The eigenvalue characterization for the constant sign Green’s functions of \((k,n-k)\) problems

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• 作者：Alberto Cabada ; Lorena Saavedra
• 关键词：34B05 ; 34B08 ; 34B09 ; 34B27 ; 34C10 ; nth order boundary value problem ; Green’s functions ; disconjugation ; maximum principles ; spectral theory
• 刊名：Boundary Value Problems
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,920 KB
• 参考文献：1. De Coster, C, Habets, P: Two-Point Boundary Value Problems: Lower and Upper Solutions. Mathematics in Science and Engineering, vol. 205. Elsevier, Amsterdam (2006)
  2. Ladde, GS, Lakshmikantham, V, Vatsala, AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985) MATH
  3. Cabada, A: Green’s Functions in the Theory of Ordinary Differential Equations. Briefs in Mathematics. Springer, Berlin (2014) CrossRef MATH
  4. Cabada, A: The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. J. Math. Anal. Appl. 185, 302-320 (1994) CrossRef MathSciNet MATH
  5. Krasnosel’skiĭ, MA: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)
  6. Cabada, A, Cid, JA: Existence and multiplicity of solutions for a periodic Hill’s equation with parametric dependence and singularities. Abstr. Appl. Anal. 2011, Article ID 545264 (2011) CrossRef MathSciNet
  7. Graef, JR, Kong, L, Wang, H: A periodic boundary value problem with vanishing Green’s function. Appl. Math. Lett. 21, 176-180 (2008) CrossRef MathSciNet MATH
  8. Graef, JR, Kong, L, Wang, H: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differ. Equ. 245, 1185-1197 (2008) CrossRef MathSciNet MATH
  9. Torres, PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 190(2), 643-662 (2003) CrossRef MATH
  10. Cabada, A, Cid, JA: Existence of a non-zero fixed point for non-decreasing operators via Krasnoselskii’s fixed point theorem. Nonlinear Anal. 71, 2114-2118 (2009) CrossRef MathSciNet MATH
  11. Cabada, A, Cid, JA, Infante, G: New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl. 2013, 125 (2013) CrossRef MathSciNet
  12. Cid, JA, Franco, D, Minhós, F: Positive fixed points and fourth-order equations. Bull. Lond. Math. Soc. 41, 72-78 (2009) CrossRef MathSciNet MATH
  13. Franco, D, Infante, G, Perán, J: A new criterion for the existence of multiple solutions in cones. Proc. R. Soc. Edinb., Sect. A 142, 1043-1050 (2012) CrossRef MATH
  14. Persson, H: A fixed point theorem for monotone functions. Appl. Math. Lett. 19, 1207-1209 (2006) CrossRef MathSciNet MATH
  15. Coppel, WA: Disconjugacy. Lecture Notes in Mathematics, vol. 220. Springer, Berlin (1971) MATH
  16. Zettl, A: A constructive characterization of disconjugacy. Bull. Am. Math. Soc. 81, 145-147 (1975) CrossRef MathSciNet MATH
  17. Zettl, A: A characterization of the factors of ordinary linear differential operators. Bull. Am. Math. Soc. 80, 498-499 (1974) CrossRef MathSciNet MATH
  18. Erbe, L: Hille-Wintner type comparison theorem for selfadjoint fourth order linear differential equations. Proc. Am. Math. Soc. 80(3), 417-422 (1980) CrossRef MathSciNet MATH
  19. Kwong, MK, Zettl, A: Asymptotically constant functions and second order linear oscillation. J. Math. Anal. Appl. 93(2), 475-494 (1983) CrossRef MathSciNet MATH
  20. Simons, W: Some disconjugacy criteria for selfadjoint linear differential equations. J. Math. Anal. Appl. 34, 445-463 (1971) CrossRef MathSciNet MATH
  21. Elias, U: Eventual disconjugacy of \(y^{(n)}+\mu p(x) y=0\) for every μ. Arch. Math. 40(2), 193-200 (2004) MathSciNet MATH
  22. Cabada, A, Cid, JA, Sanchez, L: Positivity and lower and upper solutions for fourth order boundary value problems. Nonlinear Anal. 67, 1599-1612 (2007) CrossRef MathSciNet MATH
  23. Li, H, Feng, Y, Bu, C: Non-conjugate boundary value problem of a third order differential equation. Electron. J. Qual. Theory Differ. Equ. 2015, 21 (2015) CrossRef MathSciNet
  24. Cabada, A, Cid, JA, Máquez-Villamarín, B: Computation of Green’s functions for boundary value problems with Mathematica. Appl. Math. Comput. 219, 1919-1936 (2012) CrossRef MathSciNet MATH
  25. Ma, R, Lu, Y: Disconjugacy and monotone iteration method for third-order equations. Commun. Pure Appl. Anal. 13(3), 1223-1236 (2014) CrossRef MathSciNet MATH
  26. Cabada, A, Enguiça, RR: Positive solutions of fourth order problems with clamped beam boundary conditions. Nonlinear Anal. 74, 3112-3122 (2011) CrossRef MathSciNet MATH
  27. Cabada, A, Fernández-Gómez, C: Constant sign solutions of two-point fourth order problems. Appl. Math. Comput. 263, 122-133 (2015) CrossRef MathSciNet
  28. Cabada, A, Saavedra, L: Disconjugacy characterization by means of spectral \((k,n-k)\) problems. Appl. Math. Lett. 52, 21-29 (2016) CrossRef MathSciNet
  
• 作者单位：Alberto Cabada (1)
  Lorena Saavedra (1)
  
  1. Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia, Spain
  
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770

文摘

This paper is devoted to the study of the sign of the Green’s function related to a general linear nth-order operator, depending on a real parameter, \(T_{n}[M]\), coupled with the \((k,n-k)\) boundary value conditions. If the operator \(T_{n}[\bar{M}]\) is disconjugate for a given M̄, we describe the interval of values on the real parameter M for which the Green’s function has constant sign. One of the extremes of the interval is given by the first eigenvalue of the operator \(T_{n}[\bar{M}]\) satisfying \((k,n-k)\) conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of \((k-1,n-k+1)\) and \((k+1,n-k-1)\) problems. Moreover, if \(n-k\) is even (odd) the Green’s function cannot be nonpositive (nonnegative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green’s functions for particular operators. Our method avoids the necessity of calculating the expression of the Green’s function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator \(T_{n}[\bar{M}]\) for a given M̄ cannot be eliminated. Keywords nth order boundary value problem Green’s functions disconjugation maximum principles spectral theory