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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2001 • Question 5

Question

Math 6410 • ODE Fall 2001 • Question 5

Use the implicit function theorem for mappings between Banach spaces to discuss the unique solvability of the scalar boundary value problem \[ u''+2u+u^2=h(t),\qquad u(0)=u(\pi ), \tag {6} \] for continuous functions \(h\) of small maximum norm. Results needed from Sturm-Liouville theory may be assumed but should be quoted.

Raw LaTeX

Use the implicit function theorem for mappings between Banach spaces to discuss the
unique solvability of the scalar boundary value problem
\[
  u''+2u+u^2=h(t),\qquad u(0)=u(\pi),
  \tag{6}
\]
for continuous functions \(h\) of small maximum norm. Results needed from Sturm-Liouville
theory may be assumed but should be quoted.

Topic Tags

Banach and Normed Spaces Boundary Value Problems Boundary Value Problems and Sturm-Liouville Theory Contraction Mapping and Nonlinear Equations