In this paper, we introduce a new hybrid relaxed iterative algorithm with two half-spaces to solve the fixed-point problem and split-feasibility problem involving demicontractive mappings. The strong convergence of the iterative sequence produced by our algorithm is proved under certain weak conditions. We give several numerical experiments to demonstrate the efficiency of the proposed iterative method in comparison with previous algorithms.
Citation: Yuanheng Wang, Bin Huang, Bingnan Jiang, Tiantian Xu, Ke Wang. A general hybrid relaxed CQ algorithm for solving the fixed-point problem and split-feasibility problem[J]. AIMS Mathematics, 2023, 8(10): 24310-24330. doi: 10.3934/math.20231239
Related Papers:
[1]
Hussain Gissy, Abdullah Ali H. Ahmadini, Ali H. Hakami .
The solitary wave phenomena of the fractional Calogero-Bogoyavlenskii-Schiff equation. AIMS Mathematics, 2025, 10(1): 420-437.
doi: 10.3934/math.2025020
[2]
Guowei Zhang, Jianming Qi, Qinghao Zhu .
On the study of solutions of Bogoyavlenskii equation via improved G′/G2 method and simplified tan(ϕ(ξ)/2) method. AIMS Mathematics, 2022, 7(11): 19649-19663.
doi: 10.3934/math.20221078
[3]
Li Cheng, Yi Zhang, Ying-Wu Hu .
Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation. AIMS Mathematics, 2023, 8(7): 16906-16925.
doi: 10.3934/math.2023864
[4]
Musawa Yahya Almusawa, Hassan Almusawa .
Dark and bright soliton phenomena of the generalized time-space fractional equation with gas bubbles. AIMS Mathematics, 2024, 9(11): 30043-30058.
doi: 10.3934/math.20241451
Khudhayr A. Rashedi, Musawa Yahya Almusawa, Hassan Almusawa, Tariq S. Alshammari, Adel Almarashi .
Lump-type kink wave phenomena of the space-time fractional phi-four equation. AIMS Mathematics, 2024, 9(12): 34372-34386.
doi: 10.3934/math.20241637
[7]
Shami A. M. Alsallami .
Investigating exact solutions for the (3+1)-dimensional KdV-CBS equation: A non-traveling wave approach. AIMS Mathematics, 2025, 10(3): 6853-6872.
doi: 10.3934/math.2025314
[8]
Hussain Gissy, Abdullah Ali H. Ahmadini, Ali H. Hakami .
The travelling wave phenomena of the space-time fractional Whitham-Broer-Kaup equation. AIMS Mathematics, 2025, 10(2): 2492-2508.
doi: 10.3934/math.2025116
[9]
M. Ali Akbar, Norhashidah Hj. Mohd. Ali, M. Tarikul Islam .
Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics. AIMS Mathematics, 2019, 4(3): 397-411.
doi: 10.3934/math.2019.3.397
[10]
Ali Turab, Hozan Hilmi, Juan L.G. Guirao, Shabaz Jalil, Nejmeddine Chorfi, Pshtiwan Othman Mohammed .
The Rishi Transform method for solving multi-high order fractional differential equations with constant coefficients. AIMS Mathematics, 2024, 9(2): 3798-3809.
doi: 10.3934/math.2024187
Abstract
In this paper, we introduce a new hybrid relaxed iterative algorithm with two half-spaces to solve the fixed-point problem and split-feasibility problem involving demicontractive mappings. The strong convergence of the iterative sequence produced by our algorithm is proved under certain weak conditions. We give several numerical experiments to demonstrate the efficiency of the proposed iterative method in comparison with previous algorithms.
Table .
Nomenclature.
