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In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure. [ABSTRACT FROM AUTHOR]

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In this paper, we consider a two-person zero-sum game problem for an elliptic variational inequality in which the obstacle is the sum of two functions taken by two different players. A nonlinear equation involving the obstacle operator is introduced to equivalently describe the solution to the corresponding elliptic variational inequality. Using these nonlinear equations, a necessary condition of the optimal solutions, namely the saddle points of the game problem, is established. We find that, whenever the optimal solution exists, it can be solved explicitly by such a necessary condition. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study properties of the numerical range W(T) of some nonlinear operators in Banach spaces of sequences ℓp. [ABSTRACT FROM AUTHOR]

In this paper, we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue λ F ⁢ (p , Ω) {\lambda_{F}(p,\Omega)} of the anisotropic p-Laplacian, 1 < p < + ∞ {1

In this paper, we investigate the initial value problem(IVP henceforth) associated with the higher order nonlinear dispersive equation given in Jones et al. (Int J Math Math Sci 24:371–377, 2000): ∂ t u + α ∂ x 7 u + β ∂ x 5 u + γ ∂ x 3 u + μ ∂ x u + λ u ∂ x u = 0 , x ∈ R , t ∈ R , u (x , 0) = u 0 (x) , x ∈ R with the initial data in the Sobolev space H s (R). Benefited from the ideas of Huo and Jia (Z Angew Math Phys 59:634–646, 2008), Zhang et al. (Acta Math Sci 37B(2):385–394, 2017) and Zhang and Huang (Math Methods Appl Sci 39(10):2488–2513, 2016) that is, using Fourier restriction norm method, Tao's [k, Z]-multiplier method and the contraction mapping principle, we prove that IVP is locally well-posed for the initial data u 0 ∈ H s (R) with s ≥ - 5 8 . Moreover, based on the local well-posedness and conservation law, we establish the global well-posedness for the initial data u 0 ∈ H s (R) with s = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of nonlinear variational type inclusions involving (H, η, φ)-monotone operators in real Banach spaces. Further, we define a proximal operator associated with an (H; η; φ)-monotone operator and show that it is single valued and Lipschitz continuous. Using proximal point operator techniques, we prove the existence and uniqueness of a solution and suggest an iterative algorithm for the system of nonlinear variational type inclusions. Furthermore, we discuss the convergence of the iterative sequences generated by the algorithms. [ABSTRACT FROM AUTHOR]

The primal-dual hybrid gradient method, modified (PDHGM, also known as the Chambolle-Pock method), has proved very successful for convex optimization problems involving linear operators arising in image processing and inverse problems. In this paper, we analyze an extension to nonconvex problems that arise if the operator is nonlinear. Based on the idea of testing, we derive new step-length parameter conditions for the convergence in infinite-dimensional Hilbert spaces and provide acceleration rules for suitably (locally and/or partially) monotone problems. Importantly, we prove linear convergence rates as well as global convergence in certain cases. We demonstrate the efficacy of these step-length rules for PDE-constrained optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we summarize some of the recent developments in the area of fractional equations with focus on the ideas and direct methods on fractional non-local operators. These results have more or less appeared in a series of previous literature, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and apply them to a variety of problems in this area. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a new fixed point theorem in cones of Banach spaces. Then, we apply the fixed point theorem to study the existence and uniqueness of monotone positive solutions for an elastic beam equation u(4)(t) = f(t; u(t); u'(t)) with superlinear boundary conditions. An example is given to illustrate our main result. Compared with some earlier results (cf. [10]), the biggest differences are that we consider such equation with superlinear boundary conditions and remove some restrictive conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the septendecic functional equation and prove the Hyers-Ulam stability for the septendecic functional equation in matrix normed spaces by using the fixed point technique. [ABSTRACT FROM AUTHOR]

Abstract Risk-averse model predictive control (MPC) offers a control framework that allows one to account for ambiguity in the knowledge of the underlying probability distribution and unifies stochastic and worst-case MPC. In this paper we study risk-averse MPC problems for constrained nonlinear Markovian switching systems using generic cost functions, and derive Lyapunov-type risk-averse stability conditions by leveraging the properties of risk-averse dynamic programming operators. We propose a controller design procedure to design risk-averse stabilizing terminal conditions for constrained nonlinear Markovian switching systems. Lastly, we cast the resulting risk-averse optimal control problem in a favorable form which can be solved efficiently and thus deems risk-averse MPC suitable for applications. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam stability for the following cubic functional equation with involution f(2x + y) + f(2x + σ(y)) - 2f(x + y) - 2f(x + σ(y)) - 12f(x) = 0 in modular spaces by using a fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce several extensions of Meir-Keeler contractive mappings in the structure of S-metric spaces. Then we investigate some existence, uniqueness, and generalized Ulam-Hyers stability results for the classes of MKC mappings via fixed point theory. Besides the the- oretical results, we also present some illustrative examples to verify the effectiveness and applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the functional inequality N(f(x + 2y) - 3f(x + y) + 3f(x) - f(x - y) - 6f(y); t) ≥ N(f(2x + y) + f(2x - y) - 2f(x + y) - 2f(x - y) - 12f(x); kt) for some fixed nonzero real number k and prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces by fixed point methods. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study the existence of solution of a system of differential equations using fixed point technique. In this regard, in the first part of this article, along with some properties of partial b-metric topology, we prove a common fixed point theorem for generalized Geraghty type contraction mappings in a complete partial b-metric spaces. Then in second part we apply this result to show the existence of the solution of a system of ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the mapping satisfying mixed g-monotone property in partially ordered complete Menger probabilistic G-metric spaces. By weakening the notion of Ψ, we prove some new coupled coincidence point theorems and coupled common fixed point theorems. Finally, we provide an example to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, at first, we consider the concept of fuzzy Banach algebras and fuzzy compact operators. Then we apply a fixed point theorem to solve the operator equation AxBx = x in the fuzzy Banach algebras under a nonlinear contraction. [ABSTRACT FROM AUTHOR]

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce a viscosity iterative forward-backward splitting method with errors to find zeros of the sum of two accretive operators in Banach spaces. We shall prove the strong convergence of the method under mild conditions. We also discuss applications of these methods to monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of fixed points for Kannan contraction, Chatterjea contraction and Reich contraction in setting of digital metric spaces. These digital con-tractions are the applications of metric fixed point theory contractions. [ABSTRACT FROM AUTHOR]

In this paper, by considering both F-contraction and fixed point results on ordered partial metric spaces, we introduce a pair of Círíc type ordered F-contractions on an ordered partial metric space. Then we give a common fixed point theorem for such contractions. We give an example showing that our main theorem is applicable, but both results of Durmaz et al. [11] and Wardowski [18] are not. We also discuss that this fixed point result can be applied to show the existence of solution of an integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the new idea of α-C-class function and establish new fixed point results in a complete metric space. It can be stated that the results that have come into being give substantial generalizations and improvements of several well known results in the existing comparable literature. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a solution of new type quintic functional equations and prove the Hyers-Ulam-Rassias stability for a quintic functional equation by the directed method and a subaddtive function approach and also, present a counterexample. Finally, we investigate the Hyers-Ulam-Rassias stability for a quintic functional equation with an involution by the fixed point method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete population model and obtain a couple of criteria to guarantee our model being stable, including the global attractor, the extreme stability, and the periodicity. [ABSTRACT FROM AUTHOR]

In this paper, we extend relation-theoretic contraction principle due to Alam and Im- dad to a nonlinear contraction using a relatively weaker class of continuous control functions employing a locally finitely T -transitive binary relation, which improves the corresponding fixed point theorems especially due to: Alam and Imdad (J. Fixed Point Theory Appl. 17 (2015) 693-702), Agarwal et al: (Applicable Analysis, 87 (1) (2008) 109-116), Berzig and Karapinar (Fixed Point Theory Appl. 2013:205 (2013) 18 pp), Berzig et al: (Abstr. Appl. Anal. 2014:259768 (2014) 12 pp) and Turinici (The Sci. World J. 2014:169358 (2014) 10 pp). [ABSTRACT FROM AUTHOR]

