1. Introduction

Over the past century, fixed point theory has evolved into a vital tool for analysing nonlinear problems. Its mathematical foundations support a remarkably broad spectrum of applications, ranging from engineering fields such as fluid and elastic mechanics and signal processing to image reconstruction, and extending to economics and the social sciences via game theory and the concept of Nash equilibrium. Furthermore, this theory plays a central role in computer science, particularly in optimisation, iterative algorithms, and data flow analysis. It has also proven invaluable in control theory and various branches of pure mathematics [1–7].

Let be a nonvoid subset of Banach space , and let be a self-mapping. The set of fixed points of is defined by

A wide range of iterative procedures for approximating the fixed points of non-linear operators has been proposed in the literature. One of the most well-known methods is Picard iteration [8] which starts from an initial point by

The Mann iteration [9] is generated from an initial point by

where

The Ishikawa iteration [10] is defined in a similar way, by choosing and setting

where

Of the many branches of applied mathematics, graph theory (GT) is a particularly compelling and influential field. Its wide applicability has led to a steady increase in research output over the past fifty years. Indeed, GT has been extensively employed in many disciplines, including engineering, the natural and physical sciences, genetics, computer science, sociology, operations research, economics, and linguistics [11,12]. In recent years, the study of iterative methods for approximating fixed points of mappings on abstract spaces via graphs has attracted significant attention. Jachymski [13] introduced an extension of the Banach fixed point theorem to metric spaces endowed with graphs, and also proposed the notion of a contraction, summarised below.

Let be a complete metric space, and let be a digraph with and containing all loops, that is,

for every

and there is such that

for every

If , then is said to be nonexpansive on [14].

Several contributions have been made to the literature on iterative schemes for nonexpansive mappings associated with graphs. In particular, authors [15] presented results on iteration algorithms for nonexpansive and contractive mappings, building on the fundamental ideas of Reich and Zaslavski. Tripak [14] further examined the convergence properties of the Ishikawa algorithm for nonexpansive mappings in abstract spaces equipped with a graph, and the author [16] provided additional theoretical insights into nonexpansive mappings. The authors [17] showed that a parallel monotone hybrid algorithm for a finite family of nonexpansive mappings in Hilbert spaces with graphs converges to a common fixed point and can be effectively applied to signal recovery problems. Subsequently, Khemphet et al. [18] proposed an inertial Mann-type parallel algorithm for nonexpansive mappings within Hilbert spaces involving digraphs. Chairatsiripong et al. [19] showed that the iteration method for nonexpansive mappings in uniformly convex Banach spaces with directed graphs converges both weakly and strongly. It also converges faster than the Noor and iterations and can be effectively applied to image deblurring and signal recovery problems. Yambangwai and Thianwan [20] proposed a new computational approach for identifying common fixed points of nonexpansive mappings in Hilbert spaces. They proved the approach’s weak convergence and demonstrated its effectiveness in solving signal recovery problems. In 2025, Ungchittrakool and Artsawang [21] established an inertial Krasnosel’skiĭ--Mann and Ishikawa-type iterative scheme for nonexpansive mappings in Hilbert spaces. This scheme converges strongly to a fixed point and can be effectively applied to monotone inclusion and image restoration problems. More recently, Tiammee et al. [22] gave a damped double-inertial parallel algorithm with adaptive control that converges weakly to a common fixed point, achieving improved stability and faster convergence. Notably, it outperforms existing methods, including the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), in image restoration and convex feasibility problems. On the other hand, Shahzad [23] introduced a broader class of nonexpansive mappings, known as nonexpansive mappings, and examined the best approximation results for this class in Banach spaces. Building on this work, Rhoades and Temir [24] proved the weak convergence of Mann iterates for nonexpansive mappings in Banach spaces using the Opial property. Furthermore, the authors [25] developed a refined hybrid algorithm to approximate a common fixed point for a finite family of asymptotically nonexpansive mappings.

In this direction, motivated by the concept of edge preservation in the mapping defined by the author [26], this work establishes a novel notion of nonexpansiveness, combining the concepts of graphs and nonexpansive mappings.

