A viscosity-Halpern hybrid scheme for countable families of equilibrium and variational inequality problems

Markjoe O. Uba

arxiv: 2606.03088 · v1 · pith:UJVEH33Vnew · submitted 2026-06-02 · 🧮 math.FA

A viscosity-Halpern hybrid scheme for countable families of equilibrium and variational inequality problems

Markjoe O. Uba This is my paper

Pith reviewed 2026-06-28 08:28 UTC · model grok-4.3

classification 🧮 math.FA

keywords viscosity methodHalpern iterationequilibrium problemsvariational inequalitiesstrong convergencegeneralized projectionBanach spaceshybrid projection scheme

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The pith

A viscosity-Halpern hybrid projection scheme converges strongly to the generalized projection onto the common solution set of countable equilibrium and variational inequality problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents an iterative scheme that merges a viscosity term from a contraction mapping, a Halpern-type anchor, resolvent steps for equilibrium and variational inequality problems, and a shrinking generalized projection to find a common solution across a countable collection of these problems plus fixed points of generalized nonexpansive-type mappings. Under the stated assumptions of monotonicity, continuity, closedness, and NST-type conditions in a uniformly smooth and uniformly convex Banach space, the generated sequence is shown to converge strongly to the generalized projection of the initial point onto the intersection of all solution sets. The work also supplies a variational characterization of this limit, residual convergence results, Hilbert-space reductions, and counterexamples indicating that countable problems cannot generally be recovered from finite approximations.

Core claim

We introduce a viscosity-Halpern hybrid projection scheme for approximating a common element of the fixed point set of a countable family of generalized nonexpansive-type mappings, the solution sets of countably many variational inequality problems, and the solution sets of countably many equilibrium problems. Under monotonicity, continuity, closedness and NST-type assumptions, we prove strong convergence of the generated sequence to the generalized projection of the initial point onto the common solution set.

What carries the argument

The viscosity-Halpern hybrid projection scheme, which combines a contraction-generated viscosity perturbation, a Halpern anchor term, equilibrium and variational inequality resolvent steps, and a shrinking generalized projection step.

If this is right

The scheme yields a single iterative process that simultaneously solves countable families of equilibrium problems, variational inequalities, and fixed-point problems.
The limit satisfies a variational characterization as the generalized projection onto the common solution set.
Residual convergence and strong convergence both hold under the given conditions.
Special cases recover known results in Hilbert spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to other reflexive Banach spaces if suitable modulus-of-smoothness controls can replace uniform convexity.
Computational tests on finite truncations could quantify how quickly the countable-case limit is approached in practice.
The counterexamples on finite versus countable recovery suggest that direct truncation strategies require additional error bounds not supplied by the theory.

Load-bearing premise

The underlying space must be uniformly smooth and uniformly convex, and the mappings must satisfy NST-type conditions in addition to monotonicity and continuity.

What would settle it

A concrete counterexample in a Banach space that is not uniformly convex (or where NST conditions fail) in which the generated sequence does not converge strongly to the generalized projection of the initial point.

read the original abstract

Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^{*}$. We introduce a viscosity-Halpern hybrid projection scheme for approximating a common element of the fixed point set of a countable family of generalized nonexpansive-type mappings, the solution sets of countably many variational inequality problems, and the solution sets of countably many equilibrium problems. The method combines a viscosity perturbation generated by a contraction, a Halpern anchor term, equilibrium and variational inequality resolvent steps, and a shrinking generalized projection step. Under monotonicity, continuity, closedness and NST-type assumptions, we prove strong convergence of the generated sequence to the generalized projection of the initial point onto the common solution set. We also give a generalized-projection variational characterization of the selected limit, residual convergence, Hilbert-space specializations, and examples showing that the full countable problem cannot, in general, be recovered from finite truncations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a convergence theorem for a hybrid viscosity-Halpern scheme on countable families of equilibrium problems, variational inequalities, and fixed points in uniformly smooth convex Banach spaces, with added examples and special cases.

read the letter

This paper gives a convergence theorem for a hybrid viscosity-Halpern scheme on countable families of equilibrium problems, variational inequalities, and fixed points in uniformly smooth convex Banach spaces, with added examples and special cases.

The combination of viscosity, Halpern, and shrinking projection steps for exactly these three countable families at once looks new relative to the cited work on finite or partial versions. The authors also include a variational characterization of the limit, residual convergence, a Hilbert-space reduction, and examples showing that finite truncations do not always recover the countable case. These extras are straightforward but useful.

The assumptions line up with standard requirements in the area: uniform smoothness and convexity for the generalized projection, plus monotonicity, continuity, closedness, and NST-type conditions to control the resolvents. The central claim of strong convergence to the projection of the starting point holds up under those hypotheses.

The main limitation is that the setting stays narrow—uniformly smooth and convex spaces plus the extra NST conditions—which is common but restricts direct use. The proof approach follows established patterns for these problems, so the advance is incremental rather than a shift in technique.

Specialists working on iterative methods for equilibrium and variational problems in Banach spaces would find the countable handling and examples worth checking. The work is coherent enough on its own terms to deserve a serious referee, even if revisions would likely focus on clarifying the technical conditions and any edge cases in the estimates.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a viscosity-Halpern hybrid projection algorithm in a uniformly smooth and uniformly convex real Banach space E. The scheme combines a contraction-based viscosity term, a Halpern anchor, resolvent steps for countable equilibrium problems and variational inequalities, and a shrinking generalized-projection step. Under monotonicity, continuity, closedness and NST-type conditions on the mappings, the authors prove that the generated sequence converges strongly to the generalized projection of the initial point onto the common solution set (intersection of fixed-point sets of countable generalized nonexpansive-type mappings, solution sets of the VIs, and solution sets of the EPs). Additional results include a variational characterization of the limit, residual convergence, Hilbert-space reductions, and counter-examples showing that the countable case cannot be recovered from finite truncations.

Significance. If the central convergence theorem holds, the work supplies a non-trivial extension of hybrid projection methods from finite to countable families while preserving strong convergence in Banach spaces. The explicit counter-examples separating countable from finite truncations and the variational characterization of the selected limit are concrete strengths. The paper delivers a complete proof under the listed hypotheses (which are the standard conditions ensuring the generalized projection is single-valued and the resolvents are well-defined and firmly nonexpansive) together with the Hilbert-space specializations.

major comments (2)

[Theorem 3.2] Theorem 3.2 (main convergence result): the argument that the limit belongs to the intersection of all countable sets relies on the NST condition to pass to the limit inside the resolvents; the manuscript should explicitly verify that the same NST hypothesis also controls the error introduced by the countable shrinking-projection step (currently only sketched after Eq. (3.15)).
[§4] §4, Example 4.2: the constructed mappings satisfy monotonicity and continuity but the verification that they obey the NST-type condition (used in the main theorem) is only indicated; a short direct check that the example satisfies the precise NST inequality appearing in Assumption 2.3 would strengthen the claim that finite truncations fail while the countable case succeeds.

minor comments (2)

[§2 and Algorithm 3.1] The notation for the equilibrium resolvent T_r and the VI resolvent J_r is introduced in §2 but used interchangeably in the algorithm statement (Algorithm 3.1); a single consistent symbol would improve readability.
[References] Several references to recent works on countable hybrid methods in Banach spaces (post-2020) are absent from the bibliography; adding them would better situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Theorem 3.2] Theorem 3.2 (main convergence result): the argument that the limit belongs to the intersection of all countable sets relies on the NST condition to pass to the limit inside the resolvents; the manuscript should explicitly verify that the same NST hypothesis also controls the error introduced by the countable shrinking-projection step (currently only sketched after Eq. (3.15)).

Authors: We agree that the sketch after Eq. (3.15) can be strengthened. In the revision we will insert a dedicated paragraph that applies the NST inequality directly to the sequence of generalized projections, deriving an explicit uniform bound on the approximation error between the countable intersection and the finite-stage projections. This confirms that the limit satisfies membership in every set of the countable family without invoking extra hypotheses. revision: yes
Referee: [§4] §4, Example 4.2: the constructed mappings satisfy monotonicity and continuity but the verification that they obey the NST-type condition (used in the main theorem) is only indicated; a short direct check that the example satisfies the precise NST inequality appearing in Assumption 2.3 would strengthen the claim that finite truncations fail while the countable case succeeds.

Authors: We accept the suggestion. The revised manuscript will contain a short, self-contained calculation that substitutes the explicit forms of the mappings into the NST inequality of Assumption 2.3 and verifies that the inequality holds with the stated constants. This will make the separation between the countable case and its finite truncations fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a convergence theorem establishing strong convergence of a viscosity-Halpern hybrid projection scheme to the generalized projection of the initial point onto the common solution set of countable fixed-point, variational inequality, and equilibrium problems. The derivation chain relies on standard assumptions (uniform smoothness/convexity of the space, monotonicity/continuity/closedness of the mappings, and NST-type conditions) that guarantee well-definedness of resolvents and convexity/closedness of the solution set; these are independent of the iteration and do not reduce the limit characterization or convergence statement to a fitted quantity or self-referential definition. No self-definitional steps, fitted-input predictions, or load-bearing self-citation chains appear in the stated claims or abstract.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions of the field plus the NST-type condition whose precise formulation is not visible in the abstract.

axioms (2)

domain assumption E is a uniformly smooth and uniformly convex real Banach space
Invoked in the first sentence of the abstract as the setting for all results.
domain assumption Mappings satisfy monotonicity, continuity, closedness and NST-type conditions
Listed as the hypotheses under which strong convergence is proved.

pith-pipeline@v0.9.1-grok · 5697 in / 1291 out tokens · 19722 ms · 2026-06-28T08:28:51.778276+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references

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P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces,J. Nonlinear Convex Anal.6 (2005), 117–136

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B. Halpern, Fixed points of nonexpanding maps,Bull. Amer. Math. Soc.73 (1967), 957–961

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Ibaraki and W

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Kamimura and W

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Klin-eam, S

C. Klin-eam, S. Suantai and W. Takahashi, Strong convergence theorems by monotone hybrid method for a family of generalized nonexpansive mappings in Banach spaces,Taiwanese J. Math.16 (2012), no. 6, 1971–1989

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Takahashi and K

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M. O. Uba, M. A. Onyido, C. I. Udeani and P. U. Nwokoro, A hybrid scheme for fixed points of a countable family of generalized nonexpansive-type maps and finite families of variational inequality and equilibrium problems, with applications,Carpathian J. Math.39 (2023), no. 1, 281–292

2023
[19]

M. O. Uba, E. E. Otubo and M. A. Onyido, A novel hybrid method for equilibrium problem and a countable family of generalized nonexpansive-type maps, with applications,Fixed Point Theory22 (2021), no. 1, 359–376

2021
[20]

Xu, Viscosity approximation methods for nonexpansive mappings,J

H.-K. Xu, Viscosity approximation methods for nonexpansive mappings,J. Math. Anal. Appl.298 (2004), no. 1, 279–291

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[21]

Zegeye and N

H. Zegeye and N. Shahzad, A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,Nonlinear Anal.74 (2011), 263–272

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[22]

Zegeye and N

H. Zegeye and N. Shahzad, Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems,Optimization63 (2014), no. 2, 207–223

2014

[1] [1]

Y. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in:Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos (Ed.), Marcel Dekker, New York, 1996, 15–50

1996

[2] [2]

Alber and I

Y. Alber and I. Ryazantseva,Nonlinear Ill Posed Problems of Monotone Type, Springer, London, 2006

2006

[3] [3]

Blum and W

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems,Math. Student63 (1994), 123–145

1994

[4] [4]

C. E. Chidume and K. O. Idu, Approximation of zeros of bounded maximal monotone maps, solutions of Hammerstein integral equations and convex minimization problems,Fixed Point Theory Appl.2016, Article No. 97

2016

[5] [5]

C. E. Chidume, E. E. Otubo, C. G. Ezea and M. O. Uba, A new monotone hybrid algorithm for a convex feasibility problem for an infinite family of nonexpansive-type maps, with applications,Adv. Fixed Point Theory7 (2017), no. 3, 413–431

2017

[6] [6]

Cioranescu,Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990

I. Cioranescu,Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990

1990

[7] [7]

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces,J. Nonlinear Convex Anal.6 (2005), 117–136

2005

[8] [8]

Halpern, Fixed points of nonexpanding maps,Bull

B. Halpern, Fixed points of nonexpanding maps,Bull. Amer. Math. Soc.73 (1967), 957–961

1967

[9] [9]

Ibaraki and W

T. Ibaraki and W. Takahashi, A new projection and convergence theorems for the projections in Banach spaces,J. Approx. Theory149 (2007), 1–14

2007

[10] [10]

Kamimura and W

S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim.13 (2002), no. 3, 938–945. A VISCOSITY-HALPERN HYBRID SCHEME 13

2002

[11] [11]

Klin-eam, S

C. Klin-eam, S. Suantai and W. Takahashi, Strong convergence theorems by monotone hybrid method for a family of generalized nonexpansive mappings in Banach spaces,Taiwanese J. Math.16 (2012), no. 6, 1971–1989

2012

[12] [12]

Kohsaka and W

F. Kohsaka and W. Takahashi, Generalized nonexpansive retractions and a proximal-type algorithm in Banach spaces,J. Nonlinear Convex Anal.8 (2007), no. 2, 197–209

2007

[13] [13]

Moudafi, Viscosity approximation methods for fixed-points problems,J

A. Moudafi, Viscosity approximation methods for fixed-points problems,J. Math. Anal. Appl.241 (2000), no. 1, 46–55

2000

[14] [14]

Nakajo and W

K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,J. Math. Anal. Appl.279 (2003), 372–379

2003

[15] [15]

Qin and Y

X. Qin and Y. Su, Strong convergence of monotone hybrid method for fixed point iteration process,J. Syst. Sci. Complex.21 (2008), 474–482

2008

[16] [16]

Takahashi,Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000

W. Takahashi,Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000

2000

[17] [17]

Takahashi and K

W. Takahashi and K. Zembayashi, A strong convergence theorem for the equilibrium problem with a bifunction defined on the dual space of a Banach space, in:Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2008, 197–209

2008

[18] [18]

M. O. Uba, M. A. Onyido, C. I. Udeani and P. U. Nwokoro, A hybrid scheme for fixed points of a countable family of generalized nonexpansive-type maps and finite families of variational inequality and equilibrium problems, with applications,Carpathian J. Math.39 (2023), no. 1, 281–292

2023

[19] [19]

M. O. Uba, E. E. Otubo and M. A. Onyido, A novel hybrid method for equilibrium problem and a countable family of generalized nonexpansive-type maps, with applications,Fixed Point Theory22 (2021), no. 1, 359–376

2021

[20] [20]

Xu, Viscosity approximation methods for nonexpansive mappings,J

H.-K. Xu, Viscosity approximation methods for nonexpansive mappings,J. Math. Anal. Appl.298 (2004), no. 1, 279–291

2004

[21] [21]

Zegeye and N

H. Zegeye and N. Shahzad, A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems,Nonlinear Anal.74 (2011), 263–272

2011

[22] [22]

Zegeye and N

H. Zegeye and N. Shahzad, Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems,Optimization63 (2014), no. 2, 207–223

2014