On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces

R. A. Bandaliyev; K. H.   Safarova

doi:10.37863/umzh.v74i8.6146

R. A. Bandaliyev Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan and Azerbaijan Univ. Architecture and Construction, Baku https://orcid.org/0000-0002-5038-7632
K. H. Safarova Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan and Azerbaijan Univ. Architecture and Construction, Baku

DOI: https://doi.org/10.37863/umzh.v74i8.6146

Keywords: Hardy inequality, nonlinear ordinary differential equation, grand Lebesgue spaces, absolutely continuous functions

Abstract

UDC 517.9

We study the relationship between the second-order nonlinear ordinary differential equations and the Hardy inequality in grand Lebesgue spaces. In particular, we give a characterization of the Hardy inequality by using nonlinear ordinary differential equations in grand Lebesgue spaces.

Author Biography

K. H. Safarova, Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan and Azerbaijan Univ. Architecture and Construction, Baku

References

R. A. Bandaliyev, Connection of two nonlinear differential equations with a two-dimensional Hardy operator in weighted Lebesgue spaces with mixed norm, Electron. J. Different. Equat., 2016, 1 – 10 (2016).

R. A. Bandaliyev, P. Gorka, Hausdorff operator in Lebesgue spaces, Math. Inequal. Appl., 22, 657 – 676 (2019), https://doi.org/10.7153/mia-2019-22-45 DOI: https://doi.org/10.7153/mia-2019-22-45

P. R. Beesack, Hardy’s inequality and its extensions, Pacific J. Math., 11, 39 – 61 (1961). DOI: https://doi.org/10.2140/pjm.1961.11.39

P. R. Beesack, Integral inequalities involving a function and its derivatives, Amer. Math. Monthly, 78, 705 – 741 (1971), https://doi.org/10.2307/2318009 DOI: https://doi.org/10.1080/00029890.1971.11992843

J. S. Bradley, The Hardy inequalities with mixed norms, Canad. Math. Bull., 21, 405 – 408 (1978), https://doi.org/10.4153/CMB-1978-071-7 DOI: https://doi.org/10.4153/CMB-1978-071-7

P. Drabek, A. Kufner, Note on spectra of quasilinear equations and the Hardy inequality, Nonlinear Analysis and Applications: to V. Lakshmikantham on his 80th birthday, vol. 1, Kluwer Acad. Publ., Dordrecht (2003), p. 505 – 512.

D. E. Edmunds, W. D. Evans, Hardy operators, function spaces and embeddings, Springer-Verlag, Berlin (2004), https://doi.org/10.1007/978-3-662-07731-3 DOI: https://doi.org/10.1007/978-3-662-07731-3

A. Fiorenza, M. R. Formica, A. Gogatishvili, On grand and small Lebesgue and Sobolev spaces and some applications to PDE’s, Different. Integral Equat., 10, № 1, 21 – 46 (2018), https://doi.org/10.7153/dea-2018-10-03 DOI: https://doi.org/10.7153/dea-2018-10-03

A. Fiorenza, B. Gupta, P. Jain, The maximal theorem in weighted grand Lebesgue spaces, Stud. Math., 188, № 2, 123 – 133 (2008), https://doi.org/10.4064/sm188-2-2 DOI: https://doi.org/10.4064/sm188-2-2

A. Fiorenza, J. M. Rakotoson, Compactness, interpolation inequalities for small Lebesgue – Sobolev spaces and applications, Cal. Var. Partial Different. Equat., 25, № 2, 187 – 203 (2006), https://doi.org/10.1007/s00526-005-0346-5 DOI: https://doi.org/10.1007/s00526-005-0346-5

A. Fiorenza, C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right-hand side in $L^1$, Stud. Math., 127, № 3, 223 – 231 (1998), https://doi.org/10.4064/sm-127-3-223-231 DOI: https://doi.org/10.4064/sm-127-3-223-231

L. Greco, A remark on the equality det, $Df = mathrm{Det} Df$, Different, Integral Equat., 6, 1089 – 1100 (1993).

L. Greco, T. Iwaniec, C. Sbordone, Inverting the $p$-harmonic operator, Manuscripta Math., 92, 249 – 258 (1997), https://doi.org/10.1007/BF02678192 DOI: https://doi.org/10.1007/BF02678192

P. Gurka, Generalized Hardy’s inequality, Časopis Pěst. Mat., 109, 194 – 203 (1984). DOI: https://doi.org/10.21136/CPM.1984.108498

G. H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math., 54, 150 – 156 (1925).

G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge Univ. Press (1934).