Definition 1.1. Let be a normed space, be a nonvoid subset of and be a digraph such that . Let and be mappings from to . Then is said to be nonexpansive if it satisfies the following condition for all

1. i. graph preserving, i.e., ,

ii.  .

Drawing inspiration from the work of Gunduz and Akbulut [27] and Van Dung and Trung Hieu [28], we present the following case.

Example 1.2. Let be a normed space via the usual norm for all . Let and be a digraph such that and belongs to if and only if or and belong to Define and as and respectively, for all Let such that Thus, we have and , so Therefore, C is an graph-preserving mapping. Furthermore, it is clear that is an graph-preserving mapping and that is a nonexpansive mapping.

Remark 1.3. Note that if (where is the identity mapping), then the nonexpansive mapping reduces to the nonexpansive mapping.

Let be a Banach space and a nonvoid subset of . Consider the nonexpansive mapping , where is a nonexpansive mapping. Then for , consider the following iteration method:

(1)

where

Using method (1), we derive some convergence theorems for approximating common fixed points of nonexpansive mappings and nonexpansive mappings on Banach spaces endowed with a digraph. To validate these results, we present numerical experiments on benchmark image deblurring problems, as well as simulation results for signal enhancement. Unlike TV-based methods, which can result in oversmoothing and the loss of fine textures, the proposed framework aims to preserve structural details while effectively suppressing noise.

2. Related Work

Various computational and mathematical approaches to image restoration and deblurring have been extensively studied. Classical methods such as Wiener filtering and Lucy--Richardson (LR) deconvolution are based on the principles of inverse filtering and maximum likelihood estimation [29–31]. While these methods are computationally efficient, they are susceptible to noise amplification and ringing artefacts [32], which can degrade the quality of restored images, particularly when there is a high level of noise.

To address these limitations, modern optimisation-based methods have been widely developed. In particular, total variation (TV) regularisation has proven effective in preserving edges while suppressing noise [33]. Furthermore, advanced frameworks such as the Alternating Direction Method of Multipliers (ADMM) and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) efficiently solve large-scale inverse problems and offer improved reconstruction quality and convergence speed [34,35]. Recent studies have demonstrated the effectiveness of these methods in restoring images and solving convex feasibility problems [20–22,36]. However, these approaches typically require careful parameter tuning and can result in oversmoothing and the loss of fine image textures [34,35].

In parallel, classical iterative schemes such as the Picard [8], Mann [9] and Ishikawa [10] iterations have been extensively used to solve nonlinear operator equations. Building on these foundations, the effectiveness of nonexpansive mappings [14] and graph-based settings in signal recovery and image processing applications [19,22] has been demonstrated. These approaches provide a robust theoretical basis for designing iterative algorithms with guaranteed convergence.

Recently, deep learning-based approaches have achieved remarkable success in image restoration tasks, particularly deblurring and denoising. Fayolle and Belyaev [37] introduced variants of the modified Richardson-Lucy (RL) and Image Field Reconstruction (ISRA) algorithm that accelerate convergence and improve restoration quality. This was achieved by interpreting these methods as fixed-point iterations within a variational framework and incorporating adaptive image smoothing. Aberdeen et al. [38] demonstrated the effectiveness of U-Net-based deep learning frameworks for recovering blurred images of resident space objects (RSOs), improving reconstruction quality and pose estimation accuracy. Hu et al. [39] proposed a lightweight deep learning framework combining a MobileNetV2 backbone with multi-scale transformer modelling and channel attention mechanisms to achieve robust performance in industrial scenarios. Additionally, Rong and Huang [40] developed a hybrid deblurring method integrating an unsupervised encoder-decoder network with an optimisation framework. This method provides improved restoration quality and generalisation without requiring labelled training data. However, deep learning-based methods typically require substantial training datasets and considerable computational resources, while optimisation-based approaches often necessitate precise parameter tuning and can result in the loss of fine structural details.

