Multivalued Problems Research Articles

On the approximation solvability of the abstract differential inclusion

The approximation solvability of the abstract differential inclusion du/dt∈f(t,u) is presented. The convergence of the approximation solution and the existence of the solution for abstract evolution multivalued problem are discussed.

Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

We consider quasilinear strongly resonant problems with discontinuous right hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang and a generalized version of the Ekeland variational principle. At the end of the paper we also show that the nonsmooth (PS)-condition implies the coercivity of the functionaj extending this way a well known result of the "smooth" case.

Fixed Points of Multi-valued Maps and Static Coulomb Friction Problems

The existence of solutions to the Signorini problem with Coulomb friction is a long standing open question. We prove the existence of generalized solutions that satisfy the pointwise Coulomb friction conditions on the entire interface and the normal nonpenetration condition on the complement of a subset with arbitrarily small but possibly positive measure. Furthermore, the penetration itself can also be made arbitrarily small. Although “measure zero” instead of “arbitrarily small measure” would be needed to fully resolve the issue, these generalized solutions seem to be the closest answer available to date. Their existence is proved by a suitable application of Ky Fan's fixed point theorem for multi-valued maps. The same method can be used with a number of variants involving contact of two or more elastic bodies and possible debonding phenomena.

Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

We consider quasilinear strongly resonant problems with discontinuous right‐hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais‐Smale (PS)‐condition implies the coercivity of the functional, extending this way a well‐known result of the “smooth” case.

Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems

In this paper we consider a general nonlinear boundary value problem for second-order differential inclusions. We prove two existence theorems, one for the ``convex'' problem and the other for the ``nonconvex'' problem. Then we show that the solution set of the latter is dense in the C 1 (T,R N ) -norm to the solution set of the former (relaxation theorem). Subsequently for a Dirichlet boundary value problem we prove the existence of extremal solutions and we show that they are dense in the solutions of the convexified problem for the C 1 (T,R N ) -norm . Our tools come from multivalued analysis and the theory of monotone operators and our proofs are based on the Leray—Schauder principle.

Periodic Solutions for Second Order Differential Inclusions with Nonconvex and Unbounded Multifunction

In this paper we consider a second order multivalued periodic boundary value problem with a nonconvex and unbounded orientor field (set-valued vector field). Using a directionally continuous selector, through its Filippov regularization we produce a convex-valued, bounded multifunction and with this as orientor field we introduce a new multivalued periodic problem. Using the Leray-Schauder principle, we solve the convex problem and then we show that its solutions also solve the original nonconvex problem.

Multiple solutions for nonlinear discontinuous elliptic problems near resonance

We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for

Locally lipschitzian guiding function method for ODEs

by generalizing the well known guiding function method. Such an approach can be found in several works, however, under some heavier assumptions. For example, in [M] f is assumed to be locally lipschitzian and the guiding function to be C. In [GP] f needs to be of the Caratheodory type only, while the corresponding guiding function must be still C. In fact, in [GP] the more general, multivalued problem{ x′(t) ∈ φ(t, x(t)) x(0) = x(T ). (1′)

Alternating frequency/coefficient (AFC) technique in the harmonic balance method

A solution procedure for a system of non-linear equations containing a convolution term is developed by using the harmonic balance method in conjunction with a new alternating frequency/coefficient (AFC) technique. Numerical results show that, unlike the conventional harmonic balance method, the new technique is very effective in searching for solutions for multi-valued non-linear problems, especially in the vicinity of a jump phenomenon.

Measuring the MTF for focal plane arrays using random noise targets

The previous development of a unique discrete modulation transfer function (DMTF) for infrared systems based on focal plane arrays has solved the multivalued problem which is associated with the non-isoplanatic nature of discrete systems. This paper proposes an efficient method for measuring this DMTF using random noise targets, which have a bandwidth equal to that of the collection optics but much larger than the Nyquist rate of the detector array. A microscan technique has been used in order to measure this DMTF beyond the passband of the detector array.

Nontrivial solutions for a multivalued problem with strong resonance

The Mountain-Pass Theorem of Ambrosetti and Rabinowitz (see [1]) and the Saddle Point Theorem of Rabinowitz (see [21]) are very important tools in the critical point theory of C1-functional. That is why it is natural to ask us what happens if the functional fails to be differentiable. The first who considered such a case were Aubin and Clarke (see [6]) and Chang (see [12]),who gave suitable variants of the Mountain-Pass Theorem for locally Lipschitz functionals which are denned on reflexive Banach spaces. For this aim they replaced the usual gradient with a generalized one, which was firstly defined by Clarke (see [13], [14]).As observed by Brezis (see [12, p. 114]), these abstract critical point theorems remain valid in non-reflexive Banach spaces.

A 3D vector potential formulation using edge element for electrostatic field computation

This paper presents a D-oriented formulation for 3D electrostatic field computation. In this formulation, the electric flux density D is expressed by a source field and the curl of an unknown vector potential. The edge element is used for the discretization of the vector potential. The use of gauge condition for the uniqueness of the problem is discussed. Special techniques are introduced for solving the multi-valued problem and treating the voltage boundary condition. The complementary nature between the present D-oriented formulation and a usually used E-oriented formulation is mentioned. A numerical example is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the approximation of nonmonotone multivalued problems by monotone subproblems

Structures involving zig-zag type reaction, displacement or stress-strain laws, i.e. nonmonotone possibly multivalued laws, cannot be effectively treated by the classical numerical methods for nonlinear laws. In order to calculate the arising free boundaries accurately, we propose a new approximation method of the nonmonotone problem by monotone ones. The proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities. Also the case of combining this procedure with fixed point algorithms to treat more complicated problems is considered. The numerical methods proposed are illustrated mainly by numerical examples which are described in detail.

Periodic solutions of dry friction problems

Dry friction problems lead to discontinuous differential equations, e.g. to $$x'' + xx' + \mu \operatorname{sgn} x' + \beta ^2 x = \varphi (t),$$ where sgn γ=γγ for γ ≠ 0 and sgn (0)=[ − 1. 1]. We study existence of ω-periodic solutions of (1) in case ϕ is ω-periodic. Results forx> 0 are given in the book “Multivalued Differential Equations” (K. Deimling: De Gruyter 1992). and preliminary ones forx=0 are contained in K. Deimling “Multivalued differential equations and dry friction problems” (Proc. Conf. Differential & Delay Equations. World Sci. Publ. 1992). Based on the latter and considerable additional analysis, we give a complete description of the resonant casex=0.β=1. ϕ(t)=sint. In particular, it turned out that for μ e (π 4. 1) there is a unique globally asymptotically stable 2π-periodic solutionxgm, which necessarily has deadzones (i.e.xμ (t) ≡c in certain intervals). In addition, the nonresonant case is solved by means of degree theory for multivalued maps, since in this situation a priori bounds can be found easily.

An objective MRTD for discrete infrared imaging systems

An objective method has been developed for measuring the minimum resolvable temperature difference (MRTD) of infrared imaging systems employing focal plane detector arrays. The resulting MRTD is uniquely determined by applying discrete Fourier analysis, thus avoiding the previous multi-valued problem associated with the non-isoplanatic property of discrete devices. The derivation is based on sinusoidal wave input targets but a practical measuring procedure has been developed using conventional bar patterns, alternatively the MRTD may be calculated by measuring the MTF using the line spread function.

Unitary and Hermitian phase operators for the electromagnetic field.

Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] have proposed a formalism for a Hermitian optical phase operator, constructed from the so-called phase states. In this paper we give an alternative derivation. We begin by defining a unitary phase operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ from the relationship $\stackrel{^}{a}={e}^{i\stackrel{^}{\ensuremath{\varphi}}}R(\stackrel{^}{N})$ ($\stackrel{^}{a}$ is an annihilation operator for a single-mode boson field), but borrow the idea of a finite- (but arbitrarily large) dimensional space from Pegg and Barnett. Finally, we comment on an experiment to determine quantum phase uncertainty. Our analysis uses the operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$, which we feel is a more natural choice than $\stackrel{^}{\ensuremath{\varphi}}$. The use of ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ also avoids the multivalued problems associated with $\stackrel{^}{\ensuremath{\varphi}}$.

Hopfield model with multistate neurons and its optoelectronic implementation

A Hopfield model with multistate neurons is described. This model is able to deal with multivalued problems such as restoring a degraded image with gray level equivalent to that produced by the two-states model, but needs many less neurons and interconnections. The performance of this model is compared with that of the linear model, and it is concluded that the multistate neuron model can produce convergence more quickly. Finally, a hybrid system for the implementation of this model is discussed.

New technique for solving three-dimensional multiply connected eddy-current problems

In three-dimensional eddy-current computation, when the conductors are multiply connected, the use of a scalar potential in exterior regions leads to a multivalued problem. To avoid this difficulty, in a hybrid finite-element/boundary-integral model, the authors propose to use the electric field as the state variable in both conducting and nonconducting regions. A vectorial boundary-integral method, based on the use of a current sheet on the boundary, is applied in exterior regions. A spanning tree technique is introduced to ensure the null divergence of the boundary current sheet. This technique permits also a reduction in the number of boundary unknowns.

An existence theorem for inclusions of the type and application to a multivalued boundary value problem

In this paper we deal with inclusions of the type , where each of ψ and φ is an operator from a set V into an Lp space of vector-valued functions. After establishing a general existence result, we present an application to the problem

On the solution set of differential inclusions in banach space

Let X be a separable Banach space and F(•,•) an orientor field from [0,b]xX into the nonempty, compact, convex subsets of X, which is completely Hausdorff upper semicontinuous and We show that solution set of the multivalued Cauchy problem set in Cχ