# Mathematical Collection

Monografie Matematyczne

## Volume 43

Differential inequalities

Jacek Szarski

Warszawa 1965

#### Contents

 CHAPTER I Monotone functions § 1. Zygmund's lemma § 2. A necessary and sufficient condition for a continuous function to be monotone § 3. A sufficient condition for a function to be monotone CHAPTER II Maximum and minimum solution of ordinary differential equationsv § 4. Some notations and definitions § 5. Definition of the maximum (minimum) solution § 6. Basic lemmas on strong ordinary differential inequalities § 7. Some notions and theorems on ordinary differential equations § 8. Local existence of the right-hand maximum solution § 9. Global existence of the maximum and minimum solution § 10. Continuity of the maximum and minimum solution on the initial point and on the right-hand sides of system CHAPTER III First order ordinary differential inequalities § 11. Basic theorems on first order ordinary differential inequalities § 12. Necessity of condition V+ (V_) in theorems on differential inequalities § 13. Some variants of theorems on differential inequalities § 14. Comparison systems § 15. Absolute value estimates § 16. Infinite systems of ordinary differential inequalities and systems satisfying Carathéodory's conditions CHAPTER IV Ordinary differential inequalities of higher order and some integral inequalities § 17. Preliminary remarks and definitions § 18. Maximum and minimum solution of an nth order ordinary differential equation § 19. Basic theorems on nth order ordinary differential inequalities § 20. Comparison equation of order n § 21. Absolute value estimates § 22. Some integral inequalities CHAPTER V Cauchy problem for ordinary differential equations § 23. Estimates of the solution and of its existence interval § 24. Estimates of the difference between two solutions § 25. Uniqueness criteria § 26. Estimates of the error of an approximate solution § 27. Stability of the solution § 28. Differential inequalities in the complex domain § 29. Estimates of the solution and of its radius of convergence for differential equations in the complex domain § 30. Estimates of the difference between two solutions in the complex domain § 31. Chaplygin method for ordinary differential equations § 32. Approximation of solutions of an ordinary differential equation in a Banach space CHAPTER VI Some auxiliary theorems § 33. Maximum of a continuous function of n+l variables on n-dimensional planes § 34. Maximum of the absolute value of functions of n+1 variables on n-dimensional planes § 35. Maximum of a continuous function of several variables on plane sections of a pyramid § 36. Comparison systems with right-hand sides depending on parameters CHAPTER VII Cauchy problem for first order partial differential equations § 37. Comparison theorems for systems of partial differential inequalities § 38. Comparison theorems for overdetermined systems of partial differential inequalities § 39. Estimates of the solution § 40. Estimate of the existence domain of the solution § 41. Estimates of the difference between two solutions § 42. Uniqueness criteria § 43. Continuous dependence of the solution on initial data and on right-hand sides of system § 44. Estimate of the error of an approximate solution § 45. Systems with total differentials CHAPTER VIII Mixed problems for second order partial differential equations of parabolic and hyperbolic type § 46. Ellipticity and parabolicity § 47. Mixed problems § 48. Estimates of the solution of the first mixed problem § 49. Estimates of the difference between two solutions of the first mixed problem § 50. Uniqueness criteria for the solution of the first mixed problem § 51. Continuous dependence of the solution of the first mixed problem on initial and boundary values and on the right-hand sides of system § 52. Stability of the solution of the first mixed problem § 53. Preliminary remarks and lemmas referring to the second mixed problem § 54. Sufficient conditions for the existence of sign-stabilizing factors § 55. Analogues of theorems in § 48-52 in case of the second mixed problem § 56. Energy estimates for solutions of hyperbolic equations CHAPTER IX Partial differential inequalities of first order § 57. Systems of strong first order partial differential inequalities § 58. Overdetermined systems of strong first order partial differential inequalities § 59. Systems of weak first order partial differential inequalities § 60. Overdetermined systems of weak first order partial differential inequalities § 61. Comparison systems of first order partial differential equations § 62. Estimates of solutions of first order partial differential equations and a uniqueness criterion CHAPTER X Second order partial differential inequalities of parabolic type § 63. Strong partial differential inequalities of parabolic type § 64. Weak partial differential inequalities of parabolic type § 65. Parabolic differential inequalities in unbounded regions § 66. The Chaplygin method for parabolic equations § 67. Maximum solution of the parabolic equation CHAPTER XI Differential inequalities in linear spaces § 68. Convex sets in linear topological spaces § 69. Mean value theorems § 70. Strong differential inequalities § 71. Bendixson equation and differential inequalities § 72. Linear differential inequalities in Banach spaces I § 73. Linear differential inequalities in Banach. spaces II § 74. Almost linear differential inequalities in Banach spaces Materiały redakcyjne References, list of special symbols, index, contents