SOME GENERALIZATION OF GRONWALL-BIHARI INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

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_i(x)integral from n=0 to x(h_i(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_i(x)integral from n=0 to a(h_i(s)y(s)ds g_(n 1)φ(integral from n=0 to x(h_(n 1)(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.

SOME GENERALIZATION OF GRONWALL-BIHARI INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

The interest of this paper lies in the estimates of solutions of the three kinds of Gronwan-Bihari integral inequalities: (I) $[y(x) le f(x) + sumlimits_{i = 1}^n {{g_i}(x)int_0^x {{h_i}(s)y(s)ds,} } ]$(II)$[y(x) le f(x) + g(x)psi (int_0^x {h(s)w(y(s))ds} )]$(III)$[y(x) le f(x) + sumlimits_{i = 1}^n {{g_i}(x)int_0^x {{h_i}(s)y(s)ds} + {g_{n + 1}}(x)psi (int_0^s {{h_{n + 1}}(s)w(y(t))ds} ).} ]$ The results include some modificstions and generalizations of the results of D. Willett U. D. Dhongade and Zhang Binggen. Furthermore, applying the conclusion on the aboveinequalities to a Volterra integral equation and a differential equation, the authors obtain some new better results.