# On an overdetermined weighted differential inequality of Hardy-type of second order

• Ainagul Zhanibekovna Adiyeva L. Gumilyov Eurasian National University
• Askar Oinarovich Baiarystanov L.Gumilyov Eurasian National University

### Abstract

The classical one-dimensional integral Hardy's inequality, despite its one-dimensionality, has numerous applications in many branches of mathematics. Beginning in the 1930s, weighted versions of Hardy's inequality began intensively studied, but the first successes, in the sense of fulfillment criteria, were obtained in the years 1969-1970. At present, the one-dimensional weight Hardy inequality, for almost all values of the parameters has been rather well studied. Along with integral inequality, weight differential Hardy inequality occupies an equally important place. Weighted differential Hardy inequality is studied under various boundary conditions at the boundary of a given interval. However, the specified boundary conditions depend on the behavior of the weight functions at the ends of the interval. In addition, the task depends on the finiteness or infinity of the end of the interval, since the integral behaviors of weight functions behave differently. There are various problems here, especially in the redefined case, i.e. when the given boundary conditions are greater than the order of differentiation. In this article, the problem is studied on a finite interval and it is believed that the features of the weight functions are concentrated at one end of the interval and the boundary conditions are overdetermined.

### References

[1] Adiyeva A., Oinarov R. Weighted inequality and oscillatory prop erties of a class of fourth order differential equations// Nonlinear Studies. - 2019. - Vol. 26, No. 4. -P. 741-753.
[2] Opic B. and Kufner A. Hardy-Typ e Inequalities. - Pitman Research Notes in Mathematics Series. Longman Scientific and Technical, Harlow. -1990. -344p.
[3] Абылаева А.М.,Байарыстанов А.О.,Ойнаров Р. Весовое дифференциальное неравенство Харди на множестве AC˚ ( I) // Сиб.Мат.Журнал. -2014. -Т.55,№3. -P.477-493.
[4] Kalybay A. A. One-dimensional differential Hardy inequality//J.Ineq.Appl,(2017) 2017:21 DOI 10.1186/s13660-017-1293-3.
[5] Степанов В. Д. Об одном весовом неравенстве типа Харди для производных высших порядков// Тр. МИАН СССР. -1989. -Т.187. -C.178–190.
[6] Kufner A. Higher order Hardy inequalities// Bayreuth. Math. Schr. - 1993. -Vol.44. -P.105-146.
[7] Куфнер А., Хейниг Г.П. Неравенство Харди для производных высших порядков// Тр. МИАН СССР. -1990. -T.192. -P.105–113 ( Kufner A. and Heinig H.P. Hardy’s inequality for higher order derivatives // Proc. Steklov Inst. Math. -1992. -Vol.192. -P.113-121.)
[8] Kufner A, Wannebo A. Some remarks to the Hardy inequality for higher order derivatives// in: General Inequalities (Oberwolfach, -1990), Birkhauser, Basel. -1992. -P. 33–48.
[9] Kufner A., Kuliev K. and Persson L.-E. Some higher order Hardy inequalities// J. Inequal. Appl. -2012. 2012:69. -P.14.
[10] Sinnamon G. Kufner’s conjecture for higher order Hardy inequalities// Real. Anal. Exchange. -1995. -Vol.21(2). -P.590-603.
[11] Sinnamon G. A weighted gradient inequality // Proc.Royal.Soc. Edinburg A. -1989. -Vol.111. -P.329-335.
[12] Kalybay, A. A., Persson, L.-E. Three weights higher order Hardy inequalities// Function Spaces and Applications.-2006. -Vol. 4(2). -P. 63-191.
[13] Kalybay, A. A. A Generalization of the weighted Hardy inequality for one class of integral operators// Siberian Math. J. -2004. -Vol.45, No.4. -P.100-111.
[14] Kufner A.,Simader C.G. Hardy inequalities for overdetermined classes of functions// Z. Anal.Anwendungen. -1997. No 16(2). -P.387-403.
[15] Kufner A.,Sinnamon G. Overdetermined Hardy inequalities// J.Math.Anal. Appl. -1997. -Vol.213. -P.468-486.
[16] Kufner A.,Lienfelder H. On overdetermined Hardy inequalities// Math. Bohem. -1998. -Vol.123(3), -P.279-293.
[17] Nasyrova M. and Stepanov V. D. On maximal overdetermined Hardy’s inequality of second order on a finite interval// Math. Bohem. -1999. -Vol.124. -P.293–302.
[18] Kufner A., Persson L.-E. Weighted inequalities of Hardy type. - World Scientific., New Jersey-London-Singapore-Hong Kong. - 2003.
[19] Kufner A., Persson L.-E., Samko N. Weighted inequalities of Hardy type. - World Scientific. Second Edition. - 2017.
[20] Nasyrova M., Stepanov V.D. On weighted Hardy on semiaxis for functions vanishing at the endpoints// J. Ineq. Appl. - 1997, -Vol.1, No.3. -P.223-238.
[21] Nassyrova M. Weighted inequalities involving Hardy-type and limiting Geometric Mean Operators. Doctorol thesis. Depatment of Mathematics, Lulea University of Technology,Sweeden. - 2002.
Published
2020-04-05
How to Cite
ADIYEVA, Ainagul Zhanibekovna; BAIARYSTANOV, Askar Oinarovich. On an overdetermined weighted differential inequality of Hardy-type of second order. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 105, n. 1, p. 46-58, apr. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/691>. Date accessed: 07 june 2020.
Citation Formats
Section
Mathematics
Keywords weighted differential Hardy inequality, weighted functions, the boundary value of a function, overdetermined boundary value problems, a locally absolutely continuous function