Surjectivity Result Research Articles

New surjectivity results for perturbed weakly coercive operators of monotone type in reflexive Banach spaces

Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X∗. Let T:X⊇D(T)→2X∗ be maximal monotone and S:X→2X∗ be bounded pseudomonotone. Let p be a nonnegative integer. Assume, further, that there exist nonnegative constants ai(i=1,2,…,p) such that 〈v∗+w∗,x〉≥−∑i=1pai‖x‖i−α(‖x‖)‖x‖p+2 for all x∈D(T) with sufficiently large ‖x‖, v∗∈Tx and w∗∈Sx, where α:[0,∞)→[0,∞) such that α(t)→0 as t→∞. New surjectivity results are given for the operator T+S along with weakly coercive type hypothesis on T+S. The results are new and improve the corresponding theory for coercive operators of monotone type. The theory developed herein can be suitably applied in the study of partial differential equations, variational and hemi-variational inequality problems in appropriate Sobolev spaces. To demonstrate the applicability of the theory, an example of time periodic parabolic partial differential equation, which models nonmonotone semipermeability problem, is provided.

Spectral rigidity and invariant distributions on Anosov surfaces

This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal tensors. The surjectivity results are of independent interest and imply the existence of many geometric invariant distributions on the unit sphere bundle. In particular, we show that on any Anosov surface $(M, g)$, given a smooth function $f$ on $M$, there is a distribution in the Sobolev space $H^{-1}(SM)$ that is invariant under the geodesic flow and whose projection to $M$ is the given function $f$.

Existence of anti-periodic solutions for a class of first order evolution inclusions

The existence of anti-periodic solutions for a class of first order nonlinear evolution inclusions defined in the framework of an evolution triple of spaces is considered. We study the problems under both convexity and nonconvexity conditions on the multivalued right-hand side. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions, the surjectivity result for L-pseudomonotone operators and continuous extreme selection results from multivalued analysis. An example of a nonlinear parabolic problem is given to illustrate our results.

Existence Results for Variational Inequalities with Surjectivity Consequences Related to Generalized Monotone Operators

We present existence results for variational inequalities given by generalized monotone operators. As a consequence, we deduce the existence of zeros, or even more, the surjectivity of some classes of set-valued operators. We show that by strengthening the continuity assumptions, similar surjectivity results can be obtained without any monotonicity assumption. In the framework of reflexive Banach spaces, we extend a related result due to Inoan and Kolumban (Nonlinear Anal. 68:47–53, 2008).

Dedekind zeta motives for totally real number fields

Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $\pi^{n[k:\Q]}\zeta^*_k(1-n)$, where $\zeta^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$.

Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator

The paper deals with second order nonlinear evolution inclusions and their applications. We study evolution inclusions involving a Volterra-type integral operator, which are considered within the framework of an evolution triple of spaces. First, we deliver a result on the unique solvability of the Cauchy problem for the inclusion by combining a surjectivity result for multivalued pseudomonotone operators and the Banach contraction principle. Next, we provide a theorem on the continuous dependence of the solution to the inclusion with respect to the operators involved in the problem. Finally, we consider a dynamic frictional contact problem of viscoelasticity for materials with long memory and indicate how the result on evolution inclusion is applicable to the model of the contact problem.

Some criteria for maximal abstract monotonicity

In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel’s duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators.

On the regulator of Fermat motives and generalized hypergeometric functions

We calculate the Beilinson regulator of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity results of the regulator, which support the Beilinson conjecture on special values of L -functions.

Dynamic contact problem for viscoelastic piezoelectric materials with normal damped response and friction

In this paper, we deal with a class of inequality problems for dynamic frictional contact between a piezoelectric body and a foundation. The model consists of a system of the hemivariational inequality of hyperbolic type for the displacement, the time dependent elliptic equation for the electric potential. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. The existence of a weak solution to the model is proved by embedding the problem into a class of second-order evolution inclusions and by applying a surjectivity result for multivalued operators.

Existence of anti-periodic solutions for quasilinear parabolic hemivariational inequalities

In this paper, we consider the problem of the existence of anti-periodic solutions for quasilinear parabolic hemivariational inequalities with a pseudomonotone operator. Using regularized approximating and first eigenfunction methods and the surjectivity result, we prove the existence of solutions for quasilinear parabolic hemivariational inequalities.

