Existence of positive solution to a singular second-order initial value problem with impulses

This paper deals with the existence of positive solutions to singular initial value problem with impulse effects. The right-hand side of the differential equation can be singular in its dependent variable.     

Multiple positive solutions of singularly perturbed differential systems with different orders

In this paper, we consider the multiplicity of positive solutions for a class of singular higher-order perturbed differential systems with different orders. By employing a well-known fixed point theorem, some new existence results are given under the case where nonlinearity can be sign changing.     

Boundedness of solutions for singular Hamiltonian differential systems with applications to deficiency indices

The paper gives boundedness estimation of solutions for singular Hamiltonian differential systems. As corollaries, limit-circle criteria are given and improve some previous results.     

Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems

This paper investigates the existence of positive solutions of singular multi-point boundary value problems of fourth order ordinary differential equation with p-Laplacian. A necessary and sufficient condition for the existence of C²[0,1] positive solution as well as pseudo-C³[0,1] positive solution is given by means of the fixed point theorems on cones.     

Positive solutions and eigenvalue regions of two-delay singular systems with a twin parameter

The paper is concerned with a two-delay singular differential system with a twin parameter. Applying fixed-point index theory, we show the relationship between the asymptotic behaviors of nonlinearities (at zero and infinity) and the open regions (eigenvalue regions) of parameters, which are correlated with delays, such that the system has zero, one and two positive solution(s).     

ON THE TITCHMARSH-WEYL THEORY OF ORDINARY SYMMETRIC ODD-ORDER DIFFERENTIAL EXPRESSIONS AND A DIRECT CONVERGENCE THEOREM

Abstract

In this paper the odd-order differential equation M[y] λ wy on the interval (O,∞), associated with the symmetric differential expression M of (2k-1)st order (k ≥ 2) with w a positive weight function and λ a complex number, is shown to possess k-Titchmarsh-Weyl solutions for every non-real λ in the underlying Hilbert space L² _w(O, ∞) having identical representation for every non-real λ. In terms of these solutions the Green's function associated with the singular boundary value problem is shown to possess identical representation for all non-real λ which has been further made use of in the third-order case to establish a direct convergence eigenfunction expansion theorem. The symmetric spectral matrix appearing in the expansion theorem has been characterized in terms of the Titchmarsh-Weyl m-coefficients.     

Characterization of domains of self-adjoint ordinary differential operators

The GKN (Glazman, Krein, Naimark) Theorem characterizes all self-adjoint realizations of linear symmetric (formally self-adjoint) ordinary differential equations in terms of maximal domain functions. These functions depend on the coefficients and this dependence is implicit and complicated. In the regular case an explicit characterization in terms of two-point boundary conditions can be given. In the singular case when the deficiency index d is maximal the GKN characterization can be made more explicit by replacing the maximal domain functions by a solution basis for any real or complex value of the spectral parameter λ. In the much more difficult intermediate cases, not all solutions contribute to the singular self-adjoint conditions. In 1986 Sun found a representation of the self-adjoint singular conditions in terms of certain solutions for nonreal values of λ. In this paper we give a representation in terms of certain solutions for real λ. This leads to a classification of solutions as limit-point (LP) or limit-circle (LC) in analogy with the celebrated Weyl classification in the second-order case. The LC solutions contribute to the singular boundary conditions, the LP solutions do not. The advantage of using real λ is not only because it is, in general, easier to find explicit solutions but, more importantly, it yields information about the spectrum.     

THE LEGENDRE TYPE OPERATOR IN A LEFT DEFINITE HILBERT SPACE

Abstract

The techniques for discussing linear differential operators in left definite spaces, developed earlier for regular fourth order and singular second order operators, are applied the Legendre type operator. It is shown that the operator, with its domain merely restricted to the new space, remains self-adjoint and has the same spectrum, inverse and spectral resolution (an eigenfunction expansion) as the original L ² operator.     

Positive and dead core solutions of singular BVPs for third-order differential equations

The paper discusses the existence of positive and dead core solutions of the singular differential equation (?(u^″))^′=λf(t,u,u^′,u^″) satisfying the boundary conditions u(0)=A, u(T)=A, min{u(t):t∈[0,T]}=0. Here λ is a nonnegative parameter, A is a positive constant and the Carathéodory function f(t,x,y,z) is singular at the value 0 of its space variable y.     

Inequalities for solutions of singular initial problems for Carathéodory systems via Ważewski’s principle

In this paper we give sufficient conditions for solvability of a singular initial problem formulated for Carathéodory systems of ordinary differential equations. The existence of solutions is proved by the supposition that corresponding auxiliary lower and upper singular problems have solutions. The proof technique uses a notion of a regular polyfacial subset which is developed for Carathéodory systems of ordinary differential equations and a modification of the topological method for such systems given by Palamides, Sficas and Staikos. An application concerning the existence of positive solutions for a special class of singular problems is given as well.     

Generalized singular point quantity and integrability of degenerate resonant singular point

In this paper, integrability and generalized complex resonant center condition of degenerate resonant singular point for a class of complex polynomial differential system were studied. The concept of generalized singular point quantity of degenerate resonant singular point was proposed and the construction of that was studied. Two methods of computing generalized singular point quantities were given. Furthermore, the sufficient and necessary condition of integrability of degenerate resonant singular point was discussed for the first time.     

Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations

In this paper, we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. The associated Green’s function for the boundary value problems is first given. The proofs of our main results are based upon the Leggett–Williams fixed point theorem. Finally, we give an example to demonstrate our result.     

Existence of nontrivial solutions for a nonlinear Sturm–Liouville problem with integral boundary conditions

This paper deals with the existence of nontrivial solutions for a nonlinear singular Sturm–Liouville problem with integral boundary conditions.     

Singular left-definite Sturm-Liouville problems

We study singular left-definite Sturm-Liouville problems with an indefinite weight function. The existence of eigenvalues is established based on the existence of eigenvalues of corresponding right-definite problems. Furthermore, for each singular left-definite problem with limit-circle non-oscillatory endpoints we construct a regular left-definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions. Inequalities among eigenvalues recently established for regular left-definite problems are extended to the singular case.     

Positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions

In this paper, we study the existence, multiplicity and nonexistence of positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, it generalizes and includes some known results.     

Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks

We study some properties of piecewise linear differential systems describing gene regulatory networks, where the dynamics are governed by sigmoid-type nonlinearities which are close to or coincide with the step functions. To overcome the difficulty of describing the dynamics of the system near singular stationary points (belonging to the discontinuity set of the system) we use the concept of Filippov solutions. It consists in replacing discontinuous differential equations with differential inclusions. The global existence and some other basic properties of the Filippov solutions such as continuous dependence on parameters are studied. We also study the uniqueness and non-uniqueness of the Filippov solutions in singular domains. The concept of Filippov stationary point is extensively exploited in the paper. We compare two ways of defining the singular stationary points: one is based on the Filippov theory and the other consists in replacing step functions with steep sigmoids and investigating the smooth systems thus obtained. The results are illustrated by a number of examples.     

The existence of countably many positive solutions for singular multipoint boundary value problems

In this paper, we study the existence of countably many positive solutions for a singular multipoint boundary value problem. By using fixed-point index theory and the Leggett-Williams’ fixed-point theorem, sufficient conditions for the existence of countably many positive solutions are established.     

On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators

We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x)∈L¹(0,1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.     

Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.     

Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions

The nonlinear nth-order singular nonlocal boundary value problem