Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces

We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: dv(t)/dt+V_∂φt(u(t))∋f(t), v(t)∈H_∂ψ(u(t)), 0<t<T, where H_∂ψ (respectively, V_∂φt) denotes the subdifferential operator of a proper lower semicontinuous functional ψ (respectively, φt explicitly depending on t) from a Hilbert space H (respectively, reflexive Banach space V) into (−∞,+∞] and f is given. To do so, we suppose that V?H≡H^∗?V^∗ compactly and densely, and we also assume smoothness in t, boundedness and coercivity of φt in an appropriate sense, but use neither strong monotonicity nor boundedness of H_∂ψ. The method of our proof relies on approximation problems in H and a couple of energy inequalities. We also treat the initial-boundary value problem of a non-autonomous degenerate elliptic-parabolic problem.     

Evolution inclusions governed by the difference of two subdifferentials in reflexive Banach spaces

The existence of strong solutions of Cauchy problem for the following evolution equation du(t)/dt+∂?¹(u(t))-∂?²(u(t))∋f(t) is considered in a real reflexive Banach space V, where ∂?¹ and ∂?² are subdifferential operators from V into its dual V^*. The study for this type of problems has been done by several authors in the Hilbert space setting.The scope of our study is extended to the V-V^* setting. The main tool employed here is a certain approximation argument in a Hilbert space and for this purpose we need to assume that there exists a Hilbert space H such that V⊂H≡H^*⊂V^* with densely defined continuous injections.The applicability of our abstract framework will be exemplified in discussing the existence of solutions for the nonlinear heat equation: where Ω is a bounded domain in R^N. In particular, the existence of local (in time) weak solution is shown under the subcritical growth condition q<p^* (Sobolev's critical exponent) for all initial data This fact has been conjectured but left as an open problem through many years.     

Multivalued evolution equations with nonlocal initial conditions in Banach spaces

We prove the existence of integral solutions to the nonlocal Cauchy problem in a Banach space X, where is m-accretive and such that –A generates a compact semigroup, has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and . The case when A depends on time is also considered.      

On impulsive functional differential inclusions with Hille-Yosida operators in Banach spaces

We consider a semilinear functional differential inclusion with infinite delay and impulse characteristics in a Banach space assuming that its linear part is a non-densely defined Hille-Yosida operator. We assume that the multivalued nonlinearity of upper Carathèodory or almost lower semicontinuous type satisfies a regularity condition expressed in terms of the measures of noncompactness. We apply the theory of integrated semigroups and the theory of condensing multivalued maps to obtain local and global existence results. The application to an optimization problem for an impulsive feedback control system is given.     

Evolution equations governed by the sweeping process

This paper is concerned with variants of the sweeping process introduced by J.J. Moreau in 1971. In Section 4, perturbations of the sweeping process are studied. The equation has the formX(t) -N _C(t) (X(t)) +F(t, X(t)). The dimension is finite andF is a bounded closed convex valued multifunction. WhenC(t) is the complementary of a convex set,F is globally measurable andF(t, ·) is upper semicontinuous, existence is proved (Th. 4.1). The Lipschitz constants of the solutions receive particular attention. This point is also examined for the perturbed version of the classical convex sweeping process in Th. 4.1. In Sections 5 and 6, a second-order sweeping process is considered:X (t) -N _C(X(t)) (X(t)). HereC is a bounded Lipschitzean closed convex valued multifunction defined on an open subset of a Hilbert space. Existence is proved whenC is dissipative (Th. 5.1) or when allC(x) are contained in a compact setK (Th. 5.2). In Section 6, the second-order sweeping process is solved in finite dimension whenC is continuous.     

Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces

In this paper we deal with the existence of impulsive mild solutions for semilinear differential inclusions with nonlocal conditions, where the linear part generates an evolution system and the nonlinearity satisfies the lower Scorza-Dragoni property. Our theorems extend the existence propositions proved by Fan in 2010. An example is presented.     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces

This paper deals with the existence of mild L-quasi-solutions to the initial value problem for a class of semilinear impulsive evolution equations in an ordered Banach space E. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. An example is also given.     

Second order evolution equations with time-dependent subdifferentials

We study an abstract second order nonlinear evolution equation in a real Hilbert space. We consider time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function. Introducing appropriate assumptions on the convex functions and other data, we prove the existence and uniqueness of a strong solution, and give some applications of the abstract theorem to hyperbolic variational inequalities with time-dependent constraints.      

Model of chemotaxis with threshold density and singular diffusion

A quasilinear singular parabolic system corresponding to recent models of chemotaxis in which (1) there is an impassable threshold for the density of cells and (2) the diffusion of cells becomes singular (fast or superdiffusion) when the density approaches the threshold. It is proved that for some range of parameters describing the relation between the diffusive and the chemotactic component of the cell flux there are global-in-time classical solutions which in some cases are separated from the threshold uniformly in time. Global-in-time weak solutions in the case of fast diffusion and the set of stationary states are studied as well. The applications of the general results to particular models are shown.     

Economic equilibrium problems in reflexive Banach spaces

The problem is to minimize a finite collection of objective functions over admissible sets depending on the so-called price vector. The minima in question and the price vector are linked together by a subdifferential governing law. The problem stated as a system of variational–hemivariational inequalities, defined on a nonconvex feasible set, is reduced to one variational–hemivariational inequality involving nonmonotone multivalued mapping. The existence of solutions is examined under the assumption that the constrained functions are positive homogeneous of degree one. The study has been inspired by economic issues and leads to new results concerning the existence of competitive equilibria.     

Solutions with compact support to the Cauchy problem of an equation modeling the motion of viscous droplets

The Cauchy problem to an equation arising in modeling the motion of viscous droplets is studied in the present paper. The authors prove that if the initial data has compact support, then there exists a weak solution which has compact support for all the time.     

A multiplicity theorem for critical points of functionals on reflexive Banach spaces

In this paper we establish a multiplicity theorem for critical points of functionals on reflexive Banach spaces. Precisely, we deduce the main result using a general variational principle proved by Ricceri. Moreover, we present an application to a Neumann problem which gives a positive answer to some questions formulated by the previous author.Received: 6 February 2003     

Analysis of nonlinear fractional control systems in Banach spaces

In this paper, we consider the nonlinear control systems of fractional order and its optimal controls in Banach spaces. Using the fractional calculus, Hölder’s inequality, p-mean continuity, weakly singular inequality and Leray-Schauder’s fixed point theorem with compact mapping, the sufficient condition is given for the existence and uniqueness of mild solutions for a broad class of fractional nonlinear infinite dimensional control systems. Utilizing the approximately lower semicontinuity of integral functionals and weakly compactness, we extend the existence result of optimal controls for nonlinear control systems to nonlinear fractional control systems under generally mild conditions. An example is given to illustrate the effectiveness of the results obtained.     

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.     

A nonlinear equation in Banach spaces and applications to the well-posedness of Cauchy problems

We analyze a nonlinear equation in Banach spaces, with the nonlinearity composed of multiple terms of different degrees. We prove a theorem regarding the existence of solutions for such equations. Moreover, we show how this result may be applied to obtain the well-posedness of various parabolic initial value problems.     

Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations

We consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. We then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in RN with nonlinearities depending on the gradient of the solution. We consider as well systems of reaction-diffusion equations in RN and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in RN. We further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation.     

Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions

In this paper, we prove a sufficient condition for the global existence of bounded C⁰-solutions for a class of nonlinear functional differential evolution equation of the form where X is a real Banach space, A is the infinitesimal generator of a nonlinear compact semigroup, is a nonempty, convex, weakly compact valued, and almost strongly–weakly u.s.c. multi-function, and is nonexpansive.     

Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup

We study the controllability problem for a system governed by a semilinear differential inclusion in a Banach space not assuming that the semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows us to apply the topological degree theory for condensing operators and to obtain the controllability results for both upper Carathéodory and almost lower semicontinuous types of nonlinearity. As application we consider the controllability for a system governed by a perturbed wave equation.     

An alternative approach to well-posedness of a class of differential inclusions in Hilbert spaces

We show the well-posedness of initial value problems for differential inclusions of a certain type using abstract perturbation results for maximal monotone operators in Hilbert spaces. For this purpose the time derivative is established in an exponentially weighted _L2$L_2$ space. The problem of well-posedness then reduces to show that the sum of two maximal monotone operators in time and space is again maximal monotone. The theory is exemplified by three inclusions describing phenomena in mathematical physics involving hysteresis.