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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2019 • Question 3

Question

Math 6420 • PDE Fall 2019 • Question 3

Consider the nonlinear integral equation \[ u(x)-\lambda \int _a^b w(x,y)\sin (u(y))\,dy=\phi (x), \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on the interval \([a,b]\) and \(w(x,y)\) is continuous on \([a,b]\times [a,b]\). Find a constant \(C>0\) such that \(\lvert \lambda \rvert <C\) implies that the nonlinear integral equation has a unique solution \(u\).

Hint: You may assume the inequality \(\lvert \sin (x)-\sin (y)\rvert \le \lvert x-y\rvert \), for any \(x,y\in \mathbb {R}\).

Raw LaTeX

Consider the nonlinear integral equation
\[
  u(x)-\lambda\int_a^b w(x,y)\sin(u(y))\,dy=\phi(x),
\]
where \(\lambda\in\mathbb{R}\), \(\phi\) is continuous on the interval \([a,b]\) and \(w(x,y)\)
is continuous on \([a,b]\times[a,b]\). Find a constant \(C>0\) such that
\(\lvert\lambda\rvert<C\) implies that the nonlinear integral equation has a unique
solution \(u\).

\textbf{Hint:} You may assume the inequality
\(\lvert\sin(x)-\sin(y)\rvert\le\lvert x-y\rvert\), for any \(x,y\in\mathbb{R}\).

Topic Tags

Cauchy Integral Formula and Integral Theorems Contraction Mapping and Nonlinear Equations Distributions and Weak Solutions Linear Programming and Optimization