{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T20:13:02Z","timestamp":1774383182189,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,7,22]],"date-time":"2023-07-22T00:00:00Z","timestamp":1689984000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,7,22]],"date-time":"2023-07-22T00:00:00Z","timestamp":1689984000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,10]]},"DOI":"10.1007\/s12190-023-01897-1","type":"journal-article","created":{"date-parts":[[2023,7,22]],"date-time":"2023-07-22T12:01:51Z","timestamp":1690027311000},"page":"3645-3673","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["A projection-based derivative free DFP approach for solving system of nonlinear convex constrained monotone equations with image restoration applications"],"prefix":"10.1007","volume":"69","author":[{"given":"Maaz","family":"ur Rehman","sequence":"first","affiliation":[]},{"given":"Jamilu","family":"Sabi\u2019u","sequence":"additional","affiliation":[]},{"given":"Muhammad","family":"Sohaib","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0337-1216","authenticated-orcid":false,"given":"Abdullah","family":"Shah","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,22]]},"reference":[{"issue":"5","key":"1897_CR1","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1137\/0509060","volume":"9","author":"GA Hively","year":"1978","unstructured":"Hively, G.A.: On a class of nonlinear integral equations arising in transport theory. SIAM J. Math. Anal. 9(5), 787\u2013792 (1978)","journal-title":"SIAM J. Math. Anal."},{"issue":"3","key":"1897_CR2","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1080\/02331939708844339","volume":"41","author":"NA Iusem","year":"1997","unstructured":"Iusem, N.A., Solodov, V.M.: Newton-type methods with generalized distances for constrained optimization. Optimization 41(3), 257\u2013278 (1997)","journal-title":"Optimization"},{"issue":"4","key":"1897_CR3","doi-asserted-by":"publisher","first-page":"962","DOI":"10.1137\/S1052623499357957","volume":"11","author":"YB Zhao","year":"2001","unstructured":"Zhao, Y.B., Li, D.: Monotonicity of fixed point and normal mappings associated with variational inequality and its application. SIAM J. Optim. 11(4), 962\u2013973 (2001)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"1897_CR4","doi-asserted-by":"publisher","first-page":"828","DOI":"10.1108\/HFF-07-2016-0278","volume":"28","author":"O Abu Arqub","year":"2018","unstructured":"Abu Arqub, O.: Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. Int. J. Numer. Methods Heat Fluid Flow 28(4), 828\u2013856 (2018)","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"issue":"11","key":"1897_CR5","doi-asserted-by":"publisher","first-page":"4711","DOI":"10.1108\/HFF-10-2017-0394","volume":"30","author":"O Abu Arqub","year":"2020","unstructured":"Abu Arqub, O.: Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method. Int. J. Numer. Methods Heat Fluid Flow 30(11), 4711\u20134733 (2020)","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"1897_CR6","doi-asserted-by":"crossref","unstructured":"Sweis, H., Omar Abu, A., Shawagfeh, N.: Fractional delay integro-differential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials. Int. J. Mode. Phys. C 34(4), 2350052","DOI":"10.1142\/S0129183123500523"},{"key":"1897_CR7","doi-asserted-by":"crossref","unstructured":"Omar Abu, A., Al-Smadi, M.: Numerical solutions of Riesz fractional diffusion and advection-dispersion equations in porous media using iterative reproducing kernel algorithm. J. Porous Media 23(8), 783\u2013804","DOI":"10.1615\/JPorMedia.2020025011"},{"key":"1897_CR8","unstructured":"Sun, W., Yuan, Y.X.: Optimization Theory and Methods: Nonlinear Programming, vol. 1. Springer Science & Business Media, Berlin (2006)"},{"issue":"2","key":"1897_CR9","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/s11075-018-0562-7","volume":"81","author":"N Andrei","year":"2019","unstructured":"Andrei, N.: A diagonal quasi-Newton updating method for unconstrained optimization. Numer. Algorithms 81(2), 575\u2013590 (2019)","journal-title":"Numer. Algorithms"},{"issue":"2","key":"1897_CR10","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1080\/00207160.2021.1910814","volume":"99","author":"J Sabi\u2019u","year":"2022","unstructured":"Sabi\u2019u, J., Shah, A., Waziri, M.Y.: A modified Hager\u2013Zhang conjugate gradient method with optimal choices for solving monotone nonlinear equations. Int. J. Comput. Math. 99(2), 332\u2013354 (2022)","journal-title":"Int. J. Comput. Math."