a unifying and far-reaching structural tool for the analysis, approximation, and computation of variational and differential inclusions, especially in settings where classical Lipschitz continuity or monotonicity fails. Its broad applicability to set-valued, nonsmooth, and infinite-dimensional problems underlies its prominence in modern variational analysis (Rieger et al., 2015, Beyn et al., 2013, Eberhard et al., 2019, Mordukhovich et al., 2014, Beyn et al., 2017).