{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T20:44:16Z","timestamp":1775594656783,"version":"3.50.1"},"reference-count":5,"publisher":"American Mathematical Society (AMS)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"published-print":{"date-parts":[[1990,8]]},"abstract":"<p>\n                    We consider the forced Li\u00e9nard equation\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u plus f left-parenthesis u right-parenthesis u Superscript prime Baseline plus g left-parenthesis t comma u right-parenthesis equals h left-parenthesis t right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u + f(u)u\u2019 + g(t,u) = h(t)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    together with the boundary conditions\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u left-parenthesis 0 right-parenthesis equals u left-parenthesis upper T right-parenthesis comma u prime left-parenthesis 0 right-parenthesis equals u prime left-parenthesis upper T right-parenthesis comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace width=\"1em\"\/>\n                            <mml:msup>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mo>\u2032<\/mml:mo>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u(0) = u(T),\\quad u\u2019(0) = u\u2019(T),<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n                        <mml:semantics>\n                          <mml:mi>g<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is continuous on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper R times left-parenthesis 0 comma plus normal infinity right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"bold\">R<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathbf {R}} \\times (0, + \\infty )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and becomes infinite at\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u = 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We consider classical solutions as well as generalized solutions that can go into the singularity\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>u<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">u = 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The method of approach uses upper and lower solutions and degree theory.\n                  <\/p>","DOI":"10.1090\/s0002-9939-1990-1009991-5","type":"journal-article","created":{"date-parts":[[2010,7,11]],"date-time":"2010-07-11T16:21:06Z","timestamp":1278865266000},"page":"1035-1044","source":"Crossref","is-referenced-by-count":4,"special_numbering":"262","title":["Periodic solutions of some Li\u00e9nard equations with singularities"],"prefix":"10.1090","volume":"109","author":[{"given":"Patrick","family":"Habets","sequence":"first","affiliation":[]},{"given":"Luis","family":"Sanchez","sequence":"additional","affiliation":[]}],"member":"14","reference":[{"key":"1","unstructured":"A. Adje, Sur et sous solutions dans les \u00e9quations diff\u00e9rentielles discontinues avec conditions aux limites non lineaires, These de doctorat, Louvain-la-Neuve, 1987."},{"key":"2","doi-asserted-by":"crossref","unstructured":"A. Bahri and P. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, CMS Technical Summary Report, 88-8, 1987.","DOI":"10.21236\/ADA193478"},{"issue":"6","key":"3","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.57262\/die\/1379101984","article-title":"Periodic solutions of dissipative dynamical systems with singular potentials","volume":"3","author":"Habets, P.","year":"1990","journal-title":"Differential Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0893-4983","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"109","DOI":"10.2307\/2046279","article-title":"On periodic solutions of nonlinear differential equations with singularities","volume":"99","author":"Lazer, A. C.","year":"1987","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"5","series-title":"S\\'{e}minaire de Math\\'{e}matiques Sup\\'{e}rieures [Seminar on Higher Mathematics]","isbn-type":"print","volume-title":"Points fixes, points critiques et probl\\`emes aux limites","volume":"92","author":"Mawhin, Jean","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/2760606961"}],"container-title":["Proceedings of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/proc\/1990-109-04\/S0002-9939-1990-1009991-5\/S0002-9939-1990-1009991-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/proc\/1990-109-04\/S0002-9939-1990-1009991-5\/S0002-9939-1990-1009991-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T19:44:40Z","timestamp":1775591080000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/proc\/1990-109-04\/S0002-9939-1990-1009991-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,8]]},"references-count":5,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1990,8]]}},"alternative-id":["S0002-9939-1990-1009991-5"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9939-1990-1009991-5","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6826","0002-9939"],"issn-type":[{"value":"1088-6826","type":"electronic"},{"value":"0002-9939","type":"print"}],"subject":[],"published":{"date-parts":[[1990,8]]}}}