Three positive solutions for delay differential equation BVPs with p-Laplacian on infinite interval
Tóm tắt
In this paper we investigate the existence of three positive solutions for delay differential equation boundary value problem with p-Laplacian on infinite interval. By using the fixed point theorem in a cone introduced by Avery and Petereson, the existence of at least three positive solutions is obtained under suitable growth conditions imposed on the nonlinear term. As an application, one example is given to demonstrate our main result. To some extend, our paper complements and generalize the known results.
Tài liệu tham khảo
Agarwal, R.P., O’Regan, D.: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer, Dordrecht (2001)
Agarwal, R.P., O’Regan, D.: An infinite interval problem arising in circularly symmetric deformations of shallow membrane caps. Int. J. Non-Linear Mech. 39, 779–784 (2004)
Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Amsterdam (1999)
Agarwal, R.P., Philos, Ch.G., Tsamatos, P.Ch.: Global solutions of a singular initial value problem to second order nonlinear delay differential equations. Math. Comput. Model. 43, 854–869 (2006)
Ahmad, B., Nieto, J.J.: The monotone iterative technique for three-point second-order integrodifferential boundary value problems with p-Laplacian. Bound. Value Probl. 2007, 57481 (2007) 9 p.
Avery, R.I., Peterson, A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001)
Azbelev, N., Rakhmatullina, L.: Theory of linear abstract functional differential equations and applications. Mem. Differ. Equ. Math. Phys. 8 (1996)
Azbelev, N., Maksimov, V., Rakhmatullina, L.: Introduction to the Theory of Linéar Functional-Differential Equations. Adv. Ser. Math. Sci. Eng., vol. 3. World Federation Publishers Company, Atlanta (1995)
Bai, C., Fang, J.: On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals. J. Math. Anal. Appl. 282, 711–731 (2003)
Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)
Diekmann, O., Van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)
Erbe, L.H., Kong, Q.: Boundary value problems for singular second-order functional-differential equations. J. Comput. Appl. Math. 53, 377–388 (1994)
Feng, H., Ge, W.: Existence of three positive solutions for m-point boundary value problems with one-dimensional p-Laplacian. Nonlinear Anal. 68, 2017–2026 (2008)
Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)
Jiang, D.: Multiple positive solutions for boundary value problems of second-order delay differential equations. Appl. Math. Lett. 15, 575–583 (2002)
Jiang, D., Wei, J.: Existence of positive periodic solutions for nonautonomous delay differential equations. Chin. Ann. Math., Ser. A 20, 715–720 (1999)
Jiang, D., Zhang, L.: Positive solutions for boundary value problems of second order delay differential equations. Acta Math. Sin. 46, 739–746 (2003)
Jiang, D., Xu, X., O’Regan, D., Agarwal, R.P.: Singular positone and semipositone boundary value problems of second order delay differential equations. Czechoslov. Math. J. 55, 483–498 (2005)
Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities: Theory and Applications. Functional, Partial, Abstract, and Complex Differential Equations, Math. Sci. Eng., vol. 55-II. Academic Press, New York (1969)
Lee, J.W., O’Regan, D.: Existence results for differential delay equations, II. Nonlinear Anal. 17, 683–702 (1991)
Lee, J.W., O’Regan, D.: Existence results for differential delay equations, I. J. Differ. Equ. 102, 342–359 (1993)
Lin, X., Xu, X.: Singular semipositive boundary value problems for second-order delay differential equations. Acta Math. Sci., Ser. A Chin. Ed. 25, 496–502 (2005)
Liu, Y.: Global attractivity for a class of delay differential equations with impulses. Math. Appl. (Wuhan) 14, 13–18 (2001)
Liu, Y.Sh: Boundary value problem for second order differential equations on unbounded domain. Acta Anal. Funct. Appl. 4(3), 211–216 (2002) (in Chinese)
Mavridis, K.G., Philos, Ch.G., Tsamatos, P.Ch.: Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. Arch. Math. (Basel) 86, 163–175 (2006)
Shu, X., Xu, Y.: Triple positive solutions for a class of boundary value problems for second-order functional differential equations. Acta Math. Sin. 48, 1113–1120 (2005)
Wang, Y., Ge, W.: Existence of positive solutions to the one-dimensional p-Laplacian equation with delay. Doctoral thesis, Beijing Institute of Technology, Beijing (2006)
Xiong, S.: Stability of an impulsive nonlinear functional differential system with finite delays. Pure Appl. Math. (Xian) 21, 39–45 (2005)
Xu, X.: Multiple positive solutions for singular semi-positone delay differential equation. Electron. J. Differ. Equ. 2005(70), 1–12 (2005)
Zhang, X., Feng, M., Ge, W.: Triple positive solution to the one-dimensional p-Laplacian equation with delay. Port. Math. 65, 143–155 (2008)