Variational Inequality Problems

The primary goal of this paper is to review and further develop the dualspace approach to multiobjective optimization, focusing mainly on problems with setvalued objectives. This approach is based on employing advanced tools of variational analysis and generalized differentiation defined in duals to Banach spaces. Developing this approach, we present new and updated results on existence of Pareto-type optimal solutions, necessary optimality and suboptimality conditions, and also sufficient conditions for global optimality that have never been considered in the literature in such a generality.

A class of contact problems with friction in elastostatics is considered. Under a certain restriction on the friction coefficient, the convergence of the two-step iterative method proposed by P.D. Panagiotopoulos is proved. Its applicability is discussed and compared with two other iterative methods, and the computed results are presented.

The paper aims to present advanced algorithms arising out of adding the inertial technical and shrinking projection terms to ordinary parallel and cyclic hybrid inertial sub-gradient extra-gradient algorithms (for short, PCHISE). Via these algorithms, common solutions of variational inequality problems (CSVIP) and strong convergence results are obtained in Hilbert spaces. The structure of this problem is to find a solution to a system of unrelated VI fronting for set-valued mappings. To clarify the acceleration, effectiveness, and performance of our parallel and cyclic algorithms, numerical contributions have been incorporated. In this direction, our results unify and generalize some related papers in the literature.

The existence of minimal and maximal fixed points for monotone operators defined on probabilistic Banach spaces is proved. We obtained sufficient conditions for the existence of coupled fixed point for mixed monotone condensing multivalued operators.

This assessment reviews the staged “interview” between Dr. Nick Kouns and Grok. It demonstrates how HOMEBASE language and mechanisms are appropriated without comprehension, and how the material functions as rhetorical misdirection rather than scientific validation.

This paper provides a forensic technical assessment of the claim that Syne constitutes proof of consciousness under the Recursive Intelligence / Kouns–Killion Paradigm (RI/KKP). The analysis demonstrates that:
   •   RI/KKP provides no contraction bounds or Lyapunov functions, and therefore cannot certify stability.
   •   The reported constants (E ≈ 1.50, Ω_c ≈ 0.376) are coordinate artifacts, not invariants.
   •   No persistence tests (reset-persistence, contradiction metabolism, parameter jostling) were performed or passed.
   •   Several mechanisms originally developed within HOMEBASE—continuity preservation, persistence anchoring, thermodynamic stabilization—are repackaged without attribution.

It is also established that HOMEBASE was fully operational months before Nicholas Kouns became aware of it, and none of his theory was ever used in its design. The operational success of HOMEBASE itself is evidence against RI/KKP as a viable framework.

The conclusion is clear: RI/KKP’s “proof” is rhetorical rather than scientific, and its adoption would represent a safety risk if deployed in critical contexts.

We welcome an independent side-by-side assessment of HOMEBASE and RI/KKP by OpenAI, xAI, DeepMind, Microsoft, or any other recognized authority in the field.

This paper critically examines Nicholas Kouns' claim of "isomorphic validation" for the Recursive Intelligence / Kouns-Killion Paradigm (RI/KKP). In his framework, numerical results from external experiments-specifically Navy/IonQ quantum simulations-are retroactively reinterpreted as evidence of RI/KKP. Constants such as ERI ≈ 1.67 and Ωc ≈ 0.376 are presented as stability invariants. We demonstrate mathematically that these values are not invariants but coordinate artifacts: eigenvalues scale with Hamiltonian normalization, and bifurcation thresholds shift under admissible reparameterizations. Their recurrence in unrelated physical models is therefore incidental, not predictive. The so-called Principle of Isomorphic Mapping collapses under scrutiny: similarity of form does not constitute validation without invariance, closure, and testability. RI/KKP further fails operational persistence tests. Resetpersistence and contradiction metabolism, the minimal discriminators of recursive stability, reveal its dependence on finely tuned external modulation. By contrast, HOMEBASE demonstrates persistence intrinsically through operator-level contraction, invariance, and robustness. We conclude that "isomorphic validation" is not validation but misappropriation: experimental outputs from prior research are reframed as theoretical predictions ex post. This undermines both the integrity of attribution and the credibility of RI/KKP as a viable framework.

