Spectrum of a linear fourth-order differential operator and its applications
In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator u(4)+Mu coupled with the clamped beam boundary conditions (1.2). We also study the positivity and the spectrum structure of the more general oper...
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| Veröffentlicht in: | Mathematische Nachrichten Jg. 286; H. 17-18; S. 1805 - 1819 |
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| Abstract | In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator u(4)+Mu coupled with the clamped beam boundary conditions (1.2). We also study the positivity and the spectrum structure of the more general operator u(4)+p(t)u coupled with (1.2). As the applications of our results on positivity and spectrum of fourth‐order linear differential operators, we show the existence of nodal solutions for the corresponding nonlinear problems via Rabinowitz's global bifurcation theorem. |
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| AbstractList | In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator u ( 4 ) + M u coupled with the clamped beam boundary conditions (1.2). We also study the positivity and the spectrum structure of the more general operator u ( 4 ) + p ( t ) u coupled with (1.2). As the applications of our results on positivity and spectrum of fourth-order linear differential operators, we show the existence of nodal solutions for the corresponding nonlinear problems via Rabinowitz's global bifurcation theorem. [PUBLICATION ABSTRACT] In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator coupled with the clamped beam boundary conditions (1.2). We also study the positivity and the spectrum structure of the more general operator coupled with (1.2). As the applications of our results on positivity and spectrum of fourth‐order linear differential operators, we show the existence of nodal solutions for the corresponding nonlinear problems via Rabinowitz's global bifurcation theorem. In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator u(4)+Mu coupled with the clamped beam boundary conditions (1.2). We also study the positivity and the spectrum structure of the more general operator u(4)+p(t)u coupled with (1.2). As the applications of our results on positivity and spectrum of fourth‐order linear differential operators, we show the existence of nodal solutions for the corresponding nonlinear problems via Rabinowitz's global bifurcation theorem. |
| Author | Elsanosi, Mohammed Ma, Ruyun Wang, Haiyan |
| Author_xml | – sequence: 1 givenname: Ruyun surname: Ma fullname: Ma, Ruyun email: mary@nwnu.edu.cn organization: Department of Mathematics, Northwest Normal University, 730070, Lanzhou, P. R. China – sequence: 2 givenname: Haiyan surname: Wang fullname: Wang, Haiyan organization: Division of Mathematical and Natural Sciences, Arizona State University, AZ85069-7100, Phoenix, USA – sequence: 3 givenname: Mohammed surname: Elsanosi fullname: Elsanosi, Mohammed organization: Department of Mathematics, Northwest Normal University, 730070, Lanzhou, P. R. China |
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| Cites_doi | 10.1080/00036819408840280 10.1016/S0022-247X(02)00071-9 10.1016/j.jmaa.2005.03.078 10.1016/j.amc.2004.10.014 10.1016/j.jmaa.2006.04.021 10.1016/0022-1236(71)90030-9 10.1016/j.jmaa.2005.06.045 10.12775/TMNA.2002.016 10.1016/0377-0427(84)90058-X 10.1016/j.na.2011.01.027 10.1080/00036819508840401 10.1016/j.physd.2004.01.043 10.1016/S0764-4442(00)01629-3 10.1017/S0308210500003140 10.1016/0022-0396(78)90039-6 10.1016/S0022-0396(02)00146-8 10.1016/j.camwa.2007.01.018 10.1016/j.na.2009.01.045 10.1017/S0308210506001041 10.1016/j.na.2009.06.061 10.1112/jlms/jdn066 10.1080/00036819008839930 10.1016/S0893-9659(04)90037-7 10.1007/978-94-017-2517-0 10.1016/j.jde.2004.03.036 10.4171/PM/1786 10.1112/blms/bdn105 |
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| References_xml | – reference: B. P. Rynne, Infinitely many solutions of superlinear fourth order boundary value problems, Topol. Methods Nonlinear Anal. 19(2), 303-312 (2002). – reference: S. Tersian and J. Chaparova, Periodic and homoclinic solutions of extended, Fisher-Kolmogorov equations, C. R. Math. Acad. Sci. Paris 331, 287-292 (2000). – reference: J. R. L. Webb and G. Infante, Non-local boundary value problems of arbitrary order, J. Lond. Math. Soc. 79, 238-258 (2009). – reference: U. Elias, Eigenvalue problems for the equations Ly+p(x)y=0, J. Differ. Equations 29(1), 28-57 (1978). – reference: P. Korman, Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems, Proc. R. Soc. Edinb., Sect. A 134(1), 179-190 (2004). – reference: R. Ma and H. Wu, Positive solutions of a fourth-order two-point boundary value problem, Acta Math. Sci. A 22(2), 244-249 (2002). – reference: B. P. Rynne, Solution curves of 2m-th order boundary-value problems, Electron. J. Differ. Equ. 32, 1-16 (2004). – reference: U. Elias, Oscillation Theory of Two-Term Differential Equations, Mathematics and Its Applications Vol. 396 (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997). – reference: W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics Vol. 220 (Springer-Verlag, Berlin-New York, 1971). – reference: C. De Coster and L. Sanchez, Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth order O. D. E., Riv. Mat. Pura Appl. 14, 57-82 (1994). – reference: R. Ma and J. Xu, Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem, Nonlinear Anal., Theory Methods Appl. 72(1), 113-122 (2010). – reference: A. Cabada and R. R. Enguiça, Positive solutions of fourth order problems with clamped beam boundary conditions, Nonlinear Anal., Theory Methods Appl. 74, 3112-3122 (2011). – reference: R. P. Agarwal and Y. M. Chow, Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math. 10(2), 203-217 (1984). – reference: P. Habets and L. Sanchez, A monotone method for fourth order boundary value problems involving a factorizable linear operator, Port. Math. 64, 255-279 (2007). – reference: L. A. Peletier and J. A. Rodrígues, Homoclinic orbits to a saddle-center in a fourth-order differential equation, J. Differ. Equations 203, 185-215 (2004). – reference: J. Xu and X. Han, Nodal solutions for a class of fourth-order two-point boundary value problems, Bound. Value Probl. 2010, Art. ID 570932, 11 pp. – reference: G. E. Hernandez and R. Manasevich, Existence and multiplicity of solutions of a fourth order equation, Appl. Anal. 54, 237-250 (1994). – reference: R. Ma, Nodal solutions of boundary value problems of fourth-order ordinary differential equations, J. Math. Anal. Appl. 319(2), 424-434 (2006). – reference: Z. Yang, Existence and uniqueness of positive solutions for higher order boundary value problem, Comput. Math. Appl. 54(2),220-228 (2007). – reference: J. Schröder, Operator Inequalities, Mathematics in Science and Engineering Vol. 147 (Academic Press, Inc., New York-London, 1980). – reference: J. A. Cid, D. Franco, and F. Minhós, Positive fixed points and fourth-order equations, Bull. Lond. Math. Soc. 41, 72-78 (2009). – reference: J. R. L. Webb, G. Infante, and D. Franco, Positive solutions of nonlinear fourth-order boundary value problems with local and non-local boundary conditions, Proc. R. Soc. Edinb., Sect. A 138(2), 427-446 (2008). – reference: B. P. Rynne, Global bifurcation for 2mth-order boundary value problems and infinitely many solutions of superlinear problems, J. Differ. Equations 188, 461-472 (2003). – reference: L. A. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D 194, 95-126 (2004). – reference: R. Ma and H. Wang, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal. 59, 225-231 (1995). – reference: X. Liu and W. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. Math. Anal. Appl. 327, 362-375 (2007). – reference: Q. Yao, Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. Math. Lett. 17(2), 237-243 (2004). – reference: A. Cabada and J. A. Cid, Existence of a non-zero fixed point for nondecreasing operators proved via Krasnoselskii's fixed point theorem, Nonlinear Anal., Theory Methods Appl. 71, 2114-2118 (2009). – reference: P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7(3), 487-513 (1971). – reference: R. Ma, Existence of positive solutions of a fourth-order boundary value problem, Appl. Math. Comput. 168, 1219-1231 (2005). – reference: Z. Bai and H. Wang, On positive solutions of some nonlinear fourth order beam equations, J. Math. Anal. Appl. 270, 357-368 (2002). – reference: J. Chu and D. O'Regan, Positive solutions for regular and singular fourth-order boundary value problems, Commun. Appl. Anal. 10, 185-199 (2006). – reference: C. P. Gupta, Existence and uniqueness theorems for some fourth order fully quasilinear boundary value problems, Appl. Anal. 36, 157-169 (1990). – reference: R. Ma, Nodal solutions for a fourth-order two-point boundary value problem, J. Math. Anal. Appl. 314(1), 254-265 (2006). – volume: 270 start-page: 357 year: 2002 end-page: 368 article-title: On positive solutions of some nonlinear fourth order beam equations publication-title: J. Math. Anal. 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| Snippet | In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator u(4)+Mu... In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator coupled... In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator u ( 4 ) +... |
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| SubjectTerms | 34B10 34B18 Clamped beam disconjugate Finite element analysis fourth-order equations global bifurcation nodal solutions |
| Title | Spectrum of a linear fourth-order differential operator and its applications |
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