Document Type : Research Paper


1 Department of Mathematics, University of Sargodha, Sargodha-40100, Pakistan.

2 Department of Mathematics, Government College University, Lahore 54000, Pakistan and Department of Mathematics and Applied Mathematics, University of Pretoria Hatfield 002, Pretoria, South Africa.


Berinde [V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum {\bf 9} (2004), 43-53] introduced almost contraction mappings and proved Banach contraction principle for such mappings. The aim of this paper is to introduce the notion of multivalued almost $\Theta$- contraction mappings and
to prove some best proximity point results for this new class of mappings. As applications, best proximity point and fixed point results for weak single valued $\Theta$-contraction mappings are obtained. Moreover, we give an example to support the results presented herein. An application to a nonlinear differential equation is also provided.


[1] A. Abkar and M. Gabeleh, The existence of best proximity points for multivalued non-selfmappings, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM, 107 (2013), pp. 319-325.
[2] A. Abkar and M. Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl., 153 (2012), pp. 298-305.
[3] I. Altun, H.A. Hnacer and G. Minak, On a general class of weakly picard operators, Miskolc Math. Notes, 16 (2015), pp. 25-32.
[4] M.A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal., 70 (2009), pp. 3665-3671.
[5] A. Amini-Harandi, Best proximity points for proximal generalized contractions in metric spaces, Optim. Lett., 7 (2013), pp. 913-921.
[6] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3 (1922), pp. 133-181.
[7] S.S. Basha, Best proximity points: Global optimal approximate solutions, J. Global Optim., 49 (2011), pp. 15-21.
[8] S.S. Basha and P. Veeramani, Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory, 103 (2000), pp. 119-129.
[9] V. Berinde, Approximating fixed points of weak contractions using the Picard itration, Nonlinear Anal. Forum, 9 (2004), pp. 43-53.
[10] V. Berinde, General constructive fixed point theorems for Ciric-type almost contractions in metric spaces, Carpathian J. Math., 24 (2008), pp. 10-19.
[11] M. Berinde and V. Berinde, On a general class of multivalued weakly picard mappings, J. Math. Anal. 326 (2007), pp. 772-782.
[12] S. Chandok, H. Huang and S. Radenovic, Some Fixed Point Results for the Generalized F-suzuki Type Contractions in b-metric Spaces, Sahand Commun. Math. Anal., 11 (2018), pp. 81-89.
[13] BS. Choudhurya, P. Maitya and N. Metiya, Best proximity point results in set-valued analysis, Nonlinear Anal. Model. Control, 21 (2016), pp. 293-305.
[14] B.S. Choudhury, P. Maity and P. Konar, A global optimality result using nonself mappings, Opsearch, 51 (2014), pp. 312-320.
[15] R.C. Dimri and P. Semwal, Best proximity results for multivalued mappings, Int. J. Math. Anal., 7 (2013), pp. 1355-1362.
[16] G. Durmaz, Some theorems for a new type of multivalued contractive maps on metric space, Turkish J. Math., 41 (2017), pp. 1092-1100.
[17] G. Durmaz and I. Altun, A new perspective for multivalued weakly picard operators, Publications De L'institut Mathematique, 101 (2017), pp. 197-204.
[18] A.A. Eldred, J. Anuradha and P. Veeramani, On equivalence of generalized multivalued contractions and Nadler's fixed point theorem, J. Math. Anal. Appl., 336 (2007), pp. 751-757.
[19] A.A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), pp. 1001-1006.
[20] M. Gabeleh, Best proximity points: Global minimization of multivalued non-self mappings, Optim. Lett., 8 (2014) pp. 1101-1112.
[21] H.A. Hancer, G. Mmak and I. Altun, On a broad category of multivalued weakly Picard operators, Fixed Point Theory, 18 (2017), pp. 229-236.
[22] A. Hussain, M. Adeel, T. Kanwal and N. Sultana, Set valued contraction of Suzuki-Edelstein-Wardowski type and best proximity point results, Bull. Math. Anal. Appl., 10 (2018), pp. 53-67.
[23] M. Jleli and B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), pp. 1-8.
[24] Z. Kadelburg and S. Radenovi, A note on some recent best proximity point results for non-self mappings, Gulf J. Math., 1 (2013), pp. 36-41.
[25] M.H. Labbaf Ghasemi, M.R. Haddadi and N. Eftekhari, Proximity Point Properties for Admitting Center Maps, Sahand Commun. Math. Anal., 15 (2019), pp. 159-167.
[26] A. Latif, M. Abbas and A. Husain, Coincidence best proximity point of $F_g$-weak contractive mappings in partially ordered metric spaces, J. Nonlinear Sci. Appl., 9 (2016), pp. 2448-2457.
[27] S.B. Nadler Jr, Multivalued contraction mappings, Pacific J. Math., 30 (1969), pp. 475-488.
[28] E. Nazari, Best proximity points for generalized muktivalued contractions in metric spaces, Miskolc Math. Notes, 16 (2015), pp. 1055-1062.
[29] H. Piri and S. Rahrovi, Generalized multivalued F-weak contractions on complete metric spaces, Sahand Commun. Math. Anal., 2 (2015), pp. 1-11.
[30] S. Razavi and H. Parvaneh Masiha, Generalized F-contractions in Partially Ordered Metric Spaces, Sahand Commun. Math. Anal., 16 (2019), pp. 93-104.
[31] V. Sankar Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal., 74 (2011), pp. 4804-4808.
[32] J. Von Neuman, Uber ein okonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes, Ergebn. Math. Kolloq., 8 (1937), pp. 73-83.
[33] K. Wlodarczyk, R. Plebaniak and A. Banach, Best proximity points for cyclic and noncyclic setvalued relatively quasi-asymptotic contractions in uniform spaces, Nonlinear Anal., 70 (2009), pp. 3332-3341.
[34] K. Wlodarczyk, R. Plebaniak and C. Obczynski, Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear Anal., 72 (2010), pp. 794-805.
[35] J. Zhang, Y. Su and Q. Cheng, A note on `A best proximity point theorem for Geraghty-contractions', Fixed Point Theory Appl., 2013:99 (2013).