T. Iwaniec, C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Ration. Mech. and Anal., 119, 129 – 143 (1992), https://doi.org/10.1007/BF00375119 DOI: https://doi.org/10.1007/BF00375119

T. Iwaniec, C. Sbordone, Weak minima of variational integrals, J. reine und angew. Math., 454, 143 – 161 (1994), https://doi.org/10.1515/crll.1994.454.143 DOI: https://doi.org/10.1515/crll.1994.454.143

T. Iwaniec, C. Sbordone, Riesz transforms and elliptic pde’s with VMO coefficients, J. Anal. Math., 74, 183 – 212 (1998), https://doi.org/10.1007/BF02819450 DOI: https://doi.org/10.1007/BF02819450

P. Jain, M. Singh, A. P. Singh, Hardy type operators on grand Lebesgue spaces for non-increasing functions, Trans. A. Razmadze Math. Inst., 270, 34 – 46 (2016), https://doi.org/10.1016/j.trmi.2016.02.003 DOI: https://doi.org/10.1016/j.trmi.2016.02.003

P. Jain, M. Singh, A. P. Singh, Hardy-type integral inequalities for quasi-monotone functions, Georgian Math. J., 24, № 4, 523 – 533 (2016), https://doi.org/10.1515/gmj-2016-0016 DOI: https://doi.org/10.1515/gmj-2016-0016

V. M. Kokilashvili, On Hardy’s inequality in weighted spaces, Bull. Akad. Nauk Geor. USSR, 96, 37 – 40 (1979).

A. Kufner, L. Maligranda, L. E. Persson, The Hardy inequality-about its history and some related results, Research report, Lulea Univ. Technology, Sweden (2005). ˚

A. Kufner, L.-E. Persson, Weighted inequalities of Hardy type, World Sci. Publ. Co, New Jersey etc. (2003), https://doi.org/10.1142/5129 DOI: https://doi.org/10.1142/5129

V. G. Maz’ya, Sobolev spaces, Springer-Verlag, Berlin (1985), https://doi.org/10.1007/978-3-662-09922-3 DOI: https://doi.org/10.1007/978-3-662-09922-3

B. Muckenhoupt, Hardy’s inequalities with weights, Stud. Math., 44, 31 – 38 (1972), https://doi.org/10.4064/sm-44-1-31-38 DOI: https://doi.org/10.4064/sm-44-1-31-38

S. H. Saker, R. R. Mahmoud, A connection between weighted Hardy’s inequality and half-linear dynamic equations, Adv. Different. Equat., 129, 1 – 15 (2019), https://doi.org/10.1186/s13662-019-2072-x DOI: https://doi.org/10.1186/s13662-019-2072-x

C. Sbordone, Grand Sobolev spaces and their applications to variational problems, Matematiche, 55, № 2, 335 – 347 (1996).

C. Sbordone, Nonlinear elliptic equations with right-hand side in nonstandard spaces, Rend. Semin. Mat. Fis. Modena, Supp., 46, 361 – 368 (1998).

D. T. Shum, On a class of new inequalities, Trans. Amer. Math. Soc., 204, 299 – 341 (1975), https://doi.org/10.2307/1997361 DOI: https://doi.org/10.1090/S0002-9947-1975-0357715-3

G. Talenti, Osservazione sopra una classe di disuguaglianze, Rend. Semin. Mat. Fiz. Milano, 39, 171 – 185 (1969), https://doi.org/10.1007/BF02924135 DOI: https://doi.org/10.1007/BF02924135

G. Tomaselli, A class of inequalities, Boll. Unione Mat. Ital., 2, 622 – 631 (1969).

S. M. Umarkhadzhiev, On one-dimensional and multidimensional Hardy operators in grand Lebesgue spaces on sets which may have infinite measure, Azerb. J. Math., 7, № 2, 132 – 152 (2017).

S. M. Umarkhadzhiev, On elliptic homogeneous differential operators in grand spaces, Russian Math., 64, 57 – 65 (2020). DOI: https://doi.org/10.3103/S1066369X20030056