Fixed-point existence technique

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Problem 3.4.13 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 3.4 "Systems of Differential Equations"

For λ\lambda a real number, find all solutions of the integral equations

φ(x)=ex+λ∫0xe(x−y)φ(y) dy,0≤x≤1, \varphi (x) = e^x + \lambda \int _0^x e^{(x-y)}\varphi (y) \, dy, \quad 0 \leq x \leq 1, ψ(x)=ex+λ∫01e(x−y)ψ(y) dy,0≤x≤1. \psi (x) = e^x + \lambda \int _0^1 e^{(x-y)}\psi (y) \, dy, \quad 0 \leq x \leq 1.
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References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.