Acronym
Full name
NAR
Nonlinear auto regressive
GM(1, 1)
Traditional grey forecasting model
NARMKM
Nonlinear auto regressive neural network Markov model
MAPE
Mean absolute percentage error
RMSE
Root mean square error
STD
Standard deviation of absolute percentage error
R2
Coefficient of determination
ARIMA
Auto regressive integrated moving average
FGM(1, 1)
Fractional grey model
FANGBM
Fractional accumulation nonlinear grey Bernoulli model
Population is an important indicator to show the basic conditions of national conditions and national strength and it is the primary consideration in urban development analysis and planning. The number of population directly affects the economic and social development of a country as well as the allocation and utilization of resources. Therefore, the study of population is of great significance.
Population prediction takes the law of population development as the main body to determine the parameters, and the acquisition of these data and the choice of prediction algorithm greatly affect the accuracy of prediction results. Different models produce different results for population prediction. Jia et al. [1] used a BP neural network to establish a three-layer neural network structure model to predict the total population of the country. They established the method of combining the adaptive learning rate and the additional momentum method to learn and train the network, and the prediction results showed that the method was feasible. For the inner self-adaption, the solution can reach to true value in the way of any precision, and the BP method can reflect the essence of social or natural phenomenon. Li [2] provided the BP theory to forecast the U.S. population, which expressed the superiority of estimated population. Jiang et al. [3] constructed a multi-index grey PSO-BP neural network population prediction model when forecasting and analyzing China's total population, aiming at the problem that the prediction accuracy of a single index was not high, and experiments showed that the model had high prediction and accuracy. Ren et al. [4] proposed the ARMA model of time series to estimate the future fertility rate and mortality rate, and constructed the Leslie matrix accordingly. According to the truncation or trailing property of autocorrelation function and partial autocorrelation function, the ARMA model was recognized, and the random population prediction method combined with the two methods was proven to be robust. Yu et al. [5] applied the autoregressive analysis method to determine three predictors and establishes the projection pursuit autoregressive (PPAR) model, respectively. The data fitting accuracy of the modeling sample and the prediction accuracy of the verification sample are significantly higher than that of the GM and Logistic models, which can fully meet the needs of population prediction. Wang et al. [6] combined the population actuarial recurrence and the grey dynamic GM(1, 1) model to forecast the population distribution of urban pension insurance in Shanghai. The conclusion of their study has important theoretical significance and application value for the estimation of pension fund income and expenditure gap and the formulation of new population policy. Li [7] analyzed the GM(1, 1) modeling idea and combined the advantages of the equal dimension grey number replacement method, and applied the improved dynamic GM(1, 1) model to forecast the total population of China from 2013 to 2022. Men et al. [8] explored a grey dynamic prediction model to forecast the population of China in the next 50 years. Wang et al. [9] established the Logistic two-population mortality model to predict the aging population in China, and the results manifested that the proposed model was superior to the single-population CBD model, and provided relevant suggestions for the old-age service. Aiming at the defect of low prediction accuracy of the traditional grey model, Guo et al. [10] proposed the adaptive filtering method to correct its residual error and forecast the floating population in Nanjing. The results indicated that the improved model had high prediction accuracy. Basing on chaotic operators, Zou and Xiu [11] suggested a prediction model which is composed of weighted summation of multiple chaotic operators to realize the effective prediction of population data in China. To optimize some critical parameters of the system dynamics model to reduce the subjectivity of the model construction, Li [12] offered an urban population prediction model based on a multi-objective lioness optimization algorithm and system dynamics. Moreover, population size of Xi'an from 2019 to 2050 was predicted by the model. Hou et al. [13] built a combined prediction model with GM(1, N) and SVM. The results demonstated that the combined model was more accurate and more stable than the single model. Hao and Wang [14] denoted the grey dynamic model and its application in population prediction. Wang et al. [15] explored the resident population of Xi'an by using the grey-weighted Markov prediction model. They found that the grey model had a good effect in predicting the change trend of the resident population, but due to the random volatility of the data, the prediction ability of the grey model needed to be improved. On the basis of the above research results, many scholars have carried out a large number of research models for population prediction; in fact, these models are easy to produce random errors. A NAR neural network Markov model (NARMKM) was proposed in order to capture the nonlinear trend analysis of the annual resident population data of Huizhou and obtain higher prediction accuracy.
