Showing papers on "Fixed-point theorem published in 1998"
TL;DR: In this paper, a careful reading of Krasnoselskii's fixed-point theorem reveals that a convex set M and a mapping Pz = Bz + Az such that: 1. (i) Bx + Ay ∈ M for each x, y ∈ m 2. (ii) A is continuous and compact 3. (iii) B is a contraction.
203 citations
TL;DR: Two common fixed point theorems have been proved by using minimal type commutativity and contractive conditions as mentioned in this paper, and the last theorem extends known results on compatible maps to a wider class of mappings.
172 citations
TL;DR: In this paper, the authors focus on three fixed point theorems and an integral equation and prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a T-periodic solution of (0.2) when / defines a contraction mapping, while D and g satisfy certain sign conditions independent of their magnitude.
Abstract: In this paper we focus on three fixed point theorems and an integral equation. Schaefer's fixed point theorem will yield a T-periodic solution of
(0.1) x(t)= a(t) + tt-h D(t,s)g(s,x(s))ds
if D and g satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder's theorem (known as Krasnoselskii's theorem) will yield a T-periodic solution of
(0.2) x(t) = f(t,x(t)) + tt-h D(t,s)g(s,x(s))ds
if f defines a contraction and if D and g are small enough.
We prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a T-periodic solution of (0.2) when / defines a contraction mapping, while D and g satisfy the aforementioned sign conditions.
162 citations
TL;DR: In this article, the authors studied the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions and showed that there exists a spike-like stationary solution whose global maximum lies on the boundary.
Abstract: We study the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has a nondegenerate critical point. Then we show that there exists a spike-like stationary solution whose global maximum lies on the boundary. Our method is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem.
139 citations
TL;DR: In this article, a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers was proposed, where the set which possibly includes a solution is chosen by an approximate linearization of the operator appearing in the equation.
Abstract: In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a solution, we transform it by an approximate linearization of the operator appearing in the equation. Then we calculate the radii of the transformed set in order to check sufficient conditions for Banach's fixed-point theorem. This method is applied to elliptic problems and numerical examples are given.
138 citations
TL;DR: A common fixed point theorem for a sequence of mappings in a fuzzy metric space is proved and offers a generalization of Grebiec's theorem.
83 citations
TL;DR: In this paper, some fixed-point theorems for weakly contractive operators are established for functional, differential, and integral equations, and these results are used to establish existence principles for functional and differential equations.
83 citations
TL;DR: In this article, the problem of finding an optimal point in the intersection of the fixed point sets of a family of nonexpansive mappings is a frequent problem in various areas of mathematical science and engineering.
Abstract: Finding an optimal point in the intersection of the fixed point sets of a family of nonexpansive mappings is a frequent problem in various areas of mathematical science and engineering. Let be nonexpansive mappings on a Hilbert space H, and let be a quadratic function defined by for all , where is a strongly positive bounded self-adjoint linear operator. Then, for each sequence of scalar parameters (λn) satisfying certain conditions, we propose an algorithm that generates a sequence converting strongly to a unique minimizer u* of Θ over the intersection of the fixed point sets of all the Ti’s. This generalizes some results of Halpern (1967), Lions (1977), Wittmann (1992), and Bauschke (1996). In particular, the minimization of Θ over the intersection of closed convex sets Ci can be handled by taking Ti to the metric projection onto Ci without introducing any special inner products that depends on A. We also propose an algorithm that generates a sequence converging to a unique minimizer of Θ over , where K...
78 citations
TL;DR: In this article, the Lusternik-Schnirel-man type is used to prove the three critical points theorem (TCPT) in the context of Banach spaces.
Abstract: Let φ be a C real function defined on R. We assume that φ is coercive (i.e. φ(x) → ∞ as ||x|| → ∞). It is well-known that under these assumptions φ reaches a minimum at some point x0. Let now x1 be a critical point of φ which is not a global minimum. M. A. Krasnosel’skĭı [10] made the following observations: if x1 is a nondegenerate singular point of the vector field ∇φ (i.e. the topological index ind (∇φ(x1), 0) is different from zero), then φ admits a third critical point. In the sequel this statement became known as the “Three Critical Points Theorem” (TCPT). The above result of Krasnosel’skĭı was extended to the context of Banach spaces (see [1], [4], [8], [17]). Another generalization was obtained by Chang [5], [6] using the methods of Morse theory (the condition ind (∇φ(x1), 0) 6= 0 is replaced by the weaker assumption of nontriviality of Morse critical groups at x1). Also, Brezis and Nirenberg [3] gave a very useful variant of TCPT for applications using the principle of local linking (see also [12]). In this paper we shall give a proof of TCPT based on a “strong” deformation lemma (see Lemma 2.1 below) thus avoiding standard minimax techniques. In contrast to the previous work in this field, we prove in fact the Lusternik–Schnirel’man type
76 citations
TL;DR: In this article, it was shown that the Zermelo theorem implies a restriction of the Caristi fixed point theorem to continuous functions, and this restriction is proved to be equivalent to caristi's theorem under the axiom of choice.
73 citations
TL;DR: In this article, the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state was studied and a fixed point problem related to a Poincare operator was constructed and solved using an ejective fixed point theorem.
TL;DR: In this article, the authors considered the discrete focal boundary value problem and proved the existence of three positive solutions under various assumptions on f and the integers a, t2, and b. To prove their results, they used fixed point theorems concerning cones in a Banach space.
TL;DR: In this article, the existence of positive solutions for the boundary value problem is studied for continuous operators with a fixed point invariant cone in a Banach space of continuous functions. But this approach is based on fixed point theory for completely continuous operators which leave invariant a suitable cone.
TL;DR: A sampling theorem for regular sampling in shift-invariant subspaces is established and the sufficient-necessary condition for which it holds is found.
Abstract: A sampling theorem for regular sampling in shift-invariant subspaces is established. The sufficient-necessary condition for which it holds is found. Then, the theorem is modified to the shift sampling in shift-invariant subspaces by using the Zak transform. Finally, some examples are presented to show the generality of the theorem.
Journal Article•
TL;DR: In this article, general fixed point theorems for compact multimaps in the "better" admissible class defined on admissible convex subsets (in the sense of Klee) of a topological vector space not necessarily locally convex.
Abstract: We give general fixed point theorems for compact multimaps in the "better" admissible class defined on admissible convex subsets (in the sense of Klee) of a topological vector space not necessarily locally convex. Those theorems are used to obtain results for -condensing maps. Our new theorems subsume more than seventy known or possible particular forms, and generalize them in terms of the involving spaces and the multimaps as well. Further topics closely related to our new theorems are discussed and some related problems are given in the last section.n.
TL;DR: In this article, a generalization of the classical Schur modules of GL(n) exhibiting the same interplay among algebra, geometry, and combinatorics is presented, which is valid over fields of arbitrary characteristic.
TL;DR: In this article, the authors considered the boundary value problem with fixed point theorem for operators on a cone and developed criteria for the existence of two positive solutions of the problem, and established upper and lower bounds for these positive solutions for special cases.
TL;DR: In this article, a linear and a branching denotational semantics for an abstract programming language is developed, and the main instrument for the construction of the two models and for the semantical operators involved is the classical Banach Fixed Point Theorem.
TL;DR: In this article, the authors prove some common fixed point theorems in complete metrically convex spaces which generalize results due to Assad and Kirk, Itoh, Khan, and several others.
04 May 1998
TL;DR: A general approximation theorem is proved, which uniformly envelopes both the classical Stone theorem and approximation of functions of several variables by means of superpositions and linear combinations of function of one variable.
Abstract: A general approximation theorem is proved. It uniformly envelopes both the classical Stone theorem and approximation of functions of several variables by means of superpositions and linear combinations of functions of one variable. This theorem is interpreted as a statement on universal approximating possibilities ("approximating omnipotence") of arbitrary nonlinearity. For the neural networks, our result states that the function of neuron activation must be nonlinear, and nothing else.
TL;DR: In this paper, the converse of the Lefschetz fixed-point theorem holds for all selfmaps on a compact connected Lie group, and the Nielsen number of a selfmap f coincides with the Reidemeister number of f, which can be computed algebraically.
Abstract: Let G be a compact connected Lie group, K a closed subgroup (not necessarily connected) and M = G/K the homogeneous space of left cosets. Assume that M is orientable and p * : H n ( G ) → H n ( M ) is nonzero, where n = dim M . In this paper, we employ an equivariant version of Nielsen root theory to show that the converse of the Lefschetz fixed-point theorem holds true for all selfmaps on M . Moreover, if the Lefschetz number of a selfmap f : M → M is nonzero, then the Nielsen number of f coincides with the Reidemeister number of f , which can be computed algebraically.
TL;DR: Using the finite element approximations and explicit a priori error estimates for obstacle problems, an effective verification procedure is presented that through numerical computation generates a set which includes the exact solution.
Abstract: In this paper, we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities. This technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates for obstacle problems, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Further, a numerical example for an obstacle problem is presented.
TL;DR: In this paper, the existence of positive solutions to the boundary value problem was proved based on the Leray-Schauder fixed point theorem, where the boundary values are defined in a bounded domain inRN.
TL;DR: In this article, a general version of the classical result on stable and unstable manifolds for hyperbolic fixed points of diffeomorphisms is presented, and the assumption of invertibility of the mapping is removed.
Abstract: For discrete dynamical systems the theory of invariant manifolds is well known to be of vital importance. In terms of difference equations this theory is basically concerned with autonomous equations. However, the crucial and currently most difficult questions in this field are related to non-periodic, in particular chaotic motions. Since this topic - even in the autonomous context is an intrinsically time-variant matter. There is and urgent need for a non-autonomous version of invariant manifold theory. In this paper we present we present a very general version of the classical result on stable and unstable manifolds for hyperbolic fixed points of diffeomorphisms. In fact, we drop the assumption of invertibility of the mapping, we consider non-autonomous difference equations rather than mappings In effect, we generalize the notion of invariant manifold to the concept of invariant fiber bundle.
TL;DR: In this paper, a local existence theorem was established for smooth solutions which also depend continuously on the forcing function F(t), where x[i]-t is the i-th iterate of the function x(t).
Abstract: This paper is concerned with an iterative functional differential equation x(t) = c1x(t) + c2x[2](t) + … cmχ[m](t) + F(t), where x[i](t) is the i-th iterate of the function x(t). By means of Schauder's Fixed Point Theorem, we establish a local existence theorem for smooth solutions which also depend continuously on the forcing function F(t).
TL;DR: In this article, the existence theorems of maximal elements of one-person games, qualitative games, and non-compact abstract economies are obtained in H-spaces.
Abstract: Several new fixed point theorems in H-space are first proved. Next, by applying the fixed point theorems, some minimax inequalities and existence theorems of maximal elements for $\cal{L}$$_F$ correspondences and $\cal{L}$$_F$-majorized correspondences in H-spaces are obtained. Finally, using the existence theorems of maximal elements, some equilibrium existence theorems for one-person games, qualitative games and noncompact abstract economies with $\cal{L}$$_F$-majorized correspondences in H-spaces are obtained. Our theorems improve and generalize most known results due to Border, Borglin-Keiding, Ding-Kim-Tan, Ding-Tan, Ding-Tarafdar, Mehta-Tarafdar, Shafer-Sonnenschein, Tan-Yuan, Tarafdar, Toussaint, Tulcea, Yannelis and Yannelis-Prabhakar etc.
TL;DR: In this paper, existence results for second order discrete boundary value problems in abstract spaces are presented using only Sadovskii's fixed point theorem, which is the same as the one used in this paper.
Abstract: Existence results are presented for second order discrete boundary value problems in abstract spaces. Our analysis uses only Sadovskii’s fixed point theorem.