In this research, we investigated the Riemann-Liouville fractional-order pantograph differential equation constrained by nonlocal and weighted pantograph integral constraints. We presented novel sufficient conditions for the uniqueness of the solution. Moreover, we analyzed the continuous dependence of the solution on some functions and parameters. Additionally, we proved the Hyers-Ulam stability of the problem. To demonstrate the applicability of our results, we included several examples. The present study was located in the space L1[0, T]. The techniques of Schauder’s fixed point theorem and Kolmogorov’s compactness criterion were the primary tools utilized in this work. These contributions offer a comprehensive framework for understanding the qualitative behavior of the fractional-order pantograph equation.

This paper presents theoretical proof of the existence of a unique solution to a constrained problem of the Riemann-Liouville fractional differential equation with time delay functions by utilizing the Schauder fixed point theorem. Moreover, we analyzed the continuous dependence of the solution on the initial conditions and other parameters. Further, we investigate the Hyers-Ulam stability of the problem. We introduce some examples and special cases to illustrate our results.

The study of integral equations is one of the most important topics that researchers are interested in, it arises in many scientific fields for instance engineering, and mathematical and scientific analysis. The first who mentioned the term integral equations is Du Bois-Reymond (1888). As a result, a lot of interest appeared from researchers, and the most important of these researchers are Laplace, Fourier, Poission, Liouville, and Able. Upadhyay et al., (2015) provided some special types of integral equations. The quadratic integral equation is a special form of integral equations. The initial study appeared by Chandrasekhar (1947). More appearance of Quadratic integral equation was in the theory of radiative transfer, kinetic theory of gases, in the theory of neutron transport, and in the traffic theory, see Argyros (1985), Banaś et al. (2007), El-Sayed et al. (2008).

In this paper we establish the existence of integrable solution of boundary value problems of (mixed type) Fredholm - volterra functional integro-differential equations  with nonlocal boundary conditions.

Mathematical modelling of real-life problems usually results in functional equations, like ordinary or partial differential equations, integral and integro- differential equations, stochastic equations. Many mathematical formulation of physical phenomena contain integro-differential equations, these equations arises in many fields like fluid dynamics, biological models and chemical kinetics integro-differential equations are usually difficult to solve analytically so it is required to obtain an efficient approximate solution.

In this paper we establish the existence of solution for two boundary value problems of Fredholm functional integro-differential equations with nonlocal boundary conditions.