On summability, multipliability, product integrability, and parallel translation. (English) Zbl 1337.28022
In this paper, the authors start with a well-ordered subset \(\Lambda \) of \(R \cup \{ \infty \}\), with natural order, and a unital Banach algebra \(E\). They take a family \( \{ x_{\alpha} \} \; ( \alpha \in \Lambda) \) of elements in \(E\), define their summability and multipliability in \(E\) and prove and discuss many properties which generalize some results in infinite series and products. Then they define Kurzweil, McShane and Riemann product integrals and obtain some necessary and sufficient conditions for the integrability of some special functions. They further point out some relations between these products integrals and parallel translation operators and strong solutions of linear generalized differential equations.
Reviewer: Surjit Singh Khurana (Iowa City)
MSC:
| 28B05 | Vector-valued set functions, measures and integrals |
| 46G10 | Vector-valued measures and integration |
Keywords:
well-ordered set; summability; multipliability; Kurzweil; linear generalized differential equationsReferences:
| [1] | Arnold, V. I., Mathematical Methods of Classical Mechanics (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0692.70003 |
| [2] | Bourbaki, N., General Topology (1998), Springer-Verlag: Springer-Verlag Berlin, Chapters 5-10 |
| [3] | Dollard, J. D.; Friedman, C. N., Product Integration with Applications to Differential Equations (1979), Addison-Wesley Publishing Company · Zbl 0423.34014 |
| [4] | Dudley, R. M., Real Analysis and Probability (1989), Chapman & Hall: Chapman & Hall New York-London · Zbl 0686.60001 |
| [5] | Dudley, R. M.; Norvaiša, R., Differentiability of Six Operators on Nonsmooth Functions and \(p\)-Variation (1999), Springer-Verlag: Springer-Verlag Berlin · Zbl 0973.46033 |
| [6] | Dudley, R. M.; Norvaiša, R., Concrete Functional Calculus (2011), Springer · Zbl 1218.46003 |
| [7] | Gill, R. D.; Johansen, S., A survey of product integration with a view toward application in survival analysis, Ann. Statist., 18, 4, 1501-1555 (1990) · Zbl 0718.60087 |
| [8] | Haahti, H.; Heikkilä, S., On generalized parallel translations, Acta Univ. Oulu, Ser. A, Sci. Rerum Natur. Math., 15 (1978), 39 pp · Zbl 0407.58005 |
| [9] | Heikkilä, S., On summability, integrability and impulsive differential equations in Banach spaces, Bound. Value Probl., 2013, 1/91 (2013), 23 pp · Zbl 1305.46039 |
| [10] | Heikkilä, S.; Lakshmikantham, V., Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0804.34001 |
| [11] | Jarník, J.; Kurzweil, J., A general form of the product integral and linear ordinary differential equations, Czechoslovak Math. J., 37, 642-659 (1987) · Zbl 0642.26004 |
| [12] | Kurzweil, J., Generalized Ordinary Differential Equations. Not Absolutely Continuous Solutions (2012), World Scientific · Zbl 1248.34001 |
| [13] | Laasri, H.; El-Mennaoui, O., Stability for non-autonomous linear evolution equations with \(L^p\)-maximal regularity, Czechoslovak Math. J., 63, 138, 887-908 (2013) · Zbl 1313.35203 |
| [14] | Masani, P. R., Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc., 61, 147-192 (1947) · Zbl 0037.03802 |
| [15] | Monteiro, G. A.; Slavík, A., Linear measure functional differential equations with infinite delay, Math. Nachr., 287, 11-12, 1363-1382 (2014) · Zbl 1305.34108 |
| [16] | Monteiro, G. A.; Tvrdý, M., Generalized linear differential equations in a Banach space: continuous dependence on a parameter, Discrete Contin. Dyn. Syst., 33, 283-303 (2013) · Zbl 1268.45009 |
| [17] | Schwabik, Š., Bochner product integration, Math. Bohem., 119, 305-335 (1994) · Zbl 0830.28006 |
| [18] | Schwabik, Š., Generalized Ordinary Differential Equations (1992), World Scientific: World Scientific Singapore · Zbl 0781.34003 |
| [19] | Schwabik, Š., The Perron product integral and generalized linear differential equations, Čas. Pěstování Mat., 115, 368-404 (1990) · Zbl 0724.26006 |
| [20] | Schwabik, Š.; Ye, G., Topics in Banach Space Integration (2005), World Scientific: World Scientific Singapore · Zbl 1088.28008 |
| [21] | Slavík, A., Kurzweil and McShane product integration in Banach algebras, J. Math. Anal. Appl., 424, 748-773 (2015) · Zbl 1329.46041 |
| [22] | Slavík, A., Product Integration, Its History and Applications (2007), Matfyzpress: Matfyzpress Prague · Zbl 1216.28001 |
| [23] | Slavík, A., Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl., 385, 534-550 (2012) · Zbl 1235.34247 |
| [24] | Slavík, A., Generalized differential equations: differentiability of solutions with respect to initial conditions and parameters, J. Math. Anal. Appl., 402, 261-274 (2013) · Zbl 1279.34013 |
| [25] | Slavík, A.; Schwabik, Š., Henstock-Kurzweil and McShane product integrals; descriptive definitions, Czechoslovak Math. J., 58, 133, 241-269 (2008) · Zbl 1174.28013 |
| [26] | Slavík, A.; Stehlík, P., Dynamic diffusion-type equations on discrete-space domains, J. Math. Anal. Appl., 427, 525-545 (2015) · Zbl 1338.35449 |
| [27] | Spivak, M., A Comprehensive Introduction to Differential Geometry, vol. II (1999), Publish or Perish, Inc.: Publish or Perish, Inc. Houston, Texas · Zbl 1213.53001 |
| [28] | Volterra, V.; Hostinský, B., Opérations infinitésimales linéaires (1938), Gauthier-Villars: Gauthier-Villars Paris · JFM 64.1112.04 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
