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max. integral of u^2(x)-u(x) for u satisfying inequality

Source: VJIMC 2002 2.3

July 14, 2021
calculusintegrationinequalities

Problem Statement

Let EE be the set of all continuous functions u:[0,1]→Ru:[0,1]\to\mathbb R satisfying u2(t)≤1+4∫0ts∣u(s)∣ds,∀t∈[0,1].u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].Let φ:E→R\varphi:E\to\mathbb R be defined by φ(u)=∫01(u2(x)−u(x))dx.\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.Prove that φ\varphi has a maximum value and find it.
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