On Peano's theorem in Banach spaces

We show that if X is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping f:X→X such that the autonomous differential equation x^′=f(x) has no solution at any point.     

Differential equations with non-absolutely integrable functions in ordered Banach spaces

In this paper we derive existence and comparison results for initial value problems in ordered Banach spaces. The considered problems can be implicit, singular, functional, discontinuous and nonlocal. The main tools are fixed point results in ordered spaces and theory of HL integrable vector-valued functions. Concrete examples are presented and solved.     

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.      

The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that if 0<p<1, 0<q<1, ℝ, while if 0<p<1, 1≤q<+∞, ∝, and if 1≤p<+∞, 0<q<1, ℝ. Applications to questions regarding the global interior regularity of solutions to Poisson type problems for the three-dimensional Lamé system in Lipschitz domains are presented.     

Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces

In this paper we give the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions under the assumptions of the Hausdorff measure of noncompactness in separable and uniformly smooth Banach spaces.     

Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S-iteration process recently introduced by Sahu in [14] converges strongly to a unique fixed point of a mapping T, where T is κ-strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ_i-strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980) [1] and Yamada (2001) [15] to a general class of λ-strictly pseudocontractive mappings in uniformly smooth Banach spaces.     

Stochastic partial differential equations in M-type 2 Banach spaces

We study abstract stochastic evolution equations in M-type 2 Banach spaces. Applications to stochastic partial differential equations inL ^p spaces withp2 are given. For example, solutions of such equations are Hölder continuous in the space variables.The author is an Alexander von Humboldt Stiftung fellow     

Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces

In this paper, we introduce and investigate a new class of mixed quasi-variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality due to Ding–Tan and a lemma due to Chang, we establish some existence and uniqueness results of solution for the mixed quasi-variational-like inequality. Next, by using a KKM theorem due to Fan and an auxiliary principle technique due to Cohen, we suggest two iterative algorithms and study the convergence criteria of iterative sequences generated by the iterative algorithms. Our results extend, improve and unify several known results in the literature.     

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces

In this paper, by using a monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under wide monotonicity conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions.     

Confined Banach spaces

We discuss smoothness of theWeyl functional calculus and use it to prove that every C*-algebra is a confined Banach space. Received: 17 August 2005     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

Unconditional families in Banach spaces

It is shown that for every separable Banach space X with non-separable dual, the space contains an unconditional family of size . The proof is based on Ramsey Theory for trees and finite products of perfect sets of reals. Among its consequences, it is proved that every dual Banach space has a separable quotient.     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675     

Oscillation of linear difference equations in Banach spaces

One can easily show that almost all solutions of the difference equation

     

Periodic solutions of integro-differential equations in vector-valued function spaces

Operator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue-Bochner spaces Lp, the Besov spaces (and related spaces such as the Hölder-Zygmund spaces Cs) and the Triebel-Lizorkin spaces . We note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integro-differential equations.     

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.     

A monotone iterative technique for stationary and time dependent problems in Banach spaces

An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem Lu=Nu and the nonlinear time dependent problem u^′=(L+N)u. Under appropriate assumptions on L and N we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples.