This paper explores a fractional integro-differential equation with boundary conditions that incorporate the Hilfer-Hadamard fractional derivative. We model the RLC circuit using fractional calculus and define weighted spaces of continuous functions. The existence and uniqueness of solutions are established, along with their Ulam-Hyers and Ulam-Hyers-Rassias stability. Our analysis employs Schaefer's fixed-point theorem and Banach's contraction principle. An illustrative example is presented to validate our findings.

In this paper, we investigates the sliding mode control (SMC) strategy for neutral-type systems with distributed time-varying delay using novel improved integral inequality, which has significant applications in fields such as control systems, communication networks, and biological systems. A standard Lyapunov-Krasovskii functional is introduced, complemented by improved integral inequality techniques and two types of time-delay methods (neutral type and distributed). These methodologies enabled the derivation of sufficient conditions, formulated as linear matrix inequalities, that ensured the asymptotic stability of the system utilizing the SMC technique. The proposed approach reduced conservatism in stability criteria by investigating improved integral inequalities and delay-dependent techniques, offering more accurate and efficient stability conditions. Numerical examples are presented to validate the theoretical findings with the practical application of partial element equivalent circuit (PEEC), showcasing the effectiveness and superiority of the proposed methodology over existing results.

In this work, a new class of Gould-Hopper Sheffer- $\lambda$ polynomials was introduced by combining the structural features of Gould-Hopper polynomials with the general framework of Sheffer- $\lambda$ sequences. The proposed family was defined through an appropriate exponential generating function involving trigonometric factors and was shown to possess rich algebraic and operational properties. Explicit series representations and determinant forms were derived using Riordan array techniques and Cramer's rule. By employing the monomiality principle, the associated multiplicative and derivative operators were constructed, establishing the quasi-monomial character of the introduced polynomials and leading to the corresponding differential equations. Furthermore, several important subclasses, including Gould-Hopper-Bernoulli- $\lambda$, Euler- $\lambda$, Genocchi- $\lambda$, and Laguerre- $\lambda$ polynomials, were obtained as illustrative examples. These examples demonstrated the unifying nature of the proposed framework and highlighted its potential applicability in operational calculus, special functions, and mathematical physics.