Uncountably Many Solutions for Nonlinear Helmholtz and Curl-Curl Equations

2019 ◽  
Vol 19 (3) ◽  
pp. 569-593 ◽  
Author(s):  
Rainer Mandel
Keyword(s):  
Fixed Point ◽  
Linear Operator ◽  
Essential Spectrum ◽  
Entire Space ◽  
Limiting Absorption Principle ◽  
Fixed Point Approach

Abstract We obtain uncountably many solutions of nonlinear Helmholtz and curl-curl equations on the entire space using a fixed point approach. The constructed solutions are mildly localized as they lie in the essential spectrum of the corresponding linear operator. As a new auxiliary tool a limiting absorption principle for the curl-curl operator is proved.

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

scholarly journals Results on the approximate controllability of fractional hemivariational inequalities of order $1< r<2$

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
M. Mohan Raja ◽  
V. Vijayakumar ◽  
Le Nhat Huynh ◽  
R. Udhayakumar ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Fixed Point ◽  
Control Systems ◽  
Approximate Controllability ◽  
Hemivariational Inequalities ◽  
Evolution Inclusions ◽  
Fractional Systems ◽  
Cosine Families ◽  
Fixed Point Approach ◽  
Semilinear Control Systems ◽  
Theoretical Results

AbstractIn this paper, we investigate the approximate controllability of fractional evolution inclusions with hemivariational inequalities of order $1< r<2$ 1 < r < 2 . The main results of this paper are verified by using the fractional theories, multivalued analysis, cosine families, and fixed-point approach. At first, we discuss the existence of the mild solution for the class of fractional systems. After that, we establish the approximate controllability of linear and semilinear control systems. Finally, an application is presented to illustrate our theoretical results.

scholarly journals Nonstandard Dirichlet problems with competing (p,q)-Laplacian, convection, and convolution

2021 ◽  
Vol 66 (1) ◽  
pp. 95-103
Author(s):  
Dumitru Motreanu ◽  
Viorica Venera Motreanu
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Integrable Function ◽  
Generalized Solution ◽  
Dirichlet Problems ◽  
Reaction Term ◽  
Fixed Point Approach

"The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integrable function. We prove the existence of a generalized solution through a combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak solution to the respective elliptic problem."

Sign in / Sign up

Export Citation Format

Share Document