On different generalized interpolative proximal-type contractions in metric spaces with applications

Abstract

In this work, we establish the conditions for ensuring the existence and uniqueness of common best proximity points for non-self-mappings defined on the general metric spaces. A unified theoretical framework is formulated to cover a broad class of contraction mappings. We describe the required conditions on the real-valued functions ($\aleph, \Phi$): $[0, \infty) \rightarrow \mathbb{R}$ and verify that these secure the existence of common best proximity points for ($\aleph, \Phi$)-interpolative contractions in complete metric spaces. The study further extends this concept by examining multiple forms of interpolative proximal-type contractions, such as proximal, Ćirić-Reich-Rus, Kannan, and Hardy-Rogers variants, through the use of the auxiliary functions ($\aleph, \Phi$). Several illustrated examples are included to demonstrate the applicability of our findings. Finally, we conclude with an application involving a nonlinear fractional differential equation, showing that it fully satisfies the assumption of our main result.

Keywords

Interpolative contraction / Best proximity point / Common fixed point / Metric space

References

[1] Fan K. Extensions of two FP theorems of FE Browder. Mathematische zeitschrift. 1969; 112(3): 234-40. https://link.springer.com/article/10.1007/BF01110225

[2] Banach S. On operations in abstract sets and their application to integral equations. Fundamenta mathematicae. 1922; 3(1): 133-81. https://doi.org/10.4064/FM-3-1-133-181

[3] Kannan R. Some results on fixed points. Bull Cal Math Soc. 1968; 60: 71-6. https://www.scirp.org/reference/referencespapers?referenceid=1119410

[4] Sadiq Basha S. Common best proximity points: global minimization of multi-objective functions. J Glob Optim. 2012; 54(2): 367-73. https://link.springer.com/article/10.1007/s10898-011-9760-8

[5] Deep A, Batra R. Common best proximity point theorems under proximal F-weak dominance in complete MS. J Anal. 2023; 31(4): 2513-29. https://link.springer.com/article/10.1007/s41478-023-00570-x

[6] Mondal S, Dey LK. Some common best proximity point theorems in a complete metric space. Afrika Matematika. 2017; 28(1): 85-97. https://link.springer.com/article/10.1007/s13370-016-0432-1

[7] Younis M, Abdou AA. Novel fuzzy contractions and applications to engineering science. Fractal Fract. 2023; 8(1): 28. https://doi.org/10.3390/fractalfract8010028

[8] Sadiq Basha S. Common best proximity points: global minimal solutions. Top. 2013; 21(1): 182-8. https://link.springer.com/article/10.1007/s11750-011-0171-2

[9] Shahzad N, Sadiq Basha S, Jeyaraj R. Common best proximity points: global optimal solutions. J Optim Theory Appl. 2011; 148(1): 69-78. https://link.springer.com/article/10.1007/s10957-010-9745-7

[10] Altun I, Taşdemir A. On best proximity points of interpolative proximal contractions. Quaest Math. 2021; 44(9): 1233-41. https://doi.org/10.2989/16073606.2020.1785576

[11] Adhikari N. Interpolative contraction and discontinuity at fixed point on partial MS. Nepal J Math Sci. 2025; 6(1): 51-60. https://doi.org/10.3126/njmathsci.v6i1.77378

[12] Proinov PD. Fixed point theorems for generalized contractive mappings in MS. J Fixed Point Theory Appl. 2020; 22(1): 21. https://link.springer.com/article/10.1007/s11784-020-0756-1

[13] Malkawi AA. Fixed point theorem in mr-MS via integral type contraction. WSEAS Trans Math. 2025; 24: 295-9. https://doi.org/10.37394/23206.2025.24.28

[14] Makran N, Hammouti O, Taarabti S. A common fixed point result for multi-valued mappings in Hausdorff modular fuzzy b-MS with application to integral inclusions. Analysis. 2025; 45(1): 35-50. https://www.degruyterbrill.com/document/doi/10.1515/anly-2023-0081/html

[15] Ishtiaq U, Jahangeer F, Garayev M, Popa IL. Existence and uniqueness of a solution of a boundary value problem used in chemical sciences via a fixed point approach. Symmetry. 2025; 17(1): 127. https://doi.org/10.3390/sym17010127

[16] Karapınar E. Edelstein type fixed point theorems. Fixed Point Theory Appl. 2012; 2012(1): 107. https://link.springer.com/article/10.1186/1687-1812-2012-107

[17] Younıs M, Mutlu A, Ahmad H. C’iric’ Contraction with Graphical Structure of Bipolar MS and Related Fixed Point Theorems. Hacet J Math Stat. 2024: 1-9. https://dergipark.org.tr/en/pub/hujms/issue/42398/1302743

[18] Ishtiaq U, Jahangeer F, Kattan DA, Argyros IK, Regmi S. On Orthogonal Fuzzy Interpolative Contractions with Applications to Volterra Type Integral Equations and Fractional Differential Equations. Axioms. 2023; 12(8): 725. https://doi.org/10.3390/axioms12080725

[19] Saleem N, Isik H, Khaleeq S, Park C. Interpolative Ćirić-Reich-Rus-type best proximity point results with applications. AIMS Math. 2022; 7(6): 9731-47. https://doi.org/10.3934/math.2022542

[20] Deng J, Liu XL, Sun Y, Rathour L. Some best proximity point results of several α−ψ interpolative proximal contractions. Nonlinear Funct Anal Appl. 2022; 27(3): 533-51. https://scholar.kyobobook.co.kr/article/detail/4040047209552