This paper is concerned with the initial-value problem for nonlinear first-order impulsive integro-differential equations on time scales T . We establish certain existence criteria by using a fixed-point theorem for operator on cones, under which such problems have aminimal and amaximal solution lying in a corresponding region bounded by their lower and upper solutions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability for semi-linear retarded functional integro-differential equations by using the Fredholm theory in Hilbert spaces. We no longer require the compactness of structural operators to obtain the approximate controllability for the nonlinear differential system, but instead we use the theory of interpolation spaces and the regularity of solutions of semi-linear given equations with unbounded principal operators. Finally, based on the properties of general degree theory in infinite dimensional spaces, we investigate the relation between the reachable set of trajectories of the semi-linear retarded functional integro-differential system and that of its corresponding linear system excluded by the nonlinear term. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer's fixed point theorem. Some Ulam stability results are presented. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of nonlinear homogeneous Neumann boundary p(u)-laplacian problem of the form {b(u) - diva(x, u, ∇u) = f in Ω a(x, u, ∇u).η = 0 on ∂Ω where Ω is a smooth bounded open domain in RN, N ≥ 3 and η the outer unit normal vector on ∂Ω. The existence and uniqueness results of weak solution and the structural stability result are obtained by approximation method and convergent sequences in terms of Young measure. [ABSTRACT FROM AUTHOR]

In the present paper we consider initial value problems for Hilfer fractional differential equations and for system of Hilfer fractional differential equations. By using equivalent integral equations and some fixed point theorems, we study the local existence of solutions. We extend these local existence results globally with the help of continuation theorems and generalized Gronwall inequality. [ABSTRACT FROM AUTHOR]

In this paper are given some features and important results in the fuzzy metric space. The results that are presented are related to the fixed point in the fuzzy metric space. The theorems which are elaborated provide the existence and uniqueness of the fixed point in the complete fuzzy metric space. The proof of which is done by using Cauchy sequence. Also are given results for the fixed point in the compact fuzzy metric spaces. All these results are explained by concrete examples that once also present their application. [ABSTRACT FROM AUTHOR]

Abstract: Let be a nonempty, closed and convex subset of a real Hilbert space . Let , be a finite family of generalized asymptotically nonexpansive mappings. It is our purpose, in this paper to prove strong convergence of Mann’s type method to a common fixed point of provided that the interior of common fixed points is nonempty. No compactness assumption is imposed either on or on . As a consequence, it is proved that Mann’s method converges for a fixed point of nonexpansive mapping provided that interior of . The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings. [Copyright &y& Elsevier]

Abstract: Kirk and Xu introduced the concept of asymptotic pointwise contractions. In this paper, we investigate these kinds of mappings in modular spaces. Moreover, the fixed point theorem for asymptotic pointwise nonexpansive mappings in modular spaces is also studied. The results of this paper improve and extend the results of Razani et al. [A. Razani, E. Nabizadeh, M. Beyg Mohamadi, S. Homaei Pour, Fixed points of nonlinear and asymptotic contractions in the modular spaces, Abstract and Applied Analysis, 2007 (2007)], and Kirk and Xu [W. A. Kirk, Hong-Kun Xu, Asymptotic pointwise contractions, Nonlinear Analysis 69 (2008) 4706–4712] to modular spaces. [Copyright &y& Elsevier]

Abstract: This paper deals with the second order nonlinear neutral delay difference equation By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain the existence results of uncountably many bounded positive solutions for the above equation. Examples that cannot be solved by known results are given to illustrate the results presented in this paper. [Copyright &y& Elsevier]

Abstract: The fixed point theory for cone metric spaces, which was introduced in 2007 by Huang and Zhang in the paper [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] has recently become a subject of interest for many authors. Cone metric spaces are generalizations of metric spaces where the metric is replaced by the mapping , where , and is a real Banach space. In the present paper for a cone metric space and for the family of subsets of we establish a new cone metric . Next, we introduce the concept of set-valued contraction of Nadler type and prove a fixed point theorem. Examples are provided. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to use a new hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results. [ABSTRACT FROM AUTHOR]

