scholarly journals A nonlinear complementarity problem in mathematical programming in Hilbert space

1979 ◽  
Vol 20 (2) ◽  
pp. 233-236 ◽  
Author(s):  
Sribatsa Nanda ◽  
Sudarsan Nanda
Keyword(s):  
Hilbert Space ◽  
Complementarity Problem ◽  
Existence And Uniqueness ◽  
Nonlinear Complementarity Problem ◽  
Closed Convex Cone ◽  
Existence And Uniqueness Theorem ◽  
Polar Cone ◽  
Nonlinear Complementarity ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper we prove the following existence and uniqueness theorem for the nonlinear complementarity problem by using the Banach contraction principle. If T: K → H is strongly monotone and lipschitzian with k2 < 2c < k2+1, then there is a unique y ∈ K, such that Ty ∈ K* and (Ty, y) = 0 where H is a Hilbert space, K is a closed convex cone in H, and K* the polar cone.

scholarly journals A complex nonlinear complementarity problem

1978 ◽  
Vol 19 (3) ◽  
pp. 437-444 ◽  
Author(s):  
Sribatsa Nanda ◽  
Sudarsan Nanda
Keyword(s):  
Continuous Function ◽  
Complementarity Problem ◽  
Convex Cone ◽  
Existence And Uniqueness ◽  
Nonlinear Complementarity Problem ◽  
Closed Convex Cone ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Polar Cone ◽  
Nonlinear Complementarity

In this paper we study the existence and uniqueness of solutions for the following complex nonlinear complementarity problem: find z ∈ S such that g(z) ∈ S* and re(g(z), z) = 0, where S is a closed convex cone in Cn, S* the polar cone, and g is a continuous function from Cn into itself. We show that the existence of a z ∈ S with g(z) ∈ int S* implies the existence of a solution to the nonlinear complementarity problem if g is monotone on S and the solution is unique if g is strictly monotone. We also show that the above problem has a unique solution if the mapping g is strongly monotone on S.

scholarly journals On the complex nonlinear complementary problem

1976 ◽  
Vol 14 (1) ◽  
pp. 129-136 ◽  
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Linear Complementarity ◽  
Feasible Point ◽  
Existence Of A Solution ◽  
Polar Cone ◽  
Nonlinear Complementarity ◽  
Image Position ◽  
Nonlinear Mappings

The complex nonlinear complementarity problem considered here is the following: find z such thatwhere S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. We study the extent to which the existence of a z ∈ S with g(z) ∈ S* (feasible point) implies the existence of a solution to the nonlinear complementarity problem, and extend, to nonlinear mappings, known results in the linear complementarity problem on positive semi-definite matrices.

scholarly journals Global Error Bound Estimation for the Generalized Nonlinear Complementarity Problem over a Closed Convex Cone

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hongchun Sun ◽  
Yiju Wang
Keyword(s):  
Error Bound ◽  
Complementarity Problem ◽  
Convex Cone ◽  
Nonlinear Complementarity Problem ◽  
Global Error ◽  
Closed Convex Cone ◽  
Global Error Bound ◽  
Nonlinear Complementarity ◽  
Bound Estimation

The global error bound estimation for the generalized nonlinear complementarity problem over a closed convex cone (GNCP) is considered. To obtain a global error bound for the GNCP, we first develop an equivalent reformulation of the problem. Based on this, a global error bound for the GNCP is established. The results obtained in this paper can be taken as an extension of previously known results.

scholarly journals A fixed point method to solve the nonlinear complementarity problem for a class of monotone operators

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4587-4590
Author(s):  
Dinu Teodorescu ◽  
Mohammad Khan
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Complementarity Problem ◽  
Real Hilbert Space ◽  
Nonlinear Complementarity Problem ◽  
Monotone Operators ◽  
Fixed Point Method ◽  
Banach Fixed Point Theorem ◽  
Nonlinear Complementarity

In this paper, using the classic Banach fixed point theorem, we study the nonlinear complementarity problem for a class of monotone operators in real Hilbert space.

scholarly journals Ulam Stability of n-th Order Delay Integro-Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3029
Author(s):  
Shuyi Wang ◽  
Fanwei Meng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Mathematical Induction ◽  
Ulam Stability ◽  
Existence And Uniqueness Theorem ◽  
Integro Differential Equation ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

A Global Existence and Uniqueness Theorem for Ordinary Differential Equations

1976 ◽  
Vol 19 (1) ◽  
pp. 105-107 ◽  
Author(s):  
W. Derrick ◽  
L. Janos
Keyword(s):  
Global Existence ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Contraction Principle ◽  
Existence And Uniqueness Theorem ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Global Existence And Uniqueness ◽  
The Banach Contraction Principle

As observed by A. Bielecki and others ([1], [3]) the Banach contraction principle, when applied to the theory of differential equations, provides proofs of existence and uniqueness of solutions only in a local sense. S. C. Chu and J. B. Diaz ([2]) have found that the contraction principle can be applied to operator or functional equations and even partial differential equations if the metric of the underlying function space is suitably changed.

scholarly journals Existence theory for the complex nonlinear complementarity problem

1976 ◽  
Vol 14 (3) ◽  
pp. 417-423 ◽  
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Existence Theorem ◽  
Unique Solution ◽  
Complementarity Problem ◽  
Nonlinear Complementarity Problem ◽  
Polyhedral Cone ◽  
Existence Theory ◽  
Polar Cone ◽  
Nonlinear Complementarity ◽  
Strongly Monotone ◽  
Image Position

The main result in this paper is an existence theorem for the following complex nonlinear complementarity problem: find z such thatwhere S is a polyhedral cone in Cn, S* the polar cone, and g is a mapping from Cn into itself. It is shown that the above problem has a unique solution if the mapping g is continuous and strongly monotone on the polyhedral cone S.

A Global Existence and Uniqueness Theorem for Ordinary Differential Equations of Generalized Order

1978 ◽  
Vol 21 (3) ◽  
pp. 267-271 ◽  
Author(s):  
Ahmed Z. Al-Abedeen ◽  
H. L. Arora
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Global Existence ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Existence And Uniqueness Theorem ◽  
Banach Contraction ◽  
Global Existence And Uniqueness ◽  
The Banach Contraction Principle ◽  
Generalized Order

AbstractWe extend the Picard's theorem to ordinary differential equation of generalized order α, 0 ≤ α ≤ l, and prove a global existence and uniqueness theorem by using the Banach contraction principle.

scholarly journals A nonlinear complementarity problem in Banach space

1980 ◽  
Vol 21 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Sribatsa Nanda ◽  
Sudarsan Nanda
Keyword(s):  
Banach Space ◽  
Complementarity Problem ◽  
Existence And Uniqueness ◽  
Real Banach Space ◽  
Nonlinear Complementarity Problem ◽  
Convex Cones ◽  
Uniqueness Theorems ◽  
Existence And Uniqueness Theorems ◽  
Nonlinear Complementarity

Existence and uniqueness theorems for the nonlinear complementarity problem over closed convex cones in a reflexive real Banach space are established.

scholarly journals Applications of the Bielecki renorming technique

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Mihály Bessenyei ◽  
Zsolt Páles
Keyword(s):  
Functional Analysis ◽  
Global Existence ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Existence And Uniqueness Theorem ◽  
Banach Contraction ◽  
Global Existence And Uniqueness ◽  
The Banach Contraction Principle ◽  
Elegant Proof

AbstractThe renorming technique allows one to apply the Banach Contraction Principle for maps which are not contractions with respect to the original metric. This method was invented by Bielecki and manifested in an extremely elegant proof of the Global Existence and Uniqueness Theorem for ODEs. The present paper provides further extensions and applications of Bielecki’s method to problems stemming from functional analysis and from the theory of functional equations.

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