By contrast, this study puts forward a nonexpansive mapping framework that extends classical fixed-point theory to graph-structured settings. The proposed method combines theoretical convergence guarantees with practical applicability, delivering restoration performance that rivals Wiener filtering, LR and FISTA in both image deblurring and signal enhancement.

3. Preliminaries

In this section, we establish the lemmas and definitions that will be used in the fourth part. From now on, we will show the strong and weak convergence of to a point as and , respectively.

In the sequel, we assume that is a Banach space and is a digraph. In addition, suppose that holds no parallel edges, and thus we could describe . A digraph is said an oriented graph if whenever , then If are vertices of , then a directed path from to of length is a sequence of vertices such that and for is called to be connected if there is a path among any two vertices. A digraph is called to be transitive if, for , we acquire GT terminology and notations are standard and can be found in any GT books (for details see [41–43]).

Definiton 3.1 [44]. Let be a normed space, be a nonvoid subset of and be a digraph such that . Then is said to satisfy property (G) if for any sequence such that and , there is a subsequence of such that for every

Definiton 3.2 [45]. Let be a normed space, be a nonvoid subset of and be a mapping. Then is said to be semi-compact if for with there appears a subsequence of such that

Definiton 3.3 [46]. The mappings are said to satisfy condition (A) if there is a nondecreasing function with and for such that for every ,

Definiton 3.4 [47]. A Banach space is said to have the Opial property if, for all sequences such that , the inequality holds for all in

Definiton 3.5 [48]. Let be a nonvoid subset of a Banach space and be a mapping. Then, is said to be demiclosed at if, for any sequence such that , and imply

Definiton 3.6 [28]. Let be a vector space and be a nonvoid subset of Then is said to be coordinate-convex if for all and for we hold

Lemma 3.7 [49]. Let be a Banach space satisfy the Opial property, and exist for some , and there exist and be two subsequences of which weakly converge to and respectively. Then

Lemma 3.8 [50]. Let be a uniformly convex Banach space and let be two constants with Assume that is a real sequence and Then the terms

imply that where is a constant.

4. Main results

In this section, we demonstrate some of the features of (1), which represent the progress of the outcomes in Tripak [14] by using coordinate convexity instead of the convexity of .

Proposition 4.1. Assume that is a normed space, and is a nonvoid convex subset of . is a digraph and transitive such that and is coordinate-convexity. is edge preserving, where is edge preserving and For each the sequence is described as in (1) such that Then , , , , for every .

Proof: Initially, we verify inductively that Clearly, Next, assume that belongs to . We will demonstrate In fact, as is edge preserving, we get Note that

(2)

Due to the coordinate-convexity of and adding (2) with , we obtain Then, since is edge preserving, we get . Indeed,

(3)

Again, by the coordinate-convexity of and equation (3), we get . This shows that

Next, we show that Because is edge preserving, we have Moreover,

(4)

Owing to the coordinate-convexity of and adding (4) with , , we obtain . Therefore, is edge preserving, we have . To be clear,

(5)

Again, by the coordinate-convexity of and equation (5), we get . Using a similar argument, we conclude that , for every . As is transitive, , , , , we acquire , , for every .

Lemma 4.2. Assume that is a normed space, and is a nonvoid closed convex subset of . is a digraph and transitive such that and is coordinate-convexity. nonexpansive mapping where is nonexpansive mapping and For each the sequence is described as in (1) such that and and for some . Then

1. i. is bounded and exists;
2. ii.

Proof: i. Let and From Proposition 4.1, we get , , , , for every Using nonexpansive mapping where is nonexpansive mapping, we obtain

(6)

Using (6) and nonexpansive mapping of , we get

(7)

From inequality (7), it follows that the sequence is bounded and the limit exists.

ii. Let and From Proposition 4.1, we have that , , , , for every From Lemma 4.2 (i), exists. Put .

Letting on both sides in (6),

Since is nonexpansive mapping, we can get that

Moreover, from (1), we have

Taking the limit as on both sides yields:

By Lemma 3.8, we get

(8)

Further, from (1), we obtain

Hence, by (8), we attain

Next,

which on taking the limit as implies

Given that is nonexpansive mapping and is nonexpansive mapping, we get

(9)

Letting on both side in (9),

Moreover, from (1), we have

Taking the limit as on both sides yields:

Due to Lemma 3.8, we hold

(10)

Further, we obtain

Consequently, combining (8) and (10), we find that

(11)

The proof is complete.

Motivated by Suparatulatorn et al. [48] we give the following Proposition 4.3.

Proposition 4.3. Assume that is a Banach space providing Opial’s condition, is a nonvoid subset of and hold property (G). is digraph such that . nonexpansive mapping where is nonexpansive mapping. provides the property Definition 3.5 and Lemma 4.2. Then

Proof: Assume that with and Using the property (G), there is a subsequence of such that for every Assume for contradiction that . By the Opial property, we conclude that

This is a contradiction. Therefore,

Theorem 4.4. Assume that is a uniformly convex Banach space providing Opial property, is a nonvoid closed convex subset of and hold property (G). is a digraph and transitive such that and is coordinate-convexity. nonexpansive mapping where is nonexpansive mapping and The sequence is described as in (1) such that and and for some . Then

Proof: As is a uniformly convex Banach space, we have that is a reflexive Banach space. Furthermore, as in Lemma 4.2 (i), it follows is bounded. Then there is a subsequence of such that From Lemma 4.2 (ii), we also obtain

By Proposition 4.3, we conclude that and so

Assume that there is a subsequence of such that with Using Proposition 4.3 and a similar argument, we deduce that By Lemma 4.2 (i), and exists. From Lemma 3.7, we obtain that Thus

Proposition 4.5. Suppose that is a normed space and is a nonvoid subset of providing property (G). is a digraph such that and is convex. Assume nonexpansive mapping where is nonexpansive mapping and and . Then is closed and convex.

Proof: Use the line of procedure endowed in the proof of Theorem 3.2 in [44], we can easily see that is closed and convex.

Theorem 4.6. Assume that is a uniformly convex Banach space providing Opial property, is a nonvoid closed convex subset of and hold property (G). is a digraph and transitive such that and is coordinate-convexity. nonexpansive mapping and where is nonexpansive mapping such that , and and provide the condition (A). The sequence is described as in (1) such that and and for some . Then

Proof: Let and From Proposition 4.1, we get , , , , for every From Lemma 4.2 (i) and (7),

This implies that

and so, exists. Also by Lemma 4.2 (ii), we get

The condition (A) guarantees that As is a nondecreasing function and it follows that Thus, we may receive a subsequence of and a sequence such that

(12)

Using the proof procedure of [51], we have

Thus,

We deduce that is a Cauchy sequence in . By Proposition 4.5, is closed. Hence, there exists such that

(13)

Combining expressions (12) and (13), we conclude that . Hence, by Lemma 4.2 (i), we conclude that

Theorem 4.7. Under the presumptions of Theorem 4.6, if either or is semi-compact, then

Proof: From Lemma 4.2 (i) and (ii), we obtain that is bounded and Since either or is semi-compact, there is a subsequence of such that Because is transitive and has property (G), there exists a subsequence of such that Notice that

(14)

(15)

Taking the limit as in (14) and (15), we attain , which means After all, the limit exists owing to Lemma 4.2 (i); thus,

which concludes that The proof is complete.

To demonstrate the validity of Theorem 4.7, we will apply the numerical example given below.

Example 4.8. Let be a normed space via the usual norm for every , and be a digraph such that and if and only if or Here, is coordinate-convexity and Define as and for every Let we reckon with Therefore, we have and so Then, is graph preserving. Furthermore, it is clear that is nonexpansive mapping and is nonexpansive mapping. We also have Moreover, and are semi-compact. Set and Therefore, all the assumptions of Theorem 4.7 are met. Next, we indicate that Taking and we obtain and and get from

Similarly, . All computational procedures were executed using MATLAB R2016a. We give the first five values of as in the Table 1 below for the initial term and respectively. With help of the Table 1, we deduce that This confirms the applicability of Theorem 4.7.

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Table 1. The first five iterates of for different initial values.

https://doi.org/10.1371/journal.pone.0353844.t001

5. Applications

Ethics statement. This study did not involve human participants, animals, identifiable personal data, or human-derived materials. Therefore, ethical approval and informed consent were not required.

In this section, we present the results of experiments conducted to evaluate the performance of the proposed two-step nonexpansive iterative framework in image and signal restoration tasks. Image deblurring experiments were conducted using a set of widely used benchmark images. Signal restoration results are presented separately in Section 5.2. All methods were implemented in MATLAB R2016a under the same experimental conditions.

Fig 1 shows the original images used in the experiments. These images -- Starfish, Plane, Woman, Boats, Pirate and Couple -- cover a broad range of structures and textures, providing a comprehensive basis for performance evaluation. The images were selected from the widely used Set12 benchmark dataset, which is publicly available on Kaggle under the CC0 Public Domain license.

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Fig 1. Original benchmark images used in the experiments: (A) Starfish, (B) Plane, (C) Woman, (D) Boats, (E) Pirate, and (F) Couple.

The benchmark images are from the publicly available Set12 dataset available on Kaggle under the CC0 Public Domain license.

https://doi.org/10.1371/journal.pone.0353844.g001

5.1. Experimental evaluation on benchmark images

Each image was degraded using a motion blur model with a point spread function (PSF), which was generated using linear motion of a length of 15 pixels and an angle of . This was followed by the addition of Gaussian noise with a variance of 0.001. The blur operation was implemented using circular boundary conditions. These degraded images were then used as the initial inputs for all restoration methods. To ensure reproducibility, a fixed random seed was used to generate the noise. The parameter values were selected through empirical tuning based on experimental performance, and were then examined further via sensitivity analysis. Detailed results can be found in the Supporting Information. The selected parameter values correspond to the stable regions identified in the peak signal-to-noise ratio (PSNR) and the structural similarity index measure (SSIM) response surfaces.

The restoration performance was assessed using the PSNR and the SSIM. All images were normalised to the range [0,1] and the PSNR was derived from the mean squared error, assuming a maximum pixel value of 1. SSIM values were computed using MATLAB’s built-in implementation, if available; otherwise, a simplified global SSIM formulation was used as a fallback.

The baseline methods were implemented as follows: Wiener filtering with MATLAB’s deconvwnr function, LR with 20 iterations, TV-based deblurring using an ADMM framework, and FISTA with quadratic Laplacian regularization.

As shown in Table 2, the proposed method outperforms Wiener filtering, LR and FISTA consistently across all test images, achieving PSNR values ranging from approximately 21.10 to 24.51 dB. While TV-based deblurring achieves the highest PSNR values overall, the proposed method consistently ranks as the second-best performer.

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Table 2. PSNR (dB) comparison of different methods on benchmark images.

https://doi.org/10.1371/journal.pone.0353844.t002

As shown in Table 3, the proposed method achieves SSIM values ranging from 0.52 to 0.62, outperforming Wiener, LR and FISTA again for all images. As with the PSNR results, TV produces the highest SSIM values, while the proposed method maintains the second-best performance. To improve visualisation of the quantitative performance differences between the methods under comparison, the PSNR and SSIM values are presented in Supporting information (S1 Fig and S2 Fig).

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Table 3. SSIM comparison of different methods on benchmark images.

https://doi.org/10.1371/journal.pone.0353844.t003

Fig 2 shows typical restoration results. Wiener filtering produces poor results because the inversion of its frequency domain amplifies high-frequency noise components, particularly in regions with a low signal-to-noise ratio (SNR) where the denominator of the Wiener filter approaches zero. This results in severe noise amplification throughout the restored image. The LR algorithm improves local contrast and sharpness through the iterative back-projection of residual errors. However, ringing artefacts accumulate at sharp edges as iterations proceed due to the ill-posed nature of the deconvolution problem, which degrades the overall perceptual quality. TV-based restoration effectively suppresses noise and preserves edges, yielding the highest quantitative scores. However, it tends to oversmooth fine textures and produce regions of constant intensity, a limitation known as the staircasing effect. In contrast, the proposed method offers a stable, training-free alternative that balances noise suppression and structural preservation within an Ishikawa-type iterative framework. This approach does not require the parameter-intensive regularisation tuning associated with TV-based methods. Further qualitative results for the other test images can be found in Supporting Information (S3 Fig) to demonstrate that the visual behaviour is not confined to the examples given in Fig 2.

