Existence results for impulsive hyperbolic differential inclusions

In this article we investigate the existence of solutions for second order impulsive hyperbolic differential inclusions in separable Banach spaces. By using suitable fixed point theorems, we study the case when the multi-valued map has convex and non-convex values.     

An Equivalence Theorem on Minimum Sheltering Speed for Non-convex Habitats

The paper is devoted to a new family of variational problems for differential inclusions, motivated by the protection of human and wildlife habitats when an invasive environmental disaster occurs. Indeed, the mathematical model consists of a differential inclusion describing the expansion of invaded region over time, and an artificial barrier that cannot be penetrated by the invasive agent is erected to shield the habitat, which serves as the control strategy and is characterized as a one-dimensional rectifiable set. We consider an isotropic case that the disaster spreads with uniform speed in all directions, and develop an equivalence result on the minimum construction speed required for implementing a sheltering strategy determined by a rectifiable Jordan curve. By measuring the invaded portion of barriers over time, it is shown that each connected non-convex habitat admits an equivalent habitat that contains the given habitat and their minimum speeds are equal. In particular, the boundary of an equivalent habitat can be partitioned into two arcs: an arc is locally convex, and the other is part of the boundary of illuminated area. This leads to a corollary on the existence of admissible sheltering strategies for non-convex habitats.     

On four-point boundary value problems for differential inclusions and differential equations with and without multivalued moving constraints

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri and B. Ricceri we prove existence results improving earlier theorems by Gupta and Marano.     

Nonconvex Differential Variational Inequality and State-Dependent Sweeping Process

We prove the existence of solutions of a differential variational inequality involving a prox-regular set in an infinite dimensional Hilbert space via a new existence result of a non-convex state-dependent sweeping process.     

On Periodic Viable Solutions of Differential Inclusions

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Emarks on Differential Inclusions Without Existence or Continuous Dependence

We study differential inclusions with existence which are natural approximations of differential inclusions without existence. In a Banach space setting, we also establish a multivalued version of a recent result of Colombo and Garay [3].     

Approximation and optimization of higher order discrete and differential inclusions

This paper is mainly concerned with the necessary and sufficient conditions of optimality for Cauchy problem of higher order discrete and differential inclusions. Applying optimality conditions of problems with geometric constraints, for arbitrary higher order (say s-order) discrete inclusions optimality conditions are formulated. Also some special transversality conditions, which are peculiar to problems including third order derivatives are formulated. Formulation of sufficient conditions both for convex and non-convex discrete and differential inclusions are based on the apparatus of locally adjoint mappings. Furthermore, an application of these results is demonstrated by solving the problems with third order linear discrete and differential inclusions.     

On the “bang-bang” principle for nonlinear evolution inclusions

We establish the existence of extreme solutions for a class of nonlinear evolution inclusions with non-convex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense in the solutions of the system with convexified right-hand side. Subsequently we use this density result to derive nonlinear and infinite-dimensional version of the “bang-bang” principle for control systems. An example of a nonlinear parabolic distributed parameter system is also worked out in detail. Received November 21, 1997     

Controllability for a class of second-order evolution differential inclusions without compactness

In this paper, we discuss the existence and controllability for a class of second-order evolution differential inclusions without compactness in Banach spaces. By applying the technique of weak topology and Glicksberg–Ky Fan fixed point theorem, we prove our main results without the hypotheses of compactness on the operator generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Further, we extend our study to existence and controllability of second-order evolution differential inclusions with nonlocal conditions and impulses. Finally, an example is given for the illustration of the obtained theoretical results.     

Differential mixed variational inequalities in finite dimensional spaces

In this paper, we introduce and study a class of differential mixed variational inequalities in finite dimensional Euclidean spaces. Under various conditions, we obtain linear growth and bounded linear growth of the solution set for the mixed variational inequalities. Moreover, we present some conclusions which enrich the literature on the mixed variational inequalities and generalize the corresponding results of [4]. In particular we prove existence theorems for weak solutions of a differential mixed variational inequality in the weak sense of Carathéodory by using a result on differential inclusions involving an upper semicontinuous set-valued map with closed convex values. Also by employing the results from differential inclusions we establish a convergence result on Euler time-dependent procedure for solving initial-value differential mixed variational inequalities.     

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.     

Positive Solutions of Fractional Differential Inclusions at Resonance

This paper deals with the fractional differential inclusions at resonance. By the recent Leggett-Williams theorem for coincidences of multi-valued operators due to O’Regan and Zima in [19], we present a new result on the existence of positive solutions for a class of differential inclusion of fractional order with boundary conditions at resonance. And our results improve and generalize the existing results.     

Existence of Solutions for Nonlinear Fractional q-Difference Inclusions with Nonlocal Robin (Separated) Conditions

In this paper, we show the existence of solutions for nonlinear fractional q-difference inclusions involving convex as well as non-convex valued maps with nonlocal Robin (separated) conditions. Our results are new in the present configuration and are based on some standard principles for multivalued maps. A special case for q-difference inclusions in the given setting is also discussed. Some interesting observations are presented.     

二阶微分包含的边值问题及其应用

在Banach中讨论了一类二阶微分包含的边值问题和一类反馈控制问题.结合不动点定理,对解的存在性给出了两个充分性条件,最后应用前面的结果建立了反馈控制问题解的存在性定理.     

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.     

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.     

On Perturbed Fractional Differential Inclusions with Nonlocal Multi-point Erdélyi–Kober Fractional Integral Boundary Conditions

In this paper, by using a nonlinear alternative for a sum of compact upper semicontinuous and contractive multivalued operators, we establish sufficient conditions for the existence of solutions for perturbed fractional differential inclusions with nonlocal multi-point Erdélyi–Kober fractional integral boundary conditions. For the applicability of the main result, we include an example.     

Existence of optimal controls for a class of systems governed by differential inclusions on a Banach space

Using Cesari's approach, we prove the existence of optimal controls for a class of systems governed by differential inclusions on a Banach space having the Radon-Nikodym property. Theorem 3.1 gives the existence result for optimal relaxed controls under fairly general assumptions on the system and the admissible controls. This result depends on a fundamental result (Theorem 2.1) that proves the existence of mild solutions of differential inclusions on a Banach space, which has also independent interest. Further, the preparatory results, such as Lemma 3.1 and Lemma 3.2, are also useful in the study of time-optimal and terminal control problems.For illustration of the results, we present two examples, one on distributed controls for a class of systems governed by nonlinear parabolic equations and the other on boundary controls with discontinuous boundary operator.This work is supported in part by the National Science and Engineering Council of Canada under Grant No. 7109.     

A viability theorem for set-valued states in a Hilbert space

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H.