{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:42:27Z","timestamp":1760146947189,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,25]],"date-time":"2024-12-25T00:00:00Z","timestamp":1735084800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Innovation Project of GUET Graduate Education of Yuanyuan Gong","award":["2024YCXS105","2021KY1653","2022KY0189","AD23023002"],"award-info":[{"award-number":["2024YCXS105","2021KY1653","2022KY0189","AD23023002"]}]},{"name":"Project for Enhancing Young and Middle-Aged Teachers\u2019 Essential Research Capabilities in Colleges of Guangxi","award":["2024YCXS105","2021KY1653","2022KY0189","AD23023002"],"award-info":[{"award-number":["2024YCXS105","2021KY1653","2022KY0189","AD23023002"]}]},{"name":"Science and Technology Project of Guangxi","award":["2024YCXS105","2021KY1653","2022KY0189","AD23023002"],"award-info":[{"award-number":["2024YCXS105","2021KY1653","2022KY0189","AD23023002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we propose a new notion, which we name S-contractive mapping, in a framework of vector-valued white noise functionals W\u03c9\u2297N\u2282\u0393(H)\u2297K\u2282(W\u03c9\u2297N)*. And we give concrete definitions of S-contractive mappings for vector-valued white noise functionals. We establish the fixed-point theorems of S-contractive mappings. As applications, by applying the fixed-point theorems of generalized S-contractive mappings, we prove the existence and uniqueness of a generalized form of differential equations of vector-valued white noise functionals with weak conditions and investigate Wick-type differential equations of vector-valued white noise functionals with generalized conditions.<\/jats:p>","DOI":"10.3390\/sym17010015","type":"journal-article","created":{"date-parts":[[2024,12,25]],"date-time":"2024-12-25T19:19:32Z","timestamp":1735154372000},"page":"15","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["S-Contractive Mappings on Vector-Valued White Noise Functional Space and Their Applications"],"prefix":"10.3390","volume":"17","author":[{"given":"Pengcheng","family":"Ma","sequence":"first","affiliation":[{"name":"School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China"},{"name":"Center for Applied Mathematics of Guangxi (GUET), Guilin 541004, China"},{"name":"Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanyuan","family":"Gong","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aiju","family":"Wei","sequence":"additional","affiliation":[{"name":"Guilin Institute of Information Technology, Guilin 541004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les op\u00e9rations dans les ensembles abstraits et leurs applications aux \u00e9quations int\u00e9grales","volume":"3","author":"Banach","year":"1922","journal-title":"Fundam. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Goebel, K., and Kirk, W.A. (1990). Topics in Metric Fixed Point Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511526152"},{"key":"ref_3","first-page":"194","article-title":"Existence theories and exact solutions of nonlinear PDEs dominated by singularities and time noise","volume":"28","author":"Ahmed","year":"2023","journal-title":"Nonlinear-Anal.-Model. Control"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"271","DOI":"10.3934\/ipi.2023032","article-title":"An inverse potential problem for the stochastic diffusion equation with a multiplicative white noise","volume":"18","author":"Feng","year":"2023","journal-title":"Inverse Probl. Imaging"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Baber, M.Z., Mohammed, W.W., Ahmed, N., and Iqbal, M.S. (2024). Exact solitary wave propagations for the stochastic Burgers\u2019 equation under the influence of white noise and its comparison with computational scheme. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-58553-2"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1511","DOI":"10.1007\/s10959-021-01100-w","article-title":"Mild Solutions for the Stochastic Generalized Burgers\u2013Huxley Equation","volume":"35","author":"Mohan","year":"2022","journal-title":"J. Theor. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Seok, J., and Kim, J.-S. (2021, January 16\u201318). Quantized ADAM with Monotonically Increasing Resolution of Quantization. Proceedings of the International Conference on Information and Communication Technology Convergence, Jeju Island, Republic of Korea.","DOI":"10.1109\/ICTC52510.2021.9620825"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/0022-0396(71)90004-0","article-title":"Stochastic differential equations in Hilbert space","volume":"10","author":"Curtain","year":"1971","journal-title":"J. Differ. Equations"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.jfa.2009.11.019","article-title":"Pathwise uniqueness for a class of SDE in Hilbert spaces and applications","volume":"259","author":"Flandoli","year":"2010","journal-title":"J. Funct. Anal."},{"key":"ref_10","unstructured":"Pr\u00e9v\u00f4t, C., and R\u00f6ckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1006\/jdeq.2001.4073","article-title":"Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces","volume":"181","author":"Taniguchi","year":"2002","journal-title":"J. Differ. Equations"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/BF01258530","article-title":"Quantum Ito\u2019s formula and stochastic evolutions","volume":"93","author":"Hudson","year":"1984","journal-title":"Commun. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0022-1236(02)00027-7","article-title":"Stochastic integral representation theorem for quantum semimartingales","volume":"201","author":"Ji","year":"2003","journal-title":"J. Funct. Anal."},{"key":"ref_14","unstructured":"Obata, N., Matsui, T., and Hora, A. (2002). Quantum white noise calculus. Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads, World Scientific."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/s00220-008-0702-3","article-title":"Annihilation-derivative, creation-derivative and representation of quantum martingales","volume":"286","author":"Ji","year":"2000","journal-title":"Commun. Math. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Rao, M.M. (2014). Quantum white noise calculus and applications. Real and Stochastic Analysis, World Scientific.","DOI":"10.1142\/8940"},{"key":"ref_17","unstructured":"Ji, U.C., and Obata, N. (2007). Generalized white noise operators fields and quantum white noise Derivatives. S\u00e9minaires & Congres, Soci\u00e9t\u00e9 math\u00e9matique de France."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1142\/S0219025702000912","article-title":"Analytic characterization of generalized Fock space op- erators as two-variable entire functions with growth condition","volume":"5","author":"Ji","year":"2002","journal-title":"Infin. Dimens. Anal. Quantum Probab. Relat. Top."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"667","DOI":"10.2977\/prims\/1195163920","article-title":"Generalized quantum stochastic processes on Fock space","volume":"31","author":"Obata","year":"1995","journal-title":"Publ. Res. Inst. Math. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1063\/1.3516477","article-title":"Implementation problem for the canonical commutation relation in terms of quantum white noise derivatives","volume":"51","author":"Ji","year":"2010","journal-title":"J. Math. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1063\/1.4960454","article-title":"An implementation problem for boson fields and quantum Girsanov transform","volume":"57","author":"Ji","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"116179","DOI":"10.1016\/j.cam.2024.116179","article-title":"An explicit Euler scheme for generalized n-dimensional second-order differential equations with initial value conditions driven by additive Gaussian white noises","volume":"454","author":"Ranjbar","year":"2025","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Huang, S.-Y., and Du, W.-S. (2024). New Fixed Point Theorems for Generalized Meir-Keeler Type Nonlinear Mappings with Applications to Fixed Point Theory. Symmetry, 16.","DOI":"10.3390\/sym16081088"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"073503","DOI":"10.1063\/5.0176568","article-title":"Quantum continuity equations and quantum evolution systems for white noise operators","volume":"65","author":"Ji","year":"2024","journal-title":"J. Math. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s43037-020-00088-5","article-title":"White noise differential equations for vector-valued white noise functionals","volume":"15","author":"Ji","year":"2021","journal-title":"Banach J. Math. Anal."},{"key":"ref_26","unstructured":"Heyer, H., Hirai, T., Kawazoe, T., and Sait\u00f4, K. (2000). Initial value problem for white noise operators and quantum stochastic processes. Infinite Dimensional Harmonic Analysis, Department of Mathematics, Graduate School of Science, Kyoto University."},{"key":"ref_27","first-page":"376","article-title":"Calculus on Gaussian white noise I","volume":"56A","author":"Kubo","year":"1980","journal-title":"Proc. Jpn. Acad."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/0034-4877(93)90003-W","article-title":"Spaces of white noise distributions: Constructions, descriptions, applications I","volume":"33","author":"Kondratiev","year":"1993","journal-title":"Rep. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Obata, N. (1994). White Noise Calculus and Fock Space, Springer. Lecture Notes in Mathematics 1577.","DOI":"10.1007\/BFb0073952"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1006\/jfan.1994.1047","article-title":"Operator calculus on vector-valued white noise functionals","volume":"121","author":"Obata","year":"1994","journal-title":"J. Funct. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"299","DOI":"10.32917\/hmj\/1151105703","article-title":"General characterization theorems and intrinsic topologies in white noise analysis","volume":"31","author":"Asai","year":"2001","journal-title":"Hiroshima Math. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1142\/S0219025798000053","article-title":"A new class of white noise generalized functions","volume":"1","author":"Cochran","year":"1998","journal-title":"Infin. Dimens. Anal. Quantum Probab. Relat. 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