NONLINEAR INTEGRAL INEQUALITIES INVOLVING ... - Ele-Math

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$Some more inequalities for arithmetic mean, harmonic ... - Ele-Math$

812 M. BOHNER, S. HRISTOVA AND K. STEFANOVA ∞. 2. Main results Let h > 0 be a constant and suppose t0 and T are fixed points with 0 t0 < T DEFINITION 2.1. The function α ∈ C 1 ([t0,T),R+) is said to be from the class F if it is nondecreasing and satisfies α(t) t for t ∈ [t0,T). Let αi,βj ∈ F for i = 1,2,...,n and j = 1,2,...,m. Denote J = min min 1in αi(t0), min 1 jm β j(t0) . 2.1. Constant additive term In this subsection, we discuss the case of a constant k in the inequality (2.1) below. For this purpose, we define the following functions. DEFINITION 2.2. The function ω ∈ C(R+,R+) is said to be from Ω1 if (i) ω(x) > 0 for x > 0 and ω is a nondecreasing function; (ii) ∞ dx ω(x) = ∞. In the case when the nonlinear functions under the integrals in the inequality (2.1) below are from the set Ω1 , we obtain the following result. THEOREM 2.3. Let the following conditions be fulfilled: (A 1 ) φ ∈ C([J − h,t0],R+). (A 2 ) k > 0. (A 3 ) αi,βj ∈ F for i = 1,2,...,n, j = 1,2,...,m. (A 4 ) fi,gj ∈ C([J,T),R+) for i = 1,2,...,n, j = 1,2,...,m. (A 5 ) ωi, ˜ω j ∈ Ω1 for i = 1,2,...,n, j = 1,2,...,m. (A 6 ) ψ ∈ C 1 (R+,R+) is increasing, ψ(0) = 0, and limt→∞ ψ(t) = ∞. (A 7 ) u ∈ C([J − h,T),R+) satisfies for some p 0 the inequalities ψ u(t) k+ n ∑ αi(t) i=1 αi(t0) m β j(t) + ∑ j=1 β j(t0) fi(s)u p (s)ωi g j(s)u p (s) ˜ω j u(s) ds (2.1) max ξ ∈[s−h,s] u(ξ) ds for t ∈ [t0,T), u(t) φ(t) for t ∈ [J − h,t0]. (2.2)
Then for t0 t t1 , the inequality holds, where t1 = sup NONLINEAR INTEGRAL INEQUALITIES 813 u(t) ψ −1 Ψ −1 W −1 W Ψ(M) + A(t) r Ψ(r) = r0 r W(r) = r1 (2.3) ds ψ −1 (s) p, 0 < r0 < k, (2.4) ds q ψ−1 (Ψ−1 (s)) , 0 < r1 < Ψ(M), (2.5) q(t) = max max 1in ωi(t), max 1 jm ˜ω j(t) , (2.6) A(t) = n ∑ i=1 αi(t) αi(t0) M = max fi(s)ds+ k, ψ m β j(t) ∑ j=1 β j(t0) max s∈[J−h,t0] φ(s) τ ∈ [t0,T) : W Ψ(M) + A(t) ∈ Dom W −1 , W −1 W Ψ(M) + A(t) ∈ Dom Ψ −1 and Ψ −1 W −1 W Ψ(M) + A(t) ∈ Dom ψ −1 Proof. Define a function z : [J − h,T) → R+ by g j(s)ds, (2.7) , (2.8) for t ∈ [t0,τ] ⎧ n αi(t) M + ∑ fi(s)u ⎪⎨ i=1 αi(t0) z(t) = ⎪⎩ p (s)ωi u(s) ds m β j(t) + ∑ g j(s)u j=1 β j(t0) p t ∈ [t0,T), (s) ˜ω j max u(ξ) ds, ξ ∈[s−h,s] M, t ∈ [J − h,t0]. The function z is nondecreasing. Since ψ(u(t)) ψ(max s∈[J−h,t0] φ(s)) M = z(t) for t ∈ [J −h,t0] by (2.2) and (2.8) and ψ(u(t)) z(t) for t ∈ [t0,T) by (2.1) and (2.8), the inequality u(t) ψ −1 z(t) holds for t ∈ [J − h,T). (2.9) .
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