Многозначные отображения: результаты о неподвижной точке с использованием $\beta$-функции и некоторые приложения
Ключевые слова:
неподвижная точка, многозначные отображения, $T_\beta$-многозначное сжимающее отображение, $T_{\beta}$-слабо сжимающее многозначное отображение, интегральное включение, итерационный процесс
Аннотация
С помощью так называемой $\beta$-функции получены некоторые результаты о неподвижной точке для нерасширяющих многозначных отображений. В данной работе результаты рассматриваются в контексте полных метрических пространств, которые не являются ни равномерно выпуклыми, ни компактными. Полученные результаты расширяют, объединяют и улучшают несколько недавних результатов в существующей литературе. В заключение мы применяем наши новые результаты, чтобы обеспечить существование решения для нелинейного интегрального включения. Кроме того, мы аппроксимируем неподвижную точку более быстрым итерационным процессом.
Литература
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23. Pietkun R. Structure of the solution set to Volterra integral inclusions and applications // Journal of Mathematical Analysis and Applications. 2013. Vol. 403. Issue 2. P. 643–666. https://doi.org/10.1016/j.jmaa.2013.02.059
24. Sadygov M.A. An extreme problem for a Volterra type integral inclusion // Open Access Library Journal. 2019. Vol. 6. No. 8. P. 1–8. https://doi.org/10.4236/oalib.1105605
25. Salimi P., Latif A., Hussain N. Modified $\alpha$-$\psi$-contractive mappings with applications // Fixed Point Theory and Applications. 2013. Vol. 2013. No. 1. Article number: 151. https://doi.org/10.1186/1687-1812-2013-151
26. Samet B., Vetro C., Vetro P. Fixed point theorems for $\alpha$-$\psi$-contractive type mappings // Nonlinear Analysis: Theory, Methods and Applications. 2012. Vol. 75. Issue 4. P. 2154–2165. https://doi.org/10.1016/j.na.2011.10.014
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31. Туаль Ю., Аль-Мутавакиль Д. Теоремы о неподвижной точке для новых сжимающих отображений с приложением в динамическом программировании // Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия. 2021. Т. 8. № 2. С. 338–348. https://doi.org/10.21638/spbu01.2021.213
32. Touail Y., El Moutawakil D. Some new common fixed point theorems for contractive selfmappings with applications // Asian-European Journal of Mathematics. 2021. Vol. 15. No. 4. Article number: 2250080. https://doi.org/10.1142/S1793557122500802
33. Touail Y., El Moutawakil D. Fixed point theorems on orthogonal complete metric spaces with an application // International Journal of Nonlinear Analysis and Applications. 2021. Vol. 12. Issue 2. P. 1801–1809. https://doi.org/10.22075/IJNAA.2021.23033.2464
34. Touail Y., Jaid A., El Moutawakil D. New contribution in fixed point theory via an auxiliary function with an application // Ricerche di Matematica. 2021. Vol. 72. No. 1. P. 181–191. https://doi.org/10.1007/s11587-021-00645-6
35. Touail Y. On multivalued $\perp_{\psi F}$-contractions on generalized orthogonal sets with an application to integral inclusions // Проблемы анализа — Issues of Analysis. 2022. Т. 11 (29). Вып. 3. С. 109–124. https://www.mathnet.ru/rus/pa363
36. Ullah K., Khan M.S.U., Muhammad N., Ahmad J. Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaces // Asian-European Journal of Mathematics. 2020. Vol. 13. No. 8. Article number: 2050141. https://doi.org/10.1142/S1793557120501417
37. Wilson W.A. On semi-metric spaces // American Journal of Mathematics. 1931. Vol. 53. No. 2. P. 361–373. https://doi.org/10.2307/2370790
2. Ali F., Ali J., Nieto J.J. Some observations on generalized non-expansive mappings with an application // Computational and Applied Mathematics. 2020. Vol. 39. No. 2. Article number: 74. https://doi.org/10.1007/s40314-020-1101-4
3. Asl J.H., Rezapour S., Shahzad N. On fixed points of $\alpha$-$\psi$-contractive multifunctions // Fixed Point Theory and Applications. 2012. Vol. 2012. No. 1. Article number: 212. https://doi.org/10.1186/1687-1812-2012-212
4. Banach S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales // Fundamenta Mathematicae. 1922. Vol. 3. Issue 1. P. 133–181. https://eudml.org/doc/213289
5. Berinde V. Iterative approximation of fixed points. Berlin–Heidelberg: Springer, 2007. https://doi.org/10.1007/978-3-540-72234-2
6. Browder F.E. Nonexpansive nonlinear operators in a Banach space // Proceedings of the National Academy of Sciences of the United States of America. 1965. Vol. 54. No. 4. P. 1041–1044. https://www.jstor.org/stable/73047
7. Cho Seonghoon. Fixed point theorems for generalized weakly contractive mappings in metric spaces with applications // Fixed Point Theory and Applications. 2018. Vol. 2018. No. 1. Article number: 3. https://doi.org/10.1186/s13663-018-0628-1
8. Clarkson J.A. Uniformly convex spaces // Transactions of the American Mathematical Society. 1936. Vol. 40. No. 3. P. 396–414. https://doi.org/10.2307/1989630
9. Edelstein M. On fixed and periodic points under contractive mappings // Journal of the London Mathematical Society. 1962. Vol. s1-37. Issue 1. P. 74–79. https://doi.org/10.1112/jlms/s1-37.1.74
10. El-Sayed A.M.A., Al-Issa Sh.M. On a set-valued functional integral equation of Volterra–Stieltjes type // Journal of Mathematics and Computer Science. 2020. Vol. 21. Issue 4. P. 273–285. https://doi.org/10.22436/jmcs.021.04.01
11. El Moutawakil D. A fixed point theorem for multivalued maps in symmetric spaces // Applied Mathematics E-Notes. 2004. Vol. 4. P. 26–32. https://www.emis.de/journals/AMEN/2004/030314-1.pdf
12. Göhde D. Zum Prinzip der kontraktiven Abbildung // Mathematische Nachrichten. 1965. Vol. 30. Issues 3–4. P. 251–258. https://doi.org/10.1002/mana.19650300312
13. Hajisharifi H.R. On some generalization of multivalued nonexpansive mappings // Bulletin of the Iranian Mathematical Society. 2020. Vol. 46. No. 2. P. 557–571. https://doi.org/10.1007/s41980-019-00275-7
14. Hussain N., Parvaneh V., Hoseini Ghoncheh S.J. Generalized contractive mappings and weakly $\alpha$-admissible pairs in $G$-metric spaces // The Scientific World Journal. 2014. Vol. 2014. Article ID: 941086. https://doi.org/10.1155/2014/941086
15. Ishikawa S. Fixed points by a new iteration method // Proceedings of the American Mathematical Society. 1974. Vol. 44. No. 1. P. 147–150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
16. Karapınar E., Kumam P., Salimi P. On $\alpha$-$\psi$-Meir–Keeler contractive mappings // Fixed Point Theory and Applications. 2013. Vol. 2013. No. 1. Article number: 94. https://doi.org/10.1186/1687-1812-2013-94
17. Красносельский М.А. Два замечания о методе последовательных приближений // Успехи математических наук. 1955. Т. 10. Вып. 1 (63). С. 123–127. https://www.mathnet.ru/rus/rm7954
18. Mann W.R. Mean value methods in iteration // Proceedings of the American Mathematical Society. 1953. Vol. 4. No. 3. P. 506–510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
19. Michael E. Continuous selection. I // Annals of Mathematics. 1956. Vol. 63. No. 2. P. 361–382. https://doi.org/10.2307/1969615
20. Nadler S.B. Multi-valued contraction mappings // Pacific Journal of Mathematics. 1969. Vol. 30. No. 2. P. 475–488. https://doi.org/10.2140/pjm.1969.30.475
21. Noor M.A. New approximation schemes for general variational inequalities // Journal of Mathematical Analysis and Applications. 2000. Vol. 251. Issue 1. P. 217–229. https://doi.org/10.1006/jmaa.2000.7042
