On Newton-Direction Algorithms and Diffeomorphisms*

Quasilinger hyberbolic equation and system in the “first canonic” (diagonal) from areconsidered, with both Cauchy data and boundary data “a la Cesari”, and an arbitrary number of independent variables. The aim of the paper is to develop iterative methods converging uniformly to the a.e. solution of the Cauchy or boundary value problem, based on a modified version of the proof of Cesari's original existence and uniqueness theorem and on some general results concerning contractive Lipschitz maps in the product of two Banach spaces, as derived in earlier papers. The existence proof given here is in fact founded on a new contraction map in the product of two (closed convex subsets of) Banach function spaces.

There is a plethora of k‐step solvers for equations involving operators on Banach spaces. Their convergence is estimated by adopting hypotheses on high‐order derivatives which are not even in these iterative solvers. In addition, no computable error bounds or information on the uniqueness of the solution based on Lipschitz‐type functions are given. Moreover, the choice of the initial guess is like shooting in the dark. Finally, the criteria of convergence differ from solver to solver, so no comparison can be made between their convergence domains. The novelty of our work is that we address these problems by introducing a generalized k‐step solver containing all previous k‐step solvers. Moreover, we address all previously stated problems in the earlier papers and utilizing weaker conditions that require only the continuity of the involved operator. Applications are also presented where we test the convergence criteria.

![Figure][1] In 1958 R. B. Clark and J. B. Cowey published a paper in which they presented a simple geometric model, based on the idea of a fibre-reinforced cylinder, to explain the mechanism underlying shape changes in ribbon worms and flatworms ([Clark and Cowey, 1958][2]). While their

Throughout this paper A denotes an operator function, holomorphic on a deletedneighborhood of a complex number λo, with values in the space ℒ(X,Y) of boundedlinear operators between two complex Banach spaces X and Y. In his survey article(7), I. C. Gohberg has defined for such an arbitrary operator function A the algebraic multiplicity RM(A;λo) and the reduced algebraic multiplicity RM(A;λo) of A at λo. In earlier papers (e.g., (8, 16)) these notions have been defined and studied for morerestricted classes of operator functions. In (8) Gohberg and Sigal treated the case when A is finite-meromorphic at λo, A(λ) is bijective for λ in some deleted neighbor-hood of λo and the constant term A0 in the Laurent expansion of A at λo is aFredholm operator. They proved that in this case

Much has been learned in recent years about the existence, determination, and properties of power-series-like expansions for expressing a nonlinear system's outputs in terms of its inputs. In particular, the existence and local convergence of expansions, and of certain “associated expansions,” for important large classes of systems are now well established. While the focus of attention has been on questions such that the size of the inputs for which convergence is guaranteed is not the main issue, some related material has appeared that bears on the problem of determining the extent of the region of convergence. The result most closely related to this paper is a recent theorem that gives necessary and sufficient conditions under which f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> has a generalized power-series expansion when f is an invertible locally-Lipshitz map between certain general subsets of two complex Banach spaces. In applications involving nonlinear models, ordinarily only real spaces of inputs and outputs are of direct interest. A “complexification” involving a certain solvability condition in complex spaces has to be able to be carried out to use the theorem referred to above. This paper reports on pertinent general results concerning invertible maps between subsets of real Banach spaces, with their complex extensions, and with generalized power-series expansions in both real and complex spaces. It focuses on questions concerning expansions for inverses of maps defined in real spaces. The results show that for a very large class of systems that have input/output maps, the ability to complexify is not just a useful sufficient condition for expandability, but is in fact the key condition for an input/output map to be representable by a generalized power-series expansion.

Corrigendum to “Approximation by [formula omitted]-smooth, Lipschitz functions on Banach spaces” [J. Math. Anal. Appl. 315 (2006) 599–605

We obtain existence theorems and Filippov-Wazewski type relaxation theorems for differential inclusions in Banach spaces with measurable-pseudo-Lipschitz right-hand side. For the solution sets of these differential inclusions, we also describe some properties that extend classical theorems on continuous dependence and on differentiation of solutions with respect to initial data.

