On a New Generalization of the Lax-Milgram Lemma
Abstract
We consider a new generalization of the celebrated Lax-Milgram Lemma, which is called the harmonic Lax-Milgram Lemma. Some special cases are discussed. New concepts are introduced. The auxiliary principle approach is applied to discuss the existence of the solution as well as to propose some iterative schemes for computing the approximate solution of harmonic Max-Milgram Lemma. Convergence analysis of the proposed methods is considered under some mild conditions. Ideas and techniques of this paper may stimulate further research.
References
Al-Azemi, F., & Calin, O. (2015). Asian options with harmonic average. Applied Mathematics and Information Sciences, 9, 1–9.
Anderson, G. D., Vamanamurthy, M. K., & Vuorinen, M. (2007). Generalized convexity and inequalities. Journal of Mathematical Analysis and Applications, 335, 1294–1308. https://doi.org/10.1016/j.jmaa.2007.02.016
Bers, L., John, F., & Schechter, M. (1966). Partial differential equations. New York: Academic Press.
Fechner, W. (2013). Functional inequalities motivated by the Lax-Milgram Lemma. Journal of Mathematical Analysis and Applications, 402, 411–414. https://doi.org/10.1016/j.jmaa.2013.01.020
Frechet, M. (1907). Sur les ensembles de fonctions et les opérations linéaires. C. R. Acad. Sci., Paris, 144, 1414–1416.
Fukushima, M. (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming, 53, 99–110. https://doi.org/10.1007/BF01585696
Glowinski, R., Lions, J. L., & Trémolières, R. (1981). Numerical analysis of variational inequalities. Amsterdam: North-Holland.
Lax, P. D., & Milgram, A. N. (1954). Parabolic equations. Annals of Mathematics Studies, 33, 167–190. https://doi.org/10.1515/9781400882182-010
Lions, J. L., & Stampacchia, G. (1967). Variational inequalities. Communications on Pure and Applied Mathematics, 20, 493–519. https://doi.org/10.1002/cpa.3160200302
Noor, M. A. (1971). The Riesz-Frechet theorem and monotonicity (Master's thesis, Queen’s University). Kingston, Canada.
Noor, M. A. (1975). On variational inequalities (PhD thesis, Brunel University). London, UK.
Noor, M. A. (2004). Some developments in general variational inequalities. Applied Mathematics and Computation, 251, 199–277. https://doi.org/10.1016/S0096-3003(03)00558-7
Noor, K. I., & Noor, M. A. (1980). A generalization of the Lax-Milgram lemma. Canadian Mathematical Bulletin, 23(2), 179–184. https://doi.org/10.4153/CMB-1980-024-8
Noor, M. A., & Noor, K. I. (2016). Harmonic variational inequalities. Applied Mathematics & Information Sciences, 10(5), 1811–1814. https://doi.org/10.18576/amis/100522
Noor, M. A., & Noor, K. I. (2016). Some implicit schemes for harmonic variational inequalities. International Journal of Analysis and Applications, 12(1), 10–14.
Noor, M. A., & Noor, K. I. (2020). From representation theorems to variational inequalities. In N. Daras & Th. M. Rassias (Eds.), Computational mathematics and variational analysis (Vol. 159). Springer.
Noor, M. A., & Noor, K. I. (2022). Iterative schemes for solving new system of general equations. U.P.B. Scientific Bulletin, Series A, 84(1), 59–70.
Noor, M. A., Noor, K. I., & Rassias, Th. M. (1993). Some aspects of variational inequalities. Journal of Computational and Applied Mathematics, 47, 285–312. https://doi.org/10.1016/0377-0427(93)90058-J
Noor, M. A., Noor, K. I., & Rassias, Th. M. (2020). New trends in general variational inequalities. Acta Applicandae Mathematicae, 170(1), 981–1046. https://doi.org/10.1007/s10440-020-00366-2
Noor, M. A., Noor, K. I., Rassias, M. T., & Awan, M. U. (2024). Nonlinear harmonic variational inequalities and harmonic convex functions. In P. M. Pardalos & Th. M. Rassias (Eds.), Global optimization, computation, approximation and applications. Singapore: World Scientific Publishers.
Rassias, Th. M., Noor, M. A., & Noor, K. I. (2018). Auxiliary principle technique for Lax-Milgram Lemma. Journal of Nonlinear Functional Analysis, 2018, ID 34. https://doi.org/10.23952/jnfa.2018.34
Riesz, F. (1907). Sur une espèce de géométrie analytique des systèmes de fonctions sommables. C. R. Acad. Sci., Paris, 144, 1409–1411.
Riesz, F. (1934–1935). Zur theorie des Hilbertschen raumes. Acta Scientiarum Mathematicarum, 7, 34–38.
Stampacchia, G. (1964). Formes bilinéaires coercivités sur les ensembles convexes. C. R. Acad. Sci., Paris, 258, 4413–4416.
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