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Abstract
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.
Highlights
The method of upper and lower solutions and monotone methods have been useful in the study of boundary value problems for nonlinear ordinary differential equations
The quasilinearization method has been useful in the study of boundary value problems for ordinary differential equations and we cite a number of those applications here [1, 3, 13, 15, 19, 20, 21, 22, 23, 26]
We say β ∈ C2[0, 1] is an upper solution of the boundary value problem (2.1), (2.2) if β(0) = 0, β (0) = β (1) and β (t) ≤ f (t, β(t), β (1)), 0 ≤ t ≤ 1
Summary
The method of upper and lower solutions and monotone methods have been useful in the study of boundary value problems for nonlinear ordinary differential equations. The quasilinearization method has been useful in the study of boundary value problems for ordinary differential equations and we cite a number of those applications here [1, 3, 13, 15, 19, 20, 21, 22, 23, 26] In these works, the monotonicity is obtained rather delicately and the uniqueness of solutions plays a key role in obtaining the monotonicity. The motivation and development here is different than that in [27] or [28], since uniqueness of solutions is a key feature in this work and multiplicity of solutions is key in [27] or [28]
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