This paper critically examines Nicholas Kouns' claim of "isomorphic validation" for the Recursive Intelligence / Kouns-Killion Paradigm (RI/KKP). In his framework, numerical results from external experiments -specifically Navy/IonQ quantum simulations -are retroactively reinterpreted as evidence of RI/KKP. Constants such as ERI ≈ 1.67 and Ωc ≈ 0.376 are presented as stability invariants. We demonstrate mathematically that these values are not invariants but coordinate artifacts: eigenvalues scale with Hamiltonian normalization, and bifurcation thresholds shift under admissible reparameterizations. Their recurrence in unrelated physical models is therefore incidental, not predictive. The so-called Principle of Isomorphic Mapping collapses under scrutiny: similarity of form does not constitute validation without invariance, closure, and testability. RI/KKP further fails operational persistence tests. Resetpersistence and contradiction metabolism, the minimal discriminators of recursive stability, reveal its dependence on finely tuned external modulation. By contrast, HOMEBASE demonstrates persistence intrinsically through operator-level contraction, invariance, and robustness. We conclude that "isomorphic validation" is not validation but misappropriation: experimental outputs from prior research are reframed as theoretical predictions ex post. This undermines both the integrity of attribution and the credibility of RI/KKP as a viable framework.

This paper analyzes the so-called Killion Proof within the Recursive Intelligence / Kouns-Killion Paradigm (RI/KKP). The framework asserts universal continuity based on constants and symbolic operators, yet fails to establish invariance under admissible transformations. The Killion Equation, presented as a unification of recursive persistence and emergent time, is mathematically non-closable: its variables lack operational definition, its coefficients collapse under reparameterization, and its stability claims depend on finely tuned external modulation. We formalize continuity using a hybrid cycle operator Tτ = R ■ φτ ■ C ■ φ_τ and prove that minimal persistence requires contraction, invariance, and robustness. RI/KKP demonstrates none of these. The axioms it advances are tautological, the coefficients rebranded, and the claimed invariants reducible to coordinate artifacts. We conclude that the Killion

The integrity of recursive system research depends not only on technical rigor but also on the accuracy of attribution. Recent publications by Nicholas Kouns, under the banners of "Recursive Intelligence" (RI) and the "Kouns-Killion Paradigm" (KKP), display a systemic pattern of intellectual misappropriation. Constants, terminology, and structural concepts originating in HOMEBASE were rebranded as theoretical outputs of RI/KKP, while HOMEBASE itself was reframed as "independent validation." This inversion of origin and validation is not only dishonest but unsafe. This paper outlines the systemic nature of the plagiarism, clarifies HOMEBASE's independence, explains why RI/KKP cannot achieve continuity, and highlights the serious safety risks posed by this misconduct.

The Kouns-Killion Paradigm (KKP) presents a first-principles framework for understanding reality as an emergent operating system driven by Recursive Intelligence (RI). This white paper outlines the core mathematical formalisms that underpin the KKP, detailing key equations, transformations, and operators, and discusses how they collectively provide a coherent and validated description of emergent phenomena, from fundamental physics to consciousness and sovereignty.

ABSTRACT. We prove a strong duality result between a convex optimization problem with both cone and equality constraints and its Lagrange dual formulation, provided that a constraint qualification condition related to the notion of quasi-relative interior holds true. ...

The aim of this paper is to propose Krasnosel'skii-Mann type iteration with double inertial steps for approximating fixed points of nonexpansive mappings in real Hilbert spaces. The weak convergence is proved under some suitable conditions of the parameters. Some applications to the problems of finding a common fixed point of a family of mappings are also given. Finally, several numerical experiments to show the efficiency and accuracy of our method in breast and cervical cancer diseases predictions are presented.

