Second-order Differential Inclusions Research Articles

Existence results for second-order monotone differential inclusions on the positive half-line

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t)+q(t)u′(t)∈Au(t)+f(t) a.e. in (0,∞), with the condition (B): u(0)=x∈D(A)¯, where A:D(A)⊂H→H is a (possibly set-valued) maximal monotone operator whose range contains 0; p,q∈L∞(0,∞), with essinfp>0 and q+∈L1(0,∞). More than four decades ago, V. Barbu established the existence of a unique bounded solution to (E), (B), in the particular case p≡1, q≡0 and f≡0. Subsequently the existence and uniqueness of bounded solutions in the homogeneous case (f≡0) have been further investigated by H. Brezis (1972), N. Pavel (1976), L. Véron (1974–1976), and by E.I. Poffald and S. Reich (1986) when A is an m-accretive operator in a Banach space. The non-homogeneous case has received less attention from this point of view. In this paper, we prove existence and uniqueness of bounded solutions to (E), (B) in the general case of non-constant functions p, q satisfying the mild conditions above, thus compensating for the lack of existence theory for such kind of second order problems. Note that our results open up the possibility to apply Lions' method of artificial viscosity towards approximating the solutions of some nonlinear parabolic and hyperbolic problems, as shown in the last section of the paper.

Existence and multiplicity of solutions for second-order impulsive differential inclusions

In this paper we consider a class of a second-order impulsive differential inclusions. Using a variational method based on the non-smooth critical point theory, we prove the existence and multiplicity of anti-periodic solutions.

On controllability for Sturm-Liouville type differential inclusions

We consider a second-order differential inclusion and we obtain sufficient conditions for h-local controllability along a reference trajectory.

Relaxation Problems Involving Second-Order Differential Inclusions

We present relaxation problems in control theory for the second-order differential inclusions, with four boundary conditions,u¨(t)∈F(t,u(t),u˙(t))a.e. on[0,1];u(0)=0, u(η)=u(θ)=u(1)and, withm≥3boundary conditions,u¨(t)∈F(t,u(t),u˙(t))a.e. on[0,1]; u˙(0)=0, u(1)=∑i=1m-2‍aiu(ξi), where0<η<θ<1,0<ξ1<ξ2<⋯<ξm-2<1andFis a multifunction from[0,1]×ℝn×ℝnto the nonempty compact convex subsets ofℝn. We have results that improve earlier theorems.

Existence of three anti-periodic solutions for second-order impulsive differential inclusions with two parameters

Applying two three critical points theorems, we prove the existence of at least three anti-periodic solutions for a second-order impulsive differential inclusion with a perturbed nonlinearity and two parameters.

Three anti-periodic solutions for second-order impulsive differential inclusions via nonsmooth critical point theory

A nonsmooth version of a three critical point theorem of Ricceri (due to Iannizzotto) is used to obtain three anti-periodic solutions for a second-order impulsive differential inclusions with a perturbed nonlinearity and two parameters.

Existence of solutions for second-order differential inclusions involving proximal normal cones

In this work, we prove global existence of solutions for second-order differential problems in a general framework. More precisely, we consider second-order differential inclusions involving proximal normal cone to a set-valued map. This set-valued map is supposed to take admissible values (so in particular uniformly prox-regular values, which may be non-smooth and non-convex). Moreover the solution is required to satisfy an impact law, appearing in the description of mechanical systems with inelastic shocks.

Existence results for second-order impulsive neutral functional differential inclusions in Banach spaces

In this paper, we investigate the existence of mild solutions on a compact interval to second-order impulsive neutral functional differential inclusions in Banach spaces. The results are obtained by using the theory of continuous cosine families and a fixed point theorem due to Dhage.

On second-order functional differential inclusions in Hilbert spaces

We prove the existence result of monotone solutions, in Hilbert space, for the differential inclusion ẍ(t) ∈ f (t,T (t)x, ẋ(t))+F(T (t)x, ẋ(t)) , where f is a Caratheodory single-valued mapping and F is an upper semicontinuous set-valued mapping with compact values contained in the Clarke subdifferential ∂cV(x) of a uniformly regular function V. Mathematics subject classification (2010): 34A60, 34K05, 34K25.

Existence of solutions for second-order perturbed nonconvex sweeping process

We prove a theorem on the existence solutions for a second-order differential inclusion governed by a class of nonconvex sweeping process. Our result improves and generalizes many results from the literature of sweeping process.

On the existence of solutions for a class of second-order differential inclusions and applications

In this paper, by extending the method of guiding functions, we give sufficient conditions for the existence of solutions to the problem (1.1) u ″ ∈ Q ( u ) , u ( 0 ) = u ( 1 ) = 0 , where Q is a multivalued map whose values are not necessarily convex. It is shown how abstract results can be applied to study feedback control systems and to describe a motion of a particle in a 1-D potential.

A POSITION-BASED TIME-STEPPING ALGORITHM FOR VIBRO-IMPACT PROBLEMS WITH A MOVING SET OF CONSTRAINTS

We consider a second-order measure differential inclusion describing the dynamics of a mechanical system subjected to time-dependent frictionless unilateral constraints and we assume inelastic collisions when the contraints are saturated. For this model of impact, we propose a time-stepping algorithm formulated at the position level and we establish its convergence to a solution of the Cauchy problem.

Existence Results for Boundary Value Problems of Differential Inclusions with Three-Point Integral Boundary Conditions

We discuss the existence of solutions for a boundary value problem of second-order differential inclusions with three-point integral boundary conditions involving convex and nonconvex multivalued maps. Our results are based on the nonlinear alternative of Leray-Schauder type and some suitable theorems of fixed point theory.

PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL INCLUSIONS SYSTEMS WITH (q, p)-LAPLACIAN

Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with (q, p)-Laplacian.

Second-order differential inclusions with almost convex right-hand sides

We study the existence of solutions of a boundary second order differential inclusion under conditions that are strictly weaker than the usual assumption of convexity on the values of the right-hand side.

On controllability for a class of second-order differential inclusions

By using a suitable fixed point theorem a sufficient condition for controllability is obtained for a Sturm-Liouville type differential inclusion in the case when the right hand side has convex values.

EXISTENCE RESULTS FOR NONSMOOTH SECOND-ORDER DIFFERENTIAL INCLUSIONS, CONVERGENCE RESULT FOR A NUMERICAL SCHEME AND APPLICATION TO THE MODELING OF INELASTIC COLLISIONS

We are interested in the existence results for second-order differential inclusions, involving finite number of unilateral constraints in an abstract framework. These constraints are described by a set-valued operator, more precisely a proximal normal cone to a time-dependent set. In order to prove these existence results, we study an extension of the numerical scheme introduced in [10] and prove a convergence result for this scheme.

Analysis of lumped models with contact and friction

We consider two mathematical models that describe the vibrations of spring-mass-damper systems with contact and friction. In the first model, both the contact and frictional boundary conditions are described with subdifferentials of nonconvex functions. In the second model, the contact is modeled with a Lipschitz continuous function, and the restitution force is described by a differential equation involving a Volterra integral term. The two models lead to second-order differential inclusions with and without an integral term, in which the unknowns are the positions of the masses. For each model, we prove the existence of a solution by using an abstract result for first-order differential inclusions in finite dimensional spaces. For the second model, in addition, we prove the uniqueness of the solution by using a fixed point argument. Finally, we provide examples of systems with contact and friction conditions for which our results are valid.

A control problem governed by a second-order differential inclusion

This article studies some Bolza-type problems governed by second-order differential inclusions with two boundary conditions, where the controls are Young measures.

On two-point boundary value problems for second-order differential inclusions on manifolds

We investigate the two-point boundary value problem for second-order differential inclusions of the form on a complete Riemannian manifold for a couple of points, non-conjugate along at least one geodesic of Levi-Civita connection, where is the covariant derivative of Levi-Civita connection and F(t, m, X) is a set-valued vector field (it is either convexvalued and satisfies the upper Caratheodory condition or it is almost lower semi-continuous) such that where f : [0, ∞) → [0, ∞) is an arbitrary continuous function, increasing on [0, ∞). Some conditions on certain geometric characteristics, on the distance between points and on the length of time interval, under which the problem is solvable, are found. A generalization to inclusions of the same sort subjected to a non-holonomic constraint is also presented.