Bounds of Solutions of Integrodifferential Equations

and Applied Analysis 3 Define a function m t by m t v t ∫ t 0 g s v s ds v t ∫ t 0 g s ds, 2.5 then m 0 v 0 u0, v t ≤ m t , v′ t ≤ f t m t , 2.6 m′ t 2g t v t v′ t ( 1 ∫ t 0 g s ds ) ≤ m t [ 2g t f t ( 1 ∫ t 0 g s ds )] . 2.7 Integrating 2.7 from 0 to t, we have m t ≤ u0 exp (∫ t 0 ( 2g s f s ( 1 ∫ s 0 g σ dσ )) ds ) . 2.8 Using 2.8 in 2.6 , we obtain v′ t ≤ u0f t exp (∫ t 0 ( 2g s f s ( 1 ∫ s 0 g σ dσ )) ds ) . 2.9 Integrating from 0 to t and using u t ≤ v t , we get inequality 2.2 . The proof is complete. Lemma 2.2. Let u, f , and g be nonnegative continuous functions defined on R , w t be a positive nondecreasing continuous function defined on R . If the inequality u t ≤ w t ∫ t 0 f s ( u s ∫ s 0 g τ u s u τ dτ ) ds, 2.10 holds, where u0 is a nonnegative constant, t ∈ R , then u t ≤ w t [ 1 ∫ t 0 f s exp (∫ s 0 ( 2g τ f τ ( 1 ∫ τ 0 g σ dσ )) dτ ) ds ] , 2.11