Part I
Uniqueness Theorems
Preface ......................................................... 5
§1. Introduction ................................................ 7
§2. Denotations and definitions ................................. 8
1. A fundamental identity ................................... 9
§3. Differential-functional inequalities ........................ 9
1. Auxiliary remarks ........................................ 9
2. Quasi-maximum points (Lem.1) ............................ 10
3. The weak inequalities theorem (Th.1) .................... 11
4. The uniqueness theorem (Th.2) ........................... 12
§4. Functional inequalities. (Lem.2) ........................... 13
1. A linear Lipschitz condition (Th-s.3,4) ................. 14
§5. Systems of inequalities and equations ...................... 16
1. Systems of differential-functional inequalities
(Th-s 5,6) ................................................. 16
2. Systems of functional equations (Th-s.7,8) .............. 19
§6. Certain sufficient and necessary conditions ................ 20
1. The cases for differential-functional inequalities
(Th-s 9,10) ................................................ 21
2. The cases for functional inequalities (Th-s 11, 12) ..... 23
§7. A weaker Lipschitz condition. (Th-s 13, 14) ................ 24
§8. Two simple examples ........................................ 28
§9. Certain applications ....................................... 31
1. The differential-functional inequalities a.e. in J.
(Th-s 15,16,17,18) ......................................... 31
2. The functional problem a.e. in J. (Th-s 19.20) .......... 32
3. Systems of equations a.e. in J. (Th-s 21,22,23.24,25) ... 34
Part II
Existence theorems by a certain method of monotone approximation
§10. Introduction .............................................. 37
§11. Definitions and auxiliary theorems. (Lem.3) ............... 38
1. The construction of lower and upper functions
(Th. 26) ............................................... 39
2. Theorem 26 with Assumption of type A ( Th. 27) ......... 41
§12. Assumption В as a necessary and sufficient condition.
(Lem's 4,5.Th.28) .......................................... 43
§13. The convergence of the sequences from V(L) and W(L).
(Th-s.29,29a) ............................................. 45
§14. Systems of equations. (Th.30) ............................. 48
§15. The weaker Lipschitz condition ............................ 51
1. The indirect method (Lem-s 6,7.Th.31) .................. 51
2. The main theorem (Th.32) ............................... 52
§16. Weakly continuous solutions ............................... 56
1. Denotations and definitions (Lem.8) .................... 56
2. Auxiliary Theorems (Lem.9, Th-s 33,34,35) .............. 58
3. The convergence theorems (th-s.36,36a) ................. 60
4. Functional systems (Lem-s 10,11, Th-s 37, 38) .......... 64
§17. Weakly continuous solutions to special cases .............. 68
1. Differential systems of equations
(Lem-s 12, 13, Th. 39) ................................. 68
2. Functional systems of equations (Th. 40) ............... 70
§18. Some examples ............................................. 71
References ..................................................... 81
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