Hemivariational inequalities modeling dynamic contact problems with adhesion

This paper deals with the mathematical modelling of viscoelastic frictional contact processes in mechanics which involve adhesion. The model consists of a coupled system of the hemivariational inequality of hyperbolic type for the displacement and the ordinary differential equation for the bonding field. The frictional forces are derived from nonconvex superpotentials through the generalized Clarke subdifferential. The properties of the body are described by a modified Kelvin–Voigt constitutive law. The existence of weak solution to the problem is proved by embedding it into a class of second order evolution inclusions and by applying a surjectivity result for multivalued pseudomonotone operators. We also establish a result on the regularity of weak solution to the model. Finally, examples of subdifferential boundary conditions which include the functions of d.c. type are provided.

A DSM proof of surjectivity of monotone nonlinear mappings

A simple proof is given of a basic surjectivity result for monotone operators. The proof is based on the dynamical systems method (DSM).

Dynamic contact problem for viscoelastic piezoelectric materials with slip dependent friction

In this paper, we deal with a class of inequality problems for dynamic frictional contact between a piezoelectric body and a foundation. The model consists of a system of the hemivariational inequality of hyperbolic type for the displacement, the time dependent elliptic equation for the electric potential. The friction condition is described to be the Clarke subdifferential relations of nonmonotone and multivalued character in the tangential directions on the boundary. The existence of a weak solution to the model is proved by embedding the problem into a class of second-order evolution inclusions, and by applying a surjectivity result for multivalued operators.

A surjectivity result for quasibounded operators

Using a degree theory for countably 1-contractive operators, we show a surjectivity theorem for such quasibounded operators. Moreover, the existence of an eigenvalue for these operators is presented.

Irreducibility and p-adic monodromies on the Siegel moduli spaces

We generalize the surjectivity result of the p-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any p-rank stratum. We discuss irreducibility and connectedness of some p-rank strata of the moduli spaces with parahoric level structure. Finer results are obtained on the Siegel 3-fold with Iwahori level structure.

Surjectivity for Hamiltonian 𝐺-spaces in 𝐾-theory

Let G G be a compact connected Lie group, and ( M , ω ) (M,\omega ) a Hamiltonian G G -space with proper moment map μ \mu . We give a surjectivity result which expresses the K K -theory of the symplectic quotient M / / G M /\!\!/G in terms of the equivariant K K -theory of the original manifold M M , under certain technical conditions on μ \mu . This result is a natural K K -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K K -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G G -spaces. We discuss this lemma in detail and highlight the differences between the K K -theory and rational cohomology versions of this lemma. We also introduce a K K -theoretic version of equivariant formality and prove that when the fundamental group of G G is torsion-free, every compact Hamiltonian G G -space is equivariantly formal. Under these conditions, the forgetful map K G ∗ ( M ) → K ∗ ( M ) K_{G}^{*}(M)\to K^{*}(M) is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in H 2 ( M ; Z ) H^{2}(M;\mathbb {Z}) admits an equivariant extension in H G 2 ( M ; Z ) H_{G}^{2}(M;\mathbb {Z}) .

Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion

A model of a dynamic viscoelastic adhesive contact between a piezoelectric body and a deformable foundation is described. The model consists of a system of the hemivariational inequality of hyperbolic type for the displacement, the time dependent elliptic equation for the electric potential and the ordinary differential equation for the adhesion field. In the hemivariational inequality the friction forces are derived from a nonconvex superpotential through the generalized Clarke subdifferential. The existence of a weak solution is proved by embedding the problem into a class of second-order evolution inclusions and by applying a surjectivity result for multivalued operators.

Existence of antiperiodic solutions for hemivariational inequalities

In this paper we consider the problem of existence of antiperiodic solutions for first-order and second-order hemivariational inequalities with a pseudomonotone operator. We first give the surjectivity result and then prove a existence of antiperiodic solutions for hemivariational inequalities with the surjectivity result.

Existence and Uniqueness to a Dynamic Bilateral Frictional Contact Problem in Viscoelasticity

In this paper we examine an evolution problem which describes the dynamic bilateral contact of a viscoelastic body and a foundation. The contact is modeled by a friction multivalued subdifferential boundary condition which incorporates the Coulomb law of friction, the SJK model and the orthotropic friction law. The main result concerns the existence and uniqueness of weak solutions to the hyperbolic variational inequality when the friction coefficient is sufficiently small. The proof is based on a surjectivity result for multivalued operators and a fixed point argument.

Existence Theorems for Set-valued Operators in Banach Spaces

In this paper, we study the problem of existence of zero points for set-valued operators in Banach spaces. We present a simple condition which can be used to verify that under some conditions, set-valued operators have zero points. Furthermore, we derive the surjectivity result and the Kakutani fixed point theorem by using this condition.