Measures of noncompactness in the space of regulated functions on an unbounded interval. (English) Zbl 07587352
Summary: In this paper, we formulate a criterion for relative compactness in the space of regulated functions on an unbounded interval and not necessarily bounded. Next we construct measure of noncompactness in this space and investigate its properties. The presented measure is simpler and more convenient to use than all known so far in space of regulated functions on an unbounded interval. Moreover, we show the applicability of the measure of noncompactness in proving the existence of solutions of some Volterra-type integral equation.
MSC:
| 47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
| 46E40 | Spaces of vector- and operator-valued functions |
| 45D05 | Volterra integral equations |
| 26E20 | Calculus of functions taking values in infinite-dimensional spaces |
Keywords:
space of regulated functions; criterion of relative compactness; Fréchet space; measure of noncompactnessReferences:
| [1] | Banaś, J.; Goebel, K., Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Math (1980), New York: Marcel Dekker, New York · Zbl 0441.47056 |
| [2] | Banaś, J.; Zając, T., On a measure of noncompactness in the space of regulated functions and its applications, Adv. Nonlinear Anal. (2019) · Zbl 1461.26003 · doi:10.1515/anona-2018-0024 |
| [3] | Bothe, D., Multivalued perturbation of m-accretive differential inclusions, Isr. J. Math., 108, 109-138 (1998) · Zbl 0922.47048 · doi:10.1007/BF02783044 |
| [4] | Cichoń, K.; Cichoń, M.; Metwali, MA, On some parameters in the space of regulated functions and their applications, Carpath. J. Math., 34, 1, 17-30 (2018) · Zbl 1449.26003 · doi:10.37193/CJM.2018.01.03 |
| [5] | Cichoń, K.; Cichoń, M.; Satco, B., On regulated functions, Fasc. Math. (2018) · Zbl 1418.46006 · doi:10.1515/fasmath-2018-0003 |
| [6] | Cichoń, K.; Cichoń, M.; Satco, B., Measure differential inclusions through principles in the space of regulated functions, Mediterr. J. Math., 15, 148 (2018) · Zbl 1406.54012 · doi:10.1007/s00009-018-1192-y1660-5446/18/040001-19 |
| [7] | Drewnowski, L., On Banach spaces of regulated functions, Comment. Math., 57, 2, 153-169 (2017) · Zbl 1416.46018 |
| [8] | Dudek, S., Fixed point theorems in Fréchet algebras and Fréchet spaces and applications to nonlinear integral equations, Appl. Anal. Discrete Math., 11, 340-357 (2017) · Zbl 1503.47075 · doi:10.2298/AADM1702340D |
| [9] | Dudek, S., Olszowy, L.: Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114-168 (2020) · Zbl 07259099 |
| [10] | Fraňková, D., Regulated functions, Math. Bohem., 116, 20-59 (1991) · Zbl 0724.26009 · doi:10.21136/MB.1991.126195 |
| [11] | Gabeleh, M.; Malkowsky, E.; Mursaleen, M.; Rakočević, V., A new survey of measures of noncompactness and their applications, Axioms, 11, 299 (2022) · doi:10.3390/axioms11060299 |
| [12] | Michalak, A., On superposition operators in spaces of regular and of bounded variation functions, Z. Anal. Anwend., 35, 285-308 (2016) · Zbl 1362.47051 · doi:10.4171/ZAA/1566 |
| [13] | Mönch, H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. TMA, 4, 985-999 (1980) · Zbl 0462.34041 · doi:10.1016/0362-546X(80)90010-3 |
| [14] | Olszowy, L., Fixed point theorems in the Fréchet space and functional integral equations on an unbounded interval, Appl. Math. Comput., 218, 9066-9074 (2012) · Zbl 1245.45006 |
| [15] | Olszowy, L., Measures of noncompactness in the space of regulated functions, J. Math. Anal. Appl., 476, 860-874 (2019) · Zbl 1486.47090 · doi:10.1016/j.jmaa.2019.04.024 |
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.
