• 한국수학교육학회지시리즈B:순수및응용수학
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• 제18권1호
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• pp.63-78
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• 2011

In the present paper, we obtain integral inequalities involving the Kurzweil-Stieltjes integrals which generalize Gronwall-Bellman inequality and we use the inequalities to verify existence of solutions of a certain integral equation. Such inequalities will play an important role in the study of impulsively perturbed systems [9].

• 대한수학회지
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• 제51권4호
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• pp.791-815
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• 2014

We show amongst other that if $f,g:[a,b]{\rightarrow}\mathbb{C}$ are two functions of bounded variation and such that the Riemann-Stieltjes integral $\int_a^bf(t)dg(t)$ exists, then for any continuous functions $h:[a,b]{\rightarrow}\mathbb{C}$, the Riemann-Stieltjes integral $\int_{a}^{b}h(t)d(f(t)g(t))$ exists and $${\int}_a^bh(t)d(f(t)g(t))={\int}_a^bh(t)f(t)d(g(t))+{\int}_a^bh(t)g(t)d(f(t))$$. Using this identity we then provide sharp upper bounds for the quantity $$\|\int_a^bh(t)d(f(t)g(t))\|$$ and apply them for trapezoid and Ostrowski type inequalities. Some applications for continuous functions of selfadjoint operators on complex Hilbert spaces are given as well.