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2. Nonlinear Functional Equations with One Independent Variable

2.1. Functional Equations with Quadratic Nonlinearity

  1. y(x + 1) − ay2(x) = f(x).
  2. y(2x) − ay2(x) = 0.
  3. y(2x) − 2y2(x) + a = 0.
  4. y(x)y(a − x) = b2.
  5. y(x)y(a − x) = f 2(x).
  6. y2(x) + y2(a − x) = b2.
  7. y2(x) + Ay(x)y(a − x) + By2(a − x) + Cy(x) + Dy(a − x) = f(x).
  8. y(x)y(ax) = f(x).
  9. y(x2) − ay2(x) = 0.
  10. y(x)y(xa) = f(x).
  11. y(x)y(a/x) = b2.
  12. y(x)y(a/x) = f 2(x).
  13. y2(x) + Ay(x)y(a/x) + By2(a/x) + Cy(x) + Dy(a/x) = f(x).
  14. y(x)y((a − x)/(1 + bx)) = A2.
  15. y(x)y((a − x)/(1 + bx)) = f 2(x).
  16. y2(x) + Ay(x)y((a − x)/(1 + bx)) + By(x) = f(x).
  17. y(x)y((a2 − x2)1/2) = b2.
  18. y(x)y((a2 − x2)1/2) = f 2(x).
  19. y(sin x)y(cos x) = a2.
  20. y(sin x)y(cos x) = f 2(x).
  21. y(x)y(ω(x)) = b2,   where   ω(ω(x)) = x.
  22. y(x)y(ω(x)) = f 2(x),   where   ω(ω(x)) = x.

2.2. Functional Equations with Power − Law Nonlinearity

  1. y(x + a) − byλ(x) = f(x).
  2. yλ(x)y(a − x) = f(x).
  3. y2n + 1(x) + y2n + 1(a − x) = b,    n = 1, 2, ...
  4. yλ(x)y(a/x) = f(x).
  5. yλ(x)y((a − x)/(1 + bx)) = f(x).
  6. yλ(x)y((ax − β)/(x + b)) = f(x),    β = a2 + ab + b2.
  7. yλ(x)y((bx + β)/(a − x)) = f(x),    β = a2 + ab + b2.
  8. yλ(x)y(xa) = f(x).
  9. yλ(x)y((a2 − x2)1/2) = f(x).
  10. yλ(sin x)y(cos x) = f(x).

2.3. Nonlinear Functional Equations of General Form

  1. F(x, y(x), y(x + a)) = 0.
  2. F(x, y(x), y(a − x)) = 0.
  3. F(x, y(x), y(ax)) = 0.
  4. F(x, y(x), y(a/x)) = 0.
  5. F(x, y(x), y((a − x)/(1 + bx))) = 0.
  6. F(x, y(x), y((ax − β)/(x + b))) = 0,    β = a2 + ab + b2.
  7. F(x, y(x), y((bx + β)/(a − x))) = 0,    β = a2 + ab + b2.
  8. F(x, y(x), y(xa)) = 0.
  9. F(x, y(x), y((a2 − x2)1/2)) = 0.
  10. F(x, y(sin x), y(cos x)) = 0.
  11. F(x, y(x), y(ω(x))) = 0,   where   ω(ω(x)) = x.
  12. F(x, y(x), y(x + 1), y(x + 2)) = 0.
  13. F(x, y(x), y((ax − β)/(x + b)), y((bx + β)/(a − x))) = 0,    β = a2 + ab + b2.
  14. F(x, y(x), y(x + 1), ..., y(x + n)) = 0.
  15. F(x, y(x), y[2](x), ..., y[n](x)) = 0,    y[n](x) = y(y[n − 1](x)).
  16. F(x, y(θ1(x)), y(θ2(x)), ..., y(θn(x))) = 0.

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