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    EXISTENCE AND STABILITY RESULTS FOR COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING AB-CAPUTO DERIVATIVE
    (World Scientific Publ Co Pte Ltd, 2023) Mehmood, Nayyar; Abbas, Ahsan; Akgul, Ali; Abdeljawad, Thabet; Alqudah, Manara A.
    In this paper, we use Krasnoselskii's fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative ABC(0)D(alpha)??(l) = zeta(l,??(l),P(l)), 1 < alpha & <= 2, (SIC) AB( )C(0)D(sigma)P(l) = xi(l,??(l),P(l)), 1 < sigma <= 2,f or alll is an element of [0, 1], with boundary conditions (SIC) ??(0) = 0, lambda??'(eta) = gamma??'(1), P(0) = 0,lambda'(eta) = gamma'(1).We discuss uniqueness with the help of the Banach contraction principle. The criteria for Hyers-Ulam stability of given AB-Caputo fractional-coupled boundary value problem (BVP) is also discussed. Some examples are provided to validate our results. In Example 1, we find a unique and stable solution of AB-Caputo fractional-coupled BVP. In Example 2, the analysis of approximate and exact solutions with errors of nonlinear integral equations is elaborated with graphs.
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    EXISTENCE AND STABILITY RESULTS OF FRACTIONAL DIFFERENTIAL EQUATIONS MITTAG-LEFFLER KERNEL
    (World Scientific Publ Co Pte Ltd, 2024) Abbas, Ahsan; Mehmood, Nayyar; Akgul, Ali; Amacha, Inas; Abdeljawad, Thabet
    This paper presents the following AB-Caputo fractional boundary value problem (ABC)(0)D(alpha)u(sigma) = G(sigma, u(sigma)), sigma is an element of[0, 1] with integral-type boundary conditions u(0) = 0 = u ''(0), gamma u(1) = lambda integral(1)(0) g(1)(kappa)u(kappa)d kappa, of order 2 < alpha <= 3. Schauder and Krasnoselskii's fixed point theorems are used to find existence results. Uniqueness is obtained via the Banach contraction principle. To investigate the stability of a given problem, Hyers-Ulam stability is discussed. An example is provided to validate our results.
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    EXISTENCE RESULTS FOR ABC-FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED AND INTEGRAL TYPE OF BOUNDARY CONDITIONS
    (World Scientific Publ Co Pte Ltd, 2021) Mehmood, Nayyar; Abbas, Ahsan; Abdeljawad, Thabet; Akgul, Ali
    This paper presents a study on the existence theory of fractional differential equations involving Atangana-Baleanu (AB) derivative of order 1 < alpha <= 2, with non-separated and integral type boundary conditions. An existence result for the solutions of given AB-fractional differential equation is proved using Krasnoselskii's fixed point theorem, while the uniqueness of the solution is obtained using Banach contraction principle. Some conditions are proposed under which the given boundary value problem is Hyers-Ulam stable. Examples are given to validate our results.
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    EXISTENCE RESULTS FOR MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS INVOLVING ATANGANA-BALEANU DERIVATIVE
    (World Scientific Publ Co Pte Ltd, 2023) Abbas, Ahsan; Mehmood, Nayyar; Akgul, Ali; Abdeljawad, Thabet; Alqudah, Manar A.
    In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP) (delta(2)0(ABC)D(alpha+2) + delta( 1)0(ABC)D(alpha+1) + delta (0)0(ABC)D(alpha))x(t) = zeta(t,x(t))of order 0 < alpha < 1 with nonlocal boundary conditions have been derived by using Krasnoselskii's fixed point theorem. The uniqueness of the solution is obtained with the help of Banach contraction principle. Examples are provided to confirm our obtained results.