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Fig 2. Visual comparison of restoration results on representative benchmark images: (A) Plane, (B) Starfish and (C) Boats.

From left to right: ground truth; blurred and noisy observation; proposed method; total variation (TV); FISTA; LR; and Wiener filter. Zoomed-in regions and the corresponding error maps are shown below each example. PSNR (dB) and SSIM values are reported for each method. The original benchmark images (Plane, Starfish, and Boats) are from the publicly available Set12 dataset available on Kaggle (CC0 Public Domain license). All degraded images, restoration results, zoomed-in regions, error maps, and quantitative results were generated by the author using MATLAB.

https://doi.org/10.1371/journal.pone.0353844.g002

The Wiener filtering method amplifies noise in the frequency domain due to its inverse filtering mechanism [52]. The Lucy-Richardson method improves contrast and sharpness, but it can also introduce noise amplification and artefacts such as edge ringing, particularly when the number of iterations increases [53]. The TV-based method protects edges but does not restore them well, due to problems such as excessive smoothing and staircasing [54]. The proposed method combines data-fidelity correction (τ_data) with a smoothed regularisation (τ_reg, λ_reg) step within an Ishikawa-type iterative framework to reduce noise while preserving information. This is evaluated using PSNR and SSIM metrics on six test images.

The proposed restoration method follows a two-step Ishikawa-type iterative scheme. In the deblurring application, the abstract mapping , is exemplified as the data-fidelity gradient step,

where represents the blur operator, its adjoint operator, and the distorted image. This step ensures consistency with the blurry, noisy image observed by reducing the discrepancy between the predicted and observed data. In the second step, the mapping , which is instantiated as a smoothed Laplacian-based regularisation operator, is implemented:

where represents the discrete Laplace operator, the adjustment coefficient, and a small compensator constant introduced to improve numerical robustness. This step promotes spatial smoothness while preventing oversensitivity to local fluctuations. The final update combines these operators through convex combinations controlled by the parameters

Thus, the practical application of image blur removal establishes a clear link between the theoretical formulation and the restoration algorithm by directly materialising the abstract fixed-point frame via mappings C and I.

The parameter ρ controls the contribution of the data-fidelity step within the Ishikawa iteration, whereas determines the influence of the regularisation update on the final estimate. The step sizes τ_data and τ_reg govern the gradient descent updates for the data-fidelity and regularization terms, respectively. The regularization parameter λ_reg balances noise suppression and structural preservation.

The parameter selection process involved two stages. In the first stage, the step sizes τ_data and τ_reg were set to0.8 and 0.15, respectively. These values were selected empirically to ensure stable convergence. The clipping operation , which was applied at each iteration, further enforced stability by constraining the solution to the valid intensity range. In the second stage, the regularization parameters ρ and were determined through a structured grid search over ρ and , forming a 5 × 5 search space of 25 configurations evaluated on all six test images. The configuration ρ = 0.6 and was observed to provide strong performance across all images and was selected as the final parameter setting. The regularization coefficient was fixed at λ_reg = 0.01 and the iteration count was set to .

The results demonstrate a trade-off between reconstruction accuracy and structural preservation and confirm that the chosen parameter values deliver balanced performance. As shown in Fig 3, PSNR values increase with ρ, while SSIM decreases. This indicates a trade-off between reconstruction accuracy and structural preservation. Similarly, PSNR values increase with , while SSIM first increases and then decreases slightly after reaching its maximum at moderate values (approximately 0.3–0.4). These results suggest that moderate parameter values provide the best balance between noise suppression and detail preservation. The smooth variation of PSNR and SSIM across the tested ranges indicates that the method is stable and not overly sensitive to parameter selection. Based on these observations, the parameters ρ = 0.6 and were selected, as these provide a balanced trade-off between reconstruction accuracy and structural fidelity. The sensitivity analysis is illustrated using the Plane image, but similar trends were observed across other test images. Additional sensitivity curves are provided in the Supporting Information (S4 Fig).

