Multiplicity results for a class of fourth order semipositone m-point boundary value problems

Using bifurcation techniques, we investigate the existence of multiple solutions for a class of fourth order semipositone m-point boundary value problems. Our results improve those in the literature.     

On the boundary operator of a fourth order differential problem

We study an unilateral elliptic fourth order problem and establish some uniqueness results. The expression of the boundary operation is also given     

On the deficiency indices of a differential operator of fourth order

Translated from Matematicheskie Zametki, Vol.57, No. 2, pp. 313–314, February, 1995.     

On a class of nonlinear fourth order differential equations

Summary In this paper we study the nonlinear fourth order differentiai equation u^iv±F(u, v)u=0, where F is a pssitive monotone function of u. Asymptotic behaviour of certain solutions are treated in sections 2 and 4 while a two point boundary value problem is studied in section 3. Entrata in Redazione il 26 agosto 1968.     

Extending the threshold values for inverse-positivity of a linear fourth order operator

Positivity - In this paper, we study sufficient conditions for the (strict) inverse-positivity of the linear fourth order operator corresponding to the one-dimensional beam equation with a...     

Accurate solutions of fourth order Sturm-Liouville problems

Recently we introduced a new method which we call the Extended Sampling Method to compute the eigenvalues of second order Sturm-Liouville problems with eigenvalue dependent potential. We shall see in this paper how we use this method to compute the eigenvalues of fourth order Sturm-Liouville problems and present its practical use on a few examples.     

Spectrum of a class of fourth order left-definite differential operators

The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…     

Oscillation criteria of a class of fourth order differential equations

In the present paper, we consider the oscillation of a class of self‐adjoint fourth order differential equation. Some oscillation and non‐oscillation criteria on a perturbation equation of Euler differential equation are given. In particular, we find the exact value of a constant in some oscillation criteria, which has answered the open problem presented by Do?lý in 2004. Copyright © 2011 John Wiley & Sons, Ltd.     

The Bernstein theorem for a class of fourth order equations

In this paper, we prove the 2-dimensional Bernstein theorem for a class of fourth order equations including Abreu’s equation. The main ingredients of the paper are the a priori estimates and the proof of the strict convexity.     

Blow-up phenomena for a class of fourth order parabolic equation

A class of fourth order parabolic equation is studied in this paper. Some related blow-up results are obtained by applying the potential well theory, the concavity method and a series of differential-integral inequality techniques. More precisely, under some proper assumptions, the upper and lower bounds of the blow-up time and the growth rate for blow-up solutions are estimated. Moreover, a new blow-up condition independent of the depth of the potential well is found. These results complement the recent results obtained in Han (2018).     

On some second order and fourth order elliptic overdetermined problems

This paper deals with various overdetermined problems. In Sections 2 and 3 we investigate two problems involving the electrostatic potential confined in a region of ³ bounded by two tubular conductors, and we conclude under appropriate assumptions that the only possible configuration of the two conductors is two coaxial circular cylinders. In Section 4 we prove a conjecture of Payne and Schaefer for a fourth order overdetermined problem.     

Preconditioning of saddle point problems by the method of Hermitian and skew-Hermitian splitting iterations

For a nonsingular symmetric system of linear equations with a saddle point, a Hermitian and skew-Hermitian splitting (HSS) preconditioner is considered. For the preconditioned system, symmetrizability conditions are established under which estimates are derived for the spectrum and the convergence rate of Chebyshev-type algorithms and GMRes.     

A class of operator equilibrium problems

In this paper, we consider a class of operator equilibrium problems (OEP for short) with operator solutions and derive a Minty type lemma for this class of problems. Further, using this lemma and KKM theorem, we establish some existence theorems for OEP. The theorems presented in this paper generalize, improve and unify many known results.     

Pointwise estimates in a class of fourth order nonlinear elliptic equations

Summary The subharmonicity of a function, which is defined on solutions of a fourth order nonlinear elliptic differential equation, leads to estimates on the solution and the gradient and Laplacian of the solution at interior points in terms of values on the boundary.

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Zusammenfassung Die subharmonische Eigenschaft von Funktionen, die durch die Lösungen von nichtlinearen elliptischen Differentialgleichungen vierter Ordnung bestimmt sind, führt zu punktweiser Abschätzung der Lösung, deren Gradienten, und Laplace Operatoren, abhängig von den Randwerten sind.

     

Well-posedness for a class of fourth order diffusions for image processing

A number of image denoising models based on higher order parabolic partial differential equations (PDEs) have been proposed in an effort to overcome some of the problems attendant to second order methods such as the famous Perona–Malik model. However, there is little analysis of these equations to be found in the literature. In this paper, methods of maximal regularity are used to prove the existence of unique local solutions to a class of fourth order PDEs for noise removal. The proof is laid out explicitly for two newly proposed fourth order models, and an outline is given for how to apply the techniques to other proposed models.     

The asymptotics of the eigenvalues of a fourth order differential operator with summable coefficients

A fourth order differential operator with summable coefficients and some boundary conditions is considered. Asymptotics of solutions to a fourth order differential equation is studied. The equation for eigenvalues is also studied and an asymptotics of the eigenvalues of the considered boundary value problem is obtained.     

Global uniqueness of homoclinic orbits for a class of fourth order equations

In this paper we show the global existence and uniqueness of certain orbits homoclinic to the zero stationary solution of the fourth order equation