Generalization of Inequalities for Different Types of Functions via ψ-Caputo Fractional Derivative
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Authors: A. TAQBIBT, M. CHAIB AND M. EL OMARI
DOI: 10.46793/KgJMat2607.1171T
Abstract:
In this manuscript, we present a collection of inequalities that extend and generalize the important results of the findings from [?], with the latter serving as specific cases within our study. Our research takes many forms of fractional differential inequalities, including the comprehensive structure of the Caputo fractional derivative operator with respect to a function ψ. Furthermore, we demonstrate the practical applications of σ-convex functions in the domain of fractional calculus within this framework.
Keywords:
Convex functions, σ-convex functions, fractional inequalities, ψ-Caputo fractional operators.
References:
[1] M. S. Abdo, S. K. Panachal and A. M. Saeed, Fractional boundary value problem with ψ-Caputo fractional derivative, Proc. Indian Acad. Sci. 12 (9) (2019), Article ID 65. https://doi.org/10.1007/s12044-019-0514-8
[2] M. Bombardelli and S. Varosanec, Properties of h-convex functions related to the Hermite-Hadamard-Fejer inequalities, Comput. Math. Appl. 58 (2009), 1869–1877. https://doi.org/10.1016/j.camwa.2009.07.073
[3] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal International 13(5) (1967), 529–539.
[4] N. Chefnaj, A. Taqbibt, K. Hilal and S. Melliani, Study of nonlocal boundary value problems for hybrid differential equations involving ψ-Caputo fractional derivative with measures of noncompactness, J. Math. Sci. 271 (2023), 1–10.
[5] N. Chefnaj, A. Taqbibt, K. Hilal, S. Melliani and A. Kajouni, Boundary value problems for differential equations involving the generalized Caputo-Fabrizio fractional derivative in b-metric spaces, Turkish J. Sci. 8(1) (2023), 24–36.
[6] S. S. Dragomir, Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Proyecciones 34(4) (2015), 323–341. https://doi.org/10.4067/S0716-09172015000400002
[7] S. Erdas, E. Ünlüyol and S. Salas, Some new inequalities of operator m-convex functions and applications for Synchronous-Asynchronous functions, Complex Anal. Oper. Theory 13 (8), 3871–3881.
[8] G. Farid, A. Javed and A. U. Rehman, On Hadamard inequalities for n-times differentiable functions which are relative convex via Caputo k-fractional derivatives, Nonlinear Anal. Forum 22(2) (2017), 17–28.
[9] G. Farid, A. Javed and A. U. Rehman, Fractional integral inequalities of Hadamard-type for m-convex functions via Caputo k-fractional derivatives, J. Fract. Calc. Appl. 10(1) (2019), 120–134.
[10] G. Farid, A. Javed and S. Naqvi, Hadamard and Fejer Hadamard inequalities and related results via Caputo fractional derivatives, Bull. Math. Anal. Appl. 9(3) (2017), 16–30.
[11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006.
[12] A.Taqbibt, N. Chefnaj, K. Hilal and S. Melliani, ψ-Caputo fractional differential equations with maxima on time scales, J. Math. Sci. (2024), 1–13. https://doi.org/10.1007/s10958-024-07034-y
[13] A. Taqbibt, M. Elomari and S. Melliani, Nonlocal semilinear ϕ-Caputo fractional evolution equation with a measure of noncompactness in Banach space, Filomat 37(20) (2023), 6877–6890.
[14] D. Ucar, Inequalities for different type of functions via Caputo fractional derivative, AIMS Mathematics 7(7) (2022), 12815–12826.
[15] E. Ünlüyol, Y. Erdas and S. Salas, Operator (α,m)-convex functions and applications for synchronous and asynchronous functions, Mathematical Sciences and Applications E-Notes 7(2) (2019), 225–236.
[16] S. Wu, M. U. Awan, M. A. Noor, K. I. Noor and S. Iftikhar, On a new class of convex functions and integral inequalities, J. Inequal. Appl. 2019 (2019), 1–14.
[17] J. Zuo, A. Taqbibt, M. Chaib, M., ELomari and J. V. D. C. Sousa, An existence and uniqueness of mild solutions of fractional evolution problems, Comput. Appl. Math. 43(424) (2024). https://doi.org/10.1007/s40314-024-02943-9
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