Previous |  Up |  Next


half–linear differential equation; associated functional; Picone identity; conjugate points
In this paper we study extremal properties of functional associated with the half–linear second order differential equation E$_p$. Necessary and sufficient condition for nonnegativity of this functional is given in two special cases: the first case is when both points are regular and the second is the case, when one end point is singular. The obtained results extend the theory of quadratic functionals.
[1] Bihari, I.: On the second order half-linear differential equation. Studia Sci. Math. Hungar, 3 (1968), 411–437. MR 0267190 | Zbl 0167.37403
[2] Došlá, Z., Došlý, O.: On transformations of singular quadratic functionals corresponding to equation $(py^{\prime })^{\prime }+qy=0$. Arch. Math. (Brno) 24 (1988), 75–82. MR 0983225
[3] Došlá, Z., Došlý, O.: Quadratic funtionals with general boundary conditions. Appl. Math. Opt. 36 (1997), 243–262. MR 1457870
[4] Elbert, Á.: A half-linear second order differential equation. Coll. Math. Soc. János Bolyai 30. Qual. theory of diff. eq. Szeged (Hungary) (1979), 153–179.
[5] Ioffe, A. D., Tihomirov, V. M.: Theory of extremal problems. North-Holland Publ., 1979, pp. 122–124. MR 0528295
[6] Jaroš, J., Kusano, T.: A Picone type identity for second order half-linear differential equations. (to appear). MR 1711081
[7] Kaňovský, J.: Global transformations of linear differential equations and quadratic functionals I,II. Arch. Math. 19 (1983), 161–171. MR 0725199
[8] Leighton, W., Morse, M.: Singular quadratic functionals. Trans. Amer. Math. Soc. 40 (1936), 252–286. MR 1501873
[9] Li, H. J., Yeh, Ch. Ch.: Sturmian comparison theorem for half-linear second order diff. equations. Proc. Roy. Soc. of Edinburg 125 A (1995), 1193–1204. MR 1362999
[10] Reid, W. T.: Ordinary differential equations. John Wiley and Sons, New York, 1971. MR 0273082 | Zbl 0212.10901
Partner of
EuDML logo