Abstract: We show that if is a bounded open set in a complete space , and if is nonexpansive, then always has a fixed point if there exists such that for all . It is also shown that if is a geodesically bounded closed convex subset of a complete -tree with , and if is a continuous mapping for which for some and all , then has a fixed point. It is also noted that a geodesically bounded complete -tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory.

Abstract: This paper establishes an existence theorem for a certain class of functional integral equations via a new fixed point theorem of Leray-Schauder type for operators defined on Banach algebras.

Abstract: Introduction.- Part 1. Functional Analysis Background. 1. Preliminaries. 2. Real and Vector Measures.- Part 2. Multifunctions. 3. Preliminary Notions. 4. Upper and Lower Semicontinuous multifunctions. 5. Measurable Multifunctions. 6. Caratheodory type multifunctions. 7. Fixed Points Property for Convex-Valued Mapping.- Part 3. Decomposability. 8. Decomposable Sets. 9. Selections. 10. Fixed Points Property. 11. Aumann Integrals. 12. Selections of Aumann Integrals. 13. Fixed Points for Multivalued Contractions. 14. Operator and Differential Inclusions. 15. Decomposable Analysis.- Bibliography.

Abstract: We provide sucient conditions for the existence of positive solu- tions of a boundary-value problem for a one dimensional -Laplacian ordinary dierential equation with deviating arguments, where is a sup-multiplicative- like function (in a sense introduced here) and the boundary conditions include nonlinear expressions at the end points. For this end, we use the Krasnoselskii fixed point theorem in a cone. The results obtained improve and generalize known results in (17) and elsewhere.

Abstract: In this article, it is shown that some of the hypotheses of a fixed point theorem of the present author involving three operators in a Banach algebra are redundant. Our claim is also illustrated with the applications to some nonlinear functional integral equations for proving the existence result.

Abstract: In this paper, we apply a cone theoretic fixed point theorem and obtain conditions for the existence of positive solutions to the three-point nonlinear second order boundary value problem with boun...

Abstract: Fixed point, domain invariance and coincidence results are presented for single-valued generalized contractive maps of Meir–Keeler type defined on complete metric spaces (or more generally complete gauge spaces). The maps of Caristi type are also considered. In addition the random analogue of some of these fixed point results will be presented. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Abstract: In this paper, we examine controllability problems of evolution inclusions with nonlocal conditions. Using the set-valued and single-valued Monch fixed-point theorem, we establish some sufficient conditions for the controllability under convex and nonconvex orientor fields respectively. Our approach is different from all previous approaches; we do not assume that the evolution system generates a compact semigroup; so, our method is applicable to a wide class of (impulsive) control systems and evolution inclusions in Banach spaces.

Abstract: In this paper we study the existence of multiple positive solutions for the equation ( g ( u ′))′+ e ( t ) f ( u )=0, where g ( v )≔| v | p −2 v , p >1, subject to nonlinear boundary conditions. We show the existence of at least two positive solutions by using a new three functionals fixed point theorem in cones.

Abstract: In this paper we present a local result on the existence of insensitizing controls for a semilinear heat equation when nonlinear boundary conditions of the form $\partial_n y + f(y) = 0$ are considered. The problem leads to an analysis of a special type of nonlinear null controllability problem. A sharp study of the linear case and a later application of an appropriate fixed point argument constitute the scheme of the proof of the main result. The boundary conditions we are dealing with lead us to seek a fixed point, and thus also control functions, in certain Holder spaces. The main strategy in this paper is the construction of controls with Holderian regularity starting from L2-controls in the linear case. Sufficient regularity in the data and appropriate assumptions on the right-hand side term $\xi$ of the equation are required.

Abstract: In this paper, we extend several concepts from geometry of Ba- nach spaces to modular spaces With a careful generalization, we can cover all corresponding results in the former setting Main result we prove says that if is a convex, -complete modular space satisfying the Fatou property and r-uniformly convex for all r > 0, C a convex, -closed, -bounded subset of X , T : C ! C a -nonexpansive mapping, then T has a xed point

Abstract: We study a class of nonvariational elliptic systems involving the p-Laplacian. Under sublinear and superlinear assumptions on the nonlinearities, we show the existence of solutions using fixed point theorems.