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Fig 3. Sensitivity analysis of the proposed method on the Plane image.

(A) Variation of PSNR and SSIM with respect to the parameter ρ. (B) Variation of PSNR and SSIM with respect to .

https://doi.org/10.1371/journal.pone.0353844.g003

These findings demonstrate that the two-operator Ishikawa iteration is a robust and competitive solution for image restoration under Gaussian blur and noise. Future work will investigate additional blur types such as defocus and spatially varying blur, and explore adaptive step-size strategies. Broader comparisons will also be conducted with modern iterative and deep learning-based restoration techniques.

5.2. Simulation results for 1D signal enhancement

The two-step nonexpansive mapping proposed was evaluated numerically using a signal enhancement problem. In the experimental setup, a synthetic test signal of length 1024 was blurred using a Gaussian kernel, after which it was degraded by additive Gaussian noise with a standard deviation of The algorithm was implemented in MATLAB R2016 and executed using the recommended parameter settings. To enhance computational efficiency, an early stopping criterion based primarily on the lack of further PSNR improvement was employed.

The iterative process terminated at the 201st iteration due to the early stopping condition. The best performance was achieved at the 193rd iteration, with a PSNR value of 29.73dB. The final structural similarity index (SSIM) was approximately 0.8562.

Fig 4 illustrates the convergence behaviour of the algorithm. It can be seen that the objective function steadily decreases over the iterations while the PSNR generally improves, reaching its maximum at the 193rd iteration. The SSIM initially improves, then exhibits slight variations and remains at a high level throughout the iterative process.

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Fig 4. Convergence behaviour of the proposed two-step G-I-nonexpansive mapping under Gaussian blur and additive Gaussian noise ().

(A) shows a comparison of the ground-truth signal, the blurred and noisy observation and the restored signal (best iteration = 193). (B) A zoomed view of the restoration process for samples 300–500. (C) Evolution of the objective function, showing a steady decrease. (D) PSNR (dB) progression with respect to the iteration number. The best value is attained at iteration 193 and the vertical dashed line indicates the iteration at which early stopping is initiated (201). (E) Evolution of the SSIM, which initially increases and then stabilises at a high level.

https://doi.org/10.1371/journal.pone.0353844.g004

These results demonstrate that the proposed method effectively enhances signals degraded by blur and noise, providing stable and reliable convergence in terms of quantitative performance metrics.

6. Conclusion

In this work, inspired by the notion of edge preserving of as introduced by the authors [26], we proposed the concept of nonexpansive mappings, which integrates the ideas of nonexpansive maps with graph-theoretic structures. Several convergence theorems of the iterative scheme (1) were established for nonexpansive mappings on abstract spaces under appropriate control conditions. These results extend and generalise the findings of Tripak [14]. To validate the theoretical findings, numerical experiments were conducted on benchmark image deblurring problems, involving images degraded by motion blur and additive Gaussian noise. The proposed method demonstrated robust and stable performance, consistently outperforming classical methods such as Wiener filtering, the LR and FISTA, while producing results comparable to those of TV-based approaches. Additionally, the method exhibited reliable convergence behaviour in signal enhancement tasks, achieving high PSNR and SSIM values. These results confirm that the proposed iterative framework is an effective and versatile tool for signal and image restoration. Unlike deep learning methods, which are data- and resource-intensive, this approach requires no training and provides stable performance. Future research could involve extending the method to colour and high-resolution images. It could also examine how the method performs under different blur kernels and noise models. Furthermore, broader comparisons could be made with optimisation- and deep learning-based restoration techniques.

Supporting information

Acknowledgments

This paper was presented as an abstract at the 16th Ankara Mathematics Days held at Çankaya University on June 19--20, 2025.