Although NAR neural network has strong advantages in processing short-term data in time series, it has certain limitations that make it difficult to accurately make long-term predictions and mitigate the effects of random errors, especially in capturing relationships in large data sets when nonlinear. To overcome these limitations and improve prediction accuracy, we propose the NARMKM model to capture nonlinear trends. Thus, we improve the accuracy to forecast the annual resident population data of Huizhou and various population data of Guangdong province. The major contributions of this study can be summarized as follows:
1) NARMKM model was established based on NAR neural network. The optimal delay order of NARMKM model is determined by minimizing the average relative error and reasonably determining the number of hidden layer neurons.
2) The correlation coefficient R between the output data of the training set and the original data and the Ljung-Box Q test of the training error are used to determine whether the trained neural network is reliable.
3) The specific expressions of the estimated and predicted values of the NARMKM model are constructed based on the Markov transition probability matrix and state division.
4) The validity of the model is verified by numerical examples, and the model is applied to predict the annual resident population of Huizhou. The proposed model and five comparison models are analyzed. The validity and robustness of each model are expressed by the statistics R2 and STD.
5) In order to verify the practicability and robustness of the model, the model was applied to the permanent resident population and elderly population in Guangdong Province, and the application range of the model was extended.
This paper is organized as follows: Section 2 introduces NARMKM model and NAR neural network algorithm. It also illustrates the Markov model using modified predictions. Section 3 shows the application of the model and gives the results of using NAR neural network model and Markov model to estimate and predict the energy of the resident population. In section 4, the NARMKM model is applied to the estimation and prediction of permanent resident population and elderly population of Guangdong Province. Finally, the fifth section summarizes the full text and discusses the prospect of this paper.
2.
The NAR neural network Markov model
2.1. The NAR neural network
According to the existence of feedback and memory, artificial neural networks can be divided into static neural networks and dynamic neural networks. Dynamic neural network has feedback and memory. Its output at every moment is based on the development and change law of system variables before the current moment, and it is a continuous deduction of the development of system variables. Dynamic neural networks can be divided into two categories according to the method of dynamic realization: One is the regression neural network composed of static neurons and the output feedback of the network, such as NAR dynamic regression neural network; the other is the neural network composed by neuronal feedback, such as the PID neural network, which is more commonly used. The variable relationship between the input and output of the model is not only a static mapping, but the output of each moment is synthesized based on the dynamic results of the system before the current moment, that is, it has the function of feedback and memory. The neural network is characterized by both dynamic and complete system information. An artificially generated neural network, which is established by simulating the human nervous system, can achieve specific functions. The network is constructed based on the structure connecting various neurons in the brain [16]. The network's input and output layers are determined by the background of the problem under study. The adjustment of the hidden layer and the number of neurons in the layer depends on the prediction results' accuracy [17]. During data learning and training, the NAR neural network discovers potential patterns and minimizes the difference between the final predicted outputs and actual records through a continuous and iterative fitting process [18]. Therefore, NAR neural network not only inherits the advantages of traditional time series model, but also has better adaptability and prediction effect for nonlinear data [19,20,21,22,23,24]. The model of NAR neural network can be defined as [25]:
y(t)=g(y(t−1),y(t−2),⋯,y(t−d))+εt1.
(1)
Where, y(t) is the variable value of the current moment, y(t−1),y(t−2),⋯,y(t−d) are the variable values of the historical moment, d is the delay order, and εt1 is the random error term.
