We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in $C^{\gamma }\left[0,T\right]$, where . A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be $O\left({\tau }^{\gamma }\right)$, for $u\in C^{\gamma }\left[0,T\right]$, where $\tau$ is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings.

Asymptotic stability, uniform stability, integrability, and boundedness of solutions of Volterra integro-differential equations with and without constant retardation are investigated using a new type of Lyapunov–Krasovskii functionals. An advantage of the new functionals used here is that they eliminate using Gronwall’s inequality. Compared to related results in the literature, the conditions here are more general, simple, and convenient to apply. Examples to show the application of the theorems are included.

The existence of three solutions for perturbed systems of impulsive nonlinear fractional differential equations together with Lipschitz continuous nonlinear terms is discussed. The idea relies on variational methods. Moreover, an example is presented in order to summarize the feasibility and effectiveness of the main results.

We are concerned with the following fractional Laplacian equation

where $s\in \left(0,1\right)$, . Using variational methods, we show existence and multiplicity of positive solutions.

The generalized (or coupled) Abel equations exist in many BVPs of fractional-order differential equations and play a key role during the process of converting weak solutions to the true solutions. Motivated by the analysis of double-sided fractional diffusion-advection-reaction equations, this article develops the mapping properties of generalized Abel operators ${\alpha }_a{D}_x^{-s}+{\beta }_x{D}_b^{-s}$ in fractional Sobolev spaces, where , $\alpha +\beta =1$, and ${}_a{D}_x^{-s}$, ${}_x{D}_b^{-s}$ are fractional Riemann–Liouville integrals. It is mainly concerned with the regularity property of $\left({\alpha }_a{D}_x^{-s}+{\beta }_x{D}_b^{-s}\right)u=f$ by taking into account homogeneous boundary conditions. Namely, we investigate the regularity behavior of $u\left(x\right)$ while letting $f\left(x\right)$ become smoother and imposing homogeneous boundary restrictions $u\left(a\right)=u\left(b\right)=0$.

We study a nonlinear Langevin equation involving Caputo fractional derivatives of a function with respect to another function in a Banach space. Unlike previous papers, we assume the source function has a singularity. Under a regularity assumption of a solution to the problem, we show that the problem can be transformed to a Volterra integral equation with a two parameter Mittag–Leffler function in the kernel. Based on the obtained Volterra integral equation, we investigate the existence and uniqueness of the mild solution of the problem. Moreover, we show that the mild solution of the problem depends continuously on the inputs: initial data, fractional orders, appropriate function, and friction constant. Meanwhile, a new Henry–Gronwall type inequality is established to prove the main results of the paper. Examples illustrating our results are also presented.

We work out the continuous dependence, on initial values and parameters, of solutions on maximal intervals of existence (or global solutions) to Caputo fractional order systems, benefiting from a very recent continuation result.

We study a nonlinear nonautonomous evolution inclusion of Volterra-type with time delay involving a multivalued nonlinearity. The family of operators (possibly unbounded) in the principal part of the evolution inclusion generates a noncompact evolution family. Global solvability for local and nonlocal initial value problems corresponding to the evolution inclusion is considered. The investigation in the present paper is based on two topological fixed point theorems. We impose a condition of Hausdorff-measure of noncompactness on the nonlinearity, which allows us to overcome the lack of compactness of the evolution family. Finally, we present an example to illustrate the abstract results.