A vector version of Krasnosel’skiĭ’s fixed point theorem in cones and positive periodic solutions of nonlinear systems

A new version of Krasnosel’skiĭ’s fixed point theorem in cones is established for systems of operator equations, where the compression-expansion conditions are expressed on components. In applications, this allows the nonlinear term of a system to have different behaviors both in components and in variables.     

Multiple positive solutions of non-local initial value problems for first order differential systems

The paper gives a new and natural method for the existence of multiple positive solutions for first order differential systems with non-local initial value conditions involving linear functionals. The case of higher order differential equations is also considered. The results are accompanied by numerical examples confirming the theory and proving for practice the importance of the bounds in solution localization.     

Fixed point theorems for generalized metrically inward maps

We prove fixed point results for multimaps satisfying a generalized metric (or an α-metric) inwardness condition. Our results extend, generalize or improve several known results.     

Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces

We study the convergence of two iterative algorithms for finding common fixed points of finitely many Bregman strongly nonexpansive operators in reflexive Banach spaces. Both algorithms take into account possible computational errors. We establish two strong convergence theorems and then apply them to the solution of convex feasibility, variational inequality and equilibrium problems.     

Fixed point theorems for nonexpansive maps in Banach spaces

In this work, we present some new versions of fixed point theorems for nonexpansive maps and 1-set contractions defined on closed, convex, not necessarily bounded subsets of Banach spaces. Our proofs rely on a compactness result for an approximate fixed point set. The Kuratowski measure of noncompactness is used throughout. To illustrate the results obtained, some applications to Banach algebras and Hammerstein integral equations are provided.     

Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces

Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in Kurokawa and Takahashi [Y. Kurokawa, W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 1562-1568]. We explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of Halpern’s type for our class of mappings and hence resolved in the affirmative an open problem posed by Kurokawa and Takahashi in their final remark for the case where the mapping T is averaged.     

Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces

The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi-?$?$-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.     

Strong convergence theorems for finitely many nonexpansive mappings and applications

Let E$E$ be a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Recently, Takahashi and Shimoji [W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000) 1463–1471] introduced an iterative scheme given by finitely many nonexpansive mappings in E$E$ and proved weak convergence theorems which are connected with the problem of image recovery. In this paper we introduce a new iterative scheme which includes their iterative scheme as a special case. Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gâteaux differentiable and which has a weakly continuous duality mapping, we prove strong convergence theorems which are connected with the problem of image recovery. Using the established results, we consider the problem of finding a common fixed point of finitely many nonexpansive mappings.     

Weak and strong convergence theorems for countable Lipschitzian mappings and its applications

We use Mann’s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings.     

Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings

In this paper, we introduce hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family and left amenable semigroup of non-expansive mappings in the frame work of Hilbert spaces. Our goal is to prove a result of strong convergence for hybrid pseudo-viscosity approximation schemes to approach a solution of systems of equilibrium problems which is also a common fixed point of an infinite family and left amenable semigroup of non-expansive mappings. The results presented in this paper can be treated as an extension and improvement of the corresponding results announced by Ceng et al. [L.C. Ceng, Q.H. Ansari, and J.C. Yao, Hybrid pseudo-viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many non-expansive mappings, Nonlinear Analysis 4 (2010) 743-754] and many others.     

Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces

In an infinite-dimensional Hilbert space, the normal Mann’s iteration algorithm has only weak convergence, in general, even for nonexpansive mappings. In order to get a strong convergence result, we modify the normal Mann’s iterative process for an infinite family of nonexpansive mappings in the framework of Banach spaces. Our results improve and extend the recent results announced by many others.     

Strong convergence theorems for a family of quasi-asymptotic pseudo-contractions in Hilbert spaces

In the present paper, we propose two kinds of new algorithms for a family of quasi-asymptotic pseudo-contractions in real Hilbert spaces. By using the proposed algorithms, we prove several strong convergence theorems for a family of quasi-asymptotic pseudo-contractions. The results of this paper are interesting extensions of those known results.     

Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric

In this paper, we study the existence and uniqueness of (coupled) fixed points for mixed monotone mappings in partially ordered metric spaces with semi-monotone metric. As an application, we prove the existence and uniqueness of the solution for a first-order differential equation with periodic boundary conditions.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

Polynomial inverse integrating factors for quadratic differential systems

We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.     

Limit cycles for generalized Kukles polynomial differential systems

We study the limit cycles of generalized Kukles polynomial differential systems using averaging theory of first and second order.     

Averaging methods for finding periodic orbits via Brouwer degree

We consider the problem of finding T-periodic solutions for a differential system whose vector field depend on a small parameter ε. An answer to this problem can be given using the averaging method. Our main results are in this direction, but our approach is new. We use topological methods based on Brouwer degree theory to solve operator equations equivalent to this problem. The regularity assumptions are weaker then in the known results (up to second order in ε). A result for third order averaging method is also given.As an application we provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function. A concrete example is given.     

Strong convergence theorems of modified Mann iterative process for strict pseudo-contractions in Hilbert spaces

In this paper, we modify the normal Mann’s iterative process to have strong convergence for a k$k$-strictly pseudo-contractive non-self mapping in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others.     

Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces

We establish △-convergence and strong convergence theorems for an iterative process for a finite family of generalized nonexpansive multivalued mappings in a CAT(0) space. Moreover, we present a fixed point theorem for a pair consisting of a finite family of generalized nonexpansive single valued mappings, and a generalized nonexpansive multivalued mapping in CAT(0) spaces.     

Approximation methods for common fixed points for a countable family of nonexpansive mappings

Let E=L_p or l_p space, 1<p<∞. Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ∈(0,1), define a family of nonexpansive maps by S_i?(1−δ)I+δT_i where I is the identity map of K. Let . It is proved that the iterative sequence {x_n} defined by: x₀∈K,x_n+1=α_nu+∑_i≥1σ_i,tnS_ix_n,n≥0 converges strongly to a common fixed point of the family where {α_n} and {σ_i,tn} are sequences in (0,1) satisfying appropriate conditions, in each of the following cases: (a) E=l_p,1<p<∞, and (b) E=L_p,1<p<∞ and at least one of the maps T_i’s is demicompact. Our theorems extend the results of [P. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 325 (2007) 469-479] from Hilbert spaces to l_p spaces, 1<p<∞.