This study is devoted to studying the existence and uniqueness of solutions for Hadamard implicit fractional differential equations with generalized Hadamard fractional integro-differential boundary conditions by utilizing the contraction principle of the Banach and Leray–Schauder fixed point theorems. Moreover, with two different approaches, the Hyers–Ulam stabilities are also discussed. Different ordinary differential equations of the third order with different boundary conditions (e.g., initial, anti periodic and integro-differential) can be obtained as a special case for our proposed model. Finally, for verification, an example is presented, and some graphs for the particular variables and particular functions are drawn using MATLAB.

The existence of maximal and minimal positive solutions for singular infinite-point p-Laplacian fractional differential equation is investigated in this paper. Green's function is derived, and some properties of Green's function are obtained. Based upon these properties of Green's function, the existence of maximal and minimal positive solutions is obtained, and iterative schemes are established for approximating the maximal and minimal positive solutions.

By using the method of mixed monotone operator, a unique positive solution is obtained for a singular p-Laplacian boundary value system with infinite-point boundary conditions in this paper. Green’s function is derived and some useful properties of the Green’s function are obtained. Based upon these new properties and by using mixed monotone operator, the existence results of the positive solutions for the boundary value problem are established. Moreover, the unique positive solution that we obtained in this paper is dependent on λ , μ $\lambda ,\mu $ , and an iterative sequence and convergence rate, which are important for practical application, are given. An example is given to demonstrate the application of our main results.

Based on the spectral analysis method, Gelfand’s formula, and the cones fixed point theorem, some positive solutions with their existence of a nonlinear infinite-point Hadamard fractional-order differential equation is achieved on the interval [a, b] under some conditions, and particularly the nonlinear term allows singularities for time and spatial parameters in the present study. Finally, an analysis case is carried out to reveal the principal results.

Numerical solutions and approximate solutions of fractional differential equations have been studied by mathematicians recently and approximate solutions and exact solutions of fractional differential equations are obtained in many kinds of ways, such as Lie symmetry, variational method, the optimal ADM method, and so on. In this paper, we obtain the positive solutions by iterative methods for sum operators. Green’s function and the properties of Green’s function are deduced, then based on the properties of Green’s function, the existence of iterative positive solutions for a nonlinear Caputo–Hadamard infinite-point fractional differential equation are obtained by iterative methods for sum operators; an example is proved to illustrate the main result.