{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:12:10Z","timestamp":1774624330596,"version":"3.50.1"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T00:00:00Z","timestamp":1770595200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T00:00:00Z","timestamp":1770595200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004473","name":"Sakarya University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004473","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In this manuscript, we introduce a new class of proximal contractions, termed a modified proximal contraction, to establish coincidence and best proximity point results supported by illustrative examples. These results are then specialized to recover existing best proximity point theorems and further reduced to well-known common fixed point and fixed point results, thereby highlighting the unifying power and flexibility of the proximal contraction framework. In particular, we show how classical results of Banach and Reich arise naturally as special cases, demonstrating that common fixed point theorems can be reduced to classical fixed point theorems under suitable conditions. As an application, we investigate the fourth-order Euler-Bernoulli beam equation using Green\u2019s function techniques. Several graphical representations of the derived Green\u2019s function, including surface plots, heatmaps, and slice profiles, are provided to visualize symmetry, boundary effects, and maximum midspan deflection, thereby connecting the abstract theory to the physical behavior of beams under point loads. Finally, the existence and uniqueness of solutions to the Euler-Bernoulli boundary value problem are established through the developed fixed-point results.<\/jats:p>","DOI":"10.1007\/s12190-026-02771-6","type":"journal-article","created":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T15:05:42Z","timestamp":1770649542000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A unified approach to coincidence and common fixed point results with application to Euler-Bernoulli beam equation"],"prefix":"10.1007","volume":"72","author":[{"given":"Haroon","family":"Ahmad","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5499-4272","authenticated-orcid":false,"given":"Mudasir","family":"Younis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mahpeyker","family":"\u00d6zt\u00fcrk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lili","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,9]]},"reference":[{"issue":"1","key":"2771_CR1","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1006\/jath.1999.3415","volume":"103","author":"S. Basha","year":"2000","unstructured":"Basha, S., Veeramani, P.: Best proximity pair theorems for multi-functions with open fibres. J. Approximation Theory 103(1), 119\u2013129 (2000)","journal-title":"J. Approximation Theory"},{"key":"2771_CR2","first-page":"289","volume":"63","author":"S. Basha","year":"1997","unstructured":"Basha, S., Veeramani, P.: Best approximations and best proximity pairs. Acta Scientiarum Mathematicarum (Szeged) 63, 289\u2013300 (1997)","journal-title":"Acta Scientiarum Mathematicarum (Szeged)"},{"issue":"2","key":"2771_CR3","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.1016\/j.jmaa.2005.10.081","volume":"323","author":"A.A. Eldred","year":"2006","unstructured":"Eldred, A.A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323(2), 1001\u20131006 (2006)","journal-title":"J. Math. Anal. Appl."},{"issue":"12","key":"2771_CR4","doi-asserted-by":"publisher","first-page":"3790","DOI":"10.1016\/j.na.2007.10.014","volume":"69","author":"C. Di Bari","year":"2008","unstructured":"Di Bari, C., Suzuki, T., Vetro, C.: Best proximity points for cyclic meir\u2013Keeler contractions. Nonlinear Anal. 69(12), 3790\u20133794 (2008)","journal-title":"Nonlinear Anal."},{"key":"2771_CR5","doi-asserted-by":"crossref","unstructured":"Kikkawa, M., Suzuki, T.: Some similarity between contractions and Kannan mappings. Fixed Point Theory Appl. (2008), 2008, Article ID 649749","DOI":"10.1155\/2008\/649749"},{"issue":"1","key":"2771_CR6","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1186\/s13661-025-02054-2","volume":"2025","author":"G. Mani","year":"2025","unstructured":"Mani, G., Ganesh, P., Kannan, K.: A novel approach to Caputo\u2013hadamard pantograph problems in LP-spaces. Boundary Value Probl. 2025(1), 61 (2025). https:\/\/doi.org\/10.1186\/s13661-025-02054-2","journal-title":"Boundary Value Probl."},{"issue":"18","key":"2771_CR7","doi-asserted-by":"publisher","first-page":"2942","DOI":"10.1016\/j.topol.2009.01.017","volume":"156","author":"J. Anuradha","year":"2009","unstructured":"Anuradha, J., Veeramani, P.: Proximal pointwise contraction. Topol. Its Appl. 156(18), 2942\u20132948 (2009)","journal-title":"Topol. Its Appl."},{"issue":"7\u20138","key":"2771_CR8","doi-asserted-by":"publisher","first-page":"2918","DOI":"10.1016\/j.na.2009.01.173","volume":"71","author":"T. Suzuki","year":"2009","unstructured":"Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71(7\u20138), 2918\u20132926 (2009)","journal-title":"Nonlinear Anal."},{"issue":"18","key":"2771_CR9","doi-asserted-by":"publisher","first-page":"7261","DOI":"10.1016\/j.na.2011.07.043","volume":"74","author":"A. Abkar","year":"2011","unstructured":"Abkar, A., Gabeleh, M.: Best proximity points for asymptotic cyclic contraction mappings. Nonlinear Anal. 74(18), 7261\u20137268 (2011)","journal-title":"Nonlinear Anal."},{"issue":"1","key":"2771_CR10","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s10898-009-9521-0","volume":"49","author":"S. Basha","year":"2011","unstructured":"Basha, S.: Best proximity points for global optimal approximate solutions. J. Global Optim. 49(1), 15\u201321 (2011)","journal-title":"J. Global Optim."},{"issue":"1","key":"2771_CR11","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1186\/1687-1812-2012-42","volume":"2012","author":"S. Basha","year":"2012","unstructured":"Basha, S., Shahzad, N.: Best proximity point theorems for generalized proximal contractions. Fixed Point Theory Appl. 2012(1), 42 (2012)","journal-title":"Fixed Point Theory Appl."},{"issue":"4","key":"2771_CR12","doi-asserted-by":"publisher","first-page":"2154","DOI":"10.1016\/j.na.2011.10.014","volume":"75","author":"B. Samet","year":"2012","unstructured":"Samet, B., Vetro, C., Vetro, P.: Fixed point theorem for \u03b1\u2013\u03c8\u2013contractive type mappings. Nonlinear Anal. 75(4), 2154\u20132165 (2012)","journal-title":"Nonlinear Anal."},{"key":"2771_CR13","doi-asserted-by":"crossref","unstructured":"Jleli, M., Karapinar, E., Samet, B.: Best proximity points for generalized \u03b1\u2013\u03c8\u2013proximal contractive type mappings. J. Appl. Math. (2013, 2013, Article ID 245948","DOI":"10.1155\/2013\/150970"},{"issue":"1","key":"2771_CR14","doi-asserted-by":"publisher","first-page":"2522507","DOI":"10.1080\/27684830.2025.2522507","volume":"12","author":"S.D. Rajkumar","year":"2025","unstructured":"Rajkumar, S.D., Thabet, S.T., Kedim, I.: New fixed point theorems in orthogonal-multiplicative metric spaces and applications to multiplicative and fractional integral equations. Res. Math. 12(1), 2522507 (2025)","journal-title":"Res. Math."},{"issue":"1","key":"2771_CR15","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1186\/s13663-025-00792-7","volume":"2025","author":"B.W. Samuel","year":"2025","unstructured":"Samuel, B.W., Mani, G., Ramulu, S., Thabet, S.T., Kedim, I.: Integral-type contraction on orthogonal S-metric spaces with common fixed-point results and applications to fractional integral equations. Fixed Point Theor. Alg. Sci. Eng. 2025(1), 10 (2025)","journal-title":"Fixed Point Theory Algoritms Sci. Eng."},{"key":"2771_CR16","doi-asserted-by":"publisher","first-page":"98","DOI":"10.22436\/jmcs.018.01.11","volume":"18","author":"S. Komal","year":"2018","unstructured":"Komal, S., Hussain, A., Sultana, N., Kumam, P.: Coincidence best proximity points for Geraghty type proximal cyclic contractions. J. Math. Comput. Sci. 18, 98\u2013114 (2018)","journal-title":"J. Math. Comput. Sci."},{"issue":"5","key":"2771_CR17","doi-asserted-by":"publisher","first-page":"2448","DOI":"10.22436\/jnsa.009.05.44","volume":"9","author":"A. Latif","year":"2016","unstructured":"Latif, A., Abbas, M., Hussain, A.: Coincidence best proximity point of Fg\u2013weak contractive mappings in partially ordered metric space. J. Nonlinear Sci. Appl. 9(5), 2448\u20132457 (2016)","journal-title":"J. Nonlinear Sci. Appl."},{"issue":"1","key":"2771_CR18","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1186\/1687-1812-2012-94","volume":"2012","author":"D. Wardowski","year":"2012","unstructured":"Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012(1), 94 (2012)","journal-title":"Fixed Point Theory Appl."},{"issue":"7\u20138","key":"2771_CR19","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1081\/NFA-120026380","volume":"24","author":"W.A. Kirk","year":"2003","unstructured":"Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24(7\u20138), 851\u2013862 (2003)","journal-title":"Numer. Funct. Anal. Optim."},{"issue":"14","key":"2771_CR20","doi-asserted-by":"publisher","first-page":"4804","DOI":"10.1016\/j.na.2011.04.052","volume":"74","author":"V. Sankar Raj","year":"2011","unstructured":"Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74(14), 4804\u20134808 (2011)","journal-title":"Nonlinear Anal."},{"issue":"7\u20138","key":"2771_CR21","first-page":"2918","volume":"71","author":"T. Suzuki","year":"2009","unstructured":"Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with the property UC. Nonlinear analysis: theory. Methods Appl. 71(7\u20138), 2918\u20132926 (2009)","journal-title":"Methods Appl."},{"issue":"S10","key":"2771_CR22","first-page":"110","volume":"6","author":"C. Vetro","year":"2010","unstructured":"Vetro, C., Suzuki, T.: Three existence theorems for weak contractions of Matkowski type. Int. J. Math. Stat. 6(S10), 110\u2013120 (2010)","journal-title":"Int. J. Math. Stat."},{"key":"2771_CR23","doi-asserted-by":"crossref","unstructured":"Saleem, N., Ahmad, H., Aydi, H., Gaba, Y.U.: On some coincidence best proximity point results. J. Math. (2021), 2021, Article ID 6642750","DOI":"10.1155\/2021\/8005469"},{"issue":"22","key":"2771_CR24","doi-asserted-by":"publisher","first-page":"3488","DOI":"10.3390\/math12223488","volume":"12","author":"M. Zahid","year":"2024","unstructured":"Zahid, M., Din, F.U., Younis, M., Ahmad, H., \u00d6zt\u00fcrk, M.: Some results on multivalued proximal contractions with application to integral equation. Mathematics 12(22), 3488 (2024)","journal-title":"Mathematics"},{"issue":"1","key":"2771_CR25","doi-asserted-by":"publisher","first-page":"298","DOI":"10.11948\/20230213","volume":"14","author":"M. Younis","year":"2024","unstructured":"Younis, M., Ahmad, H., Shahid, W.: Best proximity points for multivalued mappings and equation of motion. J. Appl. Anal. Computation 14(1), 298\u2013316 (2024)","journal-title":"J. Appl. Anal. Computation"},{"issue":"12","key":"2771_CR26","doi-asserted-by":"publisher","first-page":"881","DOI":"10.3390\/axioms13120881","volume":"13","author":"M. Zahid","year":"2024","unstructured":"Zahid, M., Din, F.U., Younis, M., Ahmad, H., \u00d6zt\u00fcrk, M.: Mathematical modeling of fractals via proximal F-iterated function systems. Axioms 13(12), 881 (2024)","journal-title":"Axioms"},{"key":"2771_CR27","doi-asserted-by":"crossref","unstructured":"Alam, K.H., Rohen, Y., Singh, S.S., Devi, K.M., Bishwakumar, L.: New best proximity point results for different types of nonself proximal contractions with an application. Nonlinear Funct. Anal. Appl. 581\u2013596 (2024)","DOI":"10.21203\/rs.3.rs-2925675\/v1"},{"issue":"9","key":"2771_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11227-025-07562-7","volume":"81","author":"K.H. Alam","year":"2025","unstructured":"Alam, K.H., Rohen, Y., Tomar, A.: Approximating the solutions of fractional differential equations with a novel and more efficient iteration procedure. J. Supercomput. 81(9), 1\u201322 (2025)","journal-title":"J. Supercomput."},{"key":"2771_CR29","doi-asserted-by":"crossref","unstructured":"Alam, K.H., Rohen, Y., Singh, S.S.: Analysis of a refined iterative method with a new setting and applications to various models of partial differential equations. Numer. Algoritms 1\u201326 (2025)","DOI":"10.1007\/s11075-025-02159-w"},{"issue":"1","key":"2771_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF03018603","volume":"22","author":"M. Fr\u00e9chet","year":"1906","unstructured":"Fr\u00e9chet, M.: Sur quelques points du calcul fonctionnel. Rend. del Circ. Mat. di Palermo. 