In this paper, we investigate a class of nonlinear Langevin equations involving two fractional orders with nonlocal integral and three-point boundary conditions. Using the Banach contraction principle, Krasnoselskii’s and the nonlinear alternative Leray Schauder theorems, the existence and uniqueness results of solutions are proven. The paper was appended examples which illustrate the applicability of the results.

A new sequence space related to the space $\ell _{p}$ ℓ p , $1\leq p<\infty $ 1 ≤ p < ∞ (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.

In this paper, we investigate a class of nonlinear Langevin equation involving one fractional order α ∈ (0, 1] with three-point boundary conditions. By the Banach contraction principle and Krasnoselskii's fixed point theorem, the existence and uniqueness results of solutions are obtained. Two examples are given to show the applicability of our main results.

In the current study, a new class of an infinite system of two distinct fractional orders with p-Laplacian operator is presented. Our mathematical model is introduced with the Caputo–Katugampola fractional derivative which is considered a generalization to the Caputo and Hadamard fractional derivatives. In a new sequence space associated with a tempered sequence and the sequence space c0 (the space of convergent sequences to zero), a suitable new Hausdorff measure of noncompactness form is provided. This formula is applied to discuss the existence of a solution to our infinite system through applying Darbo’s theorem which extends both the classical Banach contraction principle and the Schauder fixed point theorem.