Athanassios G. Kartsatos

New results in the perturbation theory of maximal monotone and -accretive operators in Banach spaces

Let X be a real Banach space and G a bounded, open and convex subset of X. The solvability of the fixed point problem (*) Tx + Cx 3 x in D(T) n G is considered, where T . X D D(T) -* 2X is a possibly discontinuous m-dissipative operator and C: G -X is completely continuous. It is assumed that X is uniformly convex, D(T) n G

The weak solution of a functional differential equation in a general Banach space

& 0 and (T + C)(D(T) n OG) C G. A result of Browder, concerning single-valued operators T that are either uniformly continuous or continuous with X* uniformly convex, is extended to the present case. Browders method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let IF {= 7Z+ -+ R-; /3(r) -O 0 as r -* oo}. The effect of a weak boundary condition of the type (u + Cx, x) > -/3(11xfl)flxl12 on the range of operators T + C is studied for m-accretive and maximal monotone operators T. Here, ,B E F, x C D(T) with sufficiently large norm and u C Tx. Various new eigenvalue results are given involving the solvability of Tx + ACx 3 0 with respect to (A, x) E (0, oo) x D(T). Several results do not require the continuity of the operator C. Four open problems are also given, the solution of which would improve upon certain results of the paper.

Convergence of the Kato approximants for evolution equations involving functional perturbations

Abstract The existence of a unique, weak, Lipschitz continuous solution of the abstract functional differential equation u ′( t ) + A ( t ) u ( t ) = G ( t , u t ), t ϵ [0, T ] (FDE) u ( t ) = φ ( t ), t ϵ [− r ,0] is obtained by a fixed point argument. Here X is a general Banach space, A ( t ): D ( A ( t )) ⊂ X → X is m -accretive, t ϵ [0, T ], and G is Lipschitz continuous in both variables. Stability properties of this weak solution are given, and a first order hyperbolic PDE to which the theory can be applied is considered.

A new topological degree theory for densely defined quasibounded ( S ˜ + ) -perturbations of multivalued maximal monotone operators in reflexive Banach spaces

The existence of a unique strong solution of the nonlinear abstract functional differential equation u′(t) + A(t)u(t) = F(t,ut), u0 = φeC1(¦−r,0¦,X),te¦0, T¦, (E) is established. X is a Banach space with uniformly convex dual space and, for tϵ ¦0, T¦, A(t) is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u(t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence un(t), where the functions un(t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory.

The index of a critical point for densely defined operators of type (₊)_{} in Banach spaces

Let X be an infinite-dimensional real reflexive Banach space with dual space X∗ and G⊂X open and bounded. Assume that X and X∗ are locally uniformly convex. Let T:X⊃D(T)→2X∗ be maximal monotone and C:X⊃D(C)→X∗ quasibounded and of type (S˜

Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type (S + ) L acting from a real, reflexive and separable Banach space X into X * , This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Frechet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in l p , p > 2, illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type (S + )). Applications to nonlinear Dirichlet problems have appeared elsewhere.

On the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive Banach spaces

Abstract Let X be a real reflexive Banach space and A : X→2 X ∗ be maximal monotone. Let B : X→2 X ∗ be quasibounded, finitely continuous and generalized pseudomonotone with X′⊂D(B), where X′ is a dense subspace of X such that X′∩D(A)≠∅. Let S⊂X ∗ . Conditions are given under which S⊂ R(A+B) and int S⊂int R(A+B) . Results of Browder concerning everywhere defined continuous and bounded operators B are improved. Extensions of this theory are also given using the degree theory of the last two authors concerning densely defined perturbations of nonlinear maximal monotone operators which satisfy a generalized (S+)-condition. Applications of this extended theory are given involving nonlinear parabolic problems on cylindrical domains.

Ranges of perturbed maximal monotone and

A more systematic approach is introduced in the theory of zeros of maximal monotone operators T: X D D(T) -2X, where X is a real Banach space. A basic pair of necessary and sufficient boundary conditions is given for the existence of a zero of such an operator T. These conditions are then shown to be equivalent to a certain asymptotic behavior of the resolvents or the Yosida resolvents of T. Furthermore, several interesting corollaries are given, a]nd the extendability of the necessary and sufficient conditions to the existence of zeros of locally defined, demicontinuous, monotone mappings is demonstrated. A result of Guan, about a pathwise connected set lying in the range of a morlotone operator, is improved by including non-convex domains. A partial answer to Nirenbergs problem is also given. Namely, it; is shown that a continzuous, expansive mapping T on a real Hilbert space 1t is surjective if there exists a constant ae G (0, 1) such that KTx Ty, x y) > -caX y 112, x, y E H. The methods for these results do not involve explicit use of any degree theory.

m

A more comprehensive and unified theory is developed for the solvability of the inclusions S c R(A + B), intS C R(A + B), where A: X D D(A) -+ 2y, B: X D D(B) -+ Y and S c X. Here, X is a real Banach space and Y = X or Y = X*. Mainly, A is either maximal monotone or maccretive, and B is either pseudo-monotone or compact. Cases are also considered where A has compact resolvents and B is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators A and B. Various results of Browder, Calvert and Gupta, Gupta, Gupta and Hess, and Kartsatos are improved and/or extended. The methods involve the application of a basic result of Browder, concerning pseudo-monotone perturbations of maximal monotone operators, and the Leray-Schauder degree theory.

-accretive operators in Banach spaces

where it is assumed that the “resolvent” L, = (T + (l/n)l)-i, II = 1, 2,... exists and is defined at least on the range of the operator f C. Such a class of operators T includes all m-accretive operators (cf. Kato ] 5 I). The equations (G,) can be solved by a degree theory argument if we assume, among other things, that L, is compact and C is continuous and bounded, or that L, is continuous and C compact. In most of our results we do not explicitly assume the strong continuity of the operator T. Due to thi-s fact, our theorems complement and extend various results of Browder [ 2 ], Petryshyn (6-81, Petryshyn and Tucker [9] as well as Kartsatos [4] and Ward [9]. 0 ur results are particularly related to Theorem 2 of Petryshyn [7], where operators T are studied with D(T) =X and range a Banach space Y. However, these spaces are assumed to possess