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In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions. • A fully fractional thermostat model involving the Riemann–Liouville derivatives. • The equivalence between the FDE and the corresponding integral equation is obtained. • Some new existence and uniqueness results are established. • An implicit numerical scheme based on the right product rectangle rule is derived. • Numerical simulations of implicit solutions are presented. [ABSTRACT FROM AUTHOR]

• We derive the mild solution of the damped fractional integrodifferential system in terms of Riemann-Liouville fractional resolvent, where and is a real number. • We prove the existence and uniqueness of mild solutions of the system using generalized fixed point theorem. • We prove the approximate controllability of the system using iterative and approximate technique. For this, we prove the Lemma 4.3. This article is concerned with Riemann-Liouville fractional semilinear integrodifferential systems with damping in Banach spaces. First we prove the existence of mild solutions of the system using fixed point approach, then we establish new sufficient conditions for the approximate controllability of the system by means of iterative and approximate technique. To obtain our results, we use the concept of Riemann-Liouville fractional (ϑ, φ, λ) resolvent, where 0 < φ < ϑ ≤ 1 and λ is a real number. Finally, an example is provided for the illustration of the obtained results. [ABSTRACT FROM AUTHOR]