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.     

New fixed point theorems and applications of mixed monotone operator

In this paper, we study the uniqueness and existence of fixed point of mixed monotone operators in the partially ordered Banach space. The results extend and improve recent related results.     

Monotonicity properties of the superposition operator and their applications

We establish some properties of the superposition operator which are associated with monotonicity. Those properties are expressed in terms of the notion of degree of decrease or degree of increase. An application of the obtained results to the study of solvability of a quadratic Volterra integral equation is also derived.     

A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations

This paper is concerned with an operator equation Ax+Bx+Cx=x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.     

Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications

By using relatively weakly compactness conditions, we obtain existence theorems of fixed points for discontinuous multivalued increasing operators in Banach spaces. As an application, we utilize the results obtained in Section 2 to lead to the existence principles for integral inclusions of Hammerstein type multivalued maps.     

Existence of fixed points for some convex operators and applications to multi-point boundary value problems

Existence results of fixed points for some convex operators are given by means of fixed point theorem of cone expansion and compression, then they are applied to nonlinear multi-point boundary value problems.     

Existence of fixed points for the sum of two operators

The purpose of this paper is to study the existence of fixed points for the sum of two nonlinear operators in the framework of real Banach spaces. Later on, we give some examples of applications of this type of results (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 fixed point approach to stability of functional equations

We prove a simple fixed point theorem for some (not necessarily linear) operators and derive from it several quite general results on the stability of a very wide class of functional equations in single variable.     

Krasnoselskii type fixed point theorems and applications

In this paper, we establish two fixed point theorems of Krasnoselskii type for the sum of , where is a compact operator and may not be injective. Our results extend previous ones. As an application, we apply such results to obtain some existence results of periodic solutions for delay integral equations and then give three instructive examples.

     

Existence of an elliptic system involving Pucci operator

We study the existence of solutions for the nonlinear elliptic system

where Ω is a bounded domain, f₁ is superlinear and f₂ is sublinear at zero and infinity, h₁ and h₂ are perturbation terms. We will show that the system has at least two semi-trivial solutions (u,0), (0,v) and a nontrivial solution (u^*,v^*).     

Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces

Investigations concerning the existence of dynamic processes convergent to fixed points of set-valued nonlinear contractions in cone metric spaces are initiated. The conditions guaranteeing the existence and uniqueness of fixed points of such contractions are established. Our theorems generalize recent results obtained by Huang and Zhang [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] for cone metric spaces and by Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] for metric spaces. The examples and remarks provided show an essential difference between our results and those mentioned above.     

New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation

In this paper, we obtain some new and general existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation, which extend the corresponding results in [Z.T. Zhang, New fixed point theorems of mixed monotone operators and applications, J. Math. Anal. Appl. 204 (1996) 307-319, Theorem 1, Corollaries 1 and 2]. Moreover, some applications to nonlinear integral equations on unbounded region are given.     

Fixed points of generalized e-concave (generalized e-convex) operators and their applications

In this paper, we present the definitions of generalized e-concave operators and generalized e-convex operators, which are the generalizations of e-concave operators and e-convex operators, respectively. Without compactness or continuity assumption of generalized e-concave operators and generalized e-convex operators, we have proved the existence, uniqueness and monotone iterative techniques of their fixed points. Our results are even new to e-concave operators and e-convex operators. Finally, we apply the results to the singular boundary value problems for second order differential equations.     

Existence of three periodic positive solutions for a class of integral equations with parameters

In this paper, we establish the existence of three periodic positive solutions for a class of abstract integral equations by Leggett-Williams fixed point theorem. Using the existence results for abstract integral equations, the population models are also considered.     

Hybrid fixed points of multivalued operators in metric spaces with applications

This paper deals with the existence of hybrid fixed points involving two multivalued operators in a complete metric space. Our claim is also illustrated with applications to a class of integral inclusions for proving the existence results.     

Generalization for Amann's and Leggett-Williams' three-solution theorems and applications

In this paper, some new three-solution theorems are obtained. Moreover, the famous Amann's and Leggett-Williams' three-solution theorems in nonlinear functional analysis can be seen as their special cases, namely these two famous theorems are united. So they are both improved. And that some new three-solution theorems are applied to ordinary differential equations, and some new existence results are obtained.     

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.     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.     

The generalized projection operator on reflexive Banach spaces and its applications

In this paper, we extend the definition of the generalized projection operator , where B is a reflexive Banach space with dual space B^∗ and K is a nonempty, closed and convex subset of B and we study its properties and applications to solving variational inequality.