Existence of Homoclinic Solutions for Nonlinear Second-Order Problems
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In this work, we consider the second-order discontinuous equation in the real line,$$u^{\prime \prime}(t)-ku(t) = f( t, u(t), u^{\prime}(t)), \quad a.e.t \in \mathbb {R},$$with \({k > 0}\) and \({f : \mathbb{R}^{3} \rightarrow \mathbb{R}}\) an \({L^{1}}\)-Carathéodory function. The existence of homoclinic solutions in presence of not necessarily ordered lower and upper solutions is proved, without periodicity assumptions or asymptotic conditions. Some applications to Duffing-like equations are presented in last section.KeywordsHomoclinic solutionsproblems in the real lineGreen’s functionsupper and lower solutionsfixed point theoryMathematics Subject Classification34A3434B1534B40References1.Agarwal, R.P., O’Regan, D.: Infinite Interval Problems for Differential Difference and Integral Equations. Kluwer Academic Publisher, Glasgow (2001)2.Alessio, F., Caldiroli, P., Montecchiari, P.: On the existence of Homoclinic orbits for the asymptotically periodic Duffing equation. Topological methods in nonlinear analysis. J. Juliusz Schauder Center 12, 275–292 (1998)3.Alves C., Carrio P., Faria L.: Existence of homoclinic solutions for a class of second order ordinary differential equations. Nonlinear Anal. Real World Appl. 12, 2416–2428 (2011)MathSciNetCrossRefMATHGoogle Scholar4.Aronson D., Weinberger H.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)MathSciNetCrossRefMATHGoogle Scholar5.Avramescu, C.: Sur l’existence des solutions convergentes des systmes d’quations diffrentielles ordinaires, Ann. Mat. Purra ed Appl. LXXXI(IV), pp. 147–168 (1969)6.Avramescu, C., Vladimirescu, C.: Existence of solutions to second order ordinary differential equations having finite limits at \({\pm \infty}\). Electron. J. Differ. Equ. 2004(18), 1–12 (2004)7.Bonheure D., Torres P.: Bounded and homoclinic-like solutions of a second-order singular differential equation. Bull. London Math. Soc. 44, 47–54 (2012)MathSciNetCrossRefMATHGoogle Scholar8.Champneys A., Lord G.: Computation of homoclinic solutions to periodic orbits, in a reduced water-wave problem. Physica D Nonlinear Phenom. 102, 101–124 (1997)MathSciNetCrossRefMATHGoogle Scholar9.Corduneanu, C.: Integral Equations and Stability of Feedback Systems. Academic Press, New York (1973)10.Ding Y., Lee C.: Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems. Nonlinear Anal. 71, 1395–1413 (2009)MathSciNetCrossRefMATHGoogle Scholar11.Duffing, G.: Erzwungene Schwingungen bei Veriinderlicher Eigenfrequenz. F. Vieweg u. Sohn: Braunschweig (1918)12.Fialho J., Minhós F.: On higher order fully periodic boundary value problems. J. Math. Anal. Appl. 395, 616–625 (2012)MathSciNetCrossRefMATHGoogle Scholar13.Grossinho M.R., Minhós F., Tersian S.: Positive homoclinic solutions for a class of second order differential equations. J. Math. Anal. Appl. 240, 163–173 (1999)MathSciNetCrossRefGoogle Scholar14.Grossinho M.R., Minhós F.: Santos AI a note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition. Nonlinear Anal. 70, 4027–4038 (2009)MathSciNetCrossRefMATHGoogle Scholar15.Guckenheimer J., Holmes P.: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)CrossRefMATHGoogle Scholar16.Kolmogorov A., Petrovsky I., Piskunov N.: Étude de l’équation de la diffusion avec croissance de la quantité de la matire et son application un problme biologique. Moscow Univ. Bull. Math. 1, 1–25 (1937)Google Scholar17.Izydorek M., Janczewska J.: Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differ. Equ. 219, 375–389 (2005)MathSciNetCrossRefMATHGoogle Scholar18.Minhós, F.: Location results: an under used tool in higher order boundary value problems. In: International Conference on Boundary Value Problems: Mathematical Models in Engineering, Biology and Medicine. In: American Institute of Physics Conference Proceedings, vol. 1124, pp. 244–253 (2009)19.Przeradzki B.: The existence of bounded solutions for differential equations in Hilbert spaces. Annales Polonici Mathematici LVI. 2, 103–121 (1992)MathSciNetMATHGoogle Scholar20.Salas A., Castillo J.: Exact solution to duffing equation and the pendulum equation. Appl. Math. Sci. 8(176), 8781–8789(2014)CrossRefGoogle Scholar21.Sun J., Chen H., Nieto J.J.: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373, 20–29 (2011)MathSciNetCrossRefMATHGoogle Scholar22.Tersian S., Chaparova J.: Periodic and homoclinic solutions of extended Fisher–Kolmogorov equations. J. Math. Anal. Appl. 260, 490–506 (2001)MathSciNetCrossRefMATHGoogle Scholar23.Wang J., Xu J., Zhang F.: Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Commun. Pure Appl. Anal. 10, 269–286 (2011)MathSciNetCrossRefMATHGoogle Scholar24.Yan B.Q., O’Regan D., Agarwal R.P.: Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity. J. Comput. Appl. Math. 197, 365–386 (2006)MathSciNetCrossRefMATHGoogle Scholar25.Yang, X., Liu, Y.: Existence of unbounded solutions of boundary value problems for singular differential systems on whole line. Boundary Value Probl. 2015, 42 (2015)26.Zeidler E.: Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems. Springer, New York (1986)CrossRefMATHGoogle Scholar27.Zhang L., Ge W.: Solvability of a Second Order Boundary Value Problem on an Unbounded Domain. Appl. Math. E-Notes 10, 40–46 (2010)MathSciNetMATHGoogle Scholar28.Zhang, Y.: Homoclinic solutions for a forced generalized Linard system. Adv. Differ. Equ. 2012, 94 (2012)29.Zhu C., Zhang W.: Computation of bifurcation manifolds of linearly independent homoclinic orbits. J. Differ. Equ. 245, 1975–1994 (2008)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsFeliz Minhós1Hugo Carrasco2Email authorView author's OrcID profile1.Departamento de MatemáticaEscola de Ciências e Tecnologia, Universidade de ÉvoraÉvoraPortugal2.Centro de Investigação em Matemática e Aplicações (CIMA)Universidade de ÉvoraÉvoraPortugal About this article CrossMark Print ISSN 1660-5446 Online ISSN 1660-5454 Publisher Name Springer International Publishing About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s00009-016-0718-4_Existence of Homoclinic Solutions ", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s00009-016-0718-4_Existence of Homoclinic Solutions ", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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