22(1), 1\u201372 (1906)","journal-title":"Rend. del Circ. Mat. di Palermo."},{"issue":"11","key":"2771_CR31","doi-asserted-by":"publisher","first-page":"1772","DOI":"10.1016\/j.jat.2011.06.012","volume":"163","author":"S.S. Basha","year":"2011","unstructured":"Basha, S.S.: Best proximity point theorems. J. Approximation Theory 163(11), 1772\u20131781 (2011)","journal-title":"J. Approximation Theory"},{"issue":"1","key":"2771_CR32","doi-asserted-by":"publisher","first-page":"121","DOI":"10.4153\/CMB-1971-024-9","volume":"14","author":"S. Reich","year":"1971","unstructured":"Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14(1), 121\u2013124 (1971)","journal-title":"Can. Math. Bull."},{"issue":"1","key":"2771_CR33","doi-asserted-by":"publisher","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","volume":"3","author":"S. Banach","year":"1922","unstructured":"Banach, S.: Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales. Fundamenta Mathematicae 3(1), 133\u2013181 (1922)","journal-title":"Fundamenta Mathematicae"},{"issue":"3","key":"2771_CR34","doi-asserted-by":"publisher","first-page":"66","DOI":"10.3390\/computation13030066","volume":"13","author":"A. Mannan","year":"2025","unstructured":"Mannan, A., Ul Rahman, J., Iqbal, Q., Zulfiqar, R.: Design of periodic neural networks for computational investigations of nonlinear hepatitis C virus model under boozing. Computation 13(3), 66 (2025)","journal-title":"Computation"},{"issue":"1","key":"2771_CR35","doi-asserted-by":"publisher","first-page":"14","DOI":"10.3390\/computers14010014","volume":"14","author":"A.H. Ali","year":"2025","unstructured":"Ali, A.H., Amir, M., Rahman, J.U., Raza, A., Arif, G.E.: Design of Morlet wavelet neural networks for solving the nonlinear Van der pol\u2013mathieu\u2013duffing oscillator model. Computers 14(1), 14 (2025)","journal-title":"Computers"},{"issue":"2","key":"2771_CR36","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/s40808-024-02256-0","volume":"11","author":"A. Mannan","year":"2025","unstructured":"Mannan, A., Shoket, N., Ul Rahman, J., Uwitije, R.: Dynamic analysis of ebola virus disease with non-linear incidence rate using morlet wavelet neural networks and hybrid optimization techniques. Modeling Earth Syst. Environ. 11(2), 79 (2025)","journal-title":"Modeling Earth Syst. Environ."},{"key":"2771_CR37","doi-asserted-by":"crossref","unstructured":"Amir, M., Rahman, J.U., Ali, A.H., Raza, A., Abduljabbar, Z.A., Neamah, H.A.: Deep learning-based technique for investigating the behavior of MEMS systems with multiwalled carbon nanotubes and electrically actuated microbeams. MethodsX 103337 (2025)","DOI":"10.1016\/j.mex.2025.103337"},{"issue":"23","key":"2771_CR38","doi-asserted-by":"publisher","first-page":"3716","DOI":"10.3390\/math12233716","volume":"12","author":"H. Ahmad","year":"2024","unstructured":"Ahmad, H., Younis, M., G\u00fcndo\u011fdu, H., Barley, N., Patel, V.K.: Proximal contractions for multivalued mappings with an application to 2D Volterra integral equations. Mathematics 12(23), 3716 (2024)","journal-title":"Mathematics"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02771-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-026-02771-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02771-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T14:07:41Z","timestamp":1774620461000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-026-02771-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,9]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2771"],"URL":"https:\/\/doi.org\/10.1007\/s12190-026-02771-6","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,9]]},"assertion":[{"value":"22 August 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2026","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 research complies with all relevant ethical guidelines regarding the use of human and animal subjects, as none were involved.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The author declares no competing interests that may have influenced the research or its presentation.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}}],"article-number":"124"}}