Abstract: In this paper, we investigate the existence of at least three positive solutions to a singular boundary value problem of Caputo's fractional differential equations with a boundary condition involving values at infinite number of points. Firstly, we establish Green's function and its properties. Then, the existence of multiple positive solutions is obtained by Avery–Peterson's fixed point theorem. Finally, an example is given to demonstrate the application of our main results.

Abstract: In this chapter, we first study the existence of Cauchy problems for fractional evolution equations. The suitable mild solutions of fractional Cauchy problems with Riemann-Liouville derivative and Caputo derivative are introduced, respectively. By using fixed point theorems and Hausdorff measure of noncompactness, we give existence results of mild solutions in the cases that the almost sectorial operator is compact and noncompact, respectively. In Section 2.2, we discuss the existence and uniqueness of the bounded solutions on real axis for fractional evolution equations with Liouville fractional derivative of order q∈(01) q ∈ 0 1 with the lower limit –∞. Some sufficient conditions are established for the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions, and other types of bounded solutions.

Abstract: In this paper, we present some fixed point results in the setting of a complete metric spaces by defining a new contractive condition via admissible mapping imbedded in simulation function. Our results generalize and unify several fixed point theorems in the literature.

Abstract: This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such $\boldsymbol {n}$ -neuron competitive neural networks can have $5^{\boldsymbol n}$ equilibria, among which $3^{\boldsymbol n}$ equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the $3^{\boldsymbol n}$ locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.

Abstract: This paper deals with the problem of globally asymptotic stability for nonnegative equilibrium points of genetic regulatory networks (GRNs) with mixed delays (i.e., time-varying discrete delays and constant distributed delays). Up to now, all existing stability criteria for equilibrium points of the kind of considered GRNs are in the form of the linear matrix inequalities (LMIs). In this paper, the Brouwer’s fixed point theorem is employed to obtain sufficient conditions such that the kind of GRNs under consideration here has at least one nonnegative equilibrium point. Then, by using the nonsingular M-matrix theory and the functional differential equation theory, M-matrix-based sufficient conditions are proposed to guarantee that the kind of GRNs under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. The M-matrix-based sufficient conditions derived here are to check whether a constant matrix is a nonsingular M-matrix, which can be easily verified, as there are many equivalent statements on the nonsingular M-matrices. So, in terms of computational complexity, the M-matrix-based stability criteria established in this paper are superior to the LMI-based ones in literature. To illustrate the effectiveness of the approach proposed in this paper, several numerical examples and their simulations are given.

Abstract: In this paper we introduce the concept of bipolar metric space as a type of partial distance. We explore the link between metric spaces and bipolar metric spaces, especially in the context of completeness, and prove some extensions of known fixed point theorems. c ©2016 All rights reserved.

Abstract: In this paper, we consider a class of second-order evolution differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of second-order control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of second-order evolution differential inclusions in Hilbert spaces. We use Bohnenblust–Karlin’s fixed point theorem to prove our main results. Further, we extend the result to study the approximate controllability concept with nonlocal conditions and also extend the result to study the approximate controllability for impulsive control systems with nonlocal conditions. An example is also given to illustrate our main results.

Abstract: In this paper, we study the mild solutions of a class of nonlinear fractional reaction–diffusion equations with delay and Caputo’s fractional derivatives. By the measure of noncompactness, the theory of resolvent operators, the fixed point theorem and the Banach contraction mapping principle, we develop various theorems for the existence and uniqueness of the global mild solutions for the problem.

Abstract: In this paper, we consider a new class of fully history-dependent quasivariational inequalities which arise in the study of quasistatic models of contact and involve two history-dependent operators. By using a fixed-point theorem and arguments of monotonicity and convexity, we prove an existence and uniqueness result of the solution, which includes as special cases some results already obtained in some papers. Then, the obtained result is applied to two problems of quasistatic frictional contact for viscoelastic materials and the unique weak solvability of each contact problem is obtained.

Abstract: In this article, we propose and analyze a new mixed variational formulation for the stationary Boussinesq problem. Our method, which uses a technique previously applied to the Navier–Stokes equations, is based first on the introduction of a modified pseudostress tensor depending nonlinearly on the velocity through the respective convective term. Next, the pressure is eliminated, and an augmented approach for the fluid flow, which incorporates Galerkin-type terms arising from the constitutive and equilibrium equations, and from the Dirichlet boundary condition, is coupled with a primal-mixed scheme for the main equation modeling the temperature. In this way, the only unknowns of the resulting formulation are given by the aforementioned nonlinear pseudostress, the velocity, the temperature, and the normal derivative of the latter on the boundary. An equivalent fixed-point setting is then introduced and the corresponding classical Banach Theorem, combined with the Lax–Milgram Theorem and the Babuska–Brezzi theory, are applied to prove the unique solvability of the continuous problem. In turn, the Brouwer and the Banach fixed-point theorems are used to establish existence and uniqueness of solution, respectively, of the associated Galerkin scheme. In particular, Raviart–Thomas spaces of order k for the pseudostress, continuous piecewise polynomials of degree ≤ k+1 for the velocity and the temperature, and piecewise polynomials of degree ≤ k for the boundary unknown become feasible choices. Finally, we derive optimal a priori error estimates, and provide several numerical results illustrating the good performance of the augmented mixed-primal finite element method and confirming the theoretical rates of convergence. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2015

Abstract: In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L p space.

Abstract: In this paper, the following facts are stated in the setting of b-metric spaces. (1) The contraction constant in the Banach contraction principle fully extends to [0, 1), but the contraction constants in Reich’s fixed point theorem and many other fixed point theorems do not fully extend to [0, 1), which answers the early stated question on transforming fixed point theorems in metric spaces to fixed point theorems in b-metric spaces. (2) Caristi’s theorem does not fully extend to b-metric spaces, which is a negative answer to a recent Kirk–Shahzad’s question (Remark 12.6) [Fixed Point Theory in Distance Spaces. Springer, 2014].

Abstract: In this paper, the notion of �-�-contractive mappings in the setting of w-distance is introduced and some new fixed point theorems for such mappings are established. Presented fixed point theorems gener- alize recent results of Samet et al. (Nonlinear Anal. 75 (2012), 2154-2165) and others. Moreover, some examples and an application to nonlinear fractional differential equations are given to illustrate the usability of the new theory.

Abstract: In this paper a fixed point theorem of Schaefer type involving the product of two operators in a Banach algebra is proved and it is further applied to a first order nonlinear functional differential equation for proving an existence theorem under the mixed generalized Lipschitz and Caratheodory condition.

Abstract: In this paper we introduce cone metric spaces, prove some fixed point theorems of contractive mappings on cone metric spaces. In this paper, we replace the real numbers by ordering Banach space and define cone metric spaces (X, d ). We discuss some properties of convergence of sequences. We prove some fixed point theorems for contractive mappings. Our results generalized some fixed point theorems in metric spaces.

Abstract: In this paper we introduce the notion of (α∗ − ψ)Ćirić-type contractive multivalued operator and investigate the existence and uniqueness of fixed point for such a mapping in b-metric spaces. The wellposedness of the fixed point problem and the Ulam-Hyres stability is also studied. c ©2016 All rights reserved.

Abstract: In this paper, we introduce the concept of JS-quasi-contraction and prove some fixed point results for JS-quasi-contractions in complete metric spaces under the assumption that the involving function is nondecreasing and continuous. These fixed point results extend and improve many existing results since some assumptions made there are removed or weakened. In addition, we present some examples showing the usability of our results.

Abstract: The purpose of this paper is to generalize fixed point theorems introduced by Jleli et al. (J. Inequal. Appl. 2014:38, 2014) by using the concept of triangular α-orbital admissible mappings established in Popescu (Fixed Point Theory Appl. 2014:190, 2014). Some examples are given here to illustrate the usability of the obtained results.

Abstract: In this article, we use the technique based upon measures of noncompactness in conjunction with a Darbo-type fixed point theorem with a view to studying the existence of solutions of infinite systems of second-order differential equations in the Banach sequence space $\ell_{p}$ . An illustrative example is also given in support of our existence result.

Abstract: The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder's fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.

Abstract: A quasi-variational inequality is a variational inequality, in which the constraint set is depending on the variable. However, as shown by a motivating example in electricity market, the constraint map may not be a self-map, and then, there is usually no solution. Thus, we define the concept of projected solution and, based on a fixed point theorem, we establish some results on existence of projected solution for quasi-variational inequality problem in a finite-dimensional space where the constraint map is not necessarily self-map. As an application of our results, we obtain an existence theorem for quasi-optimization problems. Finally, we introduce the concept of projected Nash equilibrium and study the existence of such equilibrium for noncooperative games as another application.

Abstract: In this article, we generalize some frequently used metrical notions such as: completeness, closedness, continuity, g-continuity and compatibility to relation-theoretic setting and utilize these relatively weaker notions to prove results on the existence and uniqueness of coincidence points involving a pair of mappings defined on a metric space endowed with an arbitrary binary relation. Particularly, under universal relation our results deduce the classical coincidence point theorems of Goebel (Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 16 (1968) 733-735) and Jungck (Int. J. Math. Math. Sci. 9 (4) (1986) 771-779). In process our results generalize, extend, modify and unify several well-known results especially those obtained in Alam and Imdad (J. Fixed Point Theory Appl. 17 (4) (2015) 693-702), Karapinar et al: (Fixed Point Theory Appl. 2014:92 (2014) 16 pp), Alam et al: (Fixed Point Theory Appl. 2014:216 (2014) 30 pp), Alam and Imdad (Fixed Point Theory, in press) and Berzig (J. Fixed Point Theory Appl. 12 (1-2) (2012) 221-238.