Uniqueness of solutions of boundary value problems at resonance

In this paper, the method of upper and lower solutions is employed to obtain uniqueness of solutions for a boundary value problem at resonance. The shift method is applied to show the existence of solutions. A monotone iteration scheme is developed and sequences of approximate solutions are constructed that converge monotonically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

Keywords:

boundary value problem at resonance; shift method; upper and lower solutions; monotone convergence,

PDF

___

R.P. Agarwal, B. Ahmad and A. Alsaedi, Method of quasilinearization for a nonlocal singular boundary value problem in weighted spaces, Bound. Value Probl. 2013, 2013:261, 17 pp.
K. Alanazi, M. Alshammari and P. Eloe, Quasilinearization and boundary value problems at resonance, Georgian Math. J., in press.
E. Akin-Bohner and F.M. Atici, A quasilinearization approach for two-point nonlinear boundary value problems on time scales, Rocky Mountain J. Math. 35 (2005), no. 1, 19--45.
S. Al Mosa and P. Eloe, Upper and lower solution method for boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equ. 2016: Paper No. 40, 13 pp.
R. Bellman, Methods of Nonlinear Analysis, Vol II, Academic Press, New York, 1973.
R. Bellman and R. Kalba , Quasilinearization and Nonlinear Boundary Value Problems, Elsevier, New York, 1965.
A. Cabada, Green's Functions in the Theory of Ordinary Differential Equations, SpringerBriefs in Mathematics. Springer, New York, 2014.
A. Cabada, P. Habvets and S. Lois, Monotone method for the Neumann problem with lower and upper solutions in the reverse order, Appl. Math. Comput. 117 (2001), no. 1, 1--14.
A. Cabada and L. Sanchez, A positive operator approach to the Neumann problem for a second order ordinary differential equation, J. Math. Anal. Appl. 204 (1996), no. 3, 774--785.
M. Cherpion, C. De Coster and P. Habets, A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions, Appl. Math. Comput. 123 (2001), no.1, 75--91.
L. Collatz, Functional Analysis and Numerical Mathematics, Functional Analysis and Numerical mathematics. Translated fro the German by Hansj\"{o}rg Oser, Academic Press, New York-London, 1966.
J. Ehme, P. Eloe and J. Henderson, Upper and lower solution methods for fully nonlinear boundary value problems, J. Differential Equations 180 (2002), no. 1, 51--64.
P. Eloe, and Y. Gao, The method of quasilinearization and a three-point boundary value problem, J. Korean Math. Soc. 39 (2002), no. 2, 319--330.
P. Eloe and L. Grimm, Monotone iteration and Green's functions for boundary value problems, Proc. Amer. Math. Soc. 78 (1980), no. 4, 533--538.
P. Eloe and Y. Zhang, A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations, Nonlinear Anal. 33 (1998), no.5, 443--453.
P. Hartman, Ordinary Differential Equations, 2nd ed. Birkh\"{a}user, Boston, 1982.
G. Infante, P. Pietramala and F.A.F. Tojo, Nontrivial solutions of local and nonlocal Neumann boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), no. 2, 337--369.
L.K. Jackson, Boundary value problems for ordinary differential equations, in: J.K. Hale (Ed.), Studies in Ordinary Differential Equations, MAA Studies in Mathematics, vol.14 Mathematical Association of America Washington DC, 1977, pp. 93--127.
R.A. Khan, R.R. Lopez, Existence and approximation of solutions of second-order nonlinear four point boundary value problems, Nonlinear Anal. 63 (2005), no. 8, 1094--1115.
V. Lakshmikantham, S. Leela and F.A. McRae, Improved generalized quasilinearization method, Nonlinear Anal. 24 (1995), no. 11, 1627--1637.
V. Lakshmikantham, S. Leela and S. Sivasundaram, Extensions of the methods of quasilinearization, J. Optim. Theory Appl. 87 (1995), no. 2, 379--401.
V. Lakshmikantham, N. Shahzad and J. J. Nieto, Methods of generalized quasilinearization for periodic boundary value problems, Nonlinear Anal. 27 (1996), no. 2, 143--151.
J. Nieto, Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2599-2604.
J. T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.
V. \u{S}eda, Two remarks on boundary value problems for ordinary differential equations, J. Differential Equations 26 (1977), 278--290.
N. Shahzad and A.S Vatsala, Improved generalized quasilinearization method for second order boundary value problems, Dynam. Systems Appl. 4 (1995), no. 1, 79--85.
N. Sveikate, Resonant problems by quasilinearization, International Journal of Differential Equations, 2014: Art. ID 564914, 8 pp.
I. Yermachenko and F. Sadyrbaev, Quasilinearization and multiple solutions of the Emden-Fowler type equations, Math. Model. Anal. 10 (1): 41--50, 2005.