Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Existence, upper and lower solutions and quasilinearization for singular differential equations

In this paper, we discuss existence theorems in the presenceof upper and lower solutions as well as the method of quasilinearization(QSL) for general non-linear second-order singular ordinarydifferential equations. We show the existence of solutions underthe assumption of weak continuity of the non-linear part. Ifthe non-linear part is monotone decreasing, a solution may beobtained by the QSL method as the strong limit of a quadraticallyconvergent sequence of approximate solutions. Under strongerassumptions on the linear and the non-linear parts, a solutionis quadratically bracketed between two monotone sequences ofapproximate solutions of certain related linear equations.     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

Generalized contractions in partially ordered metric spaces

We present some fixed point results for monotone operators in a metric space endowed with a partial order using a weak generalized contraction-type assumption.     

Regular approximation of singular self-adjoint differential operators

Given a singular self-adjoint differential operator of order 2n with real coefficients we constructtwo sequences of regular self-adjoint differential expressions^r which converge to ina generalized sense of resolvent convergence. The first constructionis suitable when no information about the real resolvent setof is available. The second is suitablewhen we know a real point of the resolvent set of .The main application of this construction is in numerical solutionof singular differential equations.     

Existence and quasilinearization in Banach spaces

We establish some existence results for the nonlinear problem Au=f in a reflexive Banach space V, without and with upper and lower solutions. We then consider the application of the quasilinearization method to the above mentioned problem. Under fairly general assumptions on the nonlinear operator A and the Banach space V, we show that this problem has a solution that can be obtained as the strong limit of two quadratically convergent monotone sequences of solutions of certain related linear equations.     

Optimal control of self-adjoint nonlinear operator equations in Hilbert spaces

In this paper, we formulate and study a general optimal control problem governed by nonlinear operator equations described by unbounded self-adjoint operators in Hilbert spaces. This problem extends various particular control models studied in the literature, while it has not been considered before in such a generality. We develop an efficient way to construct a finite-dimensional subspace extension of the given self-adjoint operator that allows us to design the corresponding adjoint system and finally derive an appropriate counterpart of the Pontryagin Maximum Principle for the constrained optimal control problem under consideration by using the obtained increment formula for the cost functional and needle type variations of optimal controls.     

正则和奇异自共轭微分算子的变分公式化(英文)

1 IntroductionVariational methods offer an appealing approach to the analysis and com-putations associated with symmetric differential equations because they areclose1y connected to the under1ying physica1 principle of energy. Such methodshave been used to obtain the existence and uniqueness results, to investigateproperties of solutions [2], I61, [5], I7], and to compute solutions as welI as ei-genvalues [11, [41 fOr linear and nonlinear equations. The first step in applyinga variational me…     

Type I operators and the approximation of singular two-point boundary value problems

A class of self adjoint operators associated with second order singular ordinary differential expressions, arises naturally when the problem is weakly formulated and integration by parts is performed. We call this class Type I operators. It turns out that this class can be successfully used to tackle numerical approximations of singular two-point boundary value problems. They can also be approximated by regular differential operators in a straightforward manner without having to bring the delicate structure of singular differential operators to the forefront of the investigation.