The interest of this paper lies in the estimates of solutions of the three kinds of Gronwail-Bihari integral inequalities:（Ⅰ） y（x）≤f（x）+sum from i=1 to n（g<sub>i</sub>（x）integral from n=0 to x（...The interest of this paper lies in the estimates of solutions of the three kinds of Gronwail-Bihari integral inequalities:（Ⅰ） y（x）≤f（x）+sum from i=1 to n（g<sub>i</sub>（x）integral from n=0 to x（h<sub>i</sub>（d）y（s）ds））,（Ⅱ） y（x）≤f（x）+g（x）φ（integral from n=0 to x（h（s）w（y（s））ds））（Ⅲ） y（x）≤f（x）+sum from i=1 to n(g<sub>i</sub>（x）integral from n=0 to a(h<sub>i</sub>（s）y（s）ds+g<sub>n+1</sub>φ(integral from n=0 to x(h<sub>n+1</sub>（s）w（y（t））ds)).The results include some modifications and generalizations of the results of D. Willett, U. D. Dhongade and Zhang Binggen. Furthermore, applying the conclusion on the above inequalities to a Volterra integral equation and a differential equation, the authors obtain some new better results.展开更多
文摘The interest of this paper lies in the estimates of solutions of the three kinds of Gronwail-Bihari integral inequalities:（Ⅰ） y（x）≤f（x）+sum from i=1 to n（g<sub>i</sub>（x）integral from n=0 to x（h<sub>i</sub>（d）y（s）ds））,（Ⅱ） y（x）≤f（x）+g（x）φ（integral from n=0 to x（h（s）w（y（s））ds））（Ⅲ） y（x）≤f（x）+sum from i=1 to n(g<sub>i</sub>（x）integral from n=0 to a(h<sub>i</sub>（s）y（s）ds+g<sub>n+1</sub>φ(integral from n=0 to x(h<sub>n+1</sub>（s）w（y（t））ds)).The results include some modifications and generalizations of the results of D. Willett, U. D. Dhongade and Zhang Binggen. Furthermore, applying the conclusion on the above inequalities to a Volterra integral equation and a differential equation, the authors obtain some new better results.
