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In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings. [ABSTRACT FROM AUTHOR]

The approach described in this research is an exemplar‐based inpainting problem that combines a two‐stage structure tensor and image sparse representation to fill in any missing pixels. An important step is to select the filling order and local intensity smoothness, as well as to ensure that the structure is not destroyed. We employ a two‐stage structure tensor‐based priority for the filling order: finding the candidate patches and determining the appropriate weight of each candidate patch under the constraint of local patch consistency, then applying a blend of a sparse linear combination of candidate patches to fill in the missing region of the image. In addition, this technique may also be used for object removal. The proposed method yields results that are visually natural and qualitative. [ABSTRACT FROM AUTHOR]

In this work, we establish the closedness and convexity of the set of fixed points of equally continuous and asymptotically demicontractive mapping in the intermediate sense. We proposed an inertial hybrid projection technique for determining an approximate common solution to three significant problems. The first is the system of generalized mixed equilibrium problems with relaxed η−ζ$$ \eta -\zeta $$ monotone mappings, the second is the problem of fixed points of a countable family of equally continuous and asymptotically demicontractive mappings in the intermediate sense, and the third is of determining a point in a null space of a countable family of inverse strongly monotone mappings in Hilbert space. Based on these problems, we formulate a theorem and establish its strong convergence to their common solution. Additionally, we studied the applications of our algorithm to variational inequality problems and convex optimization problems. Finally, we numerically demonstrate the efficiency and robustness of our scheme. Several results available in the literature can be obtained as special cases of our result. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to construct a common solution of the split null point problem associated with the maximal monotone operators and the fixed point problem associated with a finite family of k‐demicontractive operators in Hilbert spaces. We compute the optimal common solution via inertial parallel hybrid algorithm under a suitable set of control conditions. The viability of parallel implementation of the algorithm is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature. [ABSTRACT FROM AUTHOR]

RMIL conjugate gradient method originally proposed by Rivaie et al. (2012) has recently gained lots of attention. In this article, we propose a generalized conjugate gradient parameter that contains both RMIL and its variant, that is, RMIL+, as special cases. We show that the search direction generated by the new method is sufficiently descent. Under standard mild conditions, we discuss the convergence analysis of the propose method. We demonstrate the numerical efficiency of the propose method on a set of unconstrained minimization benchmark test problems as well as an image restoration problem. The results of the experiment reveal that the proposed method performs better than its main competitors. [ABSTRACT FROM AUTHOR]

Cryptography is processing for securing communication between two people. The opponent wants to know the message that is encrypted using a secret key. Although the opponent can eavesdrop the message sent between the sender and the receiver, the opponent is unable to decrypt to read the message. Therefore, the secret key is very important. The sender and the receiver agree with the secret key in an insecure channel by using key distribution protocol such as the Diffie–Hellman protocol. Since quantum computer is coming soon, Diffie–Hellman protocol is not secure. We will develop a quantum key distribution protocol. The benefit of the quantum system is the quantum state that cannot copy by no‐cloning theorem. Thus, the opponent does not copy and keeps the message that is quantum. In this paper, a novel quantum key distribution protocol between two people (Alice and Bob) based on quantum operators is developed. The opponent (Eve) wants to know the secret key. Although Eve knows this quantum key distribution protocol, Eve does not behave similarly to Alice and Bob. For example, Eve eavesdrops Alice's quantum state that was sent to Bob, and Eve sends another quantum state. Therefore, we cannot control Eve's behavior. So we give the upper bound of mutual information between the user and opponent by using Holevo's bound. We verify the usual security definition for quantum key distribution that is equality‐and‐uniformity and privacy in the mutual information sense. [ABSTRACT FROM AUTHOR]

In this article, we propose an iterative scheme for solving a generalized split common null point problem. The iterative method uses self‐adaptive stepsizes that do not depend on the Lipschitz constants of the underlying operators. Instead, the stepsizes are updated using a simple updating formula. We prove the strong convergence theorems of the sequence generated by the iterative scheme under some mild assumptions. Furthermore, we derive iterative methods for solving split generalized variational inequality, convex feasibility, and minimization problems. In addition, some numerical illustrations are provided to support the main result. [ABSTRACT FROM AUTHOR]

In this manuscript, we investigate and approximate common fixed points of two nonself asymptotically nonexpansive mappings in the setting of CAT(0) spaces. We provide three examples and conduct numerical experiments to show the implementation of the approximation schemes. Our results extend and improve the related results in the literature. [ABSTRACT FROM AUTHOR]

The main objective is to apply the concept of newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function in fluid dynamics. In this article, a newly developed idea of the fractional‐order derivative of the Rabotnov fractional–exponential function and the nonsingular kernel has been applied to study viscous fluid flow in the presence of an applied magnetic field. The flow is considered over an infinite vertical plate moving with arbitrary velocity. The modeled problem is transformed into a nondimensional form via dimensionless analysis, and then the Laplace transform method is applied for the solution of the problem. Due to the complexity in Laplace inversion, a strong numerical inversion procedure, namely, Zakian's algorithm, has been used, and the results are computed in various plots and tables. The corresponding discussion of results is included in detail. It is concluded that the generalized fractional‐order derivative is accurate and efficient for describing general fractional‐order dynamics in complex and power law phenomena. [ABSTRACT FROM AUTHOR]

In this article, an Ishikawa iteration scheme is modified for b$$ b $$‐enriched nonexpansive mapping to solve a fixed point problem and a split variational inclusion problem in real Hilbert spaces. Under some suitable conditions, we obtain convergence theorem. Moreover, to demonstrate the effectiveness and performance of proposed scheme, we apply the scheme to solve a split feasibility problem and compare it with some existing iterative schemes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a derivative‐free iterative method for finding the solutions of convex constrained nonlinear equations (CCNE) using the projection strategy. The new approach is free from gradient evaluations at each iteration. Also, the search direction generated by the proposed method satisfies the sufficient descent property, which is independent of the line search. Compared with traditional methods for solving CCNE that assumes Lipschitz continuity and monotonicity to establish the global convergence result, an advantage of our proposed method is that the global convergence result does not require the assumption of Lipschitz continuity. Moreover, the underlying operator is assumed to be pseudomonotone, which is a milder condition than monotonicity. As an applications, we solve the LASSO problem in compressed sensing. Numerical experiments illustrate the performances of our proposed algorithm and provide a comparison with related algorithms. [ABSTRACT FROM AUTHOR]

In this study, a trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström (RKN) pair with six stages is formulated, considering a previous method developed by El‐Mikkawy and Rahmo. The obtained adapted pair integrates exactly the usual test equation: y′′=−w2y$$ {y}^{\prime \prime }=-{w}^2y $$. The local truncation error of the new method is presented, showing that the algebraic order of the original method is maintained. The periodicity interval of the new method is computed, showing that the developed method is "almost" P‐stable. The numerical examples considered clearly show the superiority of the new developed embedded pair over other RKN methods of algebraic orders 6(4) with six stages appeared in the literature. [ABSTRACT FROM AUTHOR]

The generalized and optimal sequence of weights on a progressive‐iterative approximation method with memory for least square fitting (GOLSPIA) improves the MLSPIA method by extends to the multidimensional data fitting. In addition, weights of the moving average are varied between iterations, using the three optimal sequences of weights derived from the singular values of the collocation matrix. It is proved that a series of data fitting with an appropriate alternative of weights converge to the solution of least square fitting. Moreover, the convergence rate of the new method is faster than that of the MLSPIA method. Some examples and applications in this paper show the efficiency and effectiveness of the GOLSPIA method. [ABSTRACT FROM AUTHOR]

