Non-starshaped Domain Research Articles

On the sample-mean method for computing hyper-volumes

Abstract Estimating hyper-volumes of convex and non-convex sets are of interest in a number of areas. In this article we develop further a simple geometric Monte Carlo method, known also as the sample-mean method, which transforms the domain to an equivalent hyper-sphere with the same volume. We first examine the performance of the method to compute the volumes of star-convex unit balls and show that it gives accurate estimates of their volumes. We then examine the use of this method for computing the volumes of nonstar-shaped domains. In particular, we develop two algorithms, which couple the sample-mean method with algebraic and geometric techniques, to generate and compute the volumes of low-dimensional stability domains in parameter space.

Random coincidence points, invariant approximation theorems, nonstarshaped domain and q-normed spaces

We present coincidence points results satisfying a generalized contractive condition in the setting of a nonstarshaped domain of a q-normed space which is not necessarily a locally convex space. Invariant approximation results are also obtained as application.

Common random fixed point and random best approximation in non-starshaped domain of q-normed spaces

Common random fixed point and random best approximation in non-starshaped domain of q-normed spaces

COMMON FIXED POINT AND BEST APPROXIMATION FOR BANACH OPERATOR PAIRS IN NON-STARSHAPED DOMAIN

Common fixed point results for Banach operator pair with generalized nonexpansive mappings in non-starshaped domain of metric space have been obtained in the present work. As application, more general best approximation results in normed space have also been determined. These results extend and generalize various existing known results with the aid of Banach operator pair and without starshaped condition of domain. 2000 AMS Classification. 41A50, 47H10, 54H25.

Common fixed point and best approximation for Banach operator pairs in non-starshaped domain

AbstractCommon fixed point results for Banach operator pair with generalized nonexpansive mappings in non-starshaped domain of metric space have been obtained in the present work. As application, more general best approximation results in normed space have also been determined. These results extend and generalize various existing known results with the aid of Banach operator pair and without starshaped condition of domain.

Invariant approximations, generalized I-non- expansive mappings and non-convex domain

Common fixed point results for generalized I-nonexpansive mappings, and nonlin-ear map on nonstarshaped domain have been obtained in the present work. Various invariant approximation results have also been determined by its application. These results extend and generalize various existing known results in the literature. A property called property; also been introduced in it.

On pointwise R-subweakly commuting maps and best approximations

The main purpose of this paper is to prove some common fixed point theorems for pointwise R-subweakly commuting maps on non-starshaped domains in p-normed spaces and locally convex topological vector spaces. As applications, invariant approximation results are established. This work provides extension as well as substantial improvement of several results in the existing literature.

Coincidence points and invariant approximation results on q-normed spaces

We present coincidence points results for multivalued /- nonexpansive mappings in the setting of nonstarshaped domain of q-normed space which is not necessarily a locally convex space. As application, an invariant approximation result is also obtained. Our results improve and extend the results of Bano, Khan and Latif [1], Hussain [4], Latif and Tweddle [7], Rhoades [11], Sahab, Khan and Sessa [13], Shahzad [14] and Singh [15] in the setting of nonstarshaped domain. .

GeneralizedI-nonexpansive maps and invariant approximation results inp-normed spaces

We extend the concept of R-subcommuting maps due to Shahzad[17,18] to the case of non-starshaped domain and obtain a common fixed point result for this class of maps on non-starshaped domain in the setup of p-normed spaces. As applications, we establish noncommutative versions of various best approximation results for generalized I-nonexpansive maps on non-starshaped domain. Our results unify and extend that of Al-Thagafi, Dotson, Habiniak, Jungck and Sessa, Latif, Sahab, Khan and Sessa and Shahzad.

Non-existence results about [formula omitted] on non-starshaped domains

We show in this paper that there exist