Solutions to boundary Cauchy problems with infinite delay

In this paper, we investigate the existence, uniqueness of solutions to boundary Cauchy equations with infinite delay, which are more general than the previous studies. The conclusions are applied to an age dependent population with delay in the birth process.     

Generalized Cauchy problems involving nonlocal and impulsive conditions

We study the abstract Cauchy problem involving a class of nonlinear differential inclusions, with impulsive and nonlocal conditions. By using MNC estimates, the existence result and continuous dependence on initial data of the solution set are proved.     

Fractional boundary value problems with integral boundary conditions

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.     

Impulsive boundary value problems with nonlinear boundary conditions

In this paper, the upper and lower solution method and Schauder’s fixed point theorem are employed in the study of boundary value problems for a class of second-order impulsive ordinary differential equations with nonlinear boundary conditions. We prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.     

Second-order boundary value problems with integral boundary conditions

In this paper we investigate the existence of positive solutions of nonlocal second-order boundary value problems with integral boundary conditions.     

Relationship between solutions of families of two-point boundary value problems and Cauchy problems

We carry out a qualitative analysis and suggest a method for the solution of the two-point boundary value problems A _l υ = g, x ? [0, l], l ? (0, L) = B ? R ₊; $$ \alpha v'_x \left| {_{x = + 0} = F_1 (v,l)} \right|_{x = 0} , \beta v'_x \left| {_{x = l - 0} = F_2 (v,l)} \right|_{x = l} , $$ , where α, β ? R, υ is the unknown function, g = g(x) is a given real function, F ₁(y, l) and F ₂(y, l) are known real functions defined on the sets Y ₁ × B and Y ₂ × B, respectively, where Y ₁ ∪ Y ₂ ? R, and A _l is the restriction of A corresponding to the embedding parameter l. (Here A is an operator taking an arbitrary function in the set of function classes defined in the paper to C(B ₁), where B ₁ = [0, L).) The study takes into account the dependence of solutions of various versions of these two-point boundary value problems on the parameter l. We construct algorithms for the reduction of these families of two-point boundary value problems to Cauchy problems for ordinary differential equations and integro-differential equations that contain only first derivatives of the unknown functions with respect to the parameter l.     

Fourth-order problems with nonlinear boundary conditions

We develop a new method of lower and upper solutions for a fourth-order nonlinear boundary value problem where the differential equation has dependence on all lower-order derivatives. Our boundary conditions are nonlinear. We will assume the functions that define the nonlinear boundary conditions are either monotone or nonmonotone. As a result we obtain existence principles which improve recent results in the literature.     

Higher order boundary value problems with nonhomogeneous boundary conditions

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

     

Blow-up in parabolic problems under Robin boundary conditions

By means of a first-order differential inequality technique, sufficient conditions are determined which imply that blow-up of the solution does occur or does not occur for the semilinear heat equation under Robin boundary conditions. In addition, a lower bound on blow-up time is obtained when blow-up does occur.     

Patterns in parabolic problems with nonlinear boundary conditions

We obtain existence of asymptotically stable nonconstant equilibrium solutions for semilinear parabolic equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator in such domains. This information is used to show that the asymptotic dynamics of the heat equation in this domain is equivalent to the asymptotic dynamics of a system of two ordinary differential equations diffusively (weakly) coupled. The main tools employed are the invariant manifold theory and a uniform trace theorem.     

Second order singular boundary value problems with integral boundary conditions

We study the second order singular boundary value problem

     

A solution to boundary value problems with over-specified boundary conditions

Summary The problem of determining the unknown terms of a differential equation from over-specified boundary conditions is solved by means of the potential theory. The boundary values of compatibility functions adapting the differential equations to the prescribed boundary conditions are introduced as explicit unknowns in the system of linear equations. Two applications to fluid mechanics are presented, which demonstrate the efficiency of the method.

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Résumé Le problème de déterminer les termes inconnus d'une équation différentielle, à partir de conditions aux limites surdéterminées, est résolu au moyen de la théorie des potentiels. Les valeurs aux limites de fonctions de compatibilité adaptant les équations différentielles aux conditions aux limites prescrites sont introduites, comme inconnues explicites, dans le système des équations intégrales linéaires. Deux applications à la mécanique des fluides sont présentées, qui démontrent l'efficacité de la méthode.

     

Method for studying boundary value problems with nonlinear boundary conditions

A constructive approach to the determination of an approximate solution of a boundary value problem with nonlinear boundary conditions g[z (0), z (T)]=0 is proposed. Existence of the exact solution is proved, and error estimates for the constructed approximate solution are provided.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 951–957, July, 1990.     

Optimal control problems with two-point boundary conditions

We study convex as well as some nonconvex optimal control problems for infinite-dimensional first-order linear differential systems with two-point boundary conditions of antiperiodic type. The main advance of this paper over existing related papers consists in the fact that the domains of the possible nonlinear operators involved in the boundary conditions have empty interior. Applications to boundary control systems governed by partial differential equations describing incompressible flows with fading memory and steering a temperature field are given.This work was supported in part by the National Science Foundation under Grant No. DMS-91-11794.     

Optimality conditions for reflecting boundary control problems

We consider a control problem with reflecting boundary and obtain necessary optimality conditions in the form of the maximum Pontryagin principle. To derive these results we transform the constrained problem in an unconstrained one or we use penalization techniques of Morreau-Yosida type to approach the original problem by a sequence of optimal control problems with Lipschitz dynamics. Then nonsmooth analysis theory is used to study the convergence of the penalization in order to obtain optimality conditions.     

Fourth-order problems with fully nonlinear boundary conditions

Differential inequality method, bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth-order differential equations under nonlinear boundary conditions, and many existing results are complemented.