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    Conditions to guarantee the existence of solutions for a nonlinear and implicit integro-differential equation with variable coefficients
    (WILEY, 2022) Li, Chenkuan; Saadati, Reza; Allahviranloo, Tofigh
    Using Babenko's approach, multivariate Mittag-Leffler (MM-L) function, and Krasnoselskii's fixed point theorem, we first investigate the existence of solutions to a Liouville-Caputo nonlinear integro-differential equation with variable coefficients and initial conditions in a Banach space. Then the existence of a positive solution for a variant equation is studied. Finally, we provide examples to illustrate the applications of the main results obtained.
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    A novel stability study on volterra integral equations with delay (VIE-D) using the fuzzy minimum optimal controller in matrix-valued fuzzy Banach spaces
    (Springer Heidelberg, 2023) Eidinejad, Zahra; Saadati, Reza; Allahviranloo, Tofigh; Li, Chenkuan
    Our main goal in this article is to investigate the Hyers-Ulam-Rassias stability (HURS) for a type of integral equation called Volterra integral equation with delay (VIE-D). First, by considering special functions such as the Wright function (WR), Mittag-Leffler function (ML), Gauss hypergeometric function (GH), H-Fox function (H-F), and also by introducing the aggregation function, we select the best control function by performing numerical calculations to investigate the stability of the desired equation. In the following, using the selected optimal function, i.e., the minimum function, we prove the existence of a unique solution and the HURS of the VI-D equation in the matrix-valued fuzzy space (MVFS) with two different intervals. At the end of each section, we provide a numerical example of the obtained results.
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    The existence of a unique solution and stability results with numerical solutions for the fractional hybrid integro-differential equations with Dirichlet boundary conditions
    (Springer Science and Business Media Deutschland GmbH, 2024) Eidinejad, Zahra; Saadati, Reza; Vahidi, Javad; Li, Chenkuan; Allahviranloo, Tofigh
    In this paper, we investigate the fractional hybrid integro-differential equations with Dirichlet boundary conditions. We first prove the existence of a unique solution for the equation using a fixed point technique. Our main goal is to obtain the best approximation using optimal controllers. After studying the stability, we present the reproducing kernel Hilbert space numerical method to obtain approximate solutions to the equation. We finally conclude with numerical results. © The Author(s) 2024.