Thesis and Dissertations

A Study on the Strong Minimum/Maximum Principle and Liouville-Type Theorems for Partial Trace Equations with Nonlinear Gradient Terms

Tue, 01 Jul 2025 00:00:00 GMT

A Study on the Strong Minimum/Maximum Principle and Liouville-Type Theorems for Partial Trace Equations with Nonlinear Gradient Terms Bukayaw, Kindu The purpose of this dissertation is to study the strong minimum/maximum principle and to investigate various Louisville-type theorems for partial trace equations with nonlinear gradient terms. To investigate the problems and obtain the results presented in this dissertation, we will use various principles (minimum and maximum principles, a comparison principle for viscosity sub/super solutions, boundary principles, and compact support principles), concepts of duality, and different mathematical estimates and inequalities. In this study, we examine a strong minimum principle of Vazquez type for partial trace operators with gradient terms. More explicitly, given a n-tuple a = (a1, · · · , an) of non-negative real numbers with an > 0, we give sufficient conditions on a continuous function h: R × R+0 → R in order for non-negative viscosity super solutions of Pa(D2u) = h(u, |Du|) (⋆) in connected open subsets of Rn that vanish at some point to Ω vanish identically in Ω. When h depends only on the gradient, the condition is also necessary. Here Pa belongs to a class of fully nonlinear degenerate elliptic operators that includes the Min-Max operator which is defined as the sum of the minimum and the maximum eigenvalues of the Hessian matrix. Under suitable conditions on h and a = (a1, · · · , an), both the Strong Maximum Principle and the Compact Support Principle for sub solutions are also investigated. The work covers a new class of degenerate operators and a wide class of Hamiltonian's not investigated in the literature and some of the results are new even when Pa reduces to the standard Laplacian. Illustrative examples are presented for such equations. We investigate various Louisville-type theorems for partial trace equations with nonlinear gradient terms. Specifically, we establish sufficient conditions for which v their viscosity sub solutions vanish identically in Rn. For a prototype of such equations, that is, when h(u, |Du|) = f(u) + g(u)|Du|q for 0 < q < 2 in (⋆), we give necessary and sufficient conditions for viscosity sub solutions identically zero in Rn. Our results are important in the field of mathematical analysis. They serve as a fundamental tool for analyzing the existence and uniqueness, regularity, and blow-up estimate behavior of solutions for practical problems.

Iterative Algorithms for Approximating Solutions of Split Common Fixed Point Problems in Banach Spaces

Wed, 01 Jan 2025 00:00:00 GMT

Iterative Algorithms for Approximating Solutions of Split Common Fixed Point Problems in Banach Spaces Dagnachew, Jenber Many real-world physical problems are inherently nonlinear in nature. In most cases, obtaining their exact solutions is challenging or even impossible. Consequently, once the existence of solutions is assured, considerable attention has been devoted in developing various approximation methods to effectively tackle these nonlinear problems. In this dissertation, we construct different iterative algorithms to approximate solutions of such non-linear problems. First, we introduce λ-strict quasi-Gf-pseudocontractive mappings and propose a method for approximating the solution of the split common fixed point problem involving λ- strict quasi-Gf-pseudocontractive mapping in the setting of two Banach spaces using Gf(., .) functional. We prove that the proposed method converges strongly to a solution of the split common fixed point problem. In addition, we provide some applications of our method and provide numerical examples to demonstrate the applicability of the proposed method. Moreover, we introduce a new problem called the two-tuple split common fixed point problem in the setting of three Banach spaces. Then, we propose an inertial algorithm for approximating a solution for the two-tuple split common fixed point problem involving the class of strict quasi-ϕ-pseudocontractive mappings. As a consequence, we provide a method of approximating a solution to the split equality fixed point problem involving strict quasi-ϕ-pseudocontractive mappings in the setting of three Banach spaces. As an application of our result, we study the split equality equilibrium problem, the split equality variational inclusion problem and the split equality problem in the framework of Banach spaces. To verify our work, we give a numerical example. Furthermore, we introduce m-tuple split common fixed point problems. We present an inertial method to estimate solutions to the m-tuple split common fixed point problems for two kinds of strict quasi-Gf-pseudocontractive mappings in (m+1) Banach ix spaces. Our approach can also be used to solve the extended split equality fixed point problem. Applications to the solution of extended split equality issues are demonstrated in the framework of Banach spaces. To illustrate our conclusions, we end with a numerical example. Overall, the proposed inertial algorithm enhances convergence speed by incorporating previous iterative information and extends the analysis from Hilbert to Banach spaces. The results provide a broad theoretical framework with potential applications in signal and image reconstruction, optimization, and inverse problems.

Iterative Methods for Approximating Solutions of Split Equality Problems Involving Variational Inequality and Fixed Point Problems

Tue, 01 Jul 2025 00:00:00 GMT

Iterative Methods for Approximating Solutions of Split Equality Problems Involving Variational Inequality and Fixed Point Problems Gedefaw, Mekuriaw Nonlinear problems, including variational inequality problems and fixed point problems, are pervasive in various fields like physics, optimization, and economics. These problems often lack closed form solutions and pose significant challenges for direct analytical methods. As a result, iterative methods are widely used for approximating their solutions. In this dissertation, we study split equality problems involving common variational inequality and fixed point problems in both real Hilbert and real Banach spaces. The problems we consider are defined under weaker conditions on the associated operators and the norm, spectral radius and Lipschitz constant of these operators are not part of our conditions. We use strong convergent projection iterative methods to approximate the solutions of the problems. The methods combine inertial steps with Tseng’s extragradient method, both of which are recognized for enhancing the performance of iterative methods. First, we propose inertial-like Tseng’s extragradient and subgradient extragradient iterative algorithms for solving split equality common variational inequality and fixed point problems in real Hilbert spaces. The underlying operators associated with the variational inequality problems are quasi-monotone and uniformly continuous and those with the fixed point problems are quasi-nonexpansive operators. Under some mild conditions, we prove strong convergence of our methods to a solution of the problem and numerical examples are presented to illustrate the efficiency of the methods. Second, we introduce and study a new split equality problem with common variational inequality and fixed point problems associated with finite families of operators in the setting of real Hilbert spaces. We propose an inertial Tseng’s extragradient iterative method with viscosity technique and prove its strong convergence to a solution under some suitable conditions. We also present some numerical experiments to illustrate and show the efficiency of the proposed method. Thirdly, we study a split equality problem involving common Minty variational inequality and fixed point problems in real reflexive Banach spaces. Unlike the existing results on Minty variational inequality problems, we consider the operators to be both quasi-monotone and uniformly continuous. Under some suitable conditions, we establish a strong convergence of the proposed method to a solution of the problem. Finally, we present some numerical experiments to illustrate and show the efficiency of the method. These findings provide a scalable and theoretically robust framework for solving complex problems such as convex and nonconvex optimization problems, game-theory, engineering and economics.

Analysis of Thermal and Surface Roughness Effects on the Performance of Infinitely Long Plane Slider Bearings: Finite Element Methods

Mon, 01 Jul 2024 00:00:00 GMT

Analysis of Thermal and Surface Roughness Effects on the Performance of Infinitely Long Plane Slider Bearings: Finite Element Methods Girma, Desu In this dissertation, the Streamline Upwind Petrov-Galerkin finite element method is used to investigate the performance analysis of slider bearings with the effect of temperature and surface roughness on one-dimensional longitudinal and transverse roughness types. Laminar fluid films, unsteady fluid films with or without heat conduction through the pad and slider, turbulent fluid films with or without porous material, and non-Newtonian power-law fluid-type lubricants were among the fluid lubricants employed in this study. The roughness is thought to have a stochastic and Gaussian random distribution. It is also thought that for a Newtonian fluid lubricant film, viscosity and density depend on temperature. For the purpose of numerical computation, the surface roughness-induced irregularity of the domains is transformed into a regular domain. In addition to the energy equation, the continuity equation and momentum equation are utilised to derive the modified Reynolds equations for each scenario in order to assess the performance of load-carrying capacity and pressure distribution. With appropriate boundary conditions, the approach is connected to the stochastically averaged Reynolds-type equation. The Ng-Pan turbulent model was used to derive the modified Reynolds equation for turbulent lubrication fluid films. In addition, the power-law viscosity model was used to derive the modified Reynolds equation for non-Newtonian lubricant fluid films. The pressure distribution of the combined effects is lower than the thermal and surface roughness effects in the case of the one-dimensional longitudinal surface roughness model for non-parallel slider bearings (w = 0.4). However, the thermal effect is less than the combined and surface roughness effect for the one-dimensional transverse surface roughness model type. In an unsteady state with heat conduction through the solid, the bearing performance under isothermal boundary conditions is superior to that of adiabatic and exposed boundary conditions to the environment. Furthermore, we also look at the combined effect at different temperatures. As a result, for both models, a higher slider temperature than the pad temperature improves load-carrying capacity performance. A one-dimensional longitudinal surface roughness slider bearing typically has a lower pressure distribution than a one-dimensional transversal surface roughness model type. In general, taking the surface roughness effect, inertial effect, turbulent lubrication effect, the porous permeability parameter, and non-Newtonian power-law fluid properties will typically improve the bearing performance of infinitely long plane slider-bearing. The numerically obtained results were presented using tables and graphs. vi