Robin boundary value problems on arbitrary domains

We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish --estimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations.

     

Higher order boundary value problems with nonhomogeneous boundary conditions

We study the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition

     

Second order singular boundary value problems with integral boundary conditions

We study the second order singular boundary value problem

     

Higher order boundary value problems on time scales

In this paper we study the existence of solutions for Lidstone boundary value problems on time scale. Firstly, by using Schauder fixed point theorem in a cone, we obtain the existence of solutions to a Lidstone boundary value problem (LBVP). Secondly, existence result for this problem is also given by the monotone method. Finally, by using Krasnosel'skii fixed point theorem, it is proved that the LBVP has a positive solution.     

Second order nonlocal boundary value problems at resonance

We present sufficient conditions for the existence of positive solutions for some second order boundary value problems at resonance. The boundary conditions that we study are quite general, involve a Stieltjes integral and include, as particular cases, multi‐point and integral boundary conditions. Our results are based on a Leggett‐Williams norm‐type theorem due to O'Regan and Zima. We employ a general abstract approach which allows us to improve and complement recent results in the literature. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Singularly perturbed higher order periodic boundary value problems

The existence of positive periodic solutions is proved for the higher order singular differential equation

(0.1)     

Fourth order finite difference method for sixth order boundary value problems

In the present paper, finite difference method is used to construct an approximate solution for the sixth order linear boundary value problems. Numerical examples are considered to illustrate the efficiency and convergence of the method. Numerical results show that proposed method is very effective, efficient, and fourth order accurate. Also fourth order accurate numerical value of second and fourth derivatives of solution, were obtained as by product of the proposed method.     

Approximate analytical solution of fourth order boundary value problems

In this paper, the decomposition method is applied to boundary-value problems of fourth order for ordinary differential equations. AMS subject classification 65L10, 65L20, 34B15Waleed Al-Hayani: He was a professor in the Department of Mathematics, College of Science, Mosul University, Mosul, Iraq.     

Higher order multi‐point boundary value problems

We consider the boundary value problem where n ? 2 and m ? 1 are integers, t_j ∈ [0, 1] for j = 1, …, m, and f and g_i, i = 0, …, n ? 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi‐point boundary conditions studied in the literature. Schauder’s fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Iterative procedures for nonlinear second order boundary value problems

Summary Questions of the existence of positive solutions of second order nonlinear boundary value problems with separated boundary conditions are investigated. The nonlinearities are such that the linearization about the trivial solution does not exist or is trivial. The methods are thus applicable when shooting methods or ordinary bifurcation techniques cannot be applied. The conditions on the nonlinearity are quite modest and both super and sublinear problems can be included.This research was performed while the author was visiting at Emory University.Research supported by AFOSR 87-0140.     

Second order three-point boundary value problems in abstract spaces

In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundaryvalue problem We search for solutions of the above problem in the Banach space of continuous functions C（[O, 1], E） with the Pettis integrability assumptions imposed on $. Some classes of Pettis-integrable functions are described in the paper and exploited in the proofs of main results. We stress on a class of pseudo-solutions of considered problem. Our results extend previous results of the same type for both Bochner and Pettis integrability settings. Similar results are also proved for differential inclusions i.e. when f is a multivalued function.     

Nonlinear boundary value problems for second order differential inclusions

In this paper, we study a class of nonlinear value boundary problems for second order differential inclusions with nonlinear perturbations, which satisfy the generalized Hartman condition weaker than that considered in some papers. Using techniques from multivalued analysis, theory of monotone operators and fixed points, we prove the existence of solutions in both “convex” and “nonconvex” cases. Our framework can be incorporated with Dirichlet, Neumann, and mixed boundary problems.     

Uniqueness in some higher order elliptic boundary value problems

Summary The uniqueness of the solution to two boundary value problems for the linear equation ³ u –a ² u +b u –cu =F and to two boundary value problems for the quasilinear differential equation ² u +w(u) =f are proved. The proofs follow as a consequence of maximum principles for a functional which is defined on solutions to the differential equation.

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Zusammenfassung Die Eindeutigkeit der Lösung zweier Randwertaufgaben für die lineare Gleichung ³ u –a ² u +b u –cu =F und zweier Randwertaufgaben für die quasilineare Differentialgleichung ² u +w(u) =f wird bewiesen. Der Beweis folgt aus einem Maximumprinzip für ein Funktional, das für die Lösungen der Gleichung definiert ist.