A complete NAR neural network is generally composed of input layer, hidden layer and output layer. Data y(t) is input from the input layer, enters the hidden layer, reaches the output layer after training, transmission and learning, and then obtains the expected result, as shown in Figure 1 below:
In Eq 2, g is the activation function, ωij is the connection weight between the ith output delay signal and the jth neuron in the hidden layer, and bi is the threshold of the ith output delay signal, and εt2 is the random error term. Our research ideas are shown in the following flow chart (Figure 2):
Markov process is a kind of stochastic process. Its original model, the Markov chain, was proposed by the Russian mathematician A.A. Markov in 1907. The Markov process is an important random process, which combines the initial probability of different states and the transition probability between states to determine the change of its future state, which is suitable for processing data with volatility and no after-effect. In the dynamic process with large random fluctuations, the Markov modified model can adjust the change degree of the predicted value according to the transition probability between states. A large amount of literatures study Markov model and their applications in energy, fiscal expenditure and economy [26,27,28,29,30,31]. The model is suitable to be applied to analyze the fluctuation data. It has been widely applied in military, biology, meteorology and so on [32,33,34]. The steps of the Markov model are described below:
Let P be a one-step state transition probability matrix, and the state probability vectors at time t and t+1 are Qt and Qt+1, respectively. The Markov model can be expressed as:
Qt+1=PQt.
(3)
The prediction relative error of NAR model can be divided into several states. The number of states is determined according to the actual situation of the data. The state interval is divided as follows:
Gi=Ei(Gi1,Gi2],i=1,2,⋯,s.
(4)
Where, Gi1 and Gi2 are the lower and upper limits of the relative errors of the state intervals, and s is the number of intervals.
Pij is the one-step transition probability of the state Gi moving to Gj over a period of time. The one-step transition probabilities of all states form the matrix P, thus:
P=[P11⋯P1r⋮⋱⋮Pr1⋯Prr].
(5)
Where, k=1,2,⋯,s, the k-step transition probability matrix is:
P(k)=P(k−1)P=Pk.
(6)
Select the s groups of data closest to the predicted data, and determine the corresponding step q in order from near to far. The row vector of the q-step transition state matrix corresponding to each data is selected to form a new matrix, and the most likely state of the predicted data is determined by the maximum sum of each column vector of the new matrix, so as to determine the corresponding state interval, the center point of which is the Markov correction value, and the revised predicted value is
yNARMKM(t)=ˆyNAR(t)Gi1+Gi22+1.
(7)
2.3. The test of the model error
In this section, the mean absolute percentage error (MAPE) and the root mean square error (RMSE) are used to evaluate model errors, respectively. By reference [35], we calculate the statistics STD and R2, and obtain them as follows:
MAPE=1n∑nt=1|ˆy(t)−y(t)|y(t),
(8)
RMSE=√1n∑nt=1(ˆy(t)−y(t))2,
(9)
STD=√1n∑nt=1(|ˆy(t)−y(t)|y(t)−MAPE)2,
(10)
R2=1−∑nt=1(ˆy(t)−y(t))2∑nt=1(ˆy(t)−−y(t))2.
(11)
Where, −y is the mean value of the training data, and −y(t)=∑nt=1y(t).
3.
The NAR neural network Markov model
3.1. Estimation and prediction of the NAR model
The NAR neural network was established using the data of 32 Huizhou resident population (unit: 10000) from 1990 to 2021. First, determine the order, that is, the lag period. If the order is too small, the application of historical data information is insufficient, and too many orders are susceptible to distortions in historical data. When the order is three, the first three values of the data in the time series are used to predict the value of the data and the prediction is the best. Second, the weights and thresholds are generated randomly. The network is trained with 70% sample data, cross-checked with 15% data, and tested with 15% data for network performance. The number of hidden layer neurons is 10. The maximum number of iterations is 1000. In the training stage of NAR neural network, the correlation coefficient R between the output data of training set and the original data and the Ljung-Box Q test according to the training error are used to judge whether the trained neural network is reliable. The training error of NAR neural network model is shown in Figure 3.
Figure 3.
Training error of NAR neural network model.
It can be seen from Figure 5 that training data, cross-check data and test data all have a high degree of fit with the corresponding original data, and the correlation coefficient R between them and the original data is greater than 0.99. The result of the Ljung-Box Q test is 0, which indicates that the NAR neural network is reliable and can predict the population data.
