Krasnoselskii's Theorem Research Articles

On the existence of nonnegative solutions to an integral equation with applications to boundary value problems

By the use of a Guo–Krasnoselskii theorem in cones, existence of positive eigenvalues yielding nonnegative or positive solutions to an integral equation is studied. The results are applied to a variety of boundary value problems concerning ordinary differential equations.

The topological nature of Krasnoselskii’s cone fixed point theorem

In recent years, the Krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types. In this article we discuss the topological nature of the Krasnoselskii theorem and show that it can be restated in a more general form without reference to a cone structure or the norm of the underlying Banach space. This new perspective brings out a closer relation between the Krasnoselskii Theorem and the classical Brouwer fixed point theorem. It also points to some obvious extensions of the cone theorem.

On Krasnoselskii's Cone Fixed Point Theorem

AbstractIn recent years, the Krasnoselskii fixed point theorem for cone maps and its many generalizations have been successfully applied to establish the existence of multiple solutions in the study of boundary value problems of various types. In the first part of this paper, we revisit the Krasnoselskii theorem, in a more topological perspective, and show that it can be deduced in an elementary way from the classical Brouwer-Schauder theorem. This viewpoint also leads to a topology-theoretic generalization of the theorem. In the second part of the paper, we extend the cone theorem in a different direction using the notion of retraction and show that a stronger form of the often cited Leggett-Williams theorem is a special case of this extension.

Generalizations of the Krasnoselskii fixed point theorem

We obtain new generalized forms of the Krasnoselskii theorem on fixed points for a sum of two operators and show that our new results encompass a number of previously known generalizations of the theorem.

EIGENVALUES OF COUNTABLY CONDENSING ADMISSIBLE MAPS

Applying a fixed point theorem for compact admissible maps due to Gorniewicz, we prove that under certain conditions each count ably condensing admissible maps in Frechet spaces has a positive eigenvalue. This result has many consequences, including the well-known theorem of Krasnoselskii.

Fixed points for weakly inward mappings in Banach spaces

S. Hu and Y. Sun [S. Hu, Y. Sun, Fixed point index for weakly inward mappings, J. Math. Anal. Appl. 172 (1993) 266–273] defined the fixed point index for weakly inward mappings, investigated its properties and studied the fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we continue to investigate boundary conditions, under which the fixed point index for the completely continuous and weakly inward mapping, denoted by i ( A , Ω , P ) , is equal to 1 or 0. Correspondingly, we can obtain some new fixed point theorems of the completely continuous and weakly inward mappings and existence theorems of solutions for the equations A x = μ x , which extend many famous theorems such as Leray–Schauder's theorem, Rothe's two theorems, Krasnoselskii's theorem, Altman's theorem, Petryshyn's theorem, etc., to the case of weakly inward mappings. In addition, our conclusions and methods are different from the ones in many recent works.

Schaefer type theorem and periodic solutions of evolution equations

Two fixed point theorems for the sum of contractive and compact operators are obtained in this paper, which generalize and improve the corresponding results in [H. Schaefer, Über die methode der a priori-Schranken, Math. Ann. 129 (1955) 415–416; T.A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc. 124 (1996) 2383–2390; V.I. Istrǎtescu, Fixed Point Theory, an Introduction, Reidel, Dordrecht, 1981; T.A. Burton, K. Colleen, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 23–31; D.R. Smart, Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980]. As the applications for the results, we obtain the existence of periodic solutions for some evolution equations with delay, which extend the corresponding results in [T.A. Burton, B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations 90 (1991) 357–396].

Erratum to “Dynamical Sources in Information Theory: Fundamental Intervals and Word Prefixes”

In [V] the third author studied statistical properties of words generated by dynamical sources by using generalized Ruelle operators. This erratum aims to show that a supplementary condition should be added in Definition 2.1 of dynamical sources. This supplementary condition (called (d4) below) is closely related to the so-called Bounded Distortion Property and should be used at two different stages of the paper: first in the definition of the generalized operator Gs, second in the proof of Proposition 2 that deals with the Positivity Theorem of Krasnoselskii. We also prove that this condition is actually necessary and provide an example of a source which does not satisfy condition (d4) and for which Gs is not well-defined.

Positive solutions of a nonlinear four-point boundary value problems

In this paper, by using the Krasnoselskii's theorem in a cone, we study the existence of at least one or two positive solutions to the four-point boundary value problem y ″(t)+a(t)f(y(t))=0, 0<t<1, y(0)=αy(ξ), y(1)=βy(η), where 0< ξ⩽ η<1, 0<α< 1 1−ξ , 0<β< 1 η and αξ(1− β)+(1− α)(1− βη)>0. As an application, we also give some examples to demonstrate our results.

Some remarks on a fixed point theorem of Krasnoselskii

Using a particular locally convex space and Schaefer’s theorem, a generalization of Krasnoselskii’s xed point Theorem is proved. This result is further applied to certain nonlinear integral equation proving the existence of a solution on IR+ = [0; +1):

On a generalization of Krasnoselskii's theorem

AbstractIn this paper we prove a generalization of the well known theorem of Krasnoselskii on the superposition operator in which the domain of Nemytskii's operator is a product space. We also give an application of this result.

A Fixed Point Theorem of Krasnoselskii—Schaefer Type

AbstractIn 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))dsif 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))dsif 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.

A fixed-point theorem of Krasnoselskii

Krasnoselskii's fixed-point theorem asks for 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. Then P has a fixed point. A careful reading of the proof reveals that (i) need only ask that Bx + Ay ∈ M when x = Bx + Ay. The proof also yields a technique for showing that such x is in M.

Bifurcation and Multiple Solutions for Perturbations of Linear Elliptic Problems

Combining a bifurcation theorem with a local Leray–Schauder degree theorem of Krasnoselskii and Zabreiko in the case of a simple singular point, we obtain an existence result on the number of small solutions for a class of functional bifurcation equations. Since this result contains the information of local Leray–Schauder degree, we obtain new multiplicity results for the perturbations of second-order linear elliptic problems by unbounded nonlinearities as applications here, bya prioribounds essentially due to Gupta and by a Leray–Schauder degree computation.

Fixed point theorems for a sum of two mappings in locally convex spaces

Cain and Nashed generalized to locally convex spaces a well known fixed point theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. The class of asymptotically nonexpansive mappings includes properly the class of nonexpansive mappings as well as the class of contraction mappings. In this paper, we prove by using the same method some results concerning the existence of fixed points for a sum of nonexpansive and continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in locally convex spaces. These results extend a result of Cain and Nashed.

Krasnoselskii's theorem on operators compressing a cone: Application to some singular boundary value problems

Krasnoselskii's theorem on operators compressing a cone: Application to some singular boundary value problems

On a fixed point theorem of Krasnoselskii and triangle contractive operators

On a fixed point theorem of Krasnoselskii and triangle contractive operators

Einschließungsaussagen f¨r positive lösungen superlinearer randwertaufgaben

Using a modified version of Krasnoselskii's theorem on operators expanding a cone in a Banach space we proved in [11] existence and inclusion results for superlinear Uryson equations. In this paper we construct suitable cones to apply these results to inverse monotone boundary value problems. The bounds are improved by the method of cone iteration. For the construction we do not need Green's function explicitly.

Boundary value problems for partial functional differential equations

468 SAMUEL M. RANKIN, III3. A. Granas, The theory of compact vector fields and some of its applications totopology and functional spaces, I. Rozprowy Mat., 30 (1962), 93.4. J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971.5. M. Z. Nashed and J. S. W. Wong, Some variants of a fixed point theorem ofKrasnoselskii and applications to nonlinear integral equations, J. Math. Mech., 18(1969), 767-777.6. A. Pazy, Semigroups of Linear Operators and Applications to Partial DifferentialEquations, Lecture Notes #10, University of Maryland, College Park, Maryland, 1974.7. G. W. Reddien and G. F. Webb, Boundary value problems for functional differentialequations with L

A generalization of theorems of Krasnoselskii and Juberg

In this paper, we use the author's diagram proof of Riesz's theorem together with special seminorms to show that two theorems proved by the author generalize a compact interpolation theorem of Krasnoselskii, and a compact interpolation theorem of Juberg.