Abstract: A noncommutative version of a best approximation result for generalized 7-nonexpansive maps is obtained. Let E = (E, ||.||) be aBanach space and C a subset of E. Let T,I : E-+E. Then T is called 7-nonexpansive on C if ||Tx Ty\\\\ < ||Ix — Iy\\\\ for all x, y € C. The set of fixed points of T (resp. 7) is denoted by F(T) (resp. F(I)). The set C is called p-starshaped with p € C if the segment [x, p] joining x to p is contained in C for all x € C (that is, kx + (1 — k)p € C for all x 6 C and all k with 0 < k < 1). The mappings T and 7 are said to be: (1) commuting on C if ITx = Tlx for all x G C; (2) ii-weakly commuting on C [3] if there exists a real number R > 0 such that | | T l x ITx|| < R\\\\Tx Ix|| for all x € C. Suppose C is p-starshaped with p € F(I) and is both Tand /-invariant. Then T and I are called (3) -R-subcommuting on C [6, 7] if there exists a real number R > 0 such that | | T l x ITx\\\\ < (R/k)\\\\(kTx + (1 k)p) Ix|| for all x € C and all k 6 (0,1). Clearly commutativity implies i?-subcommutativity but the converse is not true in general (see, [6]). The set Pc(&) = {y € C : ||y — x|| = dist(x, C)} is called the set of best approximants to x € X out of C, where dist(x, C) = inf{||y — x|| : y € C}. The space E is said to satisfy Opial's condition [2] if for every sequence {xn} C E converging weakly to y G E, liminf ||x„ — y|| < liminf ||xn — x|| n—*oo n—>oo \" \" holds for all x T£ y.

Abstract: The authors and their colleagues have developed numerical verification methods for solutions of second-order elliptic boundary value problems based on the infinite-dimensional fixed-point theorem using the Newton-like operator with appropriate approximation and constructive a priori error estimates for Poisson's equations. Many verification results show that the authors' methods are sufficiently useful when the equation has no first-order derivative. However, in the case that the equation includes the term of a first-order derivative, there is a possibility that the verification algorithm does not work even though we adopt a sufficiently accurate approximation subspace. The purpose of this paper is to propose an alternative method to overcome this difficulty. Numerical examples which confirm the effectiveness of the new method are presented.

Abstract: Minimal predicates P satisfying a given first-order description (P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order 'PIA conditions' (P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of 'predicate intersection'. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond.

Abstract: We prove the existence of periodic solutions in a second order differential system with a singular potential of attractive or repulsive type and forced periodically. The proof is based on a Krasnoselskii fixed point theorem for absolutely continuous operators on a Banach space, and this makes possible to avoid any kind of "strong force" condition.

Abstract: In [J.Math.Anal.App.277(2003) 645-650], W.A.Kirk introduced the notion of asymptotic contractions and proved a fixed point theorem for such mappings. Using techniques from proof mining, we develop a variant of the notion of asymptotic contractions and prove a quantitative version of the corresponding fixed point theorem.

Abstract: A computer assisted proof of the existence of nontrivial steady-state solutions for the two-dimensional Rayleigh-Benard convection is described. The method is based on an infinite dimensional fixed-point theorem using a Newton-like operator. This paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact nontrivial solution. All discussed numerical examples take into account of the effects of rounding errors in the floating point computations.

Abstract: Sufficient conditions for controllability of neutral functional integrodifferential infinite delay systems in Banach spaces are established. The results are obtained by using the analytic semigroup theory and the Nussbaum fixed point theorem. An example is provided to illustrate the theory.

Abstract: In this paper, we consider a vector version of Minty's lemma and obtain existence theorems for two kinds of variational-like inequality. A fixed point theorem is also discussed.