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


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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