Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces

Let V and V* be a real reflexive Banach space and its dual space, respectively. This paper is devoted to the abstract Cauchy problem for doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations of the form: ?_V y^t (u￠(t)) + ?_V j(u(t)) + B(t, u(t)) ' f(t){\partial_V \psi^t (u{^\prime}(t)) + \partial_V \varphi(u(t)) + B(t, u(t)) \ni f(t)} in V*, 0 < t < T, u(0) = u ₀, where ?_V y^t, ?_V j: V ? 2^V*{\partial_V \psi^t, \partial_V \varphi : V \to 2^{V^*}} denote the subdifferential operators of proper, lower semicontinuous and convex functions y^t, j: V ? (-￥, +￥]{\psi^t, \varphi : V \to (-\infty, +\infty]}, respectively, for each t ? [0,T]{t \in [0,T]}, and f : (0, T) → V* and u₀ ? V{u_0 \in V} are given data. Moreover, let B be a (possibly) multi-valued operator from (0, T) × V into V*. We present sufficient conditions for the local (in time) existence of strong solutions to the Cauchy problem as well as for the global existence. Our framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e., B o 0{B \equiv 0}) might not be uniquely solved in a doubly nonlinear setting. Our proof relies on a couple of approximations for the equation and a fixed point argument with a multi-valued mapping. Moreover, the preceding abstract theory is applied to doubly nonlinear parabolic equations.     

A nonstandard density theorem for weak topologies on Banach and Bochner spaces

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ?? of subsets of E, of nets of vectors in K. We then apply this result to various class ?? and setsK in the context of E‐valued Bochner integrable functions defined on a finite measure space.     

Continuous selections and reflexive Banach spaces

Every l.s.c. mapping from a space into the non-empty closed convex subsets of a reflexive Banach space admits a continuous selection provided it satisfies a ``weak' u.s.c. condition. This result partially generalizes some known selection theorems. Also, it is successful in solving a problem concerning the set of proper lower semi-continuous convex functions on a reflexive Banach space.

     

Product integration in reflexive Banach spaces

LetX denote a reflexive Banach space and {A(t)|t[0,T]} a time dependent family of accretive operators defined onX. Conditions are placed on {A(t)|t[0,T]} which are sufficient to guarantee the existence of solutions to the Cauchy initial value problem:u(t,x)+A(t)u(t,x)=0; u(0,x)=x. These solutions are obtained via the method of product integration; however limits for the infinite product are taken with respect to the weak topology.     

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.     

A characterization of reflexive Banach spaces

A Banach space is not reflexive if and only if there exist a closed separable subspace of and a convex closed subset of with empty interior which contains translates of all compact sets in . If, moreover, is separable, then it is possible to put .

     

Hammerstein integral inclusions in reflexive Banach spaces

In this paper we examine multivalued Hammerstein integral equations defined in a separable reflexive Banach space. We prove existence theorems for both the ``convex' problem (the multifunction is convex-valued) and the ``nonconvex' problem (the multifunction is not necessarily convex-valued). We also show that the solution set of the latter is dense in the solution set of the former (relaxation theorem). Finally we present some examples illustrating the applicability of our abstract results.

     

Proximal analysis in reflexive smooth Banach spaces

In the present paper, we introduce and study a new proximal normal cone in reflexive Banach spaces in terms of a generalized projection operator. Two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth Banach spaces. The density theorem for both proximal subdifferentials has been proved in p-uniformly convex and q-uniformly smooth Banach spaces. Various important properties and applications of our concepts are also proved.     

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.     

Integral boundary value problem for impulsive integro-differential equations in Banach spaces

In this paper, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solution of the integral boundary value problem for a class of first-order impulsive integro-differential equations in a real Banach space. An explicit iterative scheme for the unique solution and an error estimate of the approximation sequence are also derived.     

Stable pseudomonotone variational inequality in reflexive Banach spaces

Stability of a generalized variational inequality with either the mapping or the set perturbed is discussed in reflexive Banach spaces, provided that the mappings are pseudomonotone in the sense of Karamardian. As a byproduct, generalized variational inequality having nonempty and bounded set is proved to be equivalent to the strictly feasibility.     

Proximal methods in reflexive Banach spaces without monotonicity

We introduce the concept of hypomonotone point-to-set operators in Banach spaces, with respect to a regularizing function. This notion coincides with the one given by Rockafellar and Wets in Hilbertian spaces, when the regularizing function is the square of the norm. We study the associated proximal mapping, which leads to a hybrid proximal-extragradient and proximal-projection methods for nonmonotone operators in reflexive Banach spaces. These methods allow for inexact solution of the proximal subproblems with relative error criteria. We then consider the notion of local hypomonotonicity and propose localized versions of the algorithms, which are locally convergent.