Nonlinear almost $ F $-contractions in relational metric space with an application to periodic boundary value problems

Ahmed Alamer; Faizan Ahmad Khan; Ahmed Alamer; Faizan Ahmad Khan

doi:10.3934/math.2026312

AIMS Mathematics

2026, Volume 11, Issue 3: 7593-7609. doi: 10.3934/math.2026312

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Nonlinear almost $ F $-contractions in relational metric space with an application to periodic boundary value problems

Ahmed Alamer ^,,
Faizan Ahmad Khan ^,

Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk-71491, Saudi Arabia

Received: 22 January 2026 Revised: 17 February 2026 Accepted: 27 February 2026 Published: 23 March 2026
MSC : 34B15, 47H10, 54H25

In this paper, we establish some fixed-point outcomes for a nonlinear almost $ F $-contraction map in a metric space combined with a locally $ \mathcal{L} $-transitive binary relation. Numerous previous insights are expanded, developed, improved, and consolidated in the outcomes reported herein. We propose a few instances in order to illustrate our findings. As an application of our findings, we explore the existence of a solution of a first-order periodic boundary value problem.
- fixed points,
- $ F $-contractions,
- $ \Gamma $-continuity,
- locally transitive relations,
- boundary value problems
Citation: Ahmed Alamer, Faizan Ahmad Khan. Nonlinear almost $ F $-contractions in relational metric space with an application to periodic boundary value problems[J]. AIMS Mathematics, 2026, 11(3): 7593-7609. doi: 10.3934/math.2026312

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Abstract

In this paper, we establish some fixed-point outcomes for a nonlinear almost $ F $-contraction map in a metric space combined with a locally $ \mathcal{L} $-transitive binary relation. Numerous previous insights are expanded, developed, improved, and consolidated in the outcomes reported herein. We propose a few instances in order to illustrate our findings. As an application of our findings, we explore the existence of a solution of a first-order periodic boundary value problem.