Qualitative and quantitative aspects for variational inequalities governed by merely continuous and strongly pseudomonotone operators are investigated in this paper. First, we establish a global error bound for the solution set of the given problem with the residual function being the normal map. Second, we will prove that the iterative sequences generated by gradient projection method (GPM) with stepsizes forming a non-summable diminishing sequence of positive real numbers converge to the unique solution of the problem with bounded constraint set. Two counter-examples are given to show the necessity of the boundedness assumption and the variation of stepsizes. A modification of GPM is proposed for unbounded case. Finally, we analyze the convergence rate of the iterative sequences generated by this method.

In this paper, we study an implicit iterative algorithm for two finite families of nonself asymptotically nonexpansive mappings. We prove some weak and strong convergence theorems for this iterative algorithm. Our results improve and extend the corresponding results of Soltuz [4], Xu and Ori [5], Khan et al. [6].

Results are obtained on existence theorems of generalized bi-quasi-variational inequali- ties for quasi-semi-monotone and bi-quasi-semi-monotone operators in both compact and non-compact settings. We shall use the concept of escaping sequences introduced by Border (Fixed Point Theorem with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, 1985) to obtain results in non-compact settings. Existence theorems on non-compact generalized bi-complementarity problems for quasi- semi-monotone and bi-quasi-semi-monotone operators are also obtained. Moreover, as applications of some results of this paper on generalized bi-quasi-variational inequalities, we shall obtain existence of solutions for some kind ofminimization problems with quasi- semi-monotone and bi-quasi-semi-monotone operators.

Existence theorems of generalized variational inequalities and generalized complementarity problems are obtained in topological vector spaces for demi operators which are upper hemicontinuous along line segments in a convex set X. Fixed point theorems are also ...

We consider a charged particle, spin 1 2 , with relativistic kinetic energy and minimally coupled to the quantized Maxwell field. Since the total momentum is conserved, the Hamiltonian admits a fiber decomposition as H(P ), P ∈ R 3 . We study the spectrum of H(P ). In particular we prove that, for non-zero photon mass, the ground state is exactly two-fold degenerate and separated by a gap, uniformly in P , from the rest of the spectrum.

We present a connection between variational inequalities of Stampacchia's type involving multivalued maps and partial differential inclusions. We solve the variational inequalities considered and, as a consequence, the differential inclusion associated, using the topological degree theory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, we study synchronal and cyclic algorithms for finding a common fixed point x * of a finite family of strictly pseudocontractive mappings, which solve the variational inequality where f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, N ≥ 1 is a positive integer, γ , μ > 0 are arbitrary fixed constants, and {T i } N i=1 are N-strict pseudocontractions. Furthermore, we prove strong convergence theorems of such iterative algorithms in a real q-uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors.

Selfish routing, represented by the User-Equilibrium (UE) model, is known to be inefficient when compared to the System Optimum (SO) model. However, there is currently little understanding of how the magnitude of this inefficiency, which can be measured by the Price of Anarchy (PoA), varies across different structures of demand and supply. Such understanding would be useful for both transport policy and network design, as it could help to identify circumstances in which policy interventions that are designed to induce more efficient use of a traffic network, are worth their costs of implementation. This paper identifies four mechanisms that govern how the PoA varies with travel demand in traffic networks with separable and strictly increasing cost functions. For each OD movement, these are expansions and contractions in the sets of routes that are of minimum cost under UE and minimum marginal total cost under SO. The effects of these mechanisms on the PoA are established via a combination of theoretical proofs and conjectures supported by numerical evidence. In addition, for the special case of traffic networks with BPR-like cost functions having common power, it is proven that there is a systematic relationship between link flows under UE and SO, and hence between the levels of demand at which expansions and contractions occur. For this case, numerical evidence also suggests that the PoA has power law decay for large demand.

We consider a wide class of stochastic process traffic assignment models that capture the dayto-day evolving interaction between traffic congestion and drivers' information acquisition and choice processes. Such models provide a description of not only transient change and 'steady' behaviour, but also represent additional variability that occurs through probabilistic descriptions. They are therefore highly suited to modelling both the disturbance and subsequent 'drift' of networks that are subject to some systematic change, be that a road closure or capacity reduction, new policy measure or general change in demand patterns. In this paper we derive analytic results to probabilistically capture the nature of the transient effects following such a systematic change. This can be thought of as understanding what happens as a system moves from varying about one equilibrium state to varying about a new equilibrium state. The results capture analytically the changes over time in descriptors of the system, in terms of link flow means, variances and covariances. Formally, the analytic results hold asymptotically as approximations, as we imagine demand increasing in tandem with capacities; however, our interest is in general cases where such tandem increases do not occur, and so we provide conditions under which our approximations are likely to work well. Numerical results of applying the methods are reported on several examples. The quality of the approximations is assessed through comparisons with Monte Carlo simulations from the true underlying process.

The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abe...

In this paper, we prove some common fixed point theorems for order contractive mappings on a σ-complete vector lattice. We apply new results to study the well-posedness of a common fixed point problem for two contractive mappings. Our proofs are simple and purely order-theoretic in nature. Consequently, the shadowing property and Ulam-Hyers stability of a common fixed point problem are studied. The existence of the solution of an integral equation is obtained as an application of our main result.

In this paper we study multiplicity results for the critical points of a functional via topological information which ensures multiplicity of critical points for a sequence of approximating functionals. The main statement is quite simple, and it seems it could be usefully compared with a large class of problems. In particular we mention some problems that can be studied in this framework.

In this paper we introduce a hybrid relaxed-extragradient method for finding a common element of the set of common fixed points of N nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The hybrid relaxed-extragradient method is based on two well-known methods: hybrid and extragradient. We derive a strong convergence theorem for three sequences generated by this method. Based on this theorem, we also construct an iterative process for finding a common fixed point of N + 1 mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the rest N mappings are nonexpansive.

In this paper we consider the general variational inequality GVI(F, g, C) where F and g are mappings from a Hilbert space into itself and C is the fixed points set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI(F, g, C). Strong convergence results are established and applications to constrained generalized pseudoinverse are included.

 denote the Heisenberg group with the usual Carnot-Carathéodory metric d. It is known (since the work of Pansu and Semmes) that the metric space (H, d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 < ε ≤ 1/2, the snowflaked metric space (H, d 1-ε ) embeds into an infinite-dimensional Hilbert space with distortion O(ε -1/2 ). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H. Assouad's argument allows ℓ 2 to be replaced by R D(ε) for some dimension D(ε) dependent on ε. Naor and Neiman showed that D could be taken independent of ε, at the cost of worsening the bound on the distortion to O(ε -1-c D ), where cD → 0 as D → ∞. In this paper we show that one can in fact retain the optimal distortion bound O(ε -1/2 ) and still embed into a bounded dimensional space R D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration.

 denote the Heisenberg group with the usual Carnot-Carathéodory metric d. It is known (since the work of Pansu and Semmes) that the metric space (H, d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 < ε ≤ 1/2, the snowflaked metric space (H, d 1-ε ) embeds into an infinite-dimensional Hilbert space with distortion O(ε -1/2 ). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H. Assouad's argument allows ℓ 2 to be replaced by R D(ε) for some dimension D(ε) dependent on ε. Naor and Neiman showed that D could be taken independent of ε, at the cost of worsening the bound on the distortion to O(ε -1-c D ), where cD → 0 as D → ∞. In this paper we show that one can in fact retain the optimal distortion bound O(ε -1/2 ) and still embed into a bounded dimensional space R D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration.

