In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series method, combined with the Laplace transform, to introduce a new effective technique called the Laplace Residual Power Series Method (LRPSM). This method is applied to derive the coefficients of the series solution for MTCFKEs in the context of Hilbert algebras. In real Hilbert algebras, we obtain approximate solutions for MTCFKEs under both exact and approximate initial conditions. We present both graphical and numerical results of the approximate analytical solutions to demonstrate the capability, efficiency, and reliability of the LRPSM. Furthermore, we compare our results with solutions obtained using the homotopy analysis method and the natural transform decomposition method.

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the $\phi -$Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra $\mathbb{R}$ and commutative Banach algebra of the collection of continuous functions in $\mathbb{R}$.

Using the monophonic paths in the theory of directed graphs, this paper constructs a new topology called the out mondirected topology, which characterizes the graphs that induce indiscrete or discrete topology. We give and study some of the relations and properties, such as the relationship between the isomorphic relation, in directed graphs and the homeomorphic property in out mondirected topological spaces, compactness, ${\mathbb{D}}_{\pm }$-connectedness, connectedness and ${\mathbb{D}}_{\pm }$-discrete properties. Finally, we apply our results of out mondirected topological spaces in the nervous system of the human body, such as in the messenger signal network, in diagrams of sensory neuron cells and in models of two distinct nicotinic receptor types based on the second messenger signal.

This paper considers a general three-dimensional ABC fractional dynamical system formulated in a bounded locally compact Hausdorff space. The locally compact Hausdorff structure ensures that the compact-open topology on the space of continuous functions coincides with the topology induced by the supremum norm, providing an appropriate Banach space framework for the analysis of the system. Within this setting we study the continuity, boundedness, and Lipschitz properties of the nonlinear operators associated with the fractional model. Based on these properties, the existence of solutions is established using Schaefer fixed point theorem, while uniqueness is obtained through Banach contraction principle under suitable conditions. Furthermore, the Hyers–Ulam stability of the system is investigated, showing that small perturbations lead to small deviations in the corresponding solutions. Finally, the theoretical results are applied to the fractional Lorenz system and the two-dimensional fractional Euler system, illustrating the applicability of the proposed framework to models arising in chaotic dynamics and fluid mechanics.

In this work, we use the monophonic eccentric open neighborhood system and the upper approximation neighborhood system to construct a new class of topologies in the theory of undirected simple graphs. We study some fundamental topological properties of this class and characterize the graphs that induce the indiscrete or discrete topology. Next, we present the openness, the connectedness and the continuity properties with isomorphic maps of graphs. Some applications concerning the topological discrete and connectedness properties for some corresponding graphs of the COVID-19 form and its diffusion are introduced.

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case $0<\alpha <1$. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems.