Approximating solutions of Hammerstein type equations in Banach spaces

Abstract

Let X be a real Banach space and X^? be its dual. Let F: X → X^? and K: X^? → X be Lipschitz monotone mappings. In this paper an explicit iterative scheme is constructed for approximating solutions of the Hammerstein type equation, 0 = u + KF u, when they exist. Strong convergence of the scheme is obtained under appropriate conditions. Our results improve and unify many of the results in the literature.     

A population dynamics model with nonautonomous past

Abstract

In this paper we study a two-phase population model, which distinguishes the population by two different stages

By the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous past

where 1 ≤ p < ∞ and I = [?r, 0] (finite delay) or I = (?∞, 0] (infinite delay), E a Banach space, Φ : W^1,p(I, E) → E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8].     

Modulus Semigroups and Perturbation Classes for Linear Delay Equations in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper we study C₀-semigroups on X × L_p( − h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1,∞). Dedicated to the memory of H. H. Schaefer     

Boundary-value problems for two-dimensional canonical systems

The two-dimensional canonical systemJy=–Hy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The formal defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ) to a trace-normed system on which is in the limit-point case at ±. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.     

Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

We consider the Tikhonov-like dynamics where A is a maximal monotone operator on a Hilbert space and the parameter function ε(t) tends to 0 as t→∞ with . When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u(t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A⁻¹(0) provided that the function ε(t) has bounded variation, and provide a counterexample when this property fails.     

Iterative construction of fixed points of strictly pseudocontractive mappings

Let (E, ‖ ? ‖) be a smooth Banach space over the real field and A a nonempty closed bounded convex subset of E. Suppose T : A → A is a uniformly continuous strictly pseudocontractive selfmapping of A. Then, if [math001]satisfies [math001]the iteration process [math001] and [math001] converges strongly to the unique fixed point x of T. This is an improvement of a result of C.E. Chidume who established strong convergence of (x ⁿ to x in case E is L ^p or l ^p with [math001] making essential use of the inepuality [math001] which is kown to hold in these spaces for all x and y     

Topological degree theory for multi-valued mappings of class (S +) L

In this paper, we introduce the multi-valued mapping of class (S₊)_L in reflexive Banach spaces, which is a generalization of the single valued mapping of class (S₊)_L in [9], then we construct a degree theory for multi-valued mappings of class (S₊)_L.Received: 28 November 2003; revised: 30 July 2004     

Subsolutions for abstract evolution equations

We study different notions of subsolutions for an abstract evolution equation du/dt+Auf where A is an m-accretive nonlinear operation in an ordered Banach space X with order-preserving resolvents. A first notion is related to the operator d/dt+A in the ordered Banach space L ¹(0, T; X); a second one uses the evolution equation du/dt+A uf where A :x{y;zy for some zAx}; other notions are also considered.     

The Iterated Aluthge Transform of an Operator

The Aluthge transform (defined below) of an operator T on Hilbert space has been studied extensively, most often in connection with p-hyponormal operators. In [6] the present authors initiated a study of various relations between an arbitrary operator T and its associated , and this study was continued in [7], in which relations between the spectral pictures of T and were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates { ⁽ⁿ⁾} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence { ⁽ⁿ⁾} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6). Submitted: December 5, 2000? Revised: August 30, 2001.     

Asymptotic properties of some non-autonomous systems in Banach spaces

Let X be a reflexive Banach space. We introduce the notion of weakly almost nonexpansive sequences (xn)_n?0 in X, and study their asymptotic behavior by showing that the nonempty weak ω-limit set of the sequence (xn/n)_n?1 always lies on a convex subset of a sphere centered at the origin of radius d=limn→∞‖xn/n‖. Subsequently we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system , where A is an accretive (possibly multivalued) operator in X×X, and f−f_∞∈Lp((0,+∞);X) for some f_∞∈X and 1?p<∞. These results extend recent results of J.S. Jung and J.S. Park [J.S. Jung, J.S. Park, Asymptotic behavior of nonexpansive sequences and mean points, Proc. Amer. Math. Soc. 124 (1996) 475-480], and J.S. Jung, J.S. Park, and E.H. Park [J.S. Jung, J.S. Park, E.H. Park, Asymptotic behaviour of generalized almost nonexpansive sequences and applications, Proc. Nonlinear Funct. Anal. 1 (1996) 65-79], as well as our results cited below containing previous results by several authors.     

Operators with Wentzell boundary conditions and the Dirichlet‐to‐Neumann operator

In this paper we relate the generator property of an operator A with (abstract) generalized Wentzell boundary conditions on a Banach space X and its associated (abstract) Dirichlet‐to‐Neumann operator N acting on a “boundary” space . Our approach is based on similarity transformations and perturbation arguments and allows to split A into an operator A₀₀ with Dirichlet‐type boundary conditions on a space X₀ of states having “zero trace” and the operator N. If A₀₀ generates an analytic semigroup, we obtain under a weak Hille–Yosida type condition that A generates an analytic semigroup on X if and only if N does so on . Here we assume that the (abstract) “trace” operator is bounded that is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators.     

Averaging methods for finding periodic orbits via Brouwer degree

We consider the problem of finding T-periodic solutions for a differential system whose vector field depend on a small parameter ε. An answer to this problem can be given using the averaging method. Our main results are in this direction, but our approach is new. We use topological methods based on Brouwer degree theory to solve operator equations equivalent to this problem. The regularity assumptions are weaker then in the known results (up to second order in ε). A result for third order averaging method is also given.As an application we provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function. A concrete example is given.     

FACTORIZATION OF UNBOUNDED THIN AND COTHIN OPERATORS

Abstract

Let X and Y be normed spaces and T: D(T) ? X → Y a linear operator. Following R.D. Neidingcr [N1] we recall the Davis, Figiel, Johnson, Pelczynski factorization of T corresponding to a parameter p (1 ≤ p ≤ ∞) and apply the corresponding factorization result in [N1] to unbounded thin operators. Properties equivalent to ubiquitous thinness arc derived. Defining an operator T to be cothin if its adjoint is thin, a dual factorization result for cothin operators is obtained, where for each 1 < p < ∞, the intermediate space in the factorization is cohereditarily l_p. This result is shown to hold more generally for the cases when T is either partially continuous or closable; in particular, such operators are strictly cosingular. A condition for a closable weakly compact operator to be strictly cosingular follows as a corollary.     

Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

Dissipative q-Dirac operator with general boundary conditions

In this paper, we consider the symmetric q-Dirac operator. We describe dissipative, accumulative, self-adjoint and the other extensions of such operators with general boundary conditions. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.     

Über Die Konvergenz Operatorwertiger Cosinusfunktionen mit Gestörtem Infinitesimalen Erzeuger

On convergence of operator cosine functions with perturbed infinitesimal generator. The question under what kind of perturbations a closed linear operatorA remains of the class of infinitesimal generators of operator cosine functions seems to be a rather difficult one and is unsolved in general. In this note we give bounds for the perturbation of operator cosine functions caused byA-bounded perturbationsT ofA under the assumption thatT + A is also a generator.

     

Inequalities for the Hadamard Weighted Geometric Mean of Positive Kernel Operators on Banach Function Spaces

Let K₁, . . . , K_n be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K₁, . . . , K_n, the operator K, satisfies the following inequalities where || · ||and r(·) denote the operator norm and the spectral radius, respectively. In the case of completely atomic measure space we show some additional results. In particular, we prove an infinite-dimensional extension of the known characterization of those functions satisfying for all non-negative matrices A₁, . . . , A_n of the same order.