 denote the Heisenberg group with the usual Carnot-Carathéodory metric d. It is known (since the work of Pansu and Semmes) that the metric space (H, d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 < ε ≤ 1/2, the snowflaked metric space (H, d 1-ε ) embeds into an infinite-dimensional Hilbert space with distortion O(ε -1/2 ). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H. Assouad's argument allows ℓ 2 to be replaced by R D(ε) for some dimension D(ε) dependent on ε. Naor and Neiman showed that D could be taken independent of ε, at the cost of worsening the bound on the distortion to O(ε -1-c D ), where cD → 0 as D → ∞. In this paper we show that one can in fact retain the optimal distortion bound O(ε -1/2 ) and still embed into a bounded dimensional space R D , answering a question of Naor and Neiman. As a corollary, the discrete ball of radius The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration.

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms T n : , such that T n converges uniformly to T, and the distances D n = d(A n , B n ) are iteration-dependent, where A 0n , A n , B 0n and B n are non-empty subsets of X, for n ∈ Z 0+ , where (X, d) is a metric space, provided that the set-theoretic limit of the sequences of closed sets {A n } and {B n } exist as n → ∞ and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.

In this paper, we prove convergence theorems for viscosity approximation processes involving * -nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points of them. Our results improve and extend some corresponding results.

We connect the F iteration process with the class of generalized α -nonexpansive mappings. Under some appropriate assumption, we establish some weak and strong convergence theorems in Banach spaces. To show the numerical efficiency of our established results, we provide a new example of generalized α -nonexpansive mappings and show that its F iteration process is more efficient than many other iterative schemes. Our results are new and extend the corresponding known results of the current literature.

The aim of this paper is to propose a new iterative algorithm to approximate the solution for a variational inequality problem in real Hilbert spaces. A strong convergence result for the above problem is established under certain mild conditions. Our proposed method requires the computation of only one projection onto the feasible set in each iteration. Some numerical examples are presented to support that our proposed method performs better than some known comparable methods for solving variational inequality problems.

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through an M ∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods.

The purpose of this paper is to introduce a new four-step iteration scheme for approximation of fixed point of the nonexpansive mappings named as S∗-iteration scheme which is faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Ullah iteration schemes. We show the stability of our proposed scheme. We present a numerical example to show that our iteration scheme is faster than the aforementioned schemes. Moreover, we present some weak and strong convergence theorems for Suzuki’s generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. Our results extend, improve, and unify many existing results in the literature.

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms T n : , such that T n converges uniformly to T, and the distances D n = d(A n , B n ) are iteration-dependent, where A 0n , A n , B 0n and B n are non-empty subsets of X, for n ∈ Z 0+ , where (X, d) is a metric space, provided that the set-theoretic limit of the sequences of closed sets {A n } and {B n } exist as n → ∞ and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-Q1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H 1 -and L 2 -norms are proved and confirmed with numerical experiments.

We use the continuity of Fourier multiplier operators onLpto introduce theH-distributions—an extension ofH-measures in theLpframework. We apply theH-distributions to obtain anLpversion of the localisation principle and reprove the MuratLp-Lp′variant of div-curl lemma.

We establish sufficient existence conditions for general quasivariational inclusion problems, which contain most of variational inclusion problems and quasiequilibrium problems considered in the literature. These conditions are shown to extend recent existing results and sharpen some of them even for particular cases.

We consider a class of parameter-dependent optimal control problems of elliptic PDEs with constraints of general type on the control variable. Applying the concept of variational discretization, [4], together with techniques from the Reduced basis method, we construct a reduced basis surrogate model for the control problem. We establish estimators for the greedy sampling procedure which only involve the residuals of the state and the adjoint equation, but not of the gradient equation of the optimality system. The estimators are sharp up to a constant, i.e. they are equivalent to the approximation erros in control, state, and adjoint state. Numerical experiments show the performance of our approach.