Nontrivial Numerical Examples Research Articles

Option pricing in the illiquid markets under the mixed fractional Brownian motion model

This paper deals with the option pricing in the illiquid markets under the mixed fractional geometric Brownian motion model with jump process. We propose a general long memory financial model, where its featuring parameters are related to demand and supply by showing also the existence of some restrictions on them. Moreover, by using the delta Hedging strategy and replicating portfolio, we obtain an integro partial differential equation (PIDE) for the option price which is solved by the spectral numerical method with suitable diagonal functions and an infinite series. In particular, by using some operational matrices and Gauss–Hermite quadrature rule, we derive a linear system of algebraic equations solved by a standard collocation method. Moreover, we study the existence and uniqueness of the solution of PIDE and prove the convergence of the numerical scheme. The applicability and efficiency of the collocation method are shown on some nontrivial numerical examples.

Riemann Liouville fractional-like integral operators, convex-like real-valued mappings and their applications over fuzzy domain

It is well-known fact that fuzzy number theory is based on the characteristic function but in the fuzzy region, the characteristic function is a membership function that depends upon the interval 01. It means that real numbers and intervals are exceptional cases of fuzzy numbers. By using this approach, another type of Riemann Liouville fractional integral operator has been introduced over a fuzzy domain which is known as Riemann Liouville fractional-like integral operator. These integrals look like Riemann Liouville fractional integral operators but classical integrals are special cases of newly defined integral operators. Some novel versions of these integrals are also obtained and explained by taking the domain's triangular and trapezoidal fuzzy numbers. We also consider the problem of computing scalar-valued. To discuss the application of new integrals, a new class of convex real-valued functions has been introduced, known as convex-like real-valued mappings. With the help of these new concepts, some applications are taken into account. By using convex-like real-valued mappings and newly proposed fractional-like integral operators, the well-known Hermite-Hadamard H.H type and related inequalities are taken into account in this work. Then by defining new differentiable real-valued functions over fuzzy regions, an identity has been obtained and with the support of this identity, certain inequalities are also acquired. Also, we clearly show the important connections of the derived outcomes with those classical integrals. Finally, some nontrivial numerical examples are also provided to verify the correctness of the presented inequalities that occur with the variation of the parameters v and β.

Optimality conditions and duality for nonsmooth multiobjective semi-infinite programming problems with vanishing constraints on Hadamard manifolds

This article is concerned with nonsmooth multiobjective semi-infinite programming problems with vanishing constraints in the setting of Hadamard manifolds (abbreviated as, (NMSIPVC)). We present the Abadie constraint qualification (abbreviated as, (ACQ)) and a modified version of (ACQ), namely, (NMSIPVC)-tailored Abadie constraint qualification (abbreviated as, (NMSIPVC-ACQ)), for (NMSIPVC). The Karush-Kuhn-Tucker (abbreviated as, KKT) type necessary criteria of optimality for (NMSIPVC) are derived by employing (ACQ) and (NMSIPVC-ACQ). Moreover, sufficient criteria of optimality for (NMSIPVC) are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Further, we formulate Wolfe type and Mond-Weir type dual models related to (NMSIPVC) and derive several duality results that relate (NMSIPVC) and the corresponding dual models. Non-trivial numerical examples are incorporated to demonstrate the validity of the results established in this paper. To the best of our knowledge, this is for the first time that constraint qualifications, optimality conditions, and duality results have been investigated for (NMSIPVC) in the setting of Hadamard manifolds.

Second-Order Symmetric Duality for Multiple Objectives Nonlinear Programming Under Generalizations of Cone-Preinvexity Functions

In this article, we present generalizations of the cone-preinvexity functions and study a pair of second-order symmetric solutions for multiple objective nonlinear programming problems under these generalizations of the cone-preinvexity functions. In addition, we establish and prove the theorems of weak duality, strong duality, strict converse duality, and self-duality by assuming the skew-symmetric functions under these generalizations of the cone-preinvexity functions. Finally, we provide four nontrivial numerical examples to demonstrate that the results of the weak and strong duality theorems are true.

Computation of solution of integral equations via fixed point results

Abstract The motive of this article is to study a modified iteration scheme for monotone nonexpansive mappings in the class of uniformly convex Banach space and establish some convergence results. We obtain weak and strong convergence results. In addition, we present a nontrivial numerical example to show the convergence of our iteration scheme. To demonstrate the utility of our scheme, we discuss the solution of nonlinear integral equations as an application, which is again supported by a nontrivial example.

Some General Fractional Integral Inequalities Involving LR–Bi-Convex Fuzzy Interval-Valued Functions

The main objective of this paper is to introduce a new class of convexity called left-right–bi-convex fuzzy interval-valued functions. We study this class from the perspective of fractional Hermite–Hadamard inequalities, involving a new fractional integral called the left-right–AB fractional integral. We discuss several special cases that demonstrate that our results are quite unifying. We provide non-trivial numerical examples regarding special means for positive real numbers in order to check the validity of our outcomes.

Estimating Fixed Points via New Iterative Scheme with an Application

In this paper, we introduce a new scheme and prove convergence results for nonexpansive mappings as well as for weak contractions in the frame of Banach spaces. Moreover, we prove analytically and numerically that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Further, we prove that the new scheme is almost stable with respect to weak contraction. For supporting the main results, we give a couple of nontrivial numerical examples, and the visualization is shown by using the Matlab program. Finally, the solution of a nonlinear fractional differential equation is approximated by operating the main result of the paper.

Convergence analysis of a novel iteration process with application to a fractional differential equation

The objective of this article is to study a three-step iteration process in the framework of Banach spaces and to obtain convergence results for Suzuki generalized nonexpansive mappings. We also provide numerical examples that support our main results and illustrate the convergence behavior of the proposed process. Further, we present a data-dependence result that is also supported by a nontrivial numerical example. Finally, we discuss the solution of a nonlinear fractional differential equation by utilizing our results.

Second-order symmetric duality in vector optimization involving (K,η)-pseudobonvexity

This paper presents new classes generalizations of second-order (K,η)-pseudobonvexity functions, strongly second-order (K,η)-pseudobonvexity functions, and establishes a pair Mond-Weir type duality multiobjective nonlinear programs over these new classes generalizations. The weak duality, strong duality, converse duality, and self-duality theorems are presents under these generalizations functions. Finally, we introduced two nontrivial numerical examples to illustrating results of the weak duality theorem and strong duality theorem.

Second-order multi-objective non–differentiable Schaible type model and its duality relation under (K xQ)–C–type–I functions

ABSTRACT In this article, we constructed second-order --type-I functions and also illustrated a non-trivial concrete numerical example for such existing types of functions. Furthermore, we have presented a second-order non-differentiable multi-objective Schaible type dual model with cone objectives as well as cone constraints. Finally, we established duality theorems under --type-I assumptions.

Convergence and stability of an iteration process and solution of a fractional differential equation

In this paper, we prove that a three-step iteration process is stable for contractive-like mappings. It is also proved analytically and numerically that the considered process converges faster than some remarkable iterative processes for contractive-like mappings. Furthermore, some convergence results are proved for the mappings satisfying Suzuki’s condition (C) in uniformly convex Banach spaces. A couple of nontrivial numerical examples are presented to support the main results and the visualization is showed by Matlab. Finally, by utilizing our main result the solution of a nonlinear fractional differential equation is approximated.

Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning

In this paper, we revisit two recently published papers on the iterative approximation of fixed points by Kumam et al. (2019) [17] and Maniu (2020) [19] and reproduce convergence, stability, and data dependency results presented in these papers by removing some strong restrictions imposed on parametric control sequences. We confirm the validity and applicability of our results through various non-trivial numerical examples. We suggest a new method based on the iteration algorithm given by Thakur et al. (2014) [28] to solve the two-point second-order boundary value problems. Furthermore, based on the above mentioned iteration algorithm and S-iteration algorithm, we propose two new gradient type projection algorithms and applied them to supervised learning. In both applications, we present some numerical examples to demonstrate the superiority of the newly introduced methods in terms of convergence, accuracy, and computational time against some earlier methods.

Estimating fixed points of non-expansive mappings with an application

<abstract> In this paper, we study a three step iterative scheme to estimate fixed points of non-expansive mappings in the framework of Banach spaces. Further, some convergence results are proved for such mappings. A nontrivial numerical example is presented to verify our assertions and main results. Finally, we approximate the solution of a boundary value problem of second order differential equation. </abstract>

A Special Issue:Recent Developments in Nonlinear Partial Differential Equations

The literature reveals that numerous real-life phenomena in the subjects of physics and engineering which are governed by highly nonlinear Partial differential equations (PDEs) with unknown analytical solutions. More precisely, the (PDEs) arise in a wide variety of physical problems such as; by way of example not exhaustive enumeration, fluid dynamics, engineering mathematics, electrostatics, plasma physics, solid mechanics, chemistry, quantum field theory, bio-mathematics, etc. Therefore, such (PDEs) have received a huge attention from mathematicians, physicists, and engineers for the sake of approximating their analytical solutions. We aimed in this special issue to publish articles focusing on recent advanced numerical studies on Differential Equations related to physics and engineering. The well-developed analysis of existing numerical algorithms in terms of efficiency, applicability, convergence, stability and accuracy is of importance. A discussion of nontrivial numerical examples is encouraged.

Optimization and discrete approximation of sweeping processes with controlled moving sets and perturbations

This paper addresses a new class of optimal control problems for perturbed sweeping processes with measurable controls in additive perturbations of the dynamics and smooth controls in polyhedral moving sets. We develop a constructive discrete approximation procedure that allows us to strongly approximate any feasible trajectory of the controlled sweeping process by feasible discrete trajectories and also establish a W1,2-strong convergence of optimal trajectories for discretized control problems to a given local minimizer of the original continuous-time sweeping control problem of the Bolza type. Employing advanced tools of first-order and second-order variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete optimal solutions under fairly general assumptions formulated entirely in terms of the given data. The obtained results give us efficient suboptimality (“almost optimality”) conditions for the original sweeping control problem that are illustrated by a nontrivial numerical example.

Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces

In this paper, we propose and study a multi-step iterative algorithm that comprises of a finite family of asymptotically \begin{document}$ k_i $\end{document} -strictly pseudocontractive mappings with respect to \begin{document}$ p, $\end{document} and a \begin{document}$ p $\end{document} -resolvent operator associated with a proper convex and lower semicontinuous function in a \begin{document}$ p $\end{document} -uniformly convex metric space. Also, we establish the \begin{document}$ \Delta $\end{document} -convergence of the proposed algorithm to a common fixed point of finite family of asymptotically \begin{document}$ k_i $\end{document} -strictly pseudocontractive mappings which is also a minimizer of a proper convex and lower semicontinuous function. Furthermore, nontrivial numerical examples of our algorithm are given to show its applicability. Our results complement a host of recent results in literature.

Higher-order symmetric duality in nondifferentiable multiobjective fractional programming problem over cone contraints

In this paper, we introduce the definition of higher-order K-(C, α, ρ, d)-convexity/pseudoconvexity over cone and discuss a nontrivial numerical examples for existing such type of functions. The purpose of the paper is to study higher order fractional symmetric duality over arbitrary cones for nondifferentiable Mond-Weir type programs under higher- order K-(C, α, ρ, d)-convexity/pseudoconvexity assumptions. Next, we prove appropriate duality relations under aforesaid assumptions.

New class of G-Wolfe-type symmetric duality model and duality relations under G_{f}-bonvexity over arbitrary cones

This paper is devoted to theoretical aspects in nonlinear optimization, in particular, duality relations for some mathematical programming problems. In this paper, we introduce a new generalized class of second-order multiobjective symmetric G-Wolfe-type model over arbitrary cones and establish duality results under G_{f}-bonvexity/G_{f}-pseudobonvexity assumptions. We construct nontrivial numerical examples which are G_{f}-bonvex/G_{f}-pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex.

Multi-objective Optimization Problems with SOS-convex Polynomials over an LMI Constraint

In this paper, we aim to find efficient solutions of a multi-objective optimization problem over a linear matrix inequality (LMI in short), in which the objective functions are SOS-convex polynomials. We do this by using two scalarization approaches, that is, the $\epsilon$-constraint method and the hybrid method. More precisely, we first transform the considered multi-objective optimization problem into their scalar forms by the $\epsilon$-constraint method and the hybrid method, respectively. Then, strong duality results, between each formulated scalar problem and its associated semidefinite programming dual problem, are given, respectively. Moreover, for each proposed scalar problem, we show that its optimal solution can be found by solving an associated single semidefinite programming problem, under a suitable regularity condition. As a consequence, we prove that finding efficient solutions to the considered problem can be done by employing any of the two scalarization approaches. Besides, we illustrate our methods through some nontrivial numerical examples.

Nondifferentiable G-Mond–Weir Type Multiobjective Symmetric Fractional Problem and Their Duality Theorems under Generalized Assumptions

In this article, a pair of nondifferentiable second-order symmetric fractional primal-dual model (G-Mond–Weir type model) in vector optimization problem is formulated over arbitrary cones. In addition, we construct a nontrivial numerical example, which helps to understand the existence of such type of functions. Finally, we prove weak, strong and converse duality theorems under aforesaid assumptions.