In a separable Banach space we consider a differential inclusion whose values are nonconvex, closed, but not necessarily bounded sets. Along with the original inclusion, we consider the inclusion with convexified right-hand side. We prove existence theorems and establish relations between solutions to the original and convexified differential inclusions. In contrast to assuming that the right-hand side of the inclusion is Lipschitz with respect to the phase variable in the Hausdorff metric, which is traditional in studying this type of questions, we use the (ρ–H) Lipschitz property. Some example is given.

In the present paper, we consider solvability conditions for the Cauchy problem for ordinary differential equations of fractional order α> 1 in Banach spaces with so-called “worsening” righthand sides. For integer-order (first- and higher-order) equations with “worsening” operators, there are classical results due to Nagumo and Ovsyannikov (e.g., see [1–6]), based on the analysis of the convergence of the successive approximation method in scales of continuously embedded Banach spaces. A number of further results in this direction was obtained in [7–9]. Similar results for fractionalorder equations with “worsening” operators were considered in [10]. Nazarov [11, 12] developed a new method for first-order differential equations in scales of compactly embedded Banach spaces on the basis of Tonelli approximations. In the present paper, we suggest a modification of his method, which results in a number of assertions on the (nonlocal!) solvability of the Cauchy problem for differential equations of integer or fractional order strictly greater than unity.

: The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.

A nonhomogeneous evolution equation the right-hand side of which is unknown is considered in a Banach space. For the determination of the unknown right-hand side, in addition to the Cauchy condition, a special overdetermination is given. In the paper a systematic theory for the investigation of the problem indicated is constructed, and sufficient conditions for the existence and uniqueness of solution are given. Special attention is paid to certain problems of perturbation theory.

We consider the Caffarelli–Silvestre extension problem, i.e., a Bessel type ODE in a Banach space X with a closed and typically unbounded operator A as right-hand side and point out a couple of consequences arising from the assumption of the well-posedness of the problem. In the end a conjecture is stated concerning the implications of analyticity of the solution of the extension problem.

The flourishing field of fixed point theory started in the early days of topology with seminal contributions by Poincare, Lefschetz-Hopf, and Leray-Schauder at the turn of the 19th and early 20th centuries. The theory vigorously developed into a dense and multifaceted body of principles, results, and methods from topology and analysis to algebra and geometry as well as discrete and computational mathematics. This interdisciplinary theory par excellence provides insight and powerful tools for the solvability aspects of central problems in many areas of current interest in mathematics where topological considerations play a crucial role. Indeed, existence for linear and nonlinear problems is commonly translated into fixed point problems; for example, the existence of solutions to elliptic partial differential equations, the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming. The classical fixed point theorems of Banach and Brouwer marked the development of the two most prominent and complementary facets of the theory, namely, the metric fixed point theory and the topological fixed point theory.Themetric theory encompasses results and methods that involve properties of an essentially isometric nature. It originates with the concept of Picard successive approximations for establishing existence and uniqueness of solutions to nonlinear initial value problems of the 1st order and goes back as far as Cauchy, Liouville, Lipschitz, Peano, Fredholm, and most particularly, Emile Picard. However, the Polish mathematician Stefan Banach is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Metric fixed point theory for important classes of mapping gained respectability and prominence to become a vast field of specialization partly and not only because many results have constructive proofs, but also because it sheds a revealing light on the geometry of normed spaces, not to mention its many applications in industrial fields such as image processing engineering, physics, computer science, economics, and telecommunications. A particular interest in fixed points for set-valued operators developed towards the mid-20th century with the celebrated extensions of the Brouwer and Lefschetz theorems by Kakutani and Eilenberg-Montgomery, respectively. The Banach contraction principle was later on extended to multivalued contractions by Nadler. The fixed point theory for multivalued maps found numerous applications in control theory, convex and nonsmooth optimization, differential inclusions, and economics. The theory is also used prominently in denotational semantics (e.g., to give meaning to recursive programs). In fact, it is still too early to truly estimate the importance and impact of set-valued fixed point theorems in mathematics in general as the theory is still growing and finding renewed outlets. This special issue adds to the development of fixed point theory by focusing on most recent contributions. It includes works on nonexpansive mappings in Banach and metric spaces, multivalued mappings in Banach and metric spaces, monotone mappings in ordered spaces, multivalued mappings in ordered spaces, and applications to such nonmetric spaces as modular spaces, as well as applications to logic programming and directed graphs.

Periodic solutions of a class of functional differential equations

Modification of the application of a contraction mapping method on a class of fractional differential equation