Existence of Solutions for a Nonlinear Dirichlet Problem Involving Gradient Dependent Lipschitz Convection Function

Convergence of composite iterative methods for finding zeros of accretive operators

In this thesis we investigate the existence of solutions to boundary value problems (BVPs) for nonlinear systems of first-order ordinary differential equations (ODEs) and difference equations. The thesis is divided into two parts. Part one is devoted to the study of difference equations where two types of problems are studied. First, we derive the existence of solutions to first-order systems of difference equations that arise when one applies the trapezoidal rule to approximate solutions of second-order scalar ODEs. Strict discrete lower solutions are used with maximum principle arguments in the discrete problem to obtain a priori bounds on solutions. The a priori bounds on difference quotients for solutions of the discrete problem are obtained from the Bernstein-Nagumo condition. We use homototopy to find solutions of the discrete amroximations. Second, motivated by Ma [60] we consider existence and uniqueness of solutions to difference equations which arise as discrete approximations to three-point BVPs for systems of first-order ODES after employing the Euler method. Several existence and uniqueness results for three-point BVPs are established, using the contraction mapping theorem and the Brouwer fixed point theorem in Rn. In part two, two types of problems for systems of first-order ODES are analyzed. First, we study the existence of a unique solution to a system of first-order differentia1 equations under multi-point BVPs with nonlinear boundary conditions. We prove the existence and uniqueness of solutions for a system of first-order three-point BVP under which the properties of linear three-point BVP are preserved under small nonlinear perturbations of both the differentia1 equation and the boundary conditions. The method of proof of the existence of solutions using the Banach contraction mapping principle is similar to that used by Rodriguez [75], who studied the discrete analogue of this BVP. Second, motivated by Cronin 120,211 we investigate the existence of solutions to three-point BVPs in perturbed systems of first-order ODEs at resonance where the associated homogeneous linear problem has nontrivial solutions. Based on the second part of the investigation of three-point BVPs in perturbed systems of first-order ODEs at resonance, four different results were obtained: (i) The existence of solutions via a version of the Brouwer Fixed Point Theorem which is due to Miranda. (ii) The existence of solutions to the problem through the application of the Implicit Function Theorem. These two results extend the work of Feng and Webb [34], and Gupta [39] for the problems at resonance in Euclidean 2-space. (iii) Existence results for three-point BVPs at resonance for general BVPs through the application of Brouwer degree theory; we show the degree is non zero through the application of Borsuk's Theorem for one problem and through a result of Cronin involving pairs of polynomials in two variables whose terms of highest order have no common factors. (iv) The existence of solutions to the entrainment of frequency problem in Euclidean 2-space. We will investigate conditions under which entrainment of frequency for three-point as well as two-point BVPs by adapting the work of Cronin in periodic case. We will describe such conditions and then show the degree is non zero through a result of Cronin involving pairs of polynomials in two variables whose terms of lowest order have no common factors.

Regularization Methods for Nonlinear Ill-Posed Problems With Accretive Operators

We present a new and feasible formulation for the inverse problem of identifying the unknown shear force acting on the inaccessible tip of the nonhomogeneous micro-cantilever plate which is a key component in most atomic force microscopes. The mathematical modelling of this phenomena leads to the inverse problem of determining the shear force g(x 2, t) acting on the inaccessible boundary x 1 = ℓ 1 in a system governed by the variable coefficient damped Kirchhoff–Love plate equation ρ(x)h(x)utt+μ(x)ut+D(x)(ux1x1+νux2x2)x1x1+D(x)(νux1x1+ux2x2)x2x2+2(1−ν)D(x)ux1x2x1x2=F(x,t) , where (x, t) ∈ Ω T ≔ Ω × (0, T), x ≔ (x 1, x 2) and Ω≔(0,ℓ1)×(0,ℓ2)⊂R2 , subject to clamped-free boundary conditions, from final time measured output (displacement) u T (x) ≔ u(x, T). We prove that under appropriate conditions, this inverse problem has a unique solution. We introduce the input–output operator and prove that it is a linear compact and Lipschitz continuous operator. Based on these results, the existence of a quasi-solution of the inverse problem, as a solution of the minimization problem for the Tikhonov functional, is proved. Furthermore, an explicit gradient formula for the Fréchet derivative of the Tikhonov functional is derived. Moreover, it is proved that the Fréchet derivative is Lipschitz continuous. These results provide a mathematical basis for gradient based computational algorithm.

In this work, we study the existence and multiplicity of positive solutions for nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator , where k > 0 is a constant, λ > 0 is a parameter and f : [0,∞) → ℝ is continuous. We apply the quadrature arguments to prove how changes in the sign of f (u) lead to multiple positive solutions of the above problem for sufficiently large λ.

In the present paper, we consider the nonlinear Dirichlet problem ‐ Δ u ( x ) u β ( x ) = 0 equation image is the unit ball and q is a continuous radially symmetric function on B which may be singular on ∂B. We state some mild conditions for the function q so that the Dirichlet problem has a positive classical solution.

Combining the implicit midpoint method and the splitting method, we present a new iterative algorithm with errors to solve the problems of finding zeros of the sum of m-accretive operators and μ-inversely strongly accretive operators in a real q-uniformly smooth and uniformly convex Banach space. We obtain some strong convergence theorems, which demonstrate the relationship between the zero of the sum of m-accretive operator and μ-inversely strongly accretive operator and the solution of one kind variational inequality. Moreover, the applications of the main results on the nonlinear problems with Neumann boundaries and Signorini boundaries are demonstrated.

In a separable real Hilbert space, we study the problem of minimizing a convex function with Lipschitz continuous gradient in the presence of noisy evaluations. To this end, we associate a stochastic Heavy Ball system, incorporating a friction coefficient, with the optimization problem. We establish existence and uniqueness of trajectory solutions for this system. Under a square integrability condition for the diffusion term, we prove almost sure convergence of the trajectory process to an optimal solution, as well as almost sure convergence of its time derivative to zero. Moreover, we derive almost sure and expected convergence rates for the function values along the trajectory towards the infimal value. Finally, we show that the stochastic Heavy Ball system is equivalent to a Su-Boyd-Candès-type system for a suitable choice of the parameter function, and we provide corresponding convergence rate results for the latter.In the second part of this paper, we extend our analysis beyond the optimization framework and investigate a monotone equation induced by a monotone and Lipschitz continuous operator, whose evaluations are assumed to be corrupted by noise. As before, we consider a stochastic Heavy Ball system with a friction coefficient and a correction term, now augmented by an additional component that accounts for the time derivative of the operator. We establish analogous convergence results for both the trajectory process and its time derivative, and derive almost sure as well as expected convergence rates for the decay of the residual and the gap function along the trajectory. As a final result, we show that a particular instance of the stochastic Heavy Ball system for monotone equations is equivalent to a stochastic second-order dynamical system with a vanishing damping term. Remarkably, this system exhibits fast convergence rates for both the residual and gap functions.

The paper is focused on solvability of an inverse boundary problem with an unknown coefficient which depends on time for a second-order parabolic equation with non-classical boundary conditions. The idea of the problem is that together with the solution it is required to determine the unknown coefficient. The problem is considered in the rectangular area. The paper introduces a classical solution of the set problem. At first, an auxiliary inverse boundary problem is examined and the equivalence (in some sense) of the original problem is proved. First, we apply a method of variable separation to analyze the auxiliary inverse boundary problem. Then, we examine a spectral problem for an ordinary second-order differential equation with integral conditions. Having used a formal scheme of the method of variable separation, the solution of a direct boundary problem (in case of specified unknown function) resolves itself into solution of Cauchy problem. After that the solution is limited to the solution of a countable system of integro-differential equations in Fourier coefficients. In its turn, the last system regarding unknown Fourier coefficients is recorded in the form of an integro-differential equation in the desired solution. Using relevant additional conditions of the auxiliary inverse boundary problem, we obtain a system of two nonlinear integral equations for defining unknown functions. Thus, the solution of the auxiliary inverse boundary problem comes down to the system of two nonlinear integro-differential equations in unknown functions. The specific Banach space is designed. Then, in the sphere made of the Banach space we with the help of contracted mapping prove the solvability of the nonlinear integro-differential equations set, which is a unique solution of the additional inverse boundary problem. Using the equivalence of problems, it is concluded about existence and uniqueness of a classical solution of the original problem.

In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3 with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space Wpl(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2).

introduces the following concept of a strictly singular operator in [3].Definition.Let X and Y be Banach spaces and let T be a bounded linear operator mapping X into F. T is said to be strictly singular if given any infinite dimensional subspace M of X, T restricted to M is not an isomorphism (i.e., linear homeomorphism).In particular, every compact operator is strictly singular.Kato proceeds to show that the space S of strictly singular operators possesses some of the important features of the space of compact operators, e.g., S is a closed subspace of the space of bounded linear operators from A to F. If X= Y, then S is a closed ideal.If X and F are Hubert spaces, then every strictly singular operator from A to F is compact.The following two questions, posed by Kato, are answered below.(1) Is every strictly singular operator compact?(2) Is the conjugate of a strictly singular operator strictly singular?Theorem.3 (a) Every bounded linear operator from k to lp or from lp to l2, l^p9i2, oo is strictly singular.(b) Let X be a Banach space which does not contain an infinite dimensional reflexive subspace.Then every bounded linear operator mapping X into a reflexive space is strictly singular.Also any bounded linear operator mapping a reflexive space into X is strictly singular.Remarks, (i) R. S. Phillips exhibited (unpublished) a strictly singular operator which is not compact.Theorem 1 shows that S can indeed be a "much larger" space than the space of compact linear operators.

In this paper, we investigate the dynamics of population inhabiting a favorable environment surrounded by a region where survival is impossible. We use a parabolic partial differential equation with nonlinearity changing signs to describe the dynamics. By using a recent result on the existence of nonzero fixed point for r-nowhere normal-outward compact maps, we establish new results on existence of nonzero nonnegative or positive steady-state solutions of such boundary value problems. Additionally, we obtain the convergence of positive solutions of such parabolic boundary value problems in the Banach space $$C^1$$, which is different from the previous results that considered weak convergence or uniform convergence.

Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem $$ \mathbb{D}_t^\alpha u(t)=Au(t)+f(t), \;t > 0,\;\;0 < \alpha\le 1, $$ where $\mathbb D_t^\alpha$ is the Caputo fractional derivative of order $\alpha$, $A$ a closed linear operator on some Banach space $X$, $f:\;[0,\infty)\to X$ is a given function. We define an operator family associated with this problem and study its regularity properties. When $A$ is the generator of a $\beta$-times integrated semigroup $(T_\beta(t))$ on a Banach space $X$, explicit representations of mild and classical solutions of the above problem in terms of the integrated semigroup are derived. The results are applied to the fractional diffusion equation with non-homogeneous, Dirichlet, Neumann and Robin boundary conditions and to the time fractional order Schrödinger equation $\mathbb{D}_t^\alpha u(t,x)=e^{i\theta}\Delta_pu(t,x)+f(t,x),$ $ t > 0,\; x\in \mathbb R ^N$ where $\pi/2\le \theta < (1-\alpha/2)\pi$ and $\Delta_p$ is a realization of the Laplace operator on $L^p(\mathbb R ^N)$, $1\le p < \infty$.

The injective and surjective hulls of the ideal of (p,q)-compact operators and their approximation properties

Approximating solution of [formula omitted] for an H-accretive operator in Banach spaces