Pachpatte在參考文獻[11]中所給的定理1與定理2的證明是較困難的，而且使用到參考文獻[7]的定理1.1.2。這篇論文主要的目的是給一個簡單而且直接的證明，而不使用[7]的定理1.1.2。此外，在[11]的定理2中，(b2)和(b3)部分的證明被省略，我們將在這論文中給出證明。

The proofs of Theorem 1 and Theorem 2 given by Pachpatte in [11] is hard and has used Theorem 1.2.1 in [7] . The main purpose of this note is to give a simple and straightforward proof without using Theorem 1.2.1 in [7] . Also the proofs of parts of (b2) and (b3) in Theorem 2 is skipped in [11] , we shall supply the proofs in this note .

目錄                    頁次

1.引言……………………………………………………………..p1
        定理1……………………….…………………………..p1
        定理2…………………….……………………………..p4
2.定理1的証明…………………..…….………………………..p6
       附註1……………….…………….……….…………..p13
3.定理2的証明………………….…  ………….……………..p13
       附註2……………….…………………..……………..p20
參考書籍……….…………...………………………………….p20
                          
                           Index                    Page

1.Introduction………………………………………………..p22
           Theorem 1………………………………………..p22
           Theorem 2………………………………………..p24
2.Proof of Theorem 1………..……………………………..p26
           Remark 1…………………………….…………..p33
3. Proof of Theorem 2……………………….……………..p33
          Remark 2…………………………..……………..p39
  References……….………………………………………….p39

References
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3. S.S.Dragomir , Some Gronwall type inequalities and Applications , Nova Science Publishers , Inc. New York , 2003 .
4. Sh.G..Gamidov , Integral inequalities for boundary value problems for differential equations , Diff.Eq. 5(3) (1969) , 463-472 (English Translation from Russian) .
5. R.K.Miller , J.A.Nohel and J.S.W.Wong , A stability theorem for nonlinear mixed integreal equations , J.Math.Appl. 25(1969) , 446-449 .
6. B.G..Pachpatte , Inequalities for Differential and Integral Equations , Academic Press , New York , 1998 .
7. B.G..Pachpatte , Inequalities for Finite Difference Equations , Marcel Dekker , Inc. , New York , 2002 .
8. B.G..Pachpatte , A note on certain integral inequality , Tamkang J. Math. 33 (2002) , 353-358 .
9. B.G..Pachpatte , On an inequality applicable to general Volterra-Fredholm integral equation , An.Sti.Univ.Al.I.Cuza Iasi , Mat. XLIX (2003) , 221-228
10. B.G..Pachpatte , Explicit bound on retarded integral inequality , Math. Ineq. Appl. 7(2004) , 7-11 .
11. B.G. Pachpatte , Explicit bounds on Gamidov type integral inequalities , TamKang J. Math. Vol 37 , No. 1 . (to appear)