Opial-Type Inequalities Năm 1960, Z. Opial [231] phát hiện ra một trong những bất bình đẳng thể tách rời cơ bản nhất liên quan đến một chức năng và dẫn xuất của nó, mà bây giờ được biết đến trong văn học là sự bất bình đẳng Opial. Trong cùng năm đó, C. Olech [230] xuất bản một lưu ý địa chỉ một bằng chứng đơn giản của sự bất bình đẳng Opial trong điều kiện yếu hơn. Bắt đầu từ các giấy tờ đi tiên phong [230231], kết quả của Opial đã nhận được sự chú ý đáng. | Chapter 3 Opial-Type Inequalities Introduction In 1960 Z. Opial 231 discovered one of the most fundamental integral inequalities involving a function and its derivative which is now known in the literature as Opial s inequality. In the same year C. Olech 230 published a note which addresses a simpler proof of Opial s inequality under weaker conditions. Starting from the pioneer papers 230 231 the result of Opial has received considerable attention and many papers have appeared which provides with the simple proofs various generalizations extensions and discrete analogues of Opial s inequality and its generalizations. The importance of Opial s inequality and its generalizations and extensions lies in successful utilization to many interesting applications in the theory of differential equations. Good surveys of the work on such inequalities together with many references are contained in monographs 4 211 215 . In the past few years numerous variants generalizations and extensions of Opial s inequality which involves functions of one and many independent variables have been found in various directions. This chapter deals with important fundamental results on Opial-type inequalities recently investigated in the literature by various investigators. Opial-Type Integral Inequalities In 231 Opial established the following interesting integral inequality ỉ u t u t dt hi u t 2dt J0 4 J0 263 264 Chapter 3. Opial-Type Inequalities where u t e C1 0 h u t 0 in 0 h such that u 0 u h 0. In the constant 4 is the best possible. The first simple proof of inequality is given by Olech 230 in his paper published along with Opial s paper 231 . In the years thereafter numerous variants generalizations and extensions of inequality have appeared in the literature see 4 211 215 and the references given therein. In this section we present a weaker form of and its simplified proof based on Olech 230 as well as variants established by various investigators