22. Okeke G.A. Convergence analysis of the Picard–Ishikawa hybrid iterative process with applications // Afrika Matematika. 2019. Vol. 30. No. 5–6. P. 817–835. https://doi.org/10.1007/s13370-019-00686-z
23. Pietkun R. Structure of the solution set to Volterra integral inclusions and applications // Journal of Mathematical Analysis and Applications. 2013. Vol. 403. Issue 2. P. 643–666. https://doi.org/10.1016/j.jmaa.2013.02.059
24. Sadygov M.A. An extreme problem for a Volterra type integral inclusion // Open Access Library Journal. 2019. Vol. 6. No. 8. P. 1–8. https://doi.org/10.4236/oalib.1105605
25. Salimi P., Latif A., Hussain N. Modified $\alpha$-$\psi$-contractive mappings with applications // Fixed Point Theory and Applications. 2013. Vol. 2013. No. 1. Article number: 151. https://doi.org/10.1186/1687-1812-2013-151
26. Samet B., Vetro C., Vetro P. Fixed point theorems for $\alpha$-$\psi$-contractive type mappings // Nonlinear Analysis: Theory, Methods and Applications. 2012. Vol. 75. Issue 4. P. 2154–2165. https://doi.org/10.1016/j.na.2011.10.014
27. Sîntămărian A. Integral inclusions of Fredlhom type relative to multivalued $\varphi$-contractions // Seminar on Fixed Point Theory Cluj-Napoca. 2002. Vol. 3. P. 361–368. https://zbmath.org/1028.47043
28. Touail Y., El Moutawakil D., Bennani S. Fixed point theorems for contractive selfmappings of a bounded metric space // Journal of Function Spaces. 2019. Vol. 2019. Article ID: 4175807. https://doi.org/10.1155/2019/4175807
29. Touail Y., El Moutawakil D. Fixed point results for new type of multivalued mappings in bounded metric spaces with an application // Ricerche di Matematica. 2020. Vol. 71. No. 2. P. 315–323. https://doi.org/10.1007/s11587-020-00498-5
30. Touail Y., El Moutawakil D. New common fixed point theorems for contractive self mappings and an application to nonlinear differential equations // International Journal of Nonlinear Analysis and Applications. 2021. Vol. 12. Issue 1. P. 903–911. https://doi.org/10.22075/IJNAA.2021.21318.2245
31. Туаль Ю., Аль-Мутавакиль Д. Теоремы о неподвижной точке для новых сжимающих отображений с приложением в динамическом программировании // Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия. 2021. Т. 8. № 2. С. 338–348. https://doi.org/10.21638/spbu01.2021.213
32. Touail Y., El Moutawakil D. Some new common fixed point theorems for contractive selfmappings with applications // Asian-European Journal of Mathematics. 2021. Vol. 15. No. 4. Article number: 2250080. https://doi.org/10.1142/S1793557122500802
33. Touail Y., El Moutawakil D. Fixed point theorems on orthogonal complete metric spaces with an application // International Journal of Nonlinear Analysis and Applications. 2021. Vol. 12. Issue 2. P. 1801–1809. https://doi.org/10.22075/IJNAA.2021.23033.2464
34. Touail Y., Jaid A., El Moutawakil D. New contribution in fixed point theory via an auxiliary function with an application // Ricerche di Matematica. 2021. Vol. 72. No. 1. P. 181–191. https://doi.org/10.1007/s11587-021-00645-6
35. Touail Y. On multivalued $\perp_{\psi F}$-contractions on generalized orthogonal sets with an application to integral inclusions // Проблемы анализа — Issues of Analysis. 2022. Т. 11 (29). Вып. 3. С. 109–124. https://www.mathnet.ru/rus/pa363
36. Ullah K., Khan M.S.U., Muhammad N., Ahmad J. Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaces // Asian-European Journal of Mathematics. 2020. Vol. 13. No. 8. Article number: 2050141. https://doi.org/10.1142/S1793557120501417
37. Wilson W.A. On semi-metric spaces // American Journal of Mathematics. 1931. Vol. 53. No. 2. P. 361–373. https://doi.org/10.2307/2370790
Поступила в редакцию
2023-04-21
Опубликована 2024-05-20
Опубликована 2024-05-20