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In this work, we investigate the existence of solutions for the system of two singular fractional q-differential equations under integral boundary conditions via the concept of Caputo fractional q-derivative and fractional Riemann-Liouville type q-integral. Some new existence results are obtained by applying Krattowski measure of non-compactness. Also, the Darbo?s fixed point theorem and the Lebesgue dominated convergence theorem are the main tools in deriving our proofs. Lastly, we present an example illustrating the primary effects.

In this work, we present a new extension of Darbo?s theorem for two different classes of altering distance functions via measure of non-compactness. Using two-variable contractions we obtain the wellknown results in this literature (see [22]). We also use these results to discuss the existence of solutions for a system of integral equations. Finally, we provide an example to confirm the results obtained.

In this work we introduce a modified version of simulation function and define a simulation type contraction mappings involving measure of non-compactness in the frame work of Banach space and derive some basic Darbo type fixed point results. Also, our theorem generalizes the Theorem 4 of [R. Arab, Some generalizations of Darbo fixed point theorem and its application, Miskolc Mathematical Notes, 18(2)(2017),595-610.] and extend some recent results. Further we show the applicability of obtained results to the theory of integral equations followed by two concrete examples.

The aim of this paper is to enlarge some known results from Fredholm and perturbation theory via measure of non-compactness. As applications, we focus on the study of the essential ascent and the essential descent spectra of an operator T defined on a given Banach space. Some perturbation results are also investigated.