This study presents two inertial type extragradient algorithms for finding a common solution to the monotone variational inequalities and fixed point problems for ρ$$ \rho $$‐demicontractive mapping in real Hilbert spaces. We provide inertial type iterative algorithms with self‐adaptive variable step size rules that do not require prior knowledge of the operator value. Our algorithms employ a basic step size rule, which is derived by certain computations at each iteration. Without previous knowledge of the operators Lipschitz constant, two strong convergence theorems were obtained. Finally, we present a number of numerical experiments to evaluate the efficacy and applicability of the proposed algorithms. The conclusions of this study on variational inequality and fixed point problems support and extend previous findings. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid conjugate gradient (CG) approach for solving nonlinear equations and signal reconstruction. The CG parameter of the approach is a convex combination of the Dai‐Yuan (DY)‐like and Hestenes‐Stiefel (HS)‐like parameters. Independent of any line search, the search direction is descent and bounded. Under some reasonable assumptions, the global convergence of the hybrid approach is proved. Numerical experiments on some benchmark test problems show that the proposed approach is efficient compared with some existing algorithms. Finally, the proposed approach is applied in signal reconstruction. [ABSTRACT FROM AUTHOR]

Split variational inclusion problems are core and useful technique that connects many important problems in nonlinear analysis. This paper proposes a resolvent‐based superiorization method for solving split variational inclusion problems, and some strong convergence theorems are proved under some mild conditions. Superiorization iterative procedures are simple yet powerful method that interlaces two algorithmic procedures under the assumption that one of the algorithmic procedure called the basic algorithm is resilient to the disturbances of its iterates caused by the activities of the other algorithm called perturbations. The numerical results in this paper show that our results are improvements and extensions of the corresponding results announced by many others. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of solutions of the convex minimization problems and common fixed problems in CAT(1) spaces under some mild conditions. For this, we propose the new modified the proximal point algorithm. Further, we give some applications for the convex minimization problem and the fixed point problem in CAT(κ) spaces with the bounded positive real number κ. Our results improve and generalize many recent important results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider an improvement of the extragradient method to figure out the numerical solution for pseudomonotone equilibrium problems in arbitrary real Hilbert space. A new method is proposed with an inertial scheme and a self adaptive step size rule that is revised on each iteration based on the previous three iterations. The weak convergence of the method is proved by assuming standard cost bifunction assumptions. We also consider the application of our results to solve different kinds of variational inequality problems and a particular class of fixed point problems. For a numerical part, we study the well‐known Nash‐Cournot equilibrium model and other test problems to support our well‐established convergence results and to ensure that our proposed method has a competitive edge over CPU time and a number of iterations. [ABSTRACT FROM AUTHOR]

In this paper, we study and investigate the ψ−Hilfer fractional differential equation with nonlocal multi‐point condition of the form: Da+q,p;ψu(t)=f(t,u(t),Da+q,p;ψu(t)),t∈[a,b],Ia+1−r;ψu(a)=∑i=1mβiu(ηi),q≤r=q+p−qp<1,ηi∈[a,b], where 0

The aim of this manuscript is to improve certain results established in a recent paper of Imdad et al on the existence of a common fixed point for contractive mappings in fuzzy metric spaces. Further, we generalize and extend some common fixed point theorems due to Ali and Imdad and some fixed point theorems due to Mihet to the case of fuzzy metric‐like spaces. We also furnish an illustrative example to demonstrate the validity of the hypotheses and degree of generality of our main result. [ABSTRACT FROM AUTHOR]

In this paper, we present iteration methods to find a solution of asymptotically quasi‐nonexpansive semigroups for a split equality common fixed point in Banach spaces. The results show weak and strong convergence theorem of iteration that improve and extend some recent results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some fixed‐point theorems for a generalized almost Hardy‐Rogers‐type F contraction in a metric‐like space and give an example to illustrate these main results. Moreover, we show the applications of electric circuit equations, second‐order differential equations, and fractional differential equations. Our results improve, generalize, and extend the corresponding results in literature. [ABSTRACT FROM AUTHOR]

The objective of this article is to establish a new modified iteration process for nonexpansive mappings in complete CAT(κ) spaces. We prove strong and Δ‐convergence theorems of the proposed method in such spaces under some standard conditions. Furthermore, numerical experiments of non‐trivial examples are also provided to show performance and comparison speed of convergence with many previously known methods. Our main result extended and improved the corresponding recent results announced by many researchers. [ABSTRACT FROM AUTHOR]

In this paper, we modify the proximal point algorithm for finding common fixed points in CAT(0) spaces for nonlinear multivalued mappings and a minimizer of a convex function and prove Δ‐convergence of the proposed algorithm. A numerical example is presented to illustrate the convergence result. Our results improve and extend the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and prove fixed point results for some classes of ℑ‐contraction mappings on Pompeiu‐Hausdorff functionals in dislocated b‐metric spaces. We illustrate an example and graph to support our main results. We present the generalized Ulam‐Hyers stability of fixed point inclusion. Finally, we have applied our results to find a solution of Fredholm‐type integral inclusion in dislocated b‐metric spaces. Our results generalize some recent results and extend several theorems in the literature. [ABSTRACT FROM AUTHOR]

The main purpose of this study is to establish some coincidence and common fixed point theorems for 2 self‐mappings via the condition of implicit function on metric spaces provided with arbitrary binary relation. In our results, the main theorems are the generalized version of the literatures, and we also give some examples to guarantee our main results. Moreover, we show that the result to guarantee a coincidence point which are more useful than the previous results of Banach's contraction which cannot be applied. Furthermore, we include some consequences for proving the corollary, which satisfied several well‐known contractions that are obtained from the main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a general Bayesian abstract fuzzy economy model of product measurable spaces, and we prove the existence of Bayesian fuzzy equilibrium for this model. Our results extend and improve the corresponding recent results announced by Patriche and many authors from the literature. It captures the idea that the uncertainties characterize the individual feature of the decisions of the agents involved in different economic activities. In this paper, the uncertainties can be described by using random fuzzy mappings. Further attention is needed for the study of applications of the established result in the game theory and the fuzzy economic field.Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We present a new approach for constructing a spectral conjugate gradient‐type method for solving nonlinear equations. The proposed method uses an approximate optimal step size together with the memoryless Broyden–Fletcher–Goldfarb–Shanno (BFGS) formula to generate a new choice of the spectral conjugate gradient‐type direction that satisfies the sufficient descent condition without line search requirement. The global convergence of the method is achieved under some mild assumptions. Numerical experiments on both nonlinear monotone equations and signal reconstruction problems reveal the efficiency of the new approach. [ABSTRACT FROM AUTHOR]

In recent times, the study of diathermal oils is an area of interest for multiple researchers because of their numerous pivotal applications in industrial and engineering operations. The core aim of this work is the formulation of a fractional model to anticipate improvement in thermal and flow characteristics of a particular kind of diathermal oils named engine oil due to the dispersion of two different types of nanoparticles. Molybdenum‐disulfide (MoS2) and iron oxide (Fe3O4) nanoparticles are considered to form hybrid nanofluid, and combined impacts of their particular features on the thermal efficiency of engine oil are investigated. The ramped movement of an unbounded vertically inclined wall initiates the flow of hybrid nanofluid and some supplementary physical phenomena such as heat radiation, uniform magnetic field, and ramped heating also influence this flow. Additionally, the significant role of nanoparticles' shape factor in augmenting the heat transfer capacity of engine oil is examined. Initially, the flow of hybrid nanofluid is described through the Brinkman‐type fluid model, which is developed in light of Maxwell equations and Boussinesq approximation. Later, this mathematical model is transmuted to a fractional framework by incorporating the time‐fractional Atangana–Baleanu derivative. Laplace transformation is employed to procure exact solutions of the generalized model. These solutions are portrayed through several graphical illustrations to analyze the influence of various involved physical parameters. Special attention is given to heat transfer rate and shear stress, and a comprehensive tabular study is performed in terms of Nusselt number and skin friction coefficient. It is concluded that the heat‐transferring potential of observed hybrid nanofluid is 17.4% higher than pure engine oil. The combination of fractional model and ramping technique is found to be more effective for increasing the heat transfer rate and reducing the shear stress. [ABSTRACT FROM AUTHOR]