Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

Springer Science and Business Media LLC - Tập 2021 - Trang 1-21 - 2021
Reena Jain1, Hemant Kumar Nashine2, Vahid Parvaneh3
1Mathematics Division, SASL, VIT Bhopal University, Madhya Pradesh, India
2Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India
3Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran

Tóm tắt

This study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ , where $\mathcal{Q}$ is an $n\times n$ Hermitian positive-definite matrix, $\mathcal{A}_{1}$ , $\mathcal{A}_{2}$ , …, $\mathcal{A}_{m}$ are $n \times n$ matrices, and $\mathcal{G}$ is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ , and visualize this through convergence analysis and a solution graph.

Tài liệu tham khảo

Abdeljawad, T., Karapinar, E., Panda, S.K., Mlaiki, N.: Solutions of boundary value problems on extended-Branciari b-distance. J. Inequal. Appl. 2020, 103 (2020)

Aliouche, A., Djoudi, A.: Common fixed point theorems for mappings satisfying an implicit relation without decreasing assumption. Hacet. J. Math. Stat. 36(1), 11–18 (2007)

Bakhtin, I.A.: The contraction mapping principle in quasi metric spaces. Funkc. Anal. Ulianowsk Gos. Ped. Inst. 30, 243–253 (1999)

Branciari, A.: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29(9), 531–536 (2002)

Chen, L., Huang, S., Li, C., Zhao, Y.: Several fixed-point theorems for F-contractions in complete Branciari b-metrics, and applications. J. Funct. Spaces 2020, 7963242 (2020)

Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav. 5, 5–11 (1993)

Czerwik, S.: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Semin. Mat. Fis. Univ. Modena 46, 263–276 (1998)

Felhi, A., Sahmim, S., Aydi, H.: Ulam–Hyers stability and well-posedness of fixed point problems for α-λ-contractions on quasi b-metric spaces. Fixed Point Theory Appl. 2016, 1 (2016)

Garai, H., Dey, L.K.: Common solution to a pair of nonlinear matrix equations via fixed point results. J. Fixed Point Theory Appl. 21, Article ID 61 (2019)

Hussain, N., Vetro, C., Vetro, F.: Fixed point results for α-implicit contractions with application to integral equations. Nonlinear Anal., Model. Control 21(3), 362–378 (2016)

Kamran, T., Samreen, M., Ain, O.U.: A generalization of b-metric space and some fixed point theorems. Mathematics 5(2), Article ID 19 (2017)

Mitrović, Z.D., Isak, H., Radenović, S.: The new results in extended b-metric spaces and applications. Int. J. Nonlinear Anal. Appl. 11(1), 473–482 (2020)

Mustafa, Z., Parvaneh, V., Roshan, J.R., Kadelburg, Z.: \(b_{2}\)-metric spaces and some fixed point theorems. Fixed Point Theory Appl. 2014, 144 (2014)

Nashine, H.K., Kadelburg, Z.: Existence of solutions of cantilever beam problem via \(\alpha -\beta -FG\)-contractions in b-metric-like spaces. Filomat 31(11), 3057–3074 (2017)

Nashine, H.K., Shil, S., Garai, H., Dey, L.K.: Common fixed point results in ordered left (right) quasi b-metric spaces and applications. J. Math. 2020, Article ID 8889453 (2020)

Păcurar, M., Rus, I.A.: Fixed point theory for cyclic ϕ-contractions. Nonlinear Anal. 72, 1181–1187 (2010)

Phiangsungnoen, S., Sintunavarat, W., Kumam, P.: Fixed point results, generalized Ulam–Hyers stability and well-posedness via α-admissible mappings in b-metric spaces. Fixed Point Theory Appl. 2014, 188 (2012)

Popa, V.: Well-posedness of fixed point problems in orbitally complete metric spaces. Stud. Cerc. St. Ser. Mat. Univ. 16 (2006), Supplement. Proceedings of ICMI 45, Bacau, Sept. 18–20, 209–214 (2006)

Popa, V.: Well-posedness of fixed point problems in compact metric spaces. Bul. Univ. Petrol-Gaze, Ploiesti, Sec. Mat. Inform. Fiz. 60(1), 1–4 (2008)

Popa, V., Mocanu, M.: Altering distance and common fixed points under implicit relations. Hacet. J. Math. Stat. 38(3), 329–337 (2009)

Ran, A.C.M., Reurings, M.C.B.: On the matrix equation \(X + A^{*}F(X)A = Q\): solutions and perturbation theory. Linear Algebra Appl. 346, 15–26 (2002)

Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)

Rus, I.A.: The theory of a metrical fixed point theorem: theoretical and applicative relevances. Fixed Point Theory 9, 541–559 (2008)

Sawangsup, K., Sintunavarat, W.: Fixed point and multidimensional fixed point theorems with applications to nonlinear matrix equations in terms of weak altering distance functions. Open Math. 15, 111–125 (2017)

Shah, M.H., Hussain, N.: Nonlinear contractions in partially ordered quasi b-metric spaces. Commun. Korean Math. Soc. 27(1), 117–128 (2012)

Shatanawi, W., Abodayeh, K., Mukheimer, A.: Some fixed point theorems in extended b-metric spaces. UPB Sci. Bull., Ser. A 80(4), 71–78 (2018)

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Tập 2021

1-21

Thông tin tác giả