Contents

1.     Integral Inequalities of Gronwall Type    1

{1.1}    Some Classical Facts    {1}
{1.2}    Other Inequalities of Gronwall Type    {5}
{1.3}    Nonlinear Generalisation    {11}
{1.4}    More Nonlinear Generalisations    {13}

2 .   Inequalities for Kernels of ( L) -Type    43

{2.1}    Integral Inequalities    {43}
    {2.1.1}    Some Generalisations    {49}
    {2.1.2}    Further Generalisations    {53}
    {2.1.3}    The Discrete Version    {60}
{2.2}    Boundedness Conditions    {67}
{2.3}    Convergence to Zero Conditions    {84}

3.    Applications to Integral Equations    93

{3.1}    Solution Estimates     {93}
{3.2}    The Case of Degenerate Kernels     {101}
{3.3}    Boundedness Conditions     {107}
{3.4}    Convergence to Zero Conditions     {110}
{3.5}    Boundedness Conditions for the Difference $x-g$     {112}
{3.6}    Asymptotic Equivalence Conditions     {117}
{3.7}    The Case of Degenerate Kernels     {119}
{3.8}    Asymptotic Equivalence Conditions     {124}
{3.9}    A Pair of Volterra Integral Equations     {129}
    {3.9.1}Estimation Theorems     {129}
    {3.9.2}Boundedness Conditions     {132}
{3.10}The Case of Discrete Equations     {140}

4.    Applications to Differential Equations        149

{4.1}    Estimates for the General Case    {150}
{4.2}    Differential Equations by First Approximation    {157}
{4.3}    Boundedness Conditions    {163}
{4.4}    The Case of Non-Homogeneous Systems    {165}
{4.5}    Theorems of Uniform Stability    {167}
{4.6}    Theorems of Uniform Asymptotic Stability    {170}
{4.7}    Theorems of Global Asymptotic Stability    {177}