The forecast curve is shown in Figure 6. The forecast zigzag shows a gentle upward trend as a whole, and is relatively smooth, which is basically consistent with the trend characteristics of the actual value, and reflects that the prediction results of the model have high reliability. The estimation results of NAR neural network on Huizhou resident population data are shown in Table 1.
In order to better reflect the superiority of the NAR model, we choose the grey model (GM(1, 1)), fractional grey model (FGM(1, 1)), Multiple regression and ARIMA model as comparison models. Each model estimates the resident population data of Huizhou from 1990 to 2016 and forecasts the original data from 2017 to 2021. It can be seen from Table 1 that the values of MAPE, RMSE and STD of the NAR model are smaller than those of the other three models. The R2 value of the NAR model is larger than that of the other three models. This shows that the NAR model is more efficient.
3.2. Markov model modified Huizhou resident population data
The state interval is divided according to the relative error of NAR model. As can be seen from Table 1, the minimum and maximum relative error of the first 27 fitting data of this model are -1.57% and 0.87%. Therefore, according to the rule of equal spacing, four state intervals are divided into:
From the above state intervals and the probability of moving from the current state to the next state, the following 1, 2, 3, 4 steps state transition probability matrixes are obtained:
From Table 2, it can be concluded that the most likely state of Huizhou resident population in 2017 is E3, because E3 has the largest value in the total. The predicted value of NAR model in 2017 is 573.99, and the predicted value of Markov model is 574.24 according to formula (7). In terms of the same method, the predicted value of Markov model for 2018–2021 can be obtained, and the specific results are shown in Table 3.
Table 3.
The comparison results between NAR, Multiple regression, FANGBM and NARMKM.
It can be seen from the results in Table 3 that the statistical values of MAPE, RMSE and STD predicted by NARMKM model are smaller than those of other models. The R2 value of NARMKM model is larger than that of other models. This shows that NARMKM model is more effective.
The predicted resident population of Huizhou in 2022–2028 is 610.59,610.00,611.56,610.72,611.29,610.68 and 610.95 (unit: 10000).
4.
Research on the prediction of permanent resident population and elderly population in Guangdong
In the same way as Part 3, NAR model and NARMKM model are established to predict the permanent resident population and the elderly population in Guangdong. The results are shown in Tables 4 and 5.
Table 4.
The comparison results between NAR, GM(1, 1), FANGBM and NARMKM of resident population in Guangdong.
As can be seen from the prediction results in Table 4, the NARMKM model's statistical values of MAPE, RMSE and STD are smaller than those of NAR, FANGBM and GM(1, 1) models. The R2 value of NARMKM model is larger than the corresponding values of these three models. This shows that NARMKM model is more efficient. The results of Figure 7 show that the curves of the NARMKM model are closer to the true values than those of the NAR model. The forecast permanent population of Guangdong from 2023 to 2026 is 12742.92, 12828.69, 12806.52 and 12800.32 (unit: 10000).
Figure 7.
The forecast results of different models for permanent population of Guangdong.
Based on the elderly population data of Guangdong Province, it can be seen from the prediction results in Table 5 that the statistical values of MAPE, RMSE and STD of NARMKM model are smaller than those of GM(1, 1) model, but basically the same as those of NAR model. The R2 value of NARMKM model is larger than the corresponding values of these three models. From the estimation results, NARMKM model is better than NAR, FANGBM and GM(1, 1) model. This indicates that NARMKM model is more effective. The results of Figure 8 testify that the curve of the NARMKM model is closer to the true values than those of the NAR model.
Figure 8.
The forecast results of different models for the elderly population of Guangdong.
The projected values of the elderly population in Guangdong from 2023 to 2026 is: 1269.55, 1266.91, 1259.83 and 1259.16 (unit: 10000).
5.
Conclusions
We construct a NAR neural network Markov model (NARMKM). The NAR neural network model is provided to estimate and forecast the resident population of Huizhou, the resident population and the elderly population of Guangdong. It does not need too much preprocessing operation on the original time series, and the convergence speed is fast and easy to use. The prediction results of NAR neural network model are greatly affected by the delay order and the number of hidden layer neurons. Therefore, we should make full use of prior information to constrain and test when making prediction. We choose reasonable parameter settings to obtain more reliable prediction results. NARMKM model is compared with NAR model. The results show that the NARMKM model has better fitting effect and estimation accuracy than the other six competing models, including FGM model, GM, multiple linear model, NAR, FANGBM and ARIMA model. Finally, the NARMKM model is used to predict some kinds of population, and the results show that the new model has more accurate and effective prediction and evaluation effect than the NAR model. According to the comparison between the predicted result and the actual population, the predicted result has a high accuracy. From the forecast results after 2023, the growth trend of these populations is not large.
In order to improve the prediction accuracy, we plan to further study the BP neural network Markov model and its application. We will use the general regression neural network Markov model (GRNNMKM) to deal with the unstable data. It is hoped that further progress can be made.
Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Acknowledgments
This research was funded by the NSF of Huizhou University (Grant No. hzu201806), the Project of Guangdong Provincial Department of Education (Grant No. 2021ZDJS080). This work was also supported by the Ministry of Education's "Blue Fire Program" (Huizhou) Industry-University-Research Joint Innovation Fund 2018 Annual Project: High-power Intelligent Dimming Control System (CXZJHZ201812), the Huizhou University School Level Undergraduate Teaching Quality Engineering Project (X-JYJG2021045), the Higher Education Teaching Reform Project of Huizhou University ([2022]163), the Huizhou Philosophy and Social Sciences Discipline Co-Construction Project (2022ZX046, 2023ZX056), as well as the Guangdong Philosophy and Social Sciences Discipline Co-Construction Project "Research on the evolution characteristics and development trends of the permanent resident population, floating population and population aging in Guangdong Province" (GD23XSH27).
Conflicts of interest
The authors declare that they have no conflicts of interest.
References
[1]
Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algor., 8 (1994), 221–239. https://doi.org/10.1007/BF02142692 doi: 10.1007/BF02142692
[2]
Y. Yao, M. Postolache, X. Qin, J. Yao, Iterative algorithms for the proximal split feasibility problem, U. P. B. Sci. Bull. A, 80 (2018), 37–44.
[3]
A. E. Al-Mazrooei, A. Latif, X. Qin, J. C. Yao, Fixed point algorithms for split feasibility problems, Fixed Point Theor., 20 (2019), 245–254. https://doi.org/10.24193/fpt-ro.2019.1.16 doi: 10.24193/fpt-ro.2019.1.16
[4]
Q. Dong, S. He, Y. Zhao, On global convergence rate of two acceleration projection algorithms for solving the multiple-sets split feasibility problem, Filomat, 30 (2016), 3243–3252. https://doi.org/10.2298/FIL1612243D doi: 10.2298/FIL1612243D
[5]
J. Zhao, D. Hou, A self-adaptive iterative algorithm for the split common fixed problems, Numer Algor., 82 (2019), 1047–1063. https://doi.org/10.1007/s11075-018-0640-x doi: 10.1007/s11075-018-0640-x
[6]
D. Hou, J. Zhao, X. Wang, Weak convergence of a primal-dual algorithm for split common fixed-point problems in Hilbert spaces, J. Appl. Numer. Optim., 2 (2020), 187–197.
[7]
C. Byrne, Iterative oblique projection onto convex set and the split feasibility problem, Inverse Probl., 18 (2002), 441–453. https://doi.org/10.1088/0266-5611/18/2/310 doi: 10.1088/0266-5611/18/2/310
[8]
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20 (2004), 103–120. https://doi.org/10.1088/0266-5611/20/1/006 doi: 10.1088/0266-5611/20/1/006
[9]
G. López, V. Martín-Márquez, F. Wang, H. K. Xu, Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl., 28 (2012), 085004. https://doi.org/10.1088/0266-5611/28/8/085004 doi: 10.1088/0266-5611/28/8/085004
[10]
M. Fukushima, A relaxed projection method for variational inequalities, Math. Program., 35 (1986), 58–70. https://doi.org/10.1007/BF01589441 doi: 10.1007/BF01589441
[11]
Q. Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl., 20 (2004), 1261–1266. https://doi.org/10.1088/0266-5611/20/4/014 doi: 10.1088/0266-5611/20/4/014
[12]
X. Qin, L. Wang, A fixed point method for solving a split feasibility problem in Hilbert spaces, RACSAM, 13 (2019), 215–325. https://doi.org/10.1007/s13398-017-0476-6 doi: 10.1007/s13398-017-0476-6
[13]
Y. H. Wang, T. T. Xu, J. C. Yao, B. N. Jiang, Self-adaptive method and inertial modification for solving the split feasibility problem and fixed-point problem of quasi-nonexpansive mapping, Mathematics, 10 (2022), 1612. https://doi.org/10.3390/math10091612 doi: 10.3390/math10091612
[14]
M. Tian, A general iterative algorithm for nonexpansive mapping in Hilbert spaces, Nonlinear Anal., 73 (2010), 689–694. https://doi.org/10.1016/j.na.2010.03.058 doi: 10.1016/j.na.2010.03.058
H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert space, New York: Springer, 2011.
[17]
S. Kesornprom, N. Pholasa, P. Cholamjiak, On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem, Numer. Alogr., 84 (2020), 997–1017. https://doi.org/10.1007/s11075-019-00790-y doi: 10.1007/s11075-019-00790-y
[18]
S. Suantai, S. Kesornprom, N. Pholasa, Y. J. Cho, P. Cholamjiak, A relaxed projections method using a new linesearch for the split feasibility problem, AIMS Mathematics, 6 (2021), 2690–2703.
[19]
Y. H. Wang, M. Y. Yuan, B. N. Jiang, Multi-step inertial hybrid and shrinking Tseng's algorithm with Meir-Keeler contractions for variational inclusion problems, Mathematics, 9 (2021), 1548. https://doi.org/10.3390/math9131548 doi: 10.3390/math9131548
[20]
W. L. Sun, G. Lu, Y. F. Jin, C. Park, Self-adaptive algorithms for the split problem of the quasi-pseudocontractive operators in Hilbert spaces, AIMS Mathematics, 7 (2022), 8715–8732. https://doi.org/10.3934/math.2022487 doi: 10.3934/math.2022487
[21]
D. V. Thong, Viscosity approximation methods for solving fixed point problems and split common fixed point problems, J. Fixed Point Theory Appl., 19 (2017), 1481–1499. https://doi.org/10.1007/s11784-016-0323-y doi: 10.1007/s11784-016-0323-y
[22]
S. He, C. Yang, Solving the variational inequality problem defined on intersection of finite level sets, Abstr. Appl. Anal., 2013 (2013), 942315. https://doi.org/10.1155/2013/942315 doi: 10.1155/2013/942315
[23]
M. Tian, M. Tong, Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems, J. Inequal. Appl., 2019 (2019), 7. https://doi.org/10.1186/s13660-019-1958-1 doi: 10.1186/s13660-019-1958-1
This article has been cited by:
1.
J. F. Torres, M. Martinez-Ballesteros, A. Troncoso, F. Martinez-Alvarez,
Advances in time series forecasting: innovative methods and applications,
2024,
9,
2473-6988,
24163,
10.3934/math.20241174
Yuanheng Wang, Bin Huang, Bingnan Jiang, Tiantian Xu, Ke Wang. A general hybrid relaxed CQ algorithm for solving the fixed-point problem and split-feasibility problem[J]. AIMS Mathematics, 2023, 8(10): 24310-24330. doi: 10.3934/math.20231239
Yuanheng Wang, Bin Huang, Bingnan Jiang, Tiantian Xu, Ke Wang. A general hybrid relaxed CQ algorithm for solving the fixed-point problem and split-feasibility problem[J]. AIMS Mathematics, 2023, 8(10): 24310-24330. doi: 10.3934/math.20231239
DownLoad:
