On $\theta$-convex contractive mappings with application to integral equations

Year 2024, Volume: 73 Issue: 1, 13 - 24, 16.03.2024

Merve Özkan , Murat Özdemir , İsa Yildirim

https://doi.org/10.31801/cfsuasmas.1302945

Abstract

The fundamental goal of our paper is to study $\theta$-convex contractive mappings in metric spaces. We demonstrate some fixed point results for such mappings. Also, we give an application to integral equations of our results. Consequently, our results encompass numerous generalizations of the Banach contraction principle on metric space.

Keywords

Integral equation, fixed point, convex contraction, θ-contraction

References

• Banach, S., Sur les operationes dans les ensembles abstraits et leur application aux equation integrale, Fundam. Math., 3(1) (1922), 133-181.
• Berinde, V., Pacurar, M., Fixed points and continuity of almost contractions, Fixed Point Theory, 9(1) (2008), 23–34.
• Berinde, V., Approximating fixed points of weak contraction using the Picard iteration, Nonlinear Anal. Forum, 9 (2004), 43–53.
• Berinde, M., Berinde, V., On general class of multivalued weakly Picard mappings, J. Nonlinear Sci. Appl., 326(2) (2007), 772–782. https://doi.org/10.1016/j.jmaa.2006.03.016
• Jleli, M., Samet, B., A new generalization of the Banach contractive principle, J. Inequal. Appl., 38 (2014). https://doi.org/10.1186/1029-242X-2014-38
• Jleli, M., Karapınar, E., Samet, B., Further generalizations of the Banach contraction principle, J. Inequal. Appl., 439 (2014). https://doi.org/10.1186/1029-242X-2014-439
• Hussain, N., Parvaneh, V., Samet, B., Vetro, C., Some fixed point theorems for generalized contractive mappings in complete metric spaces, Fixed Point Theory Appl., 17 (2015). https://doi.org/10.1186/s13663-015-0433-z
• Imdad, M., Alfaqih, W. M., Khan, I. A., Weak $\theta$−contractions and some fixed point results with applications to fractal theory, Adv. Differ. Equ., 439 (2018). https://doi.org/10.1186/s13662-018-1900-8
• Istratescu, V. I., Some fixed point theorems for convex contraction mappings and convex non-expansive mapping, Libertas Mathematica, 1 (1981), 151–163.
• Istratescu, V. I., Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters-I, Ann. Mat. Pura Appl., 130 (1982), 89–104. https://doi.org/10.1007/BF01761490
• Istratescu, V. I., Some fixed point theorems for convex contraction mappings and mappings with convex diminishing diameters-II, Ann. Mat. Pura Appl., 134 (1983), 327–362. https://doi.org/10.1007/BF01773511
• Alghamdi, M. A., Alnafei, S. H., Radenovic, S., Shahzad, N., Fixed point theorems for convex contraction mappings on cone metric spaces, Math. Comput. Model., 54 (2011), no. 9-10, 2020–2026. https://doi.org/10.1016/j.mcm.2011.05.010
• Ampadu, C. K. B., On the analogue of the convex contraction mapping theorem for tricyclic convex contraction mapping of order 2 in b-metric space, J. Global Research in Math. Archives, 4(6) (2017), 1–5.
• Ampadu, C. K. B., Some fixed point theory results for convex contraction mapping of order 2, JP J. Fixed Point Theory Appl., 12(2-3) (2017), 81–130.
• Bisht, R. K., Hussain, N., A note on convex contraction mappings and discontinuity at fixed point, J. Math. Anal., 8(4) (2017), 90-96.
• Bisht, R. K., Rakocevic, V., Fixed points of convex and generalized convex contractions, Rendiconti Del Circolo Matematico Di Palermo Series 2, 69 (2020), 21-28. https://doi.org/10.1007/s12215-018-0386-2
• Georgescu, F., IFS consisting of generalized convex contractions, An. Stiint. Univ. Ovidius Constant., 25(1) (2017), 77–86, https://doi.org/10.1515/auom-2017-0007
• Khan, M. S., Singh, Y. M., Maniu, G., Postolache, M., On generalized convex contractions of type−2 in b-metric and 2−metric spaces, J. Nonlinear Sci. Appl., 10 (2017), 2902–2913. https://doi.org/10.22436/jnsa.010.06.05
• Khan, M. S., Singh, Y. M., Maniu, G., Postolache, M., On $(\alpha, p)$−convex contraction and asymptotic regularity, J. Math. Comput. Sci., 18 (2018), 132–145. https://doi.org/10.22436/jmcs.018.02.01
• Karapınar, E., Fulga, A., Petrusel, A., On Istratescu type contractions in b−metric spaces, Mathematics, 8(3) (2020), 388. https://doi.org/10.3390/math8030388
• Latif, A., Sintunavarat, W., Ninsri, A., Approximate fixed point theorems for partial generalized convex contraction mappings in a−complete metric spaces, Taiwanese J. Math., 19(1) (2015), 315–333. https://doi.org/10.11650/tjm.19.2015.4746
• Miandarag, M. A., Postolache, M., Rezapour, S., Approximate fixed points of generalizes convex contractions, Fixed Point Theory Appl., 255 (2013), 1–8. https://doi.org/ 10.1186/1687-1812-2013-255
• Miculescu, R., Mihail, A., A generalization of Istratescu’s fixed piont theorem for convex contractions, Fixed Point Theory, 18(2) (2017), 689-702. https://doi.org/10.24193/fptro.2017.2.55
• Singh, Y. M., Khan, M. S., Kang, S. M., F-Convex contraction via admissible mapping and related fixed point theorems with an application, Mathematics, 6(6) (2018), 105. https://doi.org/10.3390/math6060105
• Ciric, L. B., Generalized contractions and fixed point theorems, Publ. Inst. Math., 12(26) (1971), 19–26, https://doi.org/10.2307/2039517
• Gabeleh, M.,Vetro, C., A best proximity point approach to existence of solutions for a system of ordinary differential equations, Bull. Belg. Math. Soc., 26(4) (2019), 493-503. https://doi.org/10.36045/bbms/1576206350
• Asadi, M., Gabeleh M., Vetro, C., A new approach to the generalization of Darbo’s fixed point problem by using simulation functions with application to integral equations, Results Math., 74(86) (2019), 1-15. https://doi.org/10.1007/s00025-019-1010-2