{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T03:12:33Z","timestamp":1772507553598,"version":"3.50.1"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2011,11,14]],"date-time":"2011-11-14T00:00:00Z","timestamp":1321228800000},"content-version":"unspecified","delay-in-days":6526,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proceedings of the Royal Society of Edinburgh: Section A Mathematics"],"published-print":{"date-parts":[[1994]]},"abstract":"<jats:p>A nonlocal eigenvalue problem of the form <jats:italic>u<\/jats:italic>\u2033 + <jats:italic>a<\/jats:italic>(<jats:italic>x<\/jats:italic>)<jats:italic>u<\/jats:italic> + <jats:italic>Bu<\/jats:italic> = <jats:italic>\u03bbu<\/jats:italic> with homogeneous Dirichlet boundary conditions is considered, where <jats:italic>B<\/jats:italic> is a rank-one bounded linear operator and <jats:italic>x<\/jats:italic> belongs to some bounded interval on the real line. The behaviour of the eigenvalues is studied using methods of linear perturbation theory. In particular, some results are given which ensure that the spectrum remains real. A Sturm-type comparison result is obtained. Finally, these results are applied to the study of some nonlocal reaction\u2013diffusion equations.<\/jats:p>","DOI":"10.1017\/s0308210500029279","type":"journal-article","created":{"date-parts":[[2011,11,14]],"date-time":"2011-11-14T13:10:53Z","timestamp":1321276253000},"page":"169-188","source":"Crossref","is-referenced-by-count":43,"title":["A nonlocal Sturm\u2013Liouville eigenvalue problem"],"prefix":"10.1017","volume":"124","author":[{"given":"Pedro","family":"Freitas","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2011,11,14]]},"reference":[{"key":"S0308210500029279_ref020","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/48.3.249"},{"key":"S0308210500029279_ref018","doi-asserted-by":"publisher","DOI":"10.1007\/BF02022519"},{"key":"S0308210500029279_ref019","doi-asserted-by":"publisher","DOI":"10.1002\/mma.1670100406"},{"key":"S0308210500029279_ref015","volume-title":"Perturbation Theory for Linear Operators","author":"Kato","year":"1980"},{"key":"S0308210500029279_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF00250508"},{"key":"S0308210500029279_ref013","volume-title":"Lectures on Ordinary Differential Equations","author":"Hille","year":"1969"},{"key":"S0308210500029279_ref012","doi-asserted-by":"publisher","DOI":"10.1063\/1.523849"},{"key":"S0308210500029279_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1969-12175-0"},{"key":"S0308210500029279_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(71)90015-2"},{"key":"S0308210500029279_ref004","unstructured":"4 Budd C. , Dold B. and Stewart A. . 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Thermal runaway due to ohmic heating: general results and a one-dimensional problem (preprint)."},{"key":"S0308210500029279_ref005","first-page":"39\u201355","article-title":"A Cauchy problem for an ordinary integro-differential equation","volume":"72","author":"Catchpole","year":"1972","journal-title":"Proc. Roy. Soc. Edinburgh Sect. 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