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Showing papers in "Journal of Fixed Point Theory and Applications in 2017"


Journal ArticleDOI
TL;DR: A survey of the existence and properties of solutions to the Choquard type equations can be found in this paper, where some variants and extensions of its variants can also be found.
Abstract: We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right) |u |^{p - 2} u \quad \text {in} \ \mathbb {R}^N, \end{aligned}$$ and some of its variants and extensions.

352 citations


Journal ArticleDOI
TL;DR: In this article, the authors generalize a series of fixed point results in the framework of b-metric spaces and exemplify it by extending Nadler's contraction principle for set-valued functions.
Abstract: In this paper, we indicate a way to generalize a series of fixed point results in the framework of b-metric spaces and we exemplify it by extending Nadler’s contraction principle for set-valued functions (see Nadler, Pac J Math 30:475–488, 1969) and a fixed point theorem for set-valued quasi-contraction functions due to Aydi et al. (see Fixed Point Theory Appl 2012:88, 2012).

85 citations


Journal ArticleDOI
TL;DR: In this paper, the authors prove a strong convergence result for finding a zero of the sum of two monotone operators, with one of the two operators being co-coercive using an iterative method which is a combination of Nesterov's acceleration scheme and Haugazeau's algorithm in real Hilbert spaces.
Abstract: Our interest in this paper is to prove a strong convergence result for finding a zero of the sum of two monotone operators, with one of the two operators being co-coercive using an iterative method which is a combination of Nesterov’s acceleration scheme and Haugazeau’s algorithm in real Hilbert spaces. Our numerical results show that the proposed algorithm converges faster than the un-accelerated Haugazeau’s algorithm.

67 citations


Journal ArticleDOI
TL;DR: Agarwal et al. as discussed by the authors proved fixed point theorems for Kannan type mappings, and used the additional conditions as compactness or asymptotic regularity or involutions.
Abstract: In this note, we prove some fixed point theorems for Kannan type mappings. We will use the additional conditions as compactness or asymptotic regularity or involutions. Our proofs are inspired by the study of Lipschitzian mappings (Agarwal et al., Fixed point theory for Lipschitzian-type mappings with applications, 2009).

52 citations


Journal ArticleDOI
Pontus Giselsson1
TL;DR: In this article, the authors show that the convergence rate for Douglas-Rachford splitting is not tight, meaning that no problem from the considered class converges exactly with that rate.
Abstract: Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.

50 citations


Journal ArticleDOI
TL;DR: In this article, the existence of the best proximity points of certain mapping defined via simulation functions in the frame of complete metric spaces is investigated and the uniqueness criteria for such mappings are considered.
Abstract: In this paper, we investigate of the existence of the best proximity points of certain mapping defined via simulation functions in the frame of complete metric spaces. We consider the uniqueness criteria for such mappings. The obtained results unify a number of the existing results on the topic in the literature.

43 citations


Journal ArticleDOI
TL;DR: In this article, a proximal split feasibility algorithm with an additional inertial extrapolation term was proposed for solving the proximal-split feasibility problem under weaker conditions on the step sizes, where the convex and lower semi continuous objective functions are assumed to be non-smooth.
Abstract: In this paper, we present a proximal split feasibility algorithm with an additional inertial extrapolation term for solving a proximal split feasibility problem under weaker conditions on the step sizes. The two convex and lower semi continuous objective functions are assumed to be non-smooth. Some applications to split inclusion problem and split equilibrium problem are given. We demonstrate the efficiency of the proposed algorithm with numerical experiments.

40 citations


Journal ArticleDOI
TL;DR: In this article, a new algorithm combining the Mann iteration and the inertial method for solving split common fixed point problems is introduced. But the weak convergence of the algorithm is established under standard assumptions imposed on cost operators.
Abstract: In this paper, we introduce a new algorithm which combines the Mann iteration and the inertial method for solving split common fixed point problems. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. As a consequence, we obtain weak convergence theorems for split variational inequality problems for inverse strongly monotone operators, and split common null point problems for maximal monotone operators. Finally, for supporting the convergence of the proposed algorithms we also consider several preliminary numerical experiments on a test problem.

40 citations


Journal ArticleDOI
TL;DR: In this paper, the concept of ''b_v(s) metric space'' was introduced as a generalization of metric space, and the Banach and Reich contraction principles in metric spaces were given.
Abstract: In this paper, the concept of \(b_v(s)\)-metric space is introduced as a generalization of metric space, rectangular metric space, b-metric space, rectangular b-metric space and v-generalized metric space. We next give proofs of the Banach and Reich contraction principles in \(b_v(s)\)-metric spaces. Using a new result, we provide short proofs which are different from of the original ones in metric spaces. The results we obtain generalize many known results in fixed point theory. We also provide a solution to an open problem.

34 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigate the effect of dispersal and spatial heterogeneity of the environment on the dynamics of a predator-prey model in spatially heterogeneous environment and show that for certain ranges of death and dispersal rates of the predator, the semi-trivial steady state of the model in the heterogeneous case could change its stability multiple times as the dispersal rate of the prey varies from small to large.
Abstract: We investigate the effect of dispersal and spatial heterogeneity of the environment on the dynamics of a predator–prey model. In contrast with the homogeneous environment, the dynamics of the model in spatially heterogeneous environment is more complex. For instance, for certain ranges of death and dispersal rates of the predator, the semi-trivial steady state of the model in the heterogeneous case could change its stability multiple times as the dispersal rate of the prey varies from small to large, whereas the stability of the semi-trivial steady state is unaffected by the dispersal rates of the predator and prey in the homogeneous case.

33 citations


Journal ArticleDOI
TL;DR: In this paper, pearly Floer trajectories are incorporated into the tranversality scheme for pseudoholomorphic maps introduced by Cieliebak-Mohnke (J Symplectic Geom 5(3): 281-356, 2007).
Abstract: We incorporate pearly Floer trajectories into the tranversality scheme for pseudoholomorphic maps introduced by Cieliebak–Mohnke (J Symplectic Geom 5(3): 281–356, 2007). By choosing generic domain-dependent almost complex structures, we obtain zero- and one-dimensional moduli spaces with the structure of cell complexes with rational fundamental classes. Integrating over these moduli spaces gives a definition of Floer cohomology over Novikov rings via stabilizing divisors for rational Lagrangians that are fixed point sets of anti-symplectic involutions satisfying certain Maslov index conditions as well as Hamiltonian Floer cohomology of compact rational symplectic manifolds.

Journal ArticleDOI
TL;DR: In this article, the authors considered families of solutions to the problem and gave a complete description of the asymptotic behavior of the families as p\rightarrow +\infty \.
Abstract: We consider families \(u_p\) of solutions to the problem Open image in new window where \(p>1\) and \(\Omega \) is a smooth bounded domain of \(\mathbb {R}^2\). Under the condition Open image in new window we give a complete description of the asymptotic behavior of \(u_p\) as \(p\rightarrow +\infty \).

Journal ArticleDOI
TL;DR: In this article, the nonlinear Schrodinger system with critical Sobolev exponent was studied in dimension 4 and the Robin function was used to construct families of positive solutions which blowup and concentrate at different points.
Abstract: In this paper we deal with the nonlinear Schrodinger system $$\begin{aligned} -\Delta u_i =\mu _i u_i^3 + \beta u_i \sum _{j e i} u_j^2 + \lambda _i u_i, \qquad u_1,\ldots , u_m\in H^1_0(\Omega ) \end{aligned}$$ in dimension 4, a problem with critical Sobolev exponent. In the competitive case ( $$\beta <0$$ fixed or $$\beta \rightarrow -\infty $$ ) or in the weakly cooperative case ( $$\beta \ge 0$$ small), we construct, under suitable assumptions on the Robin function associated to the domain $$\Omega $$ , families of positive solutions which blowup and concentrate at different points as $$\lambda _1,\ldots , \lambda _m\rightarrow 0$$ . This problem can be seen as a generalization for systems of a Brezis–Nirenberg type problem.

Journal ArticleDOI
TL;DR: In this paper, the strong convergence theorem for the viscosity approximation method for the split common fixed-point problem in Hilbert spaces was introduced, and strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators were obtained.
Abstract: In this paper, we introduce the strong convergence theorem for the viscosity approximation methods for solving the split common fixed-point problem in Hilbert spaces. As a consequence, we obtain strong convergence theorems for split variational inequality problems for Lipschitz continuous and monotone operators and split common null point problems for maximal monotone operators. Our results improve and extend the corresponding results announced by many others.

Journal ArticleDOI
TL;DR: In this article, a new algorithm for the split common fixed-point problem is proposed, which does not need any priori information of the operator norm and establishes a weak convergence theorem.
Abstract: The split common fixed-point problem is an inverse problem that consists in finding an element in a fixed-point set such that its image under a linear transformation belongs to another fixed-point set. In this paper, we propose a new algorithm for the split common fixed-point problem that does not need any priori information of the operator norm. Under standard assumptions, we establish a weak convergence theorem of the proposed algorithm.

Journal ArticleDOI
TL;DR: In this paper, the authors characterize the finite ultrametric spaces X for which every self-isometry has at least |X|−2 fixed points and some extremal properties of such spaces and related graph theoretical characterizations are also obtained.
Abstract: A metric space X is rigid if the isometry group of X is trivial. The finite ultrametric spaces X with |X| ≥ 2 are not rigid since for every such X there is a self-isometry having exactly |X|−2 fixed points. Using the representing trees we characterize the finite ultrametric spaces X for which every self-isometry has at least |X|−2 fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.

Journal ArticleDOI
TL;DR: In this paper, a lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical is given.
Abstract: We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $$S^n$$ carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact forms. We also slightly relax the hypothesis of dynamical convexity. A fundamental ingredient in our proofs is the common index jump theorem due to Y. Long and C. Zhu.

Journal ArticleDOI
TL;DR: In this paper, the authors established common fixed point theorems in an ordered complete metric space using distance functions, and extended the results of Yan et al. (2012) and some other existing results announced in the literature.
Abstract: In this paper, we establish common fixed point theorems in an ordered complete metric space using distance functions. Our main result, improves and extends the results of Yan et al. (Fixed Point Theory Appl. 2012, Article id: 152, 2012) and some other existing results announced in the literature. As applications, we discuss some fixed point theorems for contraction of integral type and some existence theorem for solution of integral equation, and for solution of first and second order ordinary differential equations with periodic boundary conditions.

Journal ArticleDOI
TL;DR: In this article, an iterative algorithm that does not require any knowledge of the operator norm was introduced for approximating a solution of a split generalised mixed equilibrium problem which is also a fixed point of a strictly pseudocontractive mapping.
Abstract: The purpose of this paper is to introduce an iterative algorithm that does not require any knowledge of the operator norm for approximating a solution of a split generalised mixed equilibrium problem which is also a fixed point of a $$\kappa $$ -strictly pseudocontractive mapping. Furthermore, a strong convergence theorem for approximating a common solution of a split generalised mixed equilibrium problem and a fixed-point problem for $$\kappa $$ -strictly pseudocontractive mapping was stated and proved in the frame work of Hilbert spaces.

Journal ArticleDOI
TL;DR: In this article, a pointwise cyclic relatively nonexpansive mapping involving orbits is introduced to investigate the existence of best proximity points using a geometric property defined on a nonempty and convex pair of subsets of a Banach space X, called weak proximal normal structure.
Abstract: We introduce a concept of pointwise cyclic relatively nonexpansive mapping involving orbits to investigate the existence of best proximity points using a geometric property defined on a nonempty and convex pair of subsets of a Banach space X, called weak proximal normal structure. Examples are given to support our main conclusions. We also introduce a notion of proximal diametral sequence and establish a characterization of proximal normal structure and show that every nonempty and convex pair in uniformly convex in every direction Banach spaces has weak proximal normal structure. As an application, we give a new existence theorem for cyclic contractions in reflexive Banach spaces without strictly convexity condition.

Journal ArticleDOI
TL;DR: In this article, the existence of small-amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean, with infinite depth, in irrotational regime, under the action of gravity and surface tension at the free boundary was shown.
Abstract: We present the result and the ideas of the recent paper (Berti and Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, http://arxiv.org/abs/1602.02411 , 2016) concerning the existence of Cantor families of small-amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean, with infinite depth, in irrotational regime, under the action of gravity and surface tension at the free boundary. These quasi-periodic solutions are linearly stable.

Journal ArticleDOI
TL;DR: In this paper, the authors introduced the notion of F-contraction, where a binary relation on its domain has not to be neither transitive nor a partial order, and established some fixed point results for such contractions in complete metric spaces that improved the Wardowski's original idea.
Abstract: In (Fixed Point Theory Appl 94:6, 2012), the author introduced a new kind of contractions, called F-contractions, that extended the Banach contractions in a newfangled way. In this work, we introduce the notion of \(F_\mathfrak {R}\)-contraction where \(\mathfrak {R}\) is a binary relation on its domain that has not to be neither transitive nor a partial order. Consequently, we establish some fixed point results for such contractions in complete metric spaces that improve the Wardowski’s original idea and we also give illustrative examples. Furthermore, we show some results to guarantee existence and uniqueness of fixed point of N-order. As an application, we apply our main result to study a class of nonlinear matrix equation.

Journal ArticleDOI
TL;DR: In this paper, the authors give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions, including refinements of known results and some new results.
Abstract: The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation $$\begin{aligned} abla \times \left( \mu (x)^{-1} abla \times u\right) - \omega ^2\varepsilon (x)u = f(x,u) \end{aligned}$$ for the field $$u:\Omega \rightarrow \mathbb {R}^3$$ in a domain $$\Omega \subset \mathbb {R}^3$$ . Here, $$\varepsilon (x) \in \mathbb {R}^{3\times 3}$$ is the (linear) permittivity tensor of the material, and $$\mu (x) \in \mathbb {R}^{3\times 3}$$ denotes the magnetic permeability tensor. The nonlinearity $$f:\Omega \times \mathbb {R}^3\rightarrow \mathbb {R}^3$$ comes from the nonlinear polarization. If $$f= abla _uF$$ is a gradient, then this equation has a variational structure. The goal of this paper is to give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions. It also contains refinements of known results and some new results.

Journal ArticleDOI
TL;DR: In this paper, the authors introduced the class of extended simulation functions, which are more large than the original class of simulation functions and obtained a $$\varphi $$¯¯ -admissibility result involving extended simulation function mappings for a new class of mappings.
Abstract: We introduce the class of extended simulation functions, which is more large than the class of simulation functions, recently introduced in (Khojasteh et al. Filomat 29(6):1189–1194, 2015). We obtain a $$\varphi $$ -admissibility result involving extended simulation functions, for a new class of mappings $$T: X\rightarrow X$$ , with respect to a lower semi-continuous function $$\varphi : X\rightarrow [0,\infty )$$ , where X is a set equipped with a certain metric d. The main theorem in this paper generalizes a recent $$\varphi $$ -admissibility result obtained in (Karapinar et al. Fixed Point Theory Appl 2015:152, 2015), and many other related results.

Journal ArticleDOI
TL;DR: In this paper, it was shown that spaces and covers can be much more general and the boundary KKM rules can be substituted by more weaker boundary assumptions, and that Shapley's KKMS theorem can be replaced by a weaker boundary assumption.
Abstract: We consider generalizations of Gale’s colored KKM lemma and Shapley’s KKMS theorem. It is shown that spaces and covers can be much more general and the boundary KKM rules can be substituted by more weaker boundary assumptions.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the nonlinear stationary Schrodinger equation and established the existence of a continuous branch with a constant positive constant which only depends on the exponent p and the shape of Q.
Abstract: We consider the nonlinear stationary Schrodinger equation $$\begin{aligned} -\Delta u -\lambda u= Q(x)|u|^{p-2}u, \qquad \text {in }\mathbb {R}^N \end{aligned}$$ in the case where $$N \ge 3$$ , p is a superlinear, subcritical exponent, Q is a bounded, nonnegative and nontrivial weight function with compact support in $$\mathbb {R}^N$$ and $$\lambda \in \mathbb {R}$$ is a parameter. Under further restrictions either on the exponent p or on the shape of Q, we establish the existence of a continuous branch $$\mathcal {C}$$ of nontrivial solutions to this equation which intersects $$\{\lambda \} \times L^{s}(\mathbb {R}^N)$$ for every $$\lambda \in (-\infty , \lambda _Q)$$ and $$s> \frac{2N}{N-1}$$ . Here, $$\lambda _Q>0$$ is an explicit positive constant which only depends on N and $$\text {diam}(\text {supp }Q)$$ . In particular, the set of values $$\lambda $$ along the branch enters the essential spectrum of the operator $$-\Delta $$ .

Journal ArticleDOI
TL;DR: In this article, the authors consider perturbed Hammerstein integral equations of the form: y(t)=\gamma_1(t)H_1\big (varphi _1(y)big )+gamma _2(t),H_2(y))+\varphi ǫ-big (β, β, ε, β) √ √ G(t,s)f(s,y(s)-big )\ \mathrm{d}s \end{aligned}
Abstract: We consider perturbed Hammerstein integral equations of the form: $$\begin{aligned} y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )+\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\ \mathrm{d}s \end{aligned}$$ in the case, where $$H_1$$ and $$H_2$$ are continuous functions, which can be either linear or nonlinear subject to some restrictions, and $$\varphi _1$$ and $$\varphi _2$$ are linear functionals We demonstrate that by introducing a specially constructed order cone, one can equip $$\varphi _1$$ and $$\varphi _2$$ with coercivity conditions that are useful in improving existence results for both the integral equation and associated boundary value problems for ODEs and elliptic PDEs on annuli with nonlocal boundary conditions We illustrate this theory with some specific examples

Journal ArticleDOI
TL;DR: In this article, the generalized hyperstability results for the Drygas functional equation were investigated on a restricted domain, motivated by the notion of Ulam stability, and the results were improved and generalizations of the main results of Piszczek and Szczawinska.
Abstract: Motivated by the notion of Ulam stability, we investigate the generalized hyperstability results for the Drygas functional equation $$\begin{aligned} f(x+y)+f(x-y)=2f(x)+f(y)+f(-y), \end{aligned}$$ on a restricted domain. The method is based on a quite recent fixed point theorem (cf. [6, Theorem 1]) in some functions spaces. We derive from them some characterizations of inner product spaces. Our results are improvements and generalizations of the main results of Piszczek and Szczawinska [29].

Journal ArticleDOI
TL;DR: In this paper, a generalization of Matkowski's fixed point theorem and Istraţescu's fixed-point theorem concerning convex contractions is presented, which is a special case of the generalization presented in this paper.
Abstract: In this paper we obtain a generalization of Matkowski’s fixed point theorem and Istraţescu’s fixed point theorem concerning convex contractions. More precisely, given a complete b-metric space (X, d) we prove that every continuous function \( f:X\rightarrow X\) is a Picard operator, provided that there exist \( m\in \mathbb {N}^{*}\) and a comparison function \(\varphi \) such that \(d(f^{[m]}(x),f^{[m]}(y))\le \varphi (\max \{d(x,y),d(f(x),f(y)),\ldots ,d(f^{[m-1]}(x),f^{[m-1]}(y))\})\) for all \(x,y\in X\). In addition, we point out that if \(m=1\), taking into account that a metric space is a b-metric space, we obtain a generalization of Matkowski’s fixed point theorem. Moreover, we prove that Istraţescu’s fixed point theorem concerning convex contractions is a particular case of our result for \(m=2\). By providing appropriate examples we show that the above-mentioned two generalizations are effective.

Journal ArticleDOI
TL;DR: This study suggests a new fixed-point MEEF (FP-MEEF) algorithm, and analyzes its convergence based on Banach’s theorem (contraction mapping theorem), which is able to converge to the optimal solution quadratically with the appropriate selection of the kernel size.
Abstract: In recent years, research on information theoretic learning (ITL) criteria has become very popular and ITL concepts are widely exploited in several applications because of their robust properties in the presence of heavy-tailed noise distributions. Minimum error entropy with fiducial points (MEEF), as one of the ITL criteria, has not yet been well investigated in the literature. In this study, we suggest a new fixed-point MEEF (FP-MEEF) algorithm, and analyze its convergence based on Banach’s theorem (contraction mapping theorem). Also, we discuss in detail the convergence rate of the proposed method, which is able to converge to the optimal solution quadratically with the appropriate selection of the kernel size. Numerical results confirm our theoretical analysis and also show the outperformance of FP-MEEF in comparison with FP-MSE in some non-Gaussian environments. In addition, the convergence rate of FP-MEEF and gradient descent-based MEEF is evaluated in some numerical examples.