Set-valued mappings: fixed point results with $\beta$-function and some applications

Y. Touail

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 63 (2024), pp. 61-79.

Résumé

Via the so-called $\beta$-function, some fixed-point results for nonexpansive set-valued mappings are obtained. In this study, the results are considered in the context of complete metric spaces which are neither uniformly convex nor compact. Our theorems extend, unify, and improve several recent results in the existing literature. In the end, we apply our new results to ensure the existence of a solution for a nonlinear integral inclusion. Moreover, we approximate the fixed point by a faster iterative process.

Détail

Keywords: fixed point, set-valued mappings, $T_{\beta}$-multivalued contractive mapping, $T_{\beta}$-weakly contractive multivalued mapping, integral inclusion, iterative process

Bibliographie

[1] Alber Ya.I., Guerre-Delabriere S., “Principle of weakly contractive maps in Hilbert spaces”, New results in operator theory and its applications, Birkhäuser, Basel, 1997, 7–22 | DOI | MR | Zbl

[2] Ali F., Ali J., Nieto J.J., “Some observations on generalized non-expansive mappings with an application”, Computational and Applied Mathematics, 39:2 (2020), 74 | DOI | MR | Zbl

[3] Asl J.H., Rezapour S., Shahzad N., “On fixed points of $\alpha$-$\psi$-contractive multifunctions”, Fixed Point Theory and Applications, 2012:1 (2012), 212 | DOI | MR | Zbl

[4] Banach S., “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”, Fundamenta Mathematicae, 3:1 (1922), 133–181 https://eudml.org/doc/213289 | DOI | MR

[5] Berinde V., Iterative approximation of fixed points, Springer, Berlin–Heidelberg, 2007 | DOI | MR | Zbl

[6] Browder F.E., “Nonexpansive nonlinear operators in a Banach space”, Proceedings of the National Academy of Sciences of the United States of America, 54:4 (1965), 1041–1044 https://www.jstor.org/stable/73047 | DOI | MR | Zbl

[7] Cho Seonghoon, “Fixed point theorems for generalized weakly contractive mappings in metric spaces with applications”, Fixed Point Theory and Applications, 2018:1 (2018), 3 | DOI | MR

[8] Clarkson J.A., “Uniformly convex spaces”, Transactions of the American Mathematical Society, 40:3 (1936), 396–414 | DOI | MR | Zbl

[9] Edelstein M., “On fixed and periodic points under contractive mappings”, Journal of the London Mathematical Society, s1–37:1 (1962), 74–79 | DOI | MR | Zbl

[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, 21:4 (2020), 273–285 | DOI | MR

[11] El Moutawakil D., “A fixed point theorem for multivalued maps in symmetric spaces”, Applied Mathematics E-Notes, 4 (2004), 26–32 https://www.emis.de/journals/AMEN/2004/030314-1.pdf | MR | Zbl

[12] Göhde D., “Zum Prinzip der kontraktiven Abbildung”, Mathematische Nachrichten, 30:3–4 (1965), 251–258 | DOI | MR | Zbl

[13] Hajisharifi H.R., “On some generalization of multivalued nonexpansive mappings”, Bulletin of the Iranian Mathematical Society, 46:2 (2020), 557–571 | DOI | MR | Zbl

[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 (2014), 941086 | DOI

[15] Ishikawa S., “Fixed points by a new iteration method”, Proceedings of the American Mathematical Society, 44:1 (1974), 147–150 | DOI | MR | Zbl

[16] Karap{\i}nar E., Kumam P., Salimi P., “On $\alpha$-$\psi$-Meir–Keeler contractive mappings”, Fixed Point Theory and Applications, 2013:1 (2013), 94 | DOI | MR | Zbl

[17] Krasnosel’skii M.A., “Two remarks on the method of successive approximations”, Uspekhi Matematicheskikh Nauk, 10:1(63) (1955), 123–127 (in Russian) | MR | Zbl

[18] Mann W.R., “Mean value methods in iteration”, Proceedings of the American Mathematical Society, 4:3 (1953), 506–510 | DOI | MR | Zbl

[19] Michael E., “Continuous selection. I”, Annals of Mathematics, 63:2 (1956), 361–382 | DOI | MR | Zbl

[20] Nadler S.B., “Multi-valued contraction mappings”, Pacific Journal of Mathematics, 30:2 (1969), 475–488 | DOI | MR | Zbl

[21] Noor M.A., “New approximation schemes for general variational inequalities”, Journal of Mathematical Analysis and Applications, 251:1 (2000), 217–229 | DOI | MR | Zbl

[22] Okeke G.A., “Convergence analysis of the Picard–Ishikawa hybrid iterative process with applications”, Afrika Matematika, 30:5–6 (2019), 817–835 | DOI | MR | Zbl

[23] Pietkun R., “Structure of the solution set to Volterra integral inclusions and applications”, Journal of Mathematical Analysis and Applications, 403:2 (2013), 643–666 | DOI | MR | Zbl

[24] Sadygov M.A., “An extreme problem for a Volterra type integral inclusion”, Open Access Library Journal, 6:8 (2019), 1–8 | DOI

[25] Salimi P., Latif A., Hussain N., “Modified $\alpha$-$\psi$-contractive mappings with applications”, Fixed Point Theory and Applications, 2013:1 (2013), 151 | DOI | MR | Zbl

[26] Samet B., Vetro C., Vetro P., “Fixed point theorems for $\alpha$-$\psi$-contractive type mappings”, Nonlinear Analysis: Theory, Methods and Applications, 75:4 (2012), 2154–2165 | DOI | MR | Zbl

[27] Sîntămărian A., “Integral inclusions of Fredlhom type relative to multivalued $\varphi$-contractions”, Seminar on Fixed Point Theory Cluj-Napoca, 3 (2002), 361–368 | MR | Zbl

[28] Touail Y., El Moutawakil D., Bennani S., “Fixed point theorems for contractive selfmappings of a bounded metric space”, Journal of Function Spaces, 2019 (2019), 4175807 | DOI | MR | Zbl

[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, 71:2 (2020), 315–323 | DOI | MR

[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, 12:1 (2021), 903–911 | DOI | MR

[31] Touail Y., El Moutawakil D., “Fixed point theorems for new contractions with application in dynamic programming”, Vestnik of Saint Petersburg University. Mathematics, 54:2 (2021), 206–212 | DOI | DOI | MR | Zbl

[32] Touail Y., El Moutawakil D., “Some new common fixed point theorems for contractive selfmappings with applications”, Asian-European Journal of Mathematics, 15:4 (2021), 2250080 | DOI | MR

[33] Touail Y., El Moutawakil D., “Fixed point theorems on orthogonal complete metric spaces with an application”, International Journal of Nonlinear Analysis and Applications, 12:2 (2021), 1801–1809 | DOI | MR

[34] Touail Y., Jaid A., El Moutawakil D., “New contribution in fixed point theory via an auxiliary function with an application”, Ricerche di Matematica, 72:1 (2021), 181–191 | DOI | MR

[35] Touail Y., “On multivalued $\perp_{\psi F}$-contractions on generalized orthogonal sets with an application to integral inclusions”, Problemy Analiza — Issues of Analysis, 11(29):3 (2022), 109–124 | MR | Zbl

[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, 13:8 (2020), 2050141 | DOI | MR

[37] Wilson W.A., “On semi-metric spaces”, American Journal of Mathematics, 53:2 (1931), 361–373 | DOI | MR | Zbl