{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T16:09:54Z","timestamp":1778602194389,"version":"3.51.4"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T00:00:00Z","timestamp":1693008000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T00:00:00Z","timestamp":1693008000000},"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":["Comp. Appl. Math."],"published-print":{"date-parts":[[2023,9]]},"DOI":"10.1007\/s40314-023-02431-6","type":"journal-article","created":{"date-parts":[[2023,8,26]],"date-time":"2023-08-26T15:02:10Z","timestamp":1693062130000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On extended, and extended rectangular, Menger probabilistic b-metric spaces: applications to the existence of solutions of integral, and fractional differential, equations"],"prefix":"10.1007","volume":"42","author":[{"given":"Reza","family":"Chaharpashlou","sequence":"first","affiliation":[]},{"given":"Ehsan Lotfali","family":"Ghasab","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio M.","family":"Lopes","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,8,26]]},"reference":[{"issue":"2","key":"2431_CR1","first-page":"11","volume":"81","author":"M Asim","year":"2019","unstructured":"Asim M, Imdad M, Radenovic S (2019) Fixed point results in extended rectangular $$b$$-metric spaces with an application. UPB Sci Bull, Ser A 81(2):11\u201320","journal-title":"UPB Sci Bull, Ser A"},{"key":"2431_CR2","first-page":"26","volume":"30","author":"I Bakhtin","year":"1989","unstructured":"Bakhtin I (1989) The contraction mapping in almost metric spaces. Funct Ana Gos Ped Inst Unianowsk 30:26\u201337","journal-title":"Funct Ana Gos Ped Inst Unianowsk"},{"issue":"1","key":"2431_CR3","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","volume":"3","author":"S Banach","year":"1922","unstructured":"Banach S (1922) Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales. Fundam Math 3(1):133\u2013181","journal-title":"Fundam Math"},{"key":"2431_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-020-01164-0","volume":"39","author":"S Banaei","year":"2020","unstructured":"Banaei S, Mursaleen M, Parvaneh V (2020) Some fixed point theorems via measure of noncompactness with applications to differential equations. Comput Appl Math 39:1\u201312","journal-title":"Comput Appl Math"},{"key":"2431_CR5","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF01706080","volume":"6","author":"A Bharucha-Raid","year":"1972","unstructured":"Bharucha-Raid A, Sehgal V (1972) Fixed point of contraction mappings on pm-spaces. Math Syst Theory 6:97\u2013100","journal-title":"Math Syst Theory"},{"key":"2431_CR6","first-page":"31","volume":"57","author":"A Branciari","year":"2000","unstructured":"Branciari A (2000) A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ Math 57:31\u201337","journal-title":"Publ Math"},{"issue":"1","key":"2431_CR7","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1090\/S0002-9939-1972-0290328-3","volume":"33","author":"DK Burke","year":"1972","unstructured":"Burke DK (1972) Cauchy sequences in semimetric spaces. Proc Am Math Soc 33(1):161\u2013164","journal-title":"Proc Am Math Soc"},{"key":"2431_CR8","first-page":"1","volume":"1","author":"R Chaharpashlou","year":"2021","unstructured":"Chaharpashlou R (2021) Best approximation of a nonlinear fractional Volterra integro-differential equation in matrix MB-space. Adv Differ Equ 1:1\u201312","journal-title":"Adv Differ Equ"},{"key":"2431_CR9","first-page":"1","volume":"1","author":"R Chaharpashlou","year":"2020","unstructured":"Chaharpashlou R, O\u2019Regan D, Park C et al (2020) (2020a) $$c^*$$-algebra valued fuzzy normed spaces with application of Hyers\u2013Ulam stability of a random integral equation. Adv Differ Equ 1:1\u20139","journal-title":"Adv Differ Equ"},{"key":"2431_CR10","first-page":"1","volume":"1","author":"R Chaharpashlou","year":"2020","unstructured":"Chaharpashlou R, Saadati R, Atangana A (2020) Ulam\u2013Hyers\u2013Rassias stability for nonlinear $$\\psi $$-Hilfer stochastic fractional differential equation with uncertainty. Adv Differ Equ 1:1\u201310","journal-title":"Adv Differ Equ"},{"issue":"8","key":"2431_CR11","doi-asserted-by":"crossref","first-page":"1379","DOI":"10.1007\/s10114-007-6509-x","volume":"24","author":"BS Choudhury","year":"2008","unstructured":"Choudhury BS, Das K (2008) A new contraction principle in Menger spaces. Acta Math Sin English Ser 24(8):1379\u20131386","journal-title":"Acta Math Sin English Ser"},{"issue":"3\u20134","key":"2431_CR12","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1016\/j.na.2009.10.001","volume":"72","author":"L \u0106iri\u0107","year":"2010","unstructured":"\u0106iri\u0107 L (2010) Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces. Nonlinear Anal Theory Methods Appl 72(3\u20134):2009\u20132018","journal-title":"Nonlinear Anal Theory Methods Appl"},{"issue":"2","key":"2431_CR13","first-page":"267","volume":"45","author":"LB \u0106iri\u0107","year":"1974","unstructured":"\u0106iri\u0107 LB (1974) A generalization of Banach\u2019s contraction principle. Proc Am Math Soc 45(2):267\u2013273","journal-title":"Proc Am Math Soc"},{"issue":"17","key":"2431_CR14","doi-asserted-by":"crossref","first-page":"2838","DOI":"10.1016\/j.topol.2009.08.029","volume":"156","author":"LB \u0106iri\u0107","year":"2009","unstructured":"\u0106iri\u0107 LB, Mihe\u0163 D, Saadati R (2009) Monotone generalized contractions in partially ordered probabilistic metric spaces. Topology and its Applications 156(17):2838\u20132844","journal-title":"Topology and its Applications"},{"issue":"1","key":"2431_CR15","first-page":"5","volume":"1","author":"S Czerwik","year":"1993","unstructured":"Czerwik S (1993) Contraction mappings in $$b$$-metric spaces. Acta Math Inform Univ Ostraviensis 1(1):5\u201311","journal-title":"Acta Math Inform Univ Ostraviensis"},{"issue":"6","key":"2431_CR16","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.22436\/jnsa.008.06.11","volume":"8","author":"R George","year":"2015","unstructured":"George R, Radenovic S, Reshma K et al (2015) Rectangular $$b$$-metric space and contraction principles. J Nonlinear Sci Appl 8(6):1005\u20131013","journal-title":"J Nonlinear Sci Appl"},{"key":"2431_CR17","first-page":"20","volume":"20","author":"EL Ghasab","year":"2020","unstructured":"Ghasab EL, Majani H, Karapinar E et al (2020) New fixed point results in-quasi-metric spaces and an application. Adv Math Phys 20:20","journal-title":"Adv Math Phys"},{"issue":"1","key":"2431_CR18","doi-asserted-by":"crossref","first-page":"70","DOI":"10.3390\/sym15010070","volume":"15","author":"EL Ghasab","year":"2022","unstructured":"Ghasab EL, Chaharpashlou R, Lopes AM (2022) Solving a system of integral equations in rectangular Menger probabilistic metric spaces and rectangular Menger probabilistic $$b$$-metric spaces. Symmetry 15(1):70","journal-title":"Symmetry"},{"key":"2431_CR19","first-page":"955","volume":"232","author":"D Gopal","year":"2014","unstructured":"Gopal D, Abbas M, Vetro C (2014) Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation. Appl Math Comput 232:955\u2013967","journal-title":"Appl Math Comput"},{"key":"2431_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-017-1560-7","volume-title":"Fixed point theory in probabilistic metric spaces","author":"O Had\u017ei\u0107","year":"2001","unstructured":"Had\u017ei\u0107 O, Pap E (2001) Fixed point theory in probabilistic metric spaces, vol 536. Springer, Dordrecht"},{"key":"2431_CR21","first-page":"1","volume":"1","author":"F Hasanvand","year":"2015","unstructured":"Hasanvand F (2015) Some fixed point theorems in Menger PbM-spaces with an application. Fixed Point Theory Appl 1:1\u201318","journal-title":"Fixed Point Theory Appl"},{"issue":"7","key":"2431_CR22","doi-asserted-by":"crossref","first-page":"2199","DOI":"10.1016\/j.na.2010.05.046","volume":"73","author":"J Jachymski","year":"2010","unstructured":"Jachymski J (2010) On probabilistic $$\\varphi $$-contractions on Menger spaces. Nonlinear Anal Theory Methods Appl 73(7):2199\u20132203","journal-title":"Nonlinear Anal Theory Methods Appl"},{"issue":"2","key":"2431_CR23","doi-asserted-by":"crossref","first-page":"19","DOI":"10.3390\/math5020019","volume":"5","author":"T Kamran","year":"2017","unstructured":"Kamran T, Samreen M, UL Ain Q (2017) A generalization of $$b$$-metric space and some fixed point theorems. Mathematics 5(2):19","journal-title":"Mathematics"},{"key":"2431_CR24","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-10927-5","volume-title":"Fixed point theory in distance spaces","author":"W Kirk","year":"2014","unstructured":"Kirk W, Shahzad N (2014) Fixed point theory in distance spaces. Springer, Cham"},{"issue":"1","key":"2431_CR25","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1515\/math-2018-0120","volume":"16","author":"Liu Xl","year":"2018","unstructured":"Xl Liu, Zhou M, Mishra LN et al (2018) Common fixed point theorem of six self-mappings in Menger spaces using (CLR_ST) property. Open Math 16(1):1423\u20131434","journal-title":"Open Math"},{"issue":"2","key":"2431_CR26","first-page":"150","volume":"14","author":"GE Lotfali","year":"2021","unstructured":"Lotfali GE, Hamid M, Soleimani RG (2021) Fixed points of set-valued $$f$$-contraction operators in quasi-ordered metric spaces with an application to integral equations. J Siberian Federal Univ Math Phys 14(2):150\u2013158","journal-title":"J Siberian Federal Univ Math Phys"},{"issue":"1","key":"2431_CR27","doi-asserted-by":"crossref","first-page":"3","DOI":"10.3390\/axioms10010003","volume":"10","author":"E Lotfali Ghasab","year":"2020","unstructured":"Lotfali Ghasab E, Majani H, De la Sen M et al (2020) $$e$$-distance in Menger PGM spaces with an application. Axioms 10(1):3","journal-title":"Axioms"},{"issue":"02","key":"2431_CR28","first-page":"113","volume":"9","author":"E Lotfali Ghasab","year":"2020","unstructured":"Lotfali Ghasab E, Majani H, Soleimani Rad G (2020) Integral type contraction and coupled fixed point theorems in ordered $$g$$-metric spaces. J Linear Topol Algebra (JLTA) 9(02):113\u2013120","journal-title":"J Linear Topol Algebra (JLTA)"},{"issue":"4","key":"2431_CR29","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1023\/A:1011254307524","volume":"14","author":"H Maagli","year":"2001","unstructured":"Maagli H (2001) On the solutions of a singular nonlinear periodic boundary value problem. Potential Anal 14(4):437\u2013447","journal-title":"Potential Anal"},{"issue":"12","key":"2431_CR30","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1073\/pnas.28.12.535","volume":"28","author":"K Menger","year":"1942","unstructured":"Menger K (1942) Statistical metrics. Proc Natl Acad Sci USA 28(12):535\u2013537","journal-title":"Proc Natl Acad Sci USA"},{"issue":"1","key":"2431_CR31","doi-asserted-by":"crossref","first-page":"372","DOI":"10.55730\/1300-0098.3365","volume":"47","author":"M Metwali","year":"2023","unstructured":"Metwali M, Mishra VN (2023) On the measure of noncompactness in $$l_p({\\mathbb{R} }^+)$$ and applications to a product of $$n$$-integral equations. Turk J Math 47(1):372\u2013386","journal-title":"Turk J Math"},{"key":"2431_CR32","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/j.jmaa.2004.09.034","volume":"304","author":"D Mihet","year":"2005","unstructured":"Mihet D (2005) Multivalued generalisations of probabilistic contractions. J Math Anal Appl 304:464\u2013472","journal-title":"J Math Anal Appl"},{"key":"2431_CR33","doi-asserted-by":"crossref","first-page":"17,486","DOI":"10.3934\/math.2022964","volume":"7","author":"LN Mishra","year":"2022","unstructured":"Mishra LN, Pathak VK, Baleanu D (2022) Approximation of solutions for nonlinear functional integral equations. AIMS Math 7:17,486-17,506","journal-title":"AIMS Math"},{"key":"2431_CR34","doi-asserted-by":"crossref","first-page":"701","DOI":"10.5890\/DNC.2023.09.015","volume":"12","author":"V Pathak","year":"2023","unstructured":"Pathak V, Mishra L (2023) Existence of solution of Erd\u00e9lyi\u2013Kober fractional integral equations using measure of non-compactness. Discontin Nonlinearity Complex 12:701\u2013714","journal-title":"Discontin Nonlinearity Complex"},{"issue":"14","key":"2431_CR35","doi-asserted-by":"crossref","first-page":"2400","DOI":"10.3390\/math10142400","volume":"10","author":"VK Pathak","year":"2022","unstructured":"Pathak VK, Mishra LN (2022) Application of fixed point theorem to solvability for non-linear fractional Hadamard functional integral equations. Mathematics 10(14):2400","journal-title":"Mathematics"},{"issue":"12","key":"2431_CR36","doi-asserted-by":"crossref","first-page":"744","DOI":"10.3390\/fractalfract6120744","volume":"6","author":"VK Pathak","year":"2022","unstructured":"Pathak VK, Mishra LN, Mishra VN et al (2022) On the solvability of mixed-type fractional-order non-linear functional integral equations in the Banach space C(I). Fract Fract 6(12):744","journal-title":"Fract Fract"},{"key":"2431_CR37","first-page":"1","volume":"10","author":"VK Pathak","year":"2023","unstructured":"Pathak VK, Mishra LN, Mishra VN (2023) On the solvability of a class of nonlinear functional integral equations involving erd\u00e9lyi-Kober fractional operator. Math Methods Appl Sci 10:1\u201313","journal-title":"Math Methods Appl Sci"},{"issue":"8","key":"2431_CR38","doi-asserted-by":"crossref","first-page":"17,448","DOI":"10.3934\/math.2023891","volume":"8","author":"SK Paul","year":"2023","unstructured":"Paul SK, Mishra LN, Mishra VN et al (2023) An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator. AIMS Math 8(8):17,448-17,469","journal-title":"AIMS Math"},{"issue":"2","key":"2431_CR39","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s11118-021-09911-6","volume":"57","author":"R Precup","year":"2022","unstructured":"Precup R, Rubbioni P (2022) Stationary solutions of Fokker\u2013Planck equations with nonlinear reaction terms in bounded domains. Potential Anal 57(2):181\u2013199","journal-title":"Potential Anal"},{"key":"2431_CR40","volume-title":"Fixed point theory in various spaces: comparison between various contractions","author":"H Rahimi","year":"2013","unstructured":"Rahimi H, Rad GS (2013) Fixed point theory in various spaces: comparison between various contractions. LAP Lambert Academic Publishing, London"},{"issue":"2","key":"2431_CR41","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/BF01301400","volume":"93","author":"IL Reilly","year":"1982","unstructured":"Reilly IL, Subrahmanyam P, Vamanamurthy M (1982) Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte f\u00fcr Mathematik 93(2):127\u2013140","journal-title":"Monatshefte f\u00fcr Mathematik"},{"issue":"3","key":"2431_CR42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11784-018-0594-6","volume":"20","author":"Z Sadeghi","year":"2018","unstructured":"Sadeghi Z, Vaezpour S (2018) Fixed point theorems for multivalued and single-valued contractive mappings on Menger PM spaces with applications. J Fixed Point Theory Appl 20(3):1\u201327","journal-title":"J Fixed Point Theory Appl"},{"issue":"1","key":"2431_CR43","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1007\/s41478-022-00469-z","volume":"31","author":"AG Sanatee","year":"2023","unstructured":"Sanatee AG, Rathour L, Mishra VN et al (2023) Some fixed point theorems in regular modular metric spaces and application to Caratheodory\u2019s type anti-periodic boundary value problem. J Anal 31(1):619\u2013632","journal-title":"J Anal"},{"key":"2431_CR44","volume-title":"Probabilistic metric spaces","author":"B Schweizer","year":"2011","unstructured":"Schweizer B, Sklar A (2011) Probabilistic metric spaces. Courier Corporation, New York"},{"key":"2431_CR45","volume-title":"Introduction to the fractional calculus of variations","author":"DF Torres","year":"2012","unstructured":"Torres DF, Malinowska AB (2012) Introduction to the fractional calculus of variations. World Scientific Publishing Company, Singapore"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02431-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-023-02431-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-023-02431-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T05:17:52Z","timestamp":1694409472000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-023-02431-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,26]]},"references-count":45,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["2431"],"URL":"https:\/\/doi.org\/10.1007\/s40314-023-02431-6","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,26]]},"assertion":[{"value":"24 March 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 August 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 August 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":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The authors declare that they have no conflict of interest.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"295"}}