References

[1]	K. Zhao, X. Zhao, X. Lv, A general framework for the multiplicity of positive solutions to higher-order Caputo and Hadamard fractional functional differential coupled Laplacian systems, Fractal Fract., 9 (2025), 701. https://doi.org/10.3390/fractalfract9110701 doi: 10.3390/fractalfract9110701
[2]	K. Zhao, A generalized stochastic Nicholson blowfly model with mixed time-varying lags and harvest control: Almost periodic oscillation and global stable behavior, Adv. Cont. Discr. Mod., 2025 (2025), 171. https://doi.org/10.1186/s13662-025-04032-5 doi: 10.1186/s13662-025-04032-5
[3]	A. Alam, M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl., 17 (2015), 693–702. https://doi.org/10.1007/s11784-015-0247-y doi: 10.1007/s11784-015-0247-y
[4]	F. Sk, F. A. Khan, Q. H. Khan, A. Alam, Relation-preserving generalized nonlinear contractions and related fixed point theorems, AIMS Mathematics, 7 (2022), 6634–6649. https://doi.org/10.3934/math.2022370 doi: 10.3934/math.2022370
[5]	A. Alamer, F. A. Khan, Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems, AIMS Mathematics, 9 (2024), 6266–6280. https://doi.org/10.3934/math.2024305 doi: 10.3934/math.2024305
[6]	D. Filali, F. A. Khan, Suzuki-type weak contractions in relational metric space and applications to boundary value problems, IEEE Access, 14 (2026), 1919–1927. https://doi.org/10.1109/ACCESS.2025.3648466 doi: 10.1109/ACCESS.2025.3648466
[7]	D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 94. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94
[8]	M. Turinici, Wardowski implicit contractions in metric spaces, arXiv: 1211.3164, 2013. https://doi.org/10.48550/arXiv.1211.3164
[9]	H. Piri, P. Kumam, Some fixed point theorems concerning $F$-contraction in complete metric spaces, Fixed Point Theory Appl., 2014 (2014), 210. https://doi.org/10.1186/1687-1812-2014-210 doi: 10.1186/1687-1812-2014-210
[10]	F. Vetro, $F$-contractions of Hardy-Rogers type and application to multistage decision processes, Nonlinear Anal. Model. Control, 21 (2016), 531–546. https://doi.org/10.15388/NA.2016.4.7 doi: 10.15388/NA.2016.4.7
[11]	D. Wardowski, Solving existence problems via $F$-contractions, Proc. Amer. Math. Soc., 146 (2018), 1585–1598.
[12]	M. Arif, M. Imdad, Fixed point results under nonlinear Suzuki $(F, \mathcal{R}^{\neq})$-contractions with an application, Filomat, 36 (2022), 3155–3165. https://doi.org/10.2298/FIL2209155A doi: 10.2298/FIL2209155A
[13]	K. Sawangsup, W. Sintunavarat, A. F. R. L. de Hierro, Fixed point theorems for $F_{\Re}$-contractions with applications to solution of nonlinear matrix equations, J. Fixed Point Theory Appl., 19 (2017), 1711–1725. https://doi.org/10.1007/s11784-016-0306-z doi: 10.1007/s11784-016-0306-z
[14]	M. Imdad, Q. H. Khan, W. M. Alfaqih, R. Gurban, A relation-theoretic $(F, \mathcal{R})$-contraction principle with applications to matrix equations, Bull. Math. Anal. Appl., 10 (2018), 1–12.
[15]	K. Sawangsup, W. Sintunavarat, New algorithm for finding the solution of nonlinear matrix equations based on the weak condition with relation-theoretic $F$-contractions, J. Fixed Point Theory Appl., 23 (2021), 20. https://doi.org/10.1007/s11784-021-00859-z doi: 10.1007/s11784-021-00859-z
[16]	E. Karapinar, A. Fulga, R. P. Agarwal, A survey: $F$-contractions with related fixed point results, J. Fixed Point Theory Appl., 22 (2020), 69. https://doi.org/10.1007/s11784-020-00803-7 doi: 10.1007/s11784-020-00803-7
[17]	V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum., 9 (2004) 43–53.
[18]	S. K. Chatterjea, Fixed point theorem, C. R. Acad. Bulg. Sci., 25 (1972), 727–730.
[19]	R. Kannan, Some results on fixed points, Bull. Cal. Math. Soc., 60 (1968), 71–76.
[20]	T. Zamfirescu, Fix point theorems in metric spaces, Arch. Math., 23 (1972), 292–298. https://doi.org/10.1007/BF01304884 doi: 10.1007/BF01304884
[21]	Lj. B. Ćirić, A generalization of Banach's contraction principle, Proc. Am. Math. Soc., 45 (1974), 267–273. https://doi.org/10.2307/2040075 doi: 10.2307/2040075
[22]	M. R. Alfuraidan, M. Bachar, M. A. Khamsi, Almost monotone contractions on weighted graphs, J. Nonlinear Sci. Appl., 9 (2016), 5189–5195. http://dx.doi.org/10.22436/jnsa.009.08.04 doi: 10.22436/jnsa.009.08.04
[23]	V. Berinde, M. Păcurar, Fixed points and continuity of almost contractions, Fixed Point Theory, 9 (2008), 23–34.
[24]	G. V. R. Babu, M. L. Sandhy, M. V. R. Kameshwari, A note on a fixed point theorem of Berinde on weak contractions, Carpathian J. Math., 24 (2008), 8–12.
[25]	M. A. Alghamdi, V. Berinde, N. Shahzad, Fixed points of non-self almost contractions, Carpathian J. Math., 30 (2014), 7–14.
[26]	S. Lipschutz, Schaum's outlines of theory and problems of set theory and related topics, 2 Eds., McGraw Hill, 1998.
[27]	A. Alam, M. Imdad, Relation-theoretic metrical coincidence theorems, Filomat, 31 (2017), 4421–4439. https://doi.org/10.2298/FIL1714421A doi: 10.2298/FIL1714421A
[28]	B. Kolman, R. Busby, S. C. Ross, Discrete mathematical structures, 6 Eds., Pearson/Prentice Hall, 2009.
[29]	A. Alam, M. Imdad, Nonlinear contractions in metric spaces under locally $T$-transitive binary relations, Fixed Point Theory, 19 (2018), 13–24. https://doi.org/10.24193/fpt-ro.2018.1.02 doi: 10.24193/fpt-ro.2018.1.02
[30]	M. Turinici, Weakly contractive maps in altering metric spaces, ROMAI J., 9 (2013), 175–183.
[31]	J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239. https://doi.org/10.1007/s11083-005-9018-5 doi: 10.1007/s11083-005-9018-5
[32]	M. Hasanuzzaman, M. Imdad, H. N. Saleh, On modi1ed $\mathcal{L}$-contraction via binary relation with an application, Fixed Point Theory, 23 (2022), 267–278. https://doi.org/10.24193/fpt-ro.2022.1.17 doi: 10.24193/fpt-ro.2022.1.17
[33]	R. Jain, H. K. Nashine, Z. Kadelburg, Positive solutions of nonlinear matrix equations via fixed point results in relational metric spaces with $w$-distance, Filomat, 36 (2022), 4811–4829. https://doi.org/10.2298/FIL2214811J doi: 10.2298/FIL2214811J
[34]	S. Shukla, N. Dubey, Some fixed point results for relation theoretic weak $\varphi$-contractions in cone metric spaces equipped with a binary relation and application to the system of Volterra type equation, Positivity, 24 (2020), 1041–1059. https://doi.org/10.1007/s11117-019-00719-8 doi: 10.1007/s11117-019-00719-8
[35]	A. Alamer, N. H. E. Eljaneid, M. S. Aldhabani, N. H. Altaweel, F. A. Khan, Geraghty type contractions in relational metric space with applications to fractional differential equations, Fractal Fract., 7 (2023), 565. https://doi.org/10.3390/fractalfract7070565 doi: 10.3390/fractalfract7070565
[36]	E. A. Algehyne, N. H. Altaweel, M. Areshi, F. A. Khan, Relation-theoretic almost $\phi$-contractions with an application to elastic beam equations, AIMS Mathematics, 8 (2023), 18919–18929. https://doi.org/10.3934/math.2023963 doi: 10.3934/math.2023963

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