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Fredholm Integral Equation in Electrochemistry
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Mirela I. Iorga, Mihai V. Putz
According to Saha Ray & Sahu (2013) integral equations found applications in several domains, such as: continuum mechanics, fluid mechanics, fracture mechanics and quantum mechanics, potential theory, electricity, magnetism, radiation, acoustics, steady state heat conduction, radiative heat transfer, kinetic theory of gases, mass transfer in astrophysics and reactor theory, genetics, hereditary phenomena in physics and biology, medicine, optimization, optimal control systems, mathematical economics, communication theory, etc.
Introduction
Published in Joseph Y.-T. Leung, Handbook of SCHEDULING, 2004
[7] C. H. Papadimitriou. Game theory and mathematical economics: A theoretical computer scientist’s introduction (Tutorial). In Proceedings of the 42th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 4–8, Las Vegas, NV, October 14–17, 2001. IEEE Computer Society Press, Los Alamitos, CA.
Best Proximity Point Theorems for Cyclic Contractions Mappings
Published in Dhananjay Gopal, Poom Kumam, Mujahid Abbas, Background and Recent Developments of Metric Fixed Point Theory, 2017
Chirasak Mongkolkeha, Poom Kumam
Many problems in mathematics such as equilibrium and variational inequalities, mathematical economics, game theory and optimization can be formulated as equations of the form Tx=x, $ T x = x, $ where T is a self mapping in some suitable framework, because of its ability to solve ordinary differential equations, integral equations, matrix equations and others. However, given nonempty subsets A and B of X. A mapping T:A→B $ T : A \rightarrow B $ (or T:A∪B→A∪B $ T : A \cup B \rightarrow A \cup B $ ) using the equation Tx=x $ Tx =x $ does not necessarily have a solution. It is worthwhile to find an approximate solution x under mapping T so that the error d(x, Tx) is minimal. This is the basis of the optimal approximation theory that includes a generalized fixed point. Whenever A coincides with B, the optimization problem known as a best proximity point of the mapping T reduces to a fixed point problem. Fan [7] introduced a classical best approximation theorem: if A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space B and T:A→B $ T : A \rightarrow B $ is a continuous mapping, there exists an element x∈A $ x\in A $ such that d(Tx,x)=d(Tx,A):=inf{d(Tx,y):y∈A}. $ d(Tx,x) = d(Tx, A):=\inf \{d(Tx,y) : y \in A\}. $ Several authors, including Prolla [15], Reich [16], Sehgal [20,21], derived extensions of Fan’s theorem in many directions.
Inertial algorithm with self-adaptive step size for split common null point and common fixed point problems for multivalued mappings in Banach spaces
Published in Optimization, 2022
T. O. Alakoya, L. O. Jolaoso, A. Taiwo, O. T. Mewomo
Let be a nonlinear mapping, a point is called a fixed point of S if We denote by the set of all fixed points of S, i.e. If S is a multivalued mapping, i.e. then is called a fixed point of S if Several problems in sciences and engineering can be formulated as finding solution of Fixed Point Problem (FPP) for a nonlinear mapping. The fixed point theory for multivalued mappings can be utilized in various areas such as game theory, control theory, mathematical economics, etc.
Linear Programming from Fibonacci to Farkas
Published in Annals of Science, 2021
In the second half of the nineteenth century the subject we now know as Linear Algebra began to take shape. In 1873, when Paul Gordan proved a ‘Theorem of the Alternative’ about the existence of non-negative solutions of a system of linear homogeneous equations, he used determinants in his paper. Twenty-five years later Gyula Farkas, inspired by Fourier’s work, proved a significant extension of Gordan’s result, about non-homogeneous equations. In the twentieth century Linear Algebra found many new applications, notably as a tool for solving complex problems of organization and planning. Thus there arose the subject of Operations Research. One of its main tools was Linear Programming, where it turned out that the ‘Farkas Lemma’ plays a fundamental role. In fact, the underlying mathematical principle also played a central role in the development of Game Theory and other areas of Mathematical Economics.
Six set scalarizations based on the oriented distance: properties and application to set optimization
Published in Optimization, 2020
B. Jiménez, V. Novo, A. Vílchez
Optimization problems with set-valued objective functions provide an important generalization and allow us to unify scalar as well as vector optimization problems. In recent years, set-valued optimization problems (see [1–5]) have received an increasing attention due to extensive applications in many areas, since numerous problems that arise in different fields can be modelled as a set-valued optimization problem. This is what happens in, for example, engineering, mathematical economics, optimal control, differential inclusions, viability theory, game theory, fuzzy programming, duality in vector optimization, and so on (see [6,7]). For a detailed introduction to set optimization and its applications, we refer to [8], and to applications in finance, we can see [9].