On the Existence of Multiple Solutions of a Class of Second-order Nonlinear Two-point Boundary Value Problems

Volume 14, Issue 2, pp 97-107 http://dx.doi.org/10.22436/jmcs.014.02.02

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Authors

E. Shivanian - Department of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran F. Abdolrazaghi - Department of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran

Abstract

A general approach is presented for proving existence of multiple solutions of the second-order nonlinear differential equation \[u'' (x) + f (u(x)) = 0,\quad x\in [0,1], \] subject to given boundary conditions: \(u(0) = B_1, u(1) = B_2\) or \(u'(0) = B'_1, u(1)=B_2\). The proof is constructive in nature, and could be used for numerical generation of the solution or closed-form analytical solution by introducing some special functions. The only restriction is about \(f(u)\) , where it is supposed to be differentiable function with continuous derivative. It is proved the problem may admit no solution, may admit unique solution or may admit multiple solutions.

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Keywords

• Closed-form solution
• exact analytical solution
• special function
• unique solution
• multiple solutions.

MSC

•  65L12
•  65L10
•  65N85

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