Existence results for hybrid Caputo fractional models for a thermostat with hybrid boundary conditions

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Abstract

In this paper, we study a generalized hybrid fractional thermostat model involving the Caputo fractional derivative and the Riemann–Liouville fractional integral under nonlocal hybrid boundary conditions with feedback control. The problem describes a class of heat regulation systems in which the controller’s response depends on both the process’s thermal memory and sensor measurements taken at interior points of the domain. Working in the Banach space of integral-type Hölder functions J_α,β, we establish sufficient conditions for the existence and uniqueness of solutions to the associated boundary value problem. This functional framework is more suitable than the classical spaces C(E₀) and C^α(E₀), since it provides stronger regularity, better stability under fractional integral operators, and improved continuity properties for the nonlinear superposition terms involved in the hybrid structure. Our analysis is based on fractional calculus techniques, an appropriate measure of noncompactness, and Darbo-type fixed-point theorem. The obtained results extend several existing thermostat and hybrid fractional models in the literature. Finally, a numerical example is presented to illustrate and verify the theoretical findings for different values of α and β.

Keywords

integral-form Hölder space / fixed-point theorem / models of thermostats / measure of noncompactness / hybrid fractional BVP

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References

[1] A. M. A. El-Sayed, H. Zahed, A. B. Humieda, E. M. A. Hamdallah, Solvability of a nonlocal integral problem of integro-mixed-differential equation constrained by a nonlinear Caputo fractional order constraint, Int. J. Anal. Appl., 23 (2025), 250. https://doi.org/10.28924/2291-8639-23-2025-250

[2] Q. Du, M. Gunzburger, R. B. Lehoucq, K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540. https://doi.org/10.1142/S0218202512500546

[3] Q. Du, M. Gunzburger, R. B. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696. https://doi.org/10.1137/110833294

[4] H. H. G. Hashem, A. M. A. El-Sayed, S. M. Al-Issa, Investigating asymptotic stability for hybrid cubic integral inclusion with fractal feedback control, Fractal Fract., 7 (2023), 449. https://doi.org/10.3390/fractalfract7060449

[5] M. M. A. Metwali, On some qualitative properties of integrable solutions for Cauchy-type problem of fractional order, J. Math. Appl., 40 (2017), 121-134. https://doi.org/10.7862/rf.2017.8

[6] R. Conti, Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Unione Mat. Ital., 22 (1967), 135-178.

[7] J. R. L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc., 74 (2006), 673-693. https://doi.org/10.1112/S0024610706023179

[8] J. R. L. Webb, G. Infante, D. Franco, Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 427-446. https://doi.org/10.1017/S0308210506001041

[9] R. Poovarasan, M. E. Samei, V. Govindaraj, An analysis of nonlinear integro-differential equations with four-point nonlocal BVP using Ψ-Caputo fractional derivative, Boundary Value Probl., 2025 (2025), 130. https://doi.org/10.1186/s13661-025-02121-8

[10] J. A. Conejero, J. Franceschi, E. Picó-Marco, Fractional vs. ordinary control systems: What does the fractional derivative provide, Mathematics, 10 (2022), 2719. https://doi.org/10.3390/math10152719

[11] J. Appell, N. Guanda, N. Merentes, J. L. Sanchez, Boundedness and continuity properties of nonlinear composition operators: A survey, Commun. Appl. Anal., 15 (2011), 153.

[12] M. A. Darwish, M. M. A. Metwali, D. O’Regan, On solvability of quadratic Hammerstein integral equations in Hölder spaces, Mat. Vesnik, 74 (2022), 242-248. https://doi.org/10.57016/MV-nuyr4938

[13] J. Appell, E. De Pascale, P. P. Zabrejko, An application of B. N. Sadovskii’s fixed point principle to nonlinear singular equations, Z. Anal. Anwend., 6 (1987), 193-208. https://doi.org/10.4171/zaa/242

[14] M. Cichoń, M. M. A. Metwali, On the Banach algebra of integral-variation type Hölder spaces and quadratic fractional integral equations, Banach J. Math. Anal., 16 (2022), 34. https://doi.org/10.1007/s43037-022-00188-4

[15] J. Banaś, M. Lecko, Fixed points of the product of operators in Banach algebra, Panam. Math. J., 12 (2002), 101-109.

[16] M. A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations, Am. Math. Mon., 75 (1968), 318-319. https://doi.org/10.2307/2315009

[17] B. C. Dhage, B. D. Karande, First order integro-differential equations in Banach algebras involving Carathéodory and discontinuous nonlinearities, Electron. J. Qual. Theory Differ. Equations, 2005 (2005), 1-16.

[18] A. El-Sayed, H. Hashem, S. Al-Issa, Existence of solutions for an ordinary second-order hybrid functional differential equation, Adv. Differ. Equations, 2020 (2020), 296. https://doi.org/10.1186/s13662-020-02742-6

[19] M. A. E. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal., 2014 (2014), 389386. https://doi.org/10.1155/2014/389386

[20] S. Sitho, S. K. Ntouyas, J. Tariboon, Existence results for hybrid fractional integro-differential equations, Boundary Value Probl., 2015 (2015), 113. https://doi.org/10.1186/s13661-015-0376-7

[21] Y. Zhao, S. R. Sun, Z. L. Han, Q. P. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl., 62 (2011), 1312-1324. https://doi.org/10.1016/j.camwa.2011.03.041

[22] L. Zheng, X. Zhang, Modeling and Analysis of Modern Fluid Problems, Elsevier/Academic Press, London, 2017.

[23] S. M. Al-Issa, N. M. Mawed, Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra, J. Nonlinear Sci. Appl., 14 (2021), 181-195. https://doi.org/10.22436/jnsa.014.04.01

[24] P. Guidotti, S. Merino, Gradual loss of positivity and hidden invariant cones in a scalar heat equation, Differ. Integr. Equations, 13 (2000), 1551-1568. https://doi.org/10.57262/die/1356061139

[25] G. Infante, J. R. L. Webb, Loss of positivity in a nonlinear scalar heat equation, Nonlinear Differ. Equations Appl., 13 (2006), 249-261. https://doi.org/10.1007/s00030-005-0039-y

[26] J. R. L. Webb, Existence of positive solutions for a thermostat model, Nonlinear Anal. Real World Appl., 13 (2012), 923-938. https://doi.org/10.1016/j.nonrwa.2011.08.027

[27] J. J. Nieto, J. Pimentel, Positive solutions of a fractional thermostat model, Boundary Value Probl., 2013 (2013), 5. https://doi.org/10.1186/1687-2770-2013-5

[28] D. Baleanu, S. Etemad, S. Rezapour, A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Boundary Value Probl., 2020 (2020), 64. https://doi.org/10.1186/s13661-020-01361-0

[29] M. A. M. Metwali, A. Alahmadi, F. M. Alotaibi, M. E. Samei, On Hadamard fractional operator and three-point fractional boundary value problem in integral-form Hölder spaces, AIMS Math., 11 (2026), 7047-7065. https://doi.org/10.3934/math.2026289

[30] J. Appell, A. Dutkiewicz, B. López, S. Reinwand, K. Sadarangani, Hölder-type spaces, singular operators, and fixed point theorems, Fixed Point Theory, 22 (2021), 31-58. https://doi.org/10.24193/fpt-ro.2021.1.03

[31] J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.

[32] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.