On α-convex operators

In this paper, the existence and uniqueness of positive fixed points for a class of convex operators is obtained by means of the properties of cone, concave operators and the monotonicity of set-valued maps. In the end, we give a simple application to certain integral equations.     

Eigenvalue problems and fixed point theorems for a class of positive nonlinear operators

We study the eigenvalue problems for a class of positive nonlinear operators defined on a cone in a Banach space. Using projective metric techniques and Schauder’s fixed-point theorem, we establish existence, uniqueness, monotonicity and continuity results for the eigensolutions. Moreover, the method leads to a result on the existence of a unique fixed point of the operator. Applications to nonlinear boundary-value problems, to differential delay equations and to matrix equations are considered.      

A NOVEL FIXED POINT THEOREM AND ITS APPLICATIONS

In this article, a novel fixed point theorem in C[0,1] space is established by using the properties of fixed point index. This theorem is then applied to prove the existence of positive solutions for three-point boundary value problems and generalizes some previous results.     

Existence of symmetric positive solutions for a class of Sturm-Liouville-like boundary value problems

This paper deals with the existence of symmetric positive solutions for a class of singular Sturm-Liouville-like boundary value problems with a one-dimensional p-Laplacian operator. By using the fixed theorem of cone expansion and compression of norm type in a cone, the existence of positive solutions is established though nonlinear term contains the first derivative of unknown function.     

On a fixed point theorem of Kirk

W.A. Kirk [J. Math. Anal. Appl. 277 (2003) 645-650] first introduced the notion of asymptotic contractions and proved the fixed point theorem for this class of mappings. In this note we present a new short and simple proof of Kirk's theorem.     

Compression-expansion fixed point theorem in two norms and applications

In this paper we present a two-norms version of Krasnoselskii's fixed point theorem in cones. The abstract result is then applied to prove the existence of positive Lp solutions of Hammerstein integral equations with better integrability properties on the kernels.     

关于一个新泛函的锥拉伸与锥压缩不动点定理及应用

引进了一个新的泛函,它包含了已有的很多类型的泛函.关于此泛函建立了一个泛函形式的锥拉伸与锥压缩不动点定理,并应用此定理研究了一类二阶两点边值问题,得到了该问题至少一个正解的存在性.     

A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

Existence of a fixed point for a semicontinuous operator and its application

We employ a partial order method and cone theory to prove an existence theorem for a fixed point for a semicontinuous operator; also the decreasing and increasing semicontinuous operators are studied.     

一类单调算子的不动点定理

研究了范围广泛的半闭1-集压缩算子及凝聚算子,获得了一些新的不动点定理,所获结果推广了已知的结论.     

The characteristic of noncompact convexity and random fixed point theorem for set-valued operators

Let (Ω, Σ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C. We prove that a set-valued nonexpansive mapping T: C → KC(C) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T: Ω × C → KC(C) has a random fixed point.     

New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.     

Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem

In this paper, two results concerning the global attractivity and global asymptotic attractivity of the solutions for a nonlinear functional integral equation are proved via a variant of the Krasnoselskii fixed point theorem due to Dhage [B.C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004) 145–155]. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A couple of examples are indicated for demonstrating the natural realizations of the abstract results presented in the paper. Our results generalize the attractivity results of Banas and Rzepka [J. Banas, B. Rzepka, An application of measures of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003) 1–6] and Banas and Dhage [J. Banas, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equations, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.07.038], under weaker conditions with a different method.     

A Perron-Frobenius theorem for positive polynomial operators in Banach lattices

In this paper, we extend the Perron-Frobenius theorem for positive polynomial operators in Banach lattices. The result obtained is applied to derive necessary and sufficient conditions for the stability of positive polynomial operators. Then we study stability radii: complex, real and positive radii of positive polynomial operators and show that in this case the three radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.     

Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems

The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

A fixed point approach to the stability of isometries

In this paper, we apply a fixed point theorem to the proof of Hyers-Ulam-Rassias stability property for isometries from a normed space into a Banach space, in which the parallelogram law holds.     

Little Grothendieck's theorem for sublinear operators

Let SB(X,Y) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y. Consider π₂(X,Y) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck's theorem in the sublinear operators case. We prove under some conditions that every operator in SB(C(K),H) is in π₂(C(K),H) for any compact K and any Hilbert H. In the noncommutative case the problem is still open.     

Some fixed point results on a metric space with a graph

Combining some branches is a typical activity in different fields of science, especially in mathematics. Naturally, it is notable in fixed point theory. Over the past few decades, there have been a lot of activity in fixed point theory and another branches in mathematics such differential equations, geometry and algebraic topology. In 2006, Espinola and Kirk made a useful contribution on combining fixed point theory and graph theory. Recently, Reich and Zaslavski studied a new inexact iterative scheme for fixed points of contractive and nonexpansive multifunctions. In this paper, by using main idea of their work and the idea of combining fixed point theory and graph theory, we present some iterative scheme results for G-contractive and G-nonexpansive mappings on graphs.     

A generalization of the Lefschetz fixed point theorem and detection of chaos

We consider the problem of existence of fixed points of a continuous map in (possibly) noninvariant subsets. A pair of subsets of induces a map given by if and elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: If is metrizable, and are compact ANRs, and is continuous, then has a fixed point in provided the Lefschetz number of is nonzero. Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic.