The primary objective of the paper is exploring the Ulam stability $(\mathcal{US})$ of a Caputo q-fractional Langevin differential equation $(\mathcal{FLDE})$ under q-fractional integral boundary conditions $(\mathcal{FIBC}s)$ . The novelty of this work stands out for its broader generality compared to the existing research focused on the Caputo q-fractional derivative. We apply the Banach contraction principle $(\mathcal{BCP})$ for checking the existence and uniqueness of solutions of Caputo $q-\mathcal {FLD}$ equations. The framework of the study integrates fundamental principles from both fractional calculus and quantum calculus. Additionally, we discuss various forms of Ulam stability, namely $\mathcal{UHS}$ , $\mathcal{GUHS}$ , $\mathcal{UHRS}$ , and $\mathcal{GUHRS}$ . We validate our theoretical findings through illustrative examples.

In this manuscript, a new family of contractions called Jaggi-type hybrid -contraction is introduced and some fixed point results in generalized metric space that are not deducible from their akin in metric space are obtained. The preeminence of this class of contractions is that its contractive inequality can be extended in a variety of manners, depending on the given parameters. Consequently, several corollaries that reduce our result to other well-known results in the literature are highlighted and analyzed. Substantial examples are constructed to validate the assumptions of our obtained theorems and to show their distinction from corresponding results. Additionally, one of our obtained corollaries is applied to set up unprecedented existence conditions for the solution of a family of integral equations.

Among various efforts in advancing fuzzy mathematics, a lot of attentions have been paid to examine novel intuitionistic fuzzy analogues of the classical fixed point results. Along this direction, the idea of intuitionistic fuzzy mapping (IFM) is used in this paper to establish some fixed point (FP) results in complex-valued b -metric spaces. Moreover, from application perspective, one of our results is rendered to provide an existence condition for a solution of Caputo-type fractional differential equations. A few nontrivial illustrations are also furnished to authenticate and indicate the usability of the presented results.

In this article, the necessary conditions on s-type Orlicz generalized difference sequence space to generate an operator ideal have been examined. Therefore, the s-type Orlicz generalized difference sequence space which fails to generate an operator ideal has been shown. We investigate the sufficient conditions on this sequence space to be premodular Banach special space of sequences, and the constructed pre-quasi operator ideal becomes small, simple, closed, Banach space and has eigenvalues identical with its s-numbers.