Sign properties of Green’s functions for a family of two-point boundary value problems

Eloe, Paul W.; Ridenhour, Jerry

doi:10.1090/S0002-9939-1994-1172948-1

Sign properties of Green’s functions for a family of two-point boundary value problems
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by Paul W. Eloe and Jerry Ridenhour PDF

Proc. Amer. Math. Soc. 120 (1994), 443-452 Request permission

Abstract:

Sign properties of and comparison theorems for Green’s functions of a family of two-point boundary value problems for a general ordinary differential operator are obtained. The results extend known results for a two-term ordinary differential operator.

References

P. W. Bates and G. B. Gustafson, Maximization of Green’s function over classes of multipoint boundary value problems, SIAM J. Math. Anal. 7 (1976), no. 6, 858–871. MR 450653, DOI 10.1137/0507069
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. MR 0460785, DOI 10.1007/BFb0058618
Uri Elias, Green’s functions for a nondisconjugate differential operator, J. Differential Equations 37 (1980), no. 3, 318–350. MR 589996, DOI 10.1016/0022-0396(80)90103-5
P. W. Eloe, Sign properties of Green’s functions for two classes of boundary value problems, Canad. Math. Bull. 30 (1987), no. 1, 28–35. MR 879867, DOI 10.4153/CMB-1987-004-x
Paul W. Eloe, Darrel Hankerson, and Johnny Henderson, Positive solutions and conjugate points for multipoint boundary value problems, J. Differential Equations 95 (1992), no. 1, 20–32. MR 1142275, DOI 10.1016/0022-0396(92)90041-K
Paul W. Eloe, Darrel Hankerson, and Johnny Henderson, Positive solutions and $J$-focal points for two-point boundary value problems, Rocky Mountain J. Math. 22 (1992), no. 4, 1283–1293. MR 1201092, DOI 10.1216/rmjm/1181072655
Paul W. Eloe and Johnny Henderson, Focal points and comparison theorems for a class of two-point boundary value problems, J. Differential Equations 103 (1993), no. 2, 375–386. MR 1221911, DOI 10.1006/jdeq.1993.1055
G. B. Gustafson, A Green’s function convergence principle, with applications to computation and norm estimates, Rocky Mountain J. Math. 6 (1976), no. 3, 457–492. MR 414978, DOI 10.1216/RMJ-1976-6-3-457
A. Ju. Levin, The non-oscillation of solutions of the equation $x^{(n)}+p_{1}(t)x^{(n-1)}+\cdots +p_{n} (t)x=0$, Uspehi Mat. Nauk 24 (1969), no. 2 (146), 43–96 (Russian). MR 0254328
James S. Muldowney, A necessary and sufficient condition for disfocality, Proc. Amer. Math. Soc. 74 (1979), no. 1, 49–55. MR 521872, DOI 10.1090/S0002-9939-1979-0521872-5
James S. Muldowney, On invertibility of linear ordinary differential boundary value problems, SIAM J. Math. Anal. 12 (1981), no. 3, 368–384. MR 613318, DOI 10.1137/0512033
Zeev Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 (1967), 500–516. MR 219781, DOI 10.1090/S0002-9947-1967-0219781-0
Zeev Nehari, Green’s functions and disconjugacy, Arch. Rational Mech. Anal. 62 (1976), no. 1, 53–76. MR 412519, DOI 10.1007/BF00251856
A. C. Peterson, On the sign of the Green’s function beyond the interval of disconjugacy, Rocky Mountain J. Math. 3 (1973), 41–51. MR 326069, DOI 10.1216/RMJ-1973-3-1-41

On the sign of the Green’s functions

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Allan C. Peterson, Green’s functions for focal type boundary value problems, Rocky Mountain J. Math. 9 (1979), no. 4, 721–732. MR 560249, DOI 10.1216/RMJ-1979-9-4-721
Allan C. Peterson, Focal Green’s functions for fourth-order differential equations, J. Math. Anal. Appl. 75 (1980), no. 2, 602–610. MR 581843, DOI 10.1016/0022-247X(80)90104-3
Allan Peterson and Jerry Ridenhour, Comparison theorems for Green’s functions for focal boundary value problems, Recent trends in differential equations, World Sci. Ser. Appl. Anal., vol. 1, World Sci. Publ., River Edge, NJ, 1992, pp. 493–506. MR 1180134

Ordinary differential equations with applications

Jerry Ridenhour, On the sign of Green’s functions for multipoint boundary value problems, Pacific J. Math. 92 (1981), no. 1, 141–150. MR 618052, DOI 10.2140/pjm.1981.92.141
Jerry Ridenhour, Green’s functions and linear boundary value problems, J. Differential Equations 39 (1981), no. 2, 259–268. MR 607785, DOI 10.1016/0022-0396(81)90076-0