Abstract
Our concern is the second order difference equation $\Delta^{2} u(t-1)+g(u(t))=h(t)$ subject to the Neumann boundary conditions $\Delta u(0)=\Delta u(T)=0$ . Under convex/concave conditions imposed on g, some results on the exact numbers of solutions and positive solutions are established based on the discussions to the maximum and minimum numbers of (positive) solutions.
Highlights
B ∈ Z with a < b, define [a, b]Z = {a, a +, a +, . . . , b, b}
Our purpose is to find the exact number of solutions and positive solutions of ( . )
The existence and multiplicity of solutions for nonlinear discrete problems subject to various boundary value conditions have been widely studied by using different abstract methods such as critical point theory, fixed point theorems, lower and upper solutions method, and Brower degree
Summary
In these last years, the existence and multiplicity of solutions for nonlinear discrete problems subject to various boundary value conditions have been widely studied by using different abstract methods such as critical point theory, fixed point theorems, lower and upper solutions method, and Brower degree (see, e.g., [ – ] and the references therein). The existence and multiplicity of solutions for nonlinear discrete problems subject to various boundary value conditions have been widely studied by using different abstract methods such as critical point theory, fixed point theorems, lower and upper solutions method, and Brower degree (see, e.g., [ – ] and the references therein) All these results are about the unique solution, or the minimum amount of solutions, and positive solutions.
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