Solutions and Ulam-Hyers stability of differential inclusions involving Suzuki type multivalued mappings on b-metric spaces

Mudžahid  Abas; Basit Ali; Talat Nazir; Nebojša Dedović; Bandar Bin-Mohsin; Stojan Radenović

doi:10.5937/vojtehg68-26718

Mudžahid Abas Vladin koledž ‒ Univerzitet, Odsek matematike, Lahore, Islamska Republika Pakistan; Univerzitet u Pretoriji, Pretorija, Južnoafrička Republika http://orcid.org/0000-0001-5528-1207
Basit Ali Univerzitet za menadžment i tehnologiju, Odsek za matematiku, Lahore, Islamska Republika Pakistan https://orcid.org/0000-0003-4111-5974
Talat Nazir Univerzitet COMSATS u Islamabadu, Odsek za matematiku, Kampus Abotabad, Islamska Republika Pakistan; Univerzitet Južne Afrike, Odsek za matematičke nauke, Naučni kampus, Johanesburg, Južnoafrička Republika http://orcid.org/0000-0001-6516-3212
Nebojša Dedović Univerzitet u Novom Sadu, Poljoprivredni fakultet, Departman za poljoprivrednu tehniku, Novi Sad, Republika Srbija https://orcid.org/0000-0002-4628-1405
Bandar Bin-Mohsin Univerzitet kralja Sauda, Koledž prirodno-matematičkih nauka, Odsek za matematiku, Rijad, Kraljevina Saudijska Arabija https://orcid.org/0000-0002-2160-4159
Stojan Radenović Univerzitet u Beogradu, Mašinski fakultet, Beograd, Republika Srbija https://orcid.org/0000-0001-8254-6688

DOI: https://doi.org/10.5937/vojtehg68-26718

Ključne reči: b-metrički prostori, višeznačno preslikavanje, fiksna tačka i problemi, Ulam-Hierova stabilnost, početni problem

Sažetak

Uvod/cilj: U radu su predstavljene koincidentne i zajedničke fiksne tačke Suzukijeve vrste višeznačnog preslikavanja na b-metričkim prostorima.

Metode: Analizirana su granična svojstva, dobra postavljenost i Ulam-Hierova stabilnost rešenja za fiksni problem višeznačnih preslikavanja.

Rezultati: Dobijena je gornja granica Hauzdorfovog rastojanja između fiksnih tačaka skupova. Navedeni su primeri koji podržavaju dobijene rezultate.

Zaključak: Primenom predstavljenih rezultata ustanovljena je egzistencija diferencijalne inkluzije.

Reference

Abbas, M., Ali, B., & Vetro, C. 2013. A Suzuki type fixed pointtheorem for a generalized multivalued mapping on partial Hausdorff metric spaces. Topology and its Applications, 160(3), pp.553-563. Available at: https://doi.org/10.1016/j.topol.2013.01.006.

Abbas, M., Ćirić, Lj., Damjanović, B., & Khan, М.А. 2012. Coupled coincidence and common fixed point theorems for hybrid pair of mappings on a test. Fixed Point Theory and Applications, art.number:4. Available at: https://doi.org/10.1186/1687-1812-2012-4.

Aleksić, S., Došenović, T., Mitrović, D.Z., & Radenović, S. 2019a. Remarks on common fixed point results for generalized α_*-ψ- contraction multivalued mappings in b-metric spaces. Advances in Fixed Point Theory, 9(1), pp.1-16. Available at: https://doi.org/10.28919/afpt/3731.

Aleksić, S., Huang, H, Mitrović, D.Z., & Radenović, S. 2018. Remarks on some fixed point results in b-metric spaces. Journal of Fixed Point Theory and Applications, 20(art.number:147). Available at: https://doi.org/10.1007/s11784-018-0626-2.

Aleksić, S., Mitrović, Z.D., & Radenović, S. 2019b. Picard sequences in b-metric spaces. Fixed Point Theory, 21(1), pp.35-46. Available at: https://doi.org/10.24193/fpt-ro.2020.1.03.

Ali, B., & Abbas, M. 2017. Existence Ulam–Hyers stability of fixed point problem of generalized Suzuki type (α_*,ψ_φ)-contractive multivalued operators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM), 111(4), pp.1129-1146. Available at: https://doi.org/10.1007/s13398-016-0351-x.

An, T.V., Dung, N.V., Kadelburg, Z., & Radenović, S. 2015a. Various generalizations of metric spaces and fixed point theorems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM), 109, pp.175-198. Available at: https://doi.org/10.1007/s13398-014-0173-7.

