CONTENTS Introduction........................................................................................................................................................................................................... 5 I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^1'*(Δ_{a,b};E)$ AND $L^{1,*}_1(Δ_{a,b};E)$......................... 7 1. Basic function classes ................................................................................................................................................................................... 7 2. The Cauchy-Darboux problem ...................................................................................................................................................................... 12 II. Comparison of solutions ............................................................................................................................................................................... 18 3. The growth estimations.................................................................................................................................................................................. 18 4. Maximal solutions............................................................................................................................................................................................ 26 5. A theorem on extension of inequalities........................................................................................................................................................ 28 6. Effective estimation in the case $M_1$, (b)................................................................................................................................................. 30 III. COMPARATIVE CRITERIA OF EXISTENCE AND UNIQUENESS OP SOLUTIONS OF THE CAUCHY-DARBOUX PROBLEM...................................................................................................................................................................................... 35 7. Basic classes of comparative functions...................................................................................................................................................... 35 8. Existence and uniqueness of solutions of the Cauchy-Darboux problem............................................................................................ 42 9. Remarks on the continuous dependence of solutions on boundary data and on the second member........................................ 47 10. Examples......................................................................................................................................................................................................... 49 BIBLIOGRAPHICAL REMARKS.......................................................................................................................................................................... 66 BIBLIOGRAPHY..................................................................................................................................................................................................... 68 INDEX OF SYMBOLS............................................................................................................................................................................................ 74