Showing papers on "Fixed-point theorem published in 2006"
TL;DR: In this paper, a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order was proved, using a weak contractivity type of assumption, which can be used to investigate a large class of problems.
Abstract: We prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order, using a weak contractivity type of assumption. Besides including several recent developments, our theorem can be used to investigate a large class of problems. As an application, we discuss the existence and uniqueness of solution for a periodic boundary value problem.
1,206 citations
TL;DR: In this paper, an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping is introduced.
Abstract: In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process
349 citations
Abstract: Two fixed point theorems for uniformly lipschitzian mappings in metric spaces, due respectively to E. Lifsic and to T.-C. Lim and H.-K. Xu, are compared within the framework of the so-called CAT(0) spaces. It is shown that both results apply in this setting, and that Lifsic’s theorem gives a sharper result. Also, a new property is introduced that yields a fixed point theorem for uniformly lipschitzian mappings in a class of hyperconvex spaces, a class which includes those possessing property ( P ) of Lim and Xu.
262 citations
TL;DR: In this article, the Banach contraction principle and Caristi's fixed point theorem are generalized to the case of multi-valued mappings, and the results are extensions of the well-known Nadler's fixed-point theorem [S.B. Nadler Jr., Multi-valued contraction mappings.
250 citations
21 Mar 2006
TL;DR: In this article, the existence and multiplicity of posi- tive solutions for the nonlinear fractional dierential equation boundary-value problem D 0+u (t) = f(t,u(t)), 0 < t < 1 u(0) + u 0 (0) = 0, u(1)+u 0 (1)) = 0 where 1 < 2 is a real number.
Abstract: In this paper, we consider the existence and multiplicity of posi- tive solutions for the nonlinear fractional dierential equation boundary-value problem D 0+u(t) = f(t,u(t)), 0 < t < 1 u(0) + u 0 (0) = 0, u(1) + u 0 (1) = 0 where 1 < 2 is a real number, and D 0+ is the Caputo's fractional deriva- tive, and f : (0,1)◊(0,+1) ! (0,+1) is continuous. By means of a fixed-point theorem on cones, some existence and multiplicity results of positive solutions are obtained.
238 citations
01 Jan 2006
TL;DR: In this article, the authors obtained two fixed point theorems for a class of operators called occasionally weakly compatible maps defined on a symmetric space and established two of the most general fixed point theorem for four maps.
Abstract: We obtain two fixed point theorems for a class of operators called occasionally weakly compatible maps defined on a symmetric space . These results establish two of the most general fixed point theorems for four maps.
227 citations
TL;DR: In this paper, the equivalence between the Meir-Keeler type contractive conditions and the contractive definitions involving gauge functions is established, and a new fixed point theorem is established.
Abstract: In this paper, we establish two general theorems for equivalence between the Meir–Keeler type contractive conditions and the contractive definitions involving gauge functions. One of these theorems is an extension of a recent result of Lim (On characterization of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120). Also, we establish the following new fixed point theorem. Suppose ϕ : R + → R + is a contractive gauge function in the sense that for any e > 0 there exists δ > e such that e t δ implies ϕ ( t ) ⩽ e , and suppose T is a continuous and asymptotically regular selfmapping on a complete metric space ( X , d ) satisfying the following: (i) d ( Tx , Ty ) ⩽ ϕ ( D ( x , y ) ) for all x , y ∈ X , and (ii) d ( Tx , Ty ) D ( x , y ) for all x , y ∈ X with x ≠ y , where D ( x , y ) = d ( x , y ) + γ . [ d ( x , Tx ) + d ( y , Ty ) ] with γ ⩾ 0 . Then T has a unique fixed point and all of the Picard iterates of T converge to this fixed point. This result includes those of Jachimski (Equivalent conditions and the Meir–Keeler type theorems, J. Math. Anal. Appl. 194 (1995) 293–303), Matkowski (Fixed point theorems for contractive mappings in metric spaces, Cas. Pest. Mat. 105 (1980) 341–344) and others.
193 citations
TL;DR: In this paper, the existence of travelling wave solutions in a class of delayed reaction-diffusion systems without monotonicity was investigated, which concludes two-species diffusion-competition models with delays.
Abstract: This paper is concerned with the existence of travelling wave solutions in a class of delayed reaction-diffusion systems without monotonicity, which concludes two-species diffusion-competition models with delays. Previous methods do not apply in solving these problems because the reaction terms do not satisfy either the so-called quasimonotonicity condition or non-quasimonotonicity condition. By using Schauder's fixed point theorem, a new cross-iteration scheme is given to establish the existence of travelling wave solutions. More precisely, by using such a new cross-iteration, we reduce the existence of travelling wave solutions to the existence of an admissible pair of upper and lower solutions which are easy to construct in practice. To illustrate our main results, we study the existence of travelling wave solutions in two delayed two-species diffusion-competition systems.
186 citations
TL;DR: In this paper, a generalized Ekeland's variational principle for lower semicontinuous from above and τ -functions was established for functions with τ -function which generalizes the concept of w-distance studied in the literature.
132 citations
TL;DR: Aamri and El Moutawakil as mentioned in this paper proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property.
132 citations
27 Mar 2006
TL;DR: In this paper, Brouwer degree theory for functions in VMO applications to ODEs and PDEs is used for set-contactive mapping in Banach spaces.
Abstract: BROUWER DEGREE THEORY Continuous and Differentiable Functions Construction of Brouwer Degree Degree Theory for Functions in VMO Applications to ODEs Exercises LERAY-SCHAUDER DEGREE THEORY Compact Mappings Leray-Schauder Degree Leray-Schauder Degree for Multi-valued Mappings Applications to Bifurcations Applications to ODEs and PDEs Exercises DEGREE THEORY FOR SET CONTRACTIVE MAPS Measure of Non-compactness and Set Contractions Degree Theory for Countably Condensing Mappings Applications to ODEs in Banach Spaces Exercises GENERALIZED DEGREE THEORY FOR A-PROPER MAPS A-Proper Mappings Generalized Degree for A-Proper Mappings Equations with Fredholm Mappings of Index Zero Equations with Fredholm Mappings of Index Zero Type Applications of the Generalized Degree Exercises COINCIDENCE DEGREE THEORY Fredholm Mappings Coincidence Degree for L-Compact Mappings Existence Theorems for Operator Equations Applications to ODEs Exercises DEGREE THEORY FOR MONOTONE TYPE MAPS Monotone Type Mappings in Reflexive Banach Spaces Degree Theory for Mappings of Class (S+) Degree for Perturbations of Monotone Type Mappings Degree Theory for Mappings of Class (S+)L Coincidence Degree for Mappings of Class L - (S+) Computation of Topological Degree Applications to PDEs and Evolution Equations Exercises FIXED POINT INDEX THEORY Cone in Normed Spaces Fixed Point Index Theory Fixed Point Theorems in Cones Perturbations of Condensing Mappings Index Theory for Nonself-Mappings Applications to Integral and Differential Equations Exercises REFERENCES SUBJECT INDEX
TL;DR: In this paper, it was shown that any commuting family of edge preserving mappings of a connected reflexive graph G that contains no cycles or infinite paths always has at least one common fixed edge.
TL;DR: In this paper, two concepts of well-posedness for quasivariational inequalities having a unique solution are introduced and some equivalent characterizations of these concepts and classes of wellposed inequalities are presented.
Abstract: In this paper, two concepts of well-posedness for quasivariational inequalities having a unique solution are introduced. Some equivalent characterizations of these concepts and classes of well-posed quasivariational inequalities are presented. The corresponding concepts of well-posedness in the generalized sense are also investigated for quasivariational inequalities having more than one solution
TL;DR: The adaptive projected subgradient method as discussed by the authors generalizes Polyak's subgradient algorithm for the convexly constrained minimization of a fixed nonsmooth function by allowing a variety of a priori information on the estimandum in the form of multiple intersecting closed convex sets.
Abstract: This paper presents an algorithmic solution, the adaptive projected subgradient method, to the problem of asymptotically minimizing a certain sequence of non-negative continuous convex functions over the fixed point set of a strongly attracting nonexpansive mapping in a real Hilbert space. The method generalizes Polyak's subgradient algorithm for the convexly constrained minimization of a fixed nonsmooth function. By generating a strongly convergent and asymptotically optimal point sequence, the proposed method not only offers unifying principles for many projection-based adaptive filtering algorithms but also enhances the adaptive filtering methods with the set theoretic estimation's armory by allowing a variety of a priori information on the estimandum in the form, for example, of multiple intersecting closed convex sets.
TL;DR: In this paper, the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem was studied. But the authors only considered the case where λ is a positive parameter.
29 Mar 2006
TL;DR: In this article, a fixed point theory is proposed to solve games of imperfect information, which is defined on the lattice of antichains of sets of states and does not involve determinization.
Abstract: In this paper, we propose a fixed point theory to solve games of imperfect information. The fixed point theory is defined on the lattice of antichains of sets of states. Contrary to the classical solution proposed by Reif [Rei84], our new solution does not involve determinization. As a consequence, it is readily applicable to classes of systems that do not admit determinization. Notable examples of such systems are timed and hybrid automata. As an application, we show that the discrete control problem for games of imperfect information defined by rectangular automata is decidable. This result extends a result by Henzinger and Kopke in [HK99].
01 Jan 2006
TL;DR: In this paper, a higher order multi-point boundary value problem is considered and existence and nonexistence results for positive solutions of the problem are obtained by using Krasnosel'skii's fixed point theorem.
TL;DR: In this article, the authors 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.
Journal Article•
TL;DR: In this paper, an existence theorem for some functional-integral equa- tions is given, which includes many key integral and functional equations that arise in nonlinear analysis and its applications.
Abstract: We give an existence theorem for some functional-integral equa- tions which includes many key integral and functional equations that arise in nonlinear analysis and its applications. In particular, we extend the class of characteristic functions appearing in Chandrasekhar's classical integral equa- tion from astrophysics and retain existence of its solutions. Extensive use is made of measures of noncompactness and abstract fixed point theorems such as Darbo's theorem.
TL;DR: In this article, the notion of asymptotic contraction of Meir-Keeler type was introduced, and a fixed-point theorem for such contractions was proved.
Abstract: In this paper, we introduce the notion of asymptotic contraction of Meir–Keeler type, and prove a fixed-point theorem for such contractions, which is a generalization of fixed-point theorems of Meir–Keeler and Kirk. In our discussion, we use the characterization of Meir–Keeler contraction proved by Lim [On characterizations of Meir–Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120]. We also give a simple proof of this characterization.
TL;DR: In this paper, the existence of solutions to the boundary value problem (1.1) −(1.2) was analyzed for isotropic scattering kernels on L p spaces for p ∈ ( 1, ∞ ).
TL;DR: Insightful fuzzy contraction map- ping is introduced and a fixed point theorem in intuitionistic fuzzy metric spaces is proved.
Abstract: In this paper, we introduce intuitionistic fuzzy contraction map- ping and prove a fixed point theorem in intuitionistic fuzzy metric spaces.
TL;DR: Sufficient conditions are obtained that guarantee the existence of at least two positive solutions for the p -Laplacian difference equation = 0 subject to boundary conditions by a simple application of a new fixed point theorem due to Avery and Henderson.
TL;DR: In this article, the existence and nonexistence of positive solutions for the integral boundary value problem was studied and sufficient conditions for the existence of either none, or one, or more positive solutions were provided.
Abstract: This paper deals with the existence and nonexistence of positive solutions for the integral boundary value problem { − ( a u ′ ) ′ + b u = g ( t ) f ( t , u ) , ( cos γ 0 ) u ( 0 ) − ( sin γ 0 ) u ′ ( 0 ) = ∫ 0 1 u ( τ ) d α ( τ ) , ( cos γ 1 ) u ( 1 ) + ( sin γ 1 ) u ′ ( 1 ) = ∫ 0 1 u ( τ ) d β ( τ ) . We provide sufficient conditions for the existence of either none, or one, or more positive solutions of the above problem. The main tool used in the proofs of existence results is a fixed point theorem in a cone, due to Krasnoselskii and Zabreiko. The results presented substantially unify and extend the existing results in the literature.
TL;DR: It is shown that the construction of spherical designs is equivalent to solution of underdetermined equations and a new verification method for underd determined equations is derived using Brouwer’s fixed point theorem.
Abstract: This paper is concerned with proving the existence of solutions to an underdetermined system of equations and with the application to existence of spherical $t$-designs with $(t+1)^2$ points on the unit sphere $S^2$ in $R^3$. We show that the construction of spherical designs is equivalent to solution of underdetermined equations. A new verification method for underdetermined equations is derived using Brouwer’s fixed point theorem. Application of the method provides spherical $t$-designs which are close to extremal (maximum determinant) points and have the optimal order $O(t^2)$ for the number of points. An error bound for the computed spherical designs is provided.
TL;DR: The existence of the solution of the finite difference schemes are proved by Leray–Schauder fixed point theorem, and the uniqueness, stability and convergence of difference solutions with order O(h2 + τ2) are proved in the energy norm.
TL;DR: In this article, the existence of travelling wave fronts of delayed reaction diffusion systems with partial quasi-monotonicity was studied and the Schauder's fixed point theorem was applied to some appropriate operator in this subset.
Abstract: This paper deals with the existence of travelling wave fronts of delayed reaction diffusion systems with partial quasi-monotonicity. We propose a concept of “desirable pair of upper-lower solutions”, through which a subset can be constructed. We then apply the Schauder’s fixed point theorem to some appropriate operator in this subset to obtain the existence of the travelling wave fronts.
TL;DR: From some existence theorem, fixed point theorems for a family of lower semicontinuous or nonexpansive multivalued mappings are established and the existence theoresms of the generalized vector quasi-equilibrium problems are established.
Abstract: In this paper, we establish the existence theorems of the generalized vector quasi-equilibrium problems. From some existence theorem, we establish fixed point theorems for a family of lower semicontinuous or nonexpansive multivalued mappings. We also obtain the existence theorems of system of mixed generalized vector variational-like inequalities and existence theorems of the Debreu vector equilibrium problems and the Nash vector equilibrium problems.
TL;DR: In this paper, some common fixed point theorems due to Aamri and El Moutawakil [M.P. Pant, V.P., and D.Moutawakhil were extended to symmetric (semi-metric) spaces.