An, T.V., Tuyen, L.Q., & Dung, N.V. 2015b. Stone-type theorem on b-metric spaces and applications. Topology and its Applications,185-186, pp.50-64. Available at: https://doi.org/10.1016/j.topol.2015.02.005.

Asl, J.H., Rezapour, Sh., & Shahzad, N. 2012. On fixed points of α-ψ-contractive multifunctions. Fixed Point Theory and Applications, art.number:212. Available at: https://doi.org/10.1186/1687-1812-2012-212.

Banach, S. 1922. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundamenta Mathematicae, 3, pp.133-181 (in French). Available at: https://doi.org/10.4064/fm-3-1-133-181.

Berinde, V. 1993. Generalized contractions in quasimetric spaces. In: Seminar on Fixed Point Theory, Babeş-Bolyai University, Cluj-Napoca, pp.3-9.

Berinde, V. 1996. Sequences of operators and fixed points in quasimetric spaces. Studia Universitatis Babeş-Bolyai Mathematica, 41(4), pp.23-27.

Berinde, V. 1997. Contracţii generalizate şi aplicaţii. Baia Mare: Editura Club Press (in Romanian).

Bhaskar, T.G., & Lakshmikantham, V. 2006. Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis: Theory, Methods &Applications, 65(7), pp.1379-1393. Available at: https://doi.org/10.1016/j.na.2005.10.017.

Bota, M.F., Ilea, V., Karapınar, E., & Mleşniţ, O. 2015. On α_*-ψ-contractive multivaalued operators in b-metric spaces and applications. Applied Mathematics & Information Sciences, 9(5), pp.2611-2620 [online]. Available at: http://www.naturalspublishing.com/Article. Asp?ArtcID=9576 [Accessed: 21 May 2020].

Chifu, C., & Petruşel, G. 2014. Fixed points for multivalued contractions in b-metric spaces with applications to fractals. Taiwanese Journal of Mathematics, 18(5), pp.1365-1375. Available at: https://doi.org/10.11650/tjm.18.2014.4137.

Czerwik, S. 1993. Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, 1(1), pp.5-11 [online]. Available at: https://dml.cz/dmlcz/120469[Accessed: 21 May 2020].

Ćirić, Lj., Abbas, M., Rajović, M., & Ali, B. 2012. Suzuki type fixed point theorems for generalized multi-valued mappings on a set endowed with two b-metric. Applied Mathematics and Computation, 219(4), pp.1712-1723. Available at: https://doi.org/10.1016/j. Amc.2012.08.011.

Fréchet, M. 1905. La notion d’écart et le calcul fonctionnel. Comptes Rendus Hebdomadaires des Scéances de l'Académie des Sciences, 140(1), pp.772-774 [online]. Available at: https://www.biodiversitylibrary.org/item/31541#page/778/mode/1up (in French) [Accessed: 21 May 2020].

Haghi, R.H., Rezapour, Sh., & Shahzad, N. 2011. Some fixed point generalisations are not real generalizations. Nonlinear Analysis: Theory, Methods & Applications, 74(5), pp.1799-1803. Available at: https://doi.org/10.1016/j.na.2010.10.052.

Hussain, N., Đorić, D., Kadelburg, Z., & Radenović, S. 2012. Suzuki-type fixed point results in metric type spaces. Fixed Point Theory and Applications, art.number:126. Available at: https://doi.org/10.1186/1687-1812-2012-126.

Hyers,D.H. 1941. On the stability of the linear functional equation. In: Proceedings of the National Academy of Sciences of the USA, 27(4), pp.222-224, April 15. Available at: https://doi.org/10.1073/pnas.27.4.222.

Karapinar, E., Mitrović, Z.D., Ozturk, A., & Radenović, S. 2020. On a theorem of Cirić in b-metric spaces. Rendiconti del Circolo Matematico di palermo Series 2. Available at: https://doi.org/10.1007/s12215-020-00491-9.

Khojasteh, F., Abbas, M., & Costache, S. 2014. Two new types of fixed point theorems in complete metric spaces. Variational Analysis, Optimization, and Fixed Point Theory, art.ID325840. Available at: https://doi.org/10.1155/2014/325840.

Kirk, W., & Shahzad, N. 2014. Fixed Point Theory in Distance Spaces. Cham, Switzerland: Springer International Publishing.

Latif, A., Parvaneh, V., Salimi, P., & Al-Mazrooei, A.E. 2015. Various Suzuki type theorems in b-metric spaces. Journal of Nonlinear Sciences and Applications (JNSA), 8(4), pp.363-377. Availableat: https://doi.org/10.22436/jnsa.008.04.09.

Lazar, V.L. 2012. Ulam-Hyers stability for partial differential inclusions. Electronic Journal of Qualitative Theory of Differential Equations, 21, pp.1-19. Available at: https://doi.org/10.14232/ejqtde.2012.1.21.

Michael, E. 1956. Continuous selections. I. Annals of Mathematics, 63(2), pp.361-382. Available at: https://doi.org/10.2307/1969615.

Mitrović, Z.D. 2019. A note on the results of Suzuki, Miculescu and Mihail. Journal of Fixed Point Theory and Applications, 21(art.number:24). Available at: https://doi.org/10.1007/s11784-019-0663-5.

Mitrović, Z.D., Parvaneh, V., Mlaiki, N., Hussain, N., & Radenović, S. 2020. On some new generalizations of Nadler contraction in b-metric spaces. Cogent Mathematics and Statistics, 7(1), art.number:1760189. Available at: https://doi.org/10.1080/25742558.2020.1760189.

Mohammadi, B., Dinu, S., & Rezapour, Sh. 2013. Fixed points of Suzuki type quasi-contractions. U.P.B. Scientific Bulletin, Series A,75(3), pp.3-12 [online]. Available at: https://www.scientificbulletin.upb.ro/rev_docs_arhiva/fulld89_199620.pdf. [Accessed: 21 May 2020].

Nadler, Jr. S.B. 1969. Multi-valued contraction mappings. Pacific Journal of Mathematics, 30(2), pp.475-488. Available at: https://doi.org/10.2140/pjm.1969.30.475.

Nieto, J.J., & Rodríguez-López, R. 2005. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order, 22, pp.223-239. Available at: https://doi.org/10.1007/s11083-005-9018-5.

Nieto, J.J., & Rodríguez-López, R. 2007. Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Mathematica Sinica, English Series, 23, pp.2205-2212. Available at: https://doi.org/10.1007/s10114-005-0769-0.

Păcurar, M. 2010. A fixed point result for φ-contractions on b-metric spaces without the boundedness assumption. Fasciculi Mathematici, 43, pp.127-137 [online]. Available at: http://www.math.put.poznan.pl/artykuly/FM43(2010)-PacurarM.pdf [Accessed: 21 May 2020].

Petru, T.P., Petrusel, A., & Yao, J.-C. 2011. Ulam-Hyers stability for operatorial equations inclusions via nonself operators. Taiwanese Journal of Mathematics, 15(5), pp.2195-2212. Available at: https://doi.org/10.11650/twjm/1500406430.

Ran, A.C.M., & Reurings, M.C.B. 2004. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society, 132(5), pp.1435-1443. Available at: https://doi.org/10.1090/S0002-9939-03-07220-4.

Rhoads, B.E. 2015. Two new fixed point theorem. Gen. Math. Notes, 27(2), pp.123-132 [online]. Available at: https://www.emis.de/journals/GMN/yahoo_site_admin/assets/docs/12_GMN-7082-V27N2.154194604.pdf[Accessed: 21 May 2020].

Rus, I. A. 2001. Generalized contractions and applications. Cluj: University Press.

Rus, I. A. 2009. Remarks on Ulam stability of the operatorial equations. Fixed Point Theory,10(2), pp.305-320 [online]. Available at: http://www.math.ubbcluj.ro/~nodeacj/vol__10(2009)_no_2.php[Accessed: 21 May 2020].

Rus, I. A., Petruşel, A., & Sîntămărian, A. 2003. Data dependence of the fixed point set of some multivalued weakly Picard operators. Nonlinear Analysis: Theory, Methods & Applications, 52(8), pp.1947-1959. Available at: https://doi.org/10.1016/S0362-546X(02)00288-2.

Samet, B., Vetro, C., & Vetro, P. 2012. Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4), pp.2154-2165. Available at: https://doi.org/10.1016/j.na.2011.10.014.

Singh, S.L., & Prasad, B. 2008. Some coincidence theorems and stability of iterative procedures. Computers & Mathematics with Applications, 55(11), pp.2512-2520. Available at: https://doi.org/10.1016/j.camwa.2007.10.026.

Suzuki, T. 2008. A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society, 136(5), pp.1861-1869. Available at: https://doi.org/10.1090/S0002-9939-07-09055-7.

Ulam, S.M. 1964. Problems in Modern Mathematics. New York, NY: John Wiley and Sons.

Rešenja i Ulam-Hierova stabilnost diferencijalnih inkluzija, uključujući Suzukijeve vrste višeznačnog preslikavanja na b-metričkim prostorima

Sažetak

Reference

Vojnotehnički glasnik omogućava otvoreni pristup i, u skladu sa preporukom CEON-a, primenjuje Creative Commons odredbe o autorskim pravima:

Autori koji objavljuju u Vojnotehničkom glasniku pristaju na sledeće uslove: