In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincar&eacute / , the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant argument of generalized type in noncritical case, that is, the unperturbed linear ordinary differential equation has not any nontrivial periodic solution, are investigated. The continuous and differential dependence of the solutions on an initial value and a parameter is considered. A new Gronwall-Bellmann type lemma is proved. Next, quasilinear differential equations with alternately advanced-retarded piecewise constant argument of generalized type is addressed. The critical case, when associated linear homogeneous system admits nontrivial periodic solutions, is considered. Using the technique of Poincar&eacute / -Malkin, criteria of existence of periodic solutions of such equations are obtained. One of the main auxiliary results is an analogue of Gronwall-Bellmann Lemma for functions with alternately advanced-retarded piecewise constant argument. Dependence of solutions on an initial value and a parameter is investigated. Finally, a new class of differential equations with state-dependent piecewise constant argument is introduced. It is an extension of systems with piecewise constant argument. Fundamental theoretical results for the equations: existence and uniqueness of solutions, the existence of the periodic solutions, the stability of the zero solution are obtained. Appropriate examples are constructed.

<p dir="ltr">In this dissertation, we investigate the potential of two-dimensional (2D) transition metal carbides, known as MXenes, as precursor materials for the development of ultra-high temperature ceramics (UHTCs), with a focus on Ti₃C₂T_<em>x</em> MXene. MXenes are distinguished by their unique combination of 2D structure, high surface area, and chemically active basal planes, making them ideal candidates for a wide range of high-performance applications. This study focuses on the phase transformation, grain growth, surface texturing, and electrocatalytic behavior of Ti₃C₂T_<em>x</em> MXene films when subjected to high-temperature annealing, along with their role as sintering aids in UHTCs.</p><p dir="ltr">We present the transformation of 2D Ti₃C₂T_<em>x</em> flakes into ordered vacancy carbides of three-dimensional (3D) TiC_y phases at temperatures above 1000°C. Using X-ray diffraction and ex-situ annealing (up to 2000°C in a tube furnace and spark plasma sintering), we investigate the resulting nano-lamellar and micron-sized cubic grain morphologies. Single-flake Ti₃C₂T_<em>x</em> films retain a lamellar morphology after annealing, while multi-layer clay-like MXene transforms into irregular cubic grains.</p><p dir="ltr">In addition to investigating the structural evolution, we examine the influence of cationic intercalation on grain growth and texture. Specifically, Ca²⁺ ions lead to highly templated growth along the (111) crystal plane, significantly altering carbon diffusion and metal atom migration during annealing. We show that this preferential growth influences properties with hydrogen evolution reactions (HER) as an example functionality. We observe that with Ca²⁺-intercalated Ti₃C₂T_<em>x</em> films, exhibit an overpotential of 594 mV and a current density of -13 mA/cm² due to increased surface area and dominant texturing.</p><p dir="ltr">Additionally, we investigate the use of MXenes in self-assembly with ceramic materials such as ZrB₂, facilitated by optimizing zeta potentials. MXenes, with their functionalized hydrophilic surfaces and negative zeta potentials, serve as sintering aids and reinforcements in UHTC composites. The introduction of Ti₃C₂T_<em>x</em> to ZrB₂ enables improved sinterability, achieving 96% relative density compared to 89% for pure ZrB₂. Furthermore, the addition of MXenes leads to a core-shell microstructure with (Zr,Ti)B₂ solid-solution interfaces, enhanced mechanical properties such as a 36% increase in hardness, and reductions in oxygen content. These findings establish MXenes as promising materials for the development of advanced UHTCs, suitable for extreme environments.</p><p dir="ltr">Through a combination of experimental techniques, and theoretical estimations, and advanced characterizations, this dissertation provides critical insights into the role of MXenes in both phase transformation and mechanical reinforcement, thereby laying the foundation for future studies and opening new avenues for applications of MXene derived carbides and the design of high-performance UHTCs.</p>