In this paper, we are dealing with an analytical study of a singular fractional order nonlinear differential equation with fractional integral and differential boundary conditions and ϕp‐operator, for existence and stability results. Our problem is based on two types of fractional order derivatives, that is, Caputo factional derivative of order σ and Riemann‐Liouville derivative of order β, where m − 1  <  σ,β  ≤  m, and m  ∈  {3,4,5,...}. The suggested problem will be converted into an equivalent integral form by the help of Green function. After the proofs for these properties, some classical fixed point theorems are employed for the existence of positive solution (EPS). For application of the results, an expressive example is included. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a priori C 2 , α estimate up to the boundary for F (D 2 u) = 0 and the regularity of the free boundary of the obstacle problem for fully nonlinear operator under specific conditions for the operator and level sets of the operator. The conditions are variations of conditions for the zero set of the operator in [7]. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce several types of Meir-Keeler contractive mappings in the structure of Ms-metric spaces. Then we study some existence and uniqueness fixed point theorems for these types of MKC mappings in Ms-metric spaces via Gupta-Saxena type contraction and other fraction version type contractions. Also, we extend and improve very recent results in fixed point theory. [ABSTRACT FROM AUTHOR]

This paper deals with improvements of the Trudinger–Moser inequality related to the operator Q V ( u ) : = − Δ n u + V ( x ) | u | n − 2 u , where n ≥ 2 and the potential V : R n → R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E : = { u ∈ W 1 , n ( R n ) : ∫ R n V ( x ) | u | n d x < ∞ } endowed with the norm ‖ u ‖ : = [ ∫ R n ( | ∇ u | n + V ( x ) | u | n ) d x ] 1 / n and we prove that if ( u k ) is a sequence in E such that ‖ u k ‖ = 1 , u k ⇀ u ≢ 0 in E and 0 < p < p n ( u ) : = β n ( 1 − ‖ u ‖ n ) − 1 / ( n − 1 ) , then ( ⁎ ) sup k ⁡ ∫ R n Ψ ( p | u k | n / ( n − 1 ) ) d x < ∞ , where Ψ ( t ) : = e t − ∑ i = 0 n − 2 t i i ! , β n : = n ω n − 1 1 / ( n − 1 ) and ω n − 1 is the measure of the unit sphere in R n . Furthermore, p n ( u ) is sharp in the sense that there exists a sequence ( u k ) ⊂ E satisfying ‖ u k ‖ = 1 and u k ⇀ u ≢ 0 in E such that the supremum (⁎) is infinite for p ≥ p n ( u ) . As an application of the previous result we prove the following sharp form of the Trudinger–Moser inequality for the subspace E . Considering ℓ ( α ) : = sup { u ∈ E : ‖ u ‖ = 1 } ⁡ ∫ R n Ψ ∘ ν α ( u ) d x , where ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) , assuming some conditions of symmetry on V it is established (1) for 0 ≤ α < λ 1 ( V ) we have ℓ ( α ) < ∞ , (2) for α ≥ λ 1 ( V ) , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < λ 1 ( V ) , an extremal function for ℓ ( α ) exists. Here λ 1 ( V ) denotes the first eigenvalue of Q V ( u ) . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish and prove some new common fixed point theorems for intuitionistic fuzzy maps in the context of (α,β)-cut sets of intuitionistic fuzzy sets on a complete metric space in association with the Hausdorff metric. Furthermore, the technique of Meir-Keeler (shortly, M-K) contraction is applied to obtain common fixed point of intuitionistic fuzzy compatible maps and fixed points of Kannan type intuitionistic fuzzy set-valued contractive mappings. Our results generalize M-K type fixed point theorem along with its various generalizations. Some nontrivial examples have been furnished in the support of the main results. [ABSTRACT FROM AUTHOR]

In the study, we propose the Associated Probability Intuitionistic Fuzzy Weighted Averaging (As-P-IFWA) and the Associated Probability Intu- itionistic Fuzzy Weighted Geometric (As-P-IFWG) aggregation operators with associated probabilities of a fuzzy measure presenting an uncertainty. Decision makers' evaluations are given as intuitionistic fuzzy values and are used as the arguments of the aggregation operators. In the paper, we prove correctness of extensions and show the conjugate connections between the constructed operators. Several versions of the new operators are successfully used in the business start-up decision making problem. [ABSTRACT FROM AUTHOR]

In this paper, we study fixed point proximity algorithms for a TVL1 restoration model recovering blurred images with impulsive noise, and image inpainting. The model that minimizes the sum of a data fidelity term in theℓ¹-norm, a term in ℓ²-norm and total-variation regularization term is strictly convex. We obtain the solution of the model through finding a fixed point of a nonlinear mapping expressed in terms of the proximity operator of the ℓ¹-norm or the ℓ²-norm, each of which is explicitly given. The non-expansivity of the mapping is also analysed theoretically. This formulation naturally leads to fixed-point algorithms for numerical treatment of the model. Numerical experiments demonstrate that the proposed algorithms perform favourably. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Suzuki type Z-contraction and study the corresponding fixed point property. This kind of contraction generalizes the Banach contraction and unifies several known type of nonlinear contractions. We consider a nonlinear operator satisfying a nonlinear contraction in a metric space and prove fixed point results. As an application, we apply our result to show the solvability of nonlinear Hammerstein integral equations. [ABSTRACT FROM AUTHOR]

Abstract: The present paper is concerned with the convergence problem of inexact Newton methods. Assuming that the nonlinear operator satisfies the γ-condition, a convergence criterion for inexact Newton methods is established which includes Smale''s type convergence criterion. The concept of an approximate zero for inexact Newton methods is proposed in this paper and the criterion for judging an initial point being an approximate zero is established. Consequently, Smale''s α-theory is generalized to inexact Newton methods. Furthermore, a numerical example is presented to illustrate the applicability of our main results. [Copyright &y& Elsevier]

In this paper, we prove weak and strong convergence of the Ishikawa iterative scheme with errors to common fixed point I-asymptotically quasi-nonexpansive mappings in a Banach space. The results obtained in this paper improve and generalize the corresponding results in the existing literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonhomogeneous multipoint boundary-value problem (BVP) of a second-order differential equation with one-dimensional p-Laplacian. Using multiple fixed-point theorems, new sufficient conditions to guarantee the existence of at least three solutions of this BVP are established. An example is presented to illustrate the main results. The first emphasis of this paper is to show that the approach to get three positive solutions of a BVP by using multiple fixed-point theorems can be extended to treat nonhomogeneous BVPs. The second emphasis is put on the nonlinear term f involved with the first-order delta operator. [ABSTRACT FROM AUTHOR]

The hybrid averaging (HA) is an aggregation operator that uses the weighted average (WA) and the ordered weighted averaging (OWA) operator in the same formulation. In this paper, we introduce several generalizations of the HA operator by using generalized and quasi-arithmetic means, fuzzy numbers and order inducing variables in the reordering step of the aggregation process. We present the fuzzy generalized hybrid averaging (FGHA) operator, the fuzzy induced generalized hybrid averaging (FIGHA) operator, the Quasi-FHA operator and the Quasi-FIHA operator. The main advantage of these operators is that they generalize a wide range of fuzzy aggregation operators that can be used in a wide range of applications such as decision making problems. For example, we could mention the fuzzy induced hybrid averaging (FIHA), the fuzzy weighted generalized mean (FWGM) and the fuzzy induced generalized OWA (FIGOWA). We end the paper with an application of the new approach in a decision making problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish some new KKM type theorems in G-convex spaces and, as applications, to obtain some new matching theorems, fixed point theorems, section theorems and minimax theorems in G-convex spaces. The results presented in this paper improve and generalize the corresponding results in [1-6, 16, 20, 23]. [ABSTRACT FROM AUTHOR]