[21] Ishtiaq U, Jahangeer F, Kattan DA, Argyros IK. Generalized Common Best Proximity Point Results in Fuzzy MS with Application. Symmetry. 2023; 15(8): 1501. https://doi.org/10.3390/sym15081501

[22] Younis M, Bahuguna D. A unique approach to graph-based MS with an application to rocket ascension. Comput Appl Math. 2023; 42(1): 44. https://link.springer.com/article/10.1007/s40314-023-02193-1

[23] Ishtiaq U, Jahangeer F, Kattan DA, De la Sen M. Generalized common best proximity point results in fuzzy multiplicative MS. Aims Math. 2023; 8: 25454-76. https://doi.org/10.3934/math.20231299

[24] Heidary Joonaghany G, Farajzadeh A, Azhini M, Khojasteh F. A New Common Fixed Point Theorem for Suzuki Type Contractions via Generalized Ψ-simulation Functions. Sahand Commun Math Anal. 2019; 16(1): 129-48. https://doi.org/10.22130/scma.2018.78315.359

[25] Ramaswamy R, Murthy PP, Sahu P, Alkhowaiter RA, Abdelnaby OA, Mani G. Application of fixed point result to the boundary value problem using the M-type generalized contraction condition for best proximity point considerations. AIMS Math. 2025; 10(6): 13622-39. https://doi.org/10.3934/math.2025613

[26] Janardhanan G, Mani G, Mitrović ZD, Aloqaily A, Mlaiki N. Best proximity point results on R-MS with applications to fractional differential equation and production-consumption equilibrium. J Math Comput Sci. 2025; 38(1): 45-55. https://dx.doi.org/10.22436/jmcs.038.01.04

[27] Gnanaprakasam AJ, Nallaselli G, Haq AU, Mani G, Baloch IA, Nonlaopon K. Common fixed-points technique for the existence of a solution to fractional integro-differential equations via orthogonal Branciari MS. Symmetry. 2022; 14(9): 1859. https://doi.org/10.3390/sym14091859

[28] Younis M, Öztürk M. Some novel proximal point results and applications. Univ J Math Appl. 2025; 8(1): 8-20. https://doi.org/10.32323/ujma.1597874

[29] Karapınar E, Alqahtani O, Aydi H. On interpolative Hardy-Rogers type contractions. Symmetry. 2018; 11(1): 8. https://doi.org/10.3390/sym11010008

[30] Babu DR, Koduru NK. Interpolative Contractions for b-MS and Their Applications. Eur J Pure Appl Math. 2025; 18(3): 6113-. https://doi.org/10.29020/nybg.ejpam.v18i3.6113

[31] Younis M, Ahmad H, Asmat F, Öztürk M. Analyzing Helmholtz phenomena for mixed boundary values via graphically controlled contractions. Math Model Anal. 2025; 30(2): 342-61. https://doi.org/10.3846/mma.2025.22546

[32] Ma C, You D, Liu J, Li M, He J, Totis G. Topology optimization of cooling elements for worm wheel gear grinding machine tool bed under non-uniform heat sources. Appl Thermal Eng. 2025: 128739. https://doi.org/10.1016/j.applthermaleng.2025.128739

[33] Xu K, Fan L, Chen C, Shen C, Jiang Z, Wei Y. Analysis of dynamic coupling characteristics and multi-constraint optimization of a proton exchange membrane fuel cell considering membrane degradation. Fuel. 2026; 404: 136275. https://doi.org/10.1016/j.fuel.2025.136275

[34] Yue T. Some results on the nonuniform polynomial dichotomy of discrete evolution families. Hiroshima Math J. 2025; 55(2): 183-201. https://doi.org/10.32917/h2024003

[35] Jahangeer F, Alshaikey S, Ishtiaq U, Lazăr TA, Lazăr VL, Guran L. Certain Interpolative Proximal Contractions, Best Proximity Point Theorems in Bipolar MS with Applications. Fractal Fract. 2023; 7(10): 766. https://doi.org/10.3390/fractalfract7100766

[36] Ahmad H, Din FU, Younis M. A novel Ćirić-Reich-Rus fixed point approach for the existence and uniqueness criterion of a fractional-order Aizawa chaotic system. Chaos, Solitons & Fractals. 2025; 200: 116932. https://doi.org/10.1016/j.chaos.2025.116932

[37] Ali MU, Din Fu, Kamran T., Houmani H. Best proximity points of F-proximal contractions under the influence of an α-FUNCTION. Univ Politehn Buch Sci Bull A Appl Math Phys. 2017; 79(4): 3-18. https://www.scientificbulletin.upb.ro/rev-docsarhiva/full21e-394676

[38] Unni AS, Pragadeeswarar V, De la Sen M. Common best proximity point theorems for proximally weak reciprocal continuous mappings. AIMS Math. 2023; 8(12): 28176-87. https://doi.org/10.3934/math.20231442

[39] Vaithilingam SR, Anisha K. A common best proximity point theorem for relatively nonexpansive mappings: SR Vaithilingam, K. Anisha. J Anal. 2025; 33(6): 2897-907. https://link.springer.com/article/10.1007/s41478-025-00950-5

[40] Pragadeeswarar V, Gopi R. Existence of common best proximity point for single and multivalued non-self mappings. Carpath J Math. 2021; 37(2): 273-85. https://www.jstor.org/stable/27082105

[41] Girgin E. Enhancing Generalized Interpolative Contraction Through Simulation Functions. Math Sci Appl E-Notes. 2025; 13(1): 54-64. https://doi.org/10.36753/mathenot.1573566