},{"issue":"1","key":"1897_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/0801001","volume":"1","author":"WC Davidon","year":"1991","unstructured":"Davidon, W.C.: Variable metric method for minimization. SIAM J. Optim. 1(1), 1\u201317 (1991)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"1897_CR12","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1093\/comjnl\/6.2.163","volume":"6","author":"R Fletcher","year":"1963","unstructured":"Fletcher, R., Powell, M.J.: A rapidly convergent descent method for minimization. Comput. J. 6(2), 163\u2013168 (1963)","journal-title":"Comput. J."},{"issue":"2","key":"1897_CR13","first-page":"93","volume":"14","author":"P Dingguo","year":"1997","unstructured":"Dingguo, P., Weiwen, T.: A class of Broyden algorithms with revised search directions. Asia-Pac. J. Oper. Res. 14(2), 93 (1997)","journal-title":"Asia-Pac. J. Oper. Res."},{"issue":"1","key":"1897_CR14","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1023\/A:1013004914923","volume":"112","author":"D Pu","year":"2002","unstructured":"Pu, D.: Convergence of the DFP algorithm without exact line search. J. Optim. Theory Appl. 112(1), 187\u2013211 (2002)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"1897_CR15","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1016\/S0377-0427(02)00856-7","volume":"154","author":"D Pu","year":"2003","unstructured":"Pu, D., Tian, W.: The revised DFP algorithm without exact line search. J. Comput. Appl. Math. 154(2), 319\u2013339 (2003)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"1897_CR16","doi-asserted-by":"publisher","first-page":"246","DOI":"10.3844\/jmssp.2010.246.252","volume":"6","author":"MY Waziri","year":"2010","unstructured":"Waziri, M.Y., Leong, W.J., Hassan, M.A., Monsi, M.: A new Newton\u2019s method with diagonal Jacobian approximation for systems of nonlinear equations. J. Math. Stat. 6(3), 246\u2013252 (2010)","journal-title":"J. Math. Stat."},{"issue":"1","key":"1897_CR17","first-page":"501","volume":"18","author":"AB Abubakar","year":"2020","unstructured":"Abubakar, A.B., Rilwan, J., Yimer, S.E., Ibrahim, A.H., Ahmed, I.: Spectral three-term conjugate descent method for solving nonlinear monotone equations with convex constraints. Thai J. Math. 18(1), 501\u2013517 (2020)","journal-title":"Thai J. Math."},{"issue":"1\u20132","key":"1897_CR18","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/S0377-0427(00)00433-7","volume":"124","author":"FA Potra","year":"2000","unstructured":"Potra, F.A., Wright, S.J.: Interior-point methods. J. Comput. Appl. Math. 124(1\u20132), 281\u2013302 (2000)","journal-title":"J. Comput. Appl. Math."},{"key":"1897_CR19","doi-asserted-by":"crossref","unstructured":"Solodov, M.V., Svaiter, B.F.: A globally convergent inexact Newton method for systems of monotone equations. In: Reformulation: Nonsmooth, Piecewise Smooth, Semi-smooth and Smoothing Methods (pp. 355-369). Springer, Boston, MA (1998)","DOI":"10.1007\/978-1-4757-6388-1_18"},{"issue":"1","key":"1897_CR20","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/s10957-004-1721-7","volume":"125","author":"G Zhou","year":"2005","unstructured":"Zhou, G., Toh, K.C.: Superlinear convergence of a Newton-type algorithm for monotone equations. J. Optim. Theory Appl. 125(1), 205\u2013221 (2005)","journal-title":"J. Optim. Theory Appl."},{"issue":"264","key":"1897_CR21","doi-asserted-by":"publisher","first-page":"2231","DOI":"10.1090\/S0025-5718-08-02121-2","volume":"77","author":"WJ Zhou","year":"2008","unstructured":"Zhou, W.J., Li, D.H.: A globally convergent BFGS method for nonlinear monotone equations without any merit functions. Math. Comput. 77(264), 2231\u20132240 (2008)","journal-title":"Math. Comput."},{"key":"1897_CR22","unstructured":"Zhou, W., Li, D.: Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 89\u201396 (2007)"},{"issue":"1","key":"1897_CR23","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s00186-006-0140-y","volume":"66","author":"C Wang","year":"2007","unstructured":"Wang, C., Wang, Y., Xu, C.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Methods Oper. Res. 66(1), 33\u201346 (2007)","journal-title":"Math. Methods Oper. Res."},{"issue":"2","key":"1897_CR24","doi-asserted-by":"publisher","first-page":"478","DOI":"10.1016\/j.cam.2005.10.002","volume":"196","author":"L Zhang","year":"2006","unstructured":"Zhang, L., Zhou, W.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196(2), 478\u2013484 (2006)","journal-title":"J. Comput. Appl. Math."},{"issue":"10","key":"1897_CR25","doi-asserted-by":"publisher","first-page":"2416","DOI":"10.1016\/j.apnum.2009.04.004","volume":"59","author":"Z Yu","year":"2009","unstructured":"Yu, Z., Lin, J., Sun, J., Xiao, Y., Liu, L., Li, Z.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59(10), 2416\u20132423 (2009)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"1897_CR26","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1016\/j.jmaa.2013.04.017","volume":"405","author":"Y Xiao","year":"2013","unstructured":"Xiao, Y., Zhu, H.: A conjugate gradient method to solve convex-constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405(1), 310\u2013319 (2013)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1897_CR27","doi-asserted-by":"publisher","first-page":"170","DOI":"10.1137\/030601880","volume":"16","author":"WW Hager","year":"2005","unstructured":"Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16(1), 170\u2013192 (2005)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1897_CR28","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1145\/1132973.1132979","volume":"32","author":"WW Hager","year":"2006","unstructured":"Hager, W.W., Zhang, H.: Algorithm 851: CG DESCENT, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. (TOMS) 32(1), 113\u2013137 (2006)","journal-title":"ACM Trans. Math. Softw. (TOMS)"},{"key":"1897_CR29","unstructured":"Muhammed, A.A., Kumam, P., Abubakar, A.B., Wakili, A., Pakkaranang, N.: A new hybrid spectral gradient projection method for monotone system of nonlinear equations with convex constraints. Thai J. Math., 125\u2013147 (2018)"},{"issue":"04","key":"1897_CR30","doi-asserted-by":"publisher","first-page":"2050043","DOI":"10.1142\/S0219876220500437","volume":"18","author":"J Sabi\u2019u","year":"2021","unstructured":"Sabi\u2019u, J., Shah, A., Waziri, M.Y., Ahmed, K.: Modified Hager\u2013Zhang conjugate gradient methods via singular value analysis for solving monotone nonlinear equations with convex constraint. Int. J. Comput. Methods 18(04), 2050043 (2021)","journal-title":"Int. J. Comput. Methods"},{"issue":"8","key":"1897_CR31","doi-asserted-by":"publisher","first-page":"8792","DOI":"10.3934\/math.2021510","volume":"6","author":"AM Awwal","year":"2021","unstructured":"Awwal, A.M., Kumam, P., Sitthithakerngkiet, K., Bakoji, A.M., Halilu, A.S., Sulaiman, I.M.: Derivative-free method based on DFP updating formula for solving convex constrained nonlinear monotone equations and application. AIMS Math. 6(8), 8792\u20138814 (2021)","journal-title":"AIMS Math."},{"key":"1897_CR32","doi-asserted-by":"crossref","unstructured":"Rao, J., Huang, N.: A derivative-free scaling memoryless DFP method for solving large scale nonlinear monotone equations. J. Glob. Optim. 1\u201337 (2022)","DOI":"10.1007\/s10898-022-01215-2"},{"key":"1897_CR33","doi-asserted-by":"crossref","unstructured":"Ullah, N., Shah, A., Sabi\u2019u, J., Jiao, X., Awwal, A. M., Pakkaranang, N., Panyanak, B.: A one-parameter memoryless DFP algorithm for solving system of monotone nonlinear equations with application in image processing. Mathematics, 11(5), 1221 (2023)","DOI":"10.3390\/math11051221"},{"key":"1897_CR34","doi-asserted-by":"crossref","unstructured":"Ali, R., Pan, K.: New generalized Gauss\u2013Seidel iteration methods for solving absolute value equations. Math. Methods Appl. Sci. 1\u20138 (2023)","DOI":"10.1002\/mma.9062"},{"key":"1897_CR35","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s13160-022-00526-x","volume":"40","author":"R Ali","year":"2023","unstructured":"Ali, R., Pan, K.: Two new fixed point iterative schemes for absolute value equations. Jpn. J. Ind. Appl. Math. 40, 303\u2013314 (2023)","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"1897_CR36","doi-asserted-by":"publisher","first-page":"S1113","DOI":"10.1051\/ro\/2020061","volume":"55","author":"J Sabi\u2019u","year":"2021","unstructured":"Sabi\u2019u, J., Shah, A.: An efficient three-term conjugate gradient-type algorithm for monotone nonlinear equations. RAIRO Oper. Res. 55, S1113\u2013S1127 (2021)","journal-title":"RAIRO Oper. Res."},{"issue":"5","key":"1897_CR37","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1287\/mnsc.20.5.845","volume":"20","author":"SS Oren","year":"1974","unstructured":"Oren, S.S., Luenberger, D.G.: Self-scaling variable metric (SSVM) algorithms: Part i: criteria and sufficient conditions for scaling a class of algorithms. Manag. Sci. 20(5), 845\u2013862 (1974)","journal-title":"Manag. Sci."},{"issue":"1","key":"1897_CR38","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1007\/BF01580654","volume":"10","author":"SS Oren","year":"1976","unstructured":"Oren, S.S., Spedicato, E.: Optimal conditioning of self-scaling variable metric algorithms. Math. Program. 10(1), 70\u201390 (1976)","journal-title":"Math. Program."},{"issue":"10","key":"1897_CR39","doi-asserted-by":"crossref","first-page":"2442","DOI":"10.1016\/j.camwa.2015.09.014","volume":"70","author":"JK Liu","year":"2015","unstructured":"Liu, J.K., Li, S.J.: A projection method for convex-constrained monotone nonlinear equations with applications. Comput. Math. Appl. 70(10), 2442\u20132453 (2015)","journal-title":"Comput. Math. Appl."},{"key":"1897_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.112781","volume":"375","author":"L Zheng","year":"2020","unstructured":"Zheng, L., Yang, L., Liang, Y.: A conjugate gradient projection method for solving equations with convex constraints. J. Comput. Appl. Math. 375, 112781 (2020)","journal-title":"J. Comput. Appl. Math."},{"issue":"5","key":"1897_CR41","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2374","volume":"28","author":"N Ullah","year":"2021","unstructured":"Ullah, N., Sabi\u2019u, J., Shah, A.: A derivative-free scaling memoryless Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno method for solving a system of monotone nonlinear equations. Numer. Linear Algebra Appl. 28(5), e2374 (2021)","journal-title":"Numer. Linear Algebra Appl."},{"key":"1897_CR42","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/j.apnum.2020.02.017","volume":"153","author":"J Sabi\u2019u","year":"2020","unstructured":"Sabi\u2019u, J., Shah, A., Waziri, M.Y.: Two optimal Hager\u2013Zhang conjugate gradient methods for solving monotone nonlinear equations. Appl. Numer. Math. 153, 217\u2013233 (2020)","journal-title":"Appl. Numer. Math."},{"key":"1897_CR43","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan, E.D., Mor\u00e9, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201\u2013213 (2002)","journal-title":"Math. Program."},{"issue":"4","key":"1897_CR44","doi-asserted-by":"publisher","first-page":"586","DOI":"10.1109\/JSTSP.2007.910281","volume":"1","author":"MA Figueiredo","year":"2007","unstructured":"Figueiredo, M.A., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4), 586\u2013597 (2007)","journal-title":"IEEE J. Sel. Top. Signal Process."},{"issue":"1","key":"1897_CR45","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1007\/BF02591989","volume":"36","author":"JS Pang","year":"1986","unstructured":"Pang, J.S.: Inexact Newton methods for the nonlinear complementarity problem. Math. Program. 36(1), 54\u201371 (1986)","journal-title":"Math. Program."},{"issue":"11","key":"1897_CR46","doi-asserted-by":"publisher","first-page":"3570","DOI":"10.1016\/j.na.2011.02.040","volume":"74","author":"Y Xiao","year":"2011","unstructured":"Xiao, Y., Wang, Q., Hu, Q.: Non-smooth equations-based method for l1 norm problems with applications to compressed sensing. Nonlinear Anal. Theory Methods Appl. 74(11), 3570\u20133577 (2011)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"issue":"5\u20136","key":"1897_CR47","doi-asserted-by":"publisher","first-page":"1231","DOI":"10.1080\/02331934.2020.1808647","volume":"70","author":"AM Awwal","year":"2021","unstructured":"Awwal, A.M., Kumam, P., Mohammad, H., Watthayu, W., Abubakar, A.B.: A Perry-type derivative-free algorithm for solving nonlinear system of equations and minimizing $$l_{1}$$ regularized problem. Optimization 70(5\u20136), 1231\u20131259 (2021)","journal-title":"Optimization"},{"key":"1897_CR48","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18576\/sjm\/040203","volume":"4","author":"MY Waziri","year":"2017","unstructured":"Waziri, M.Y., Muhammad, L.: An accelerated three-term conjugate gradient algorithm for solving large-scale systems of nonlinear equations. Sohag J. Math 4, 1\u20138 (2017)","journal-title":"Sohag J. Math"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01897-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01897-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01897-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,17]],"date-time":"2023-12-17T15:46:14Z","timestamp":1702827974000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01897-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,22]]},"references-count":48,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["1897"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01897-1","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,22]]},"assertion":[{"value":"2 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 July 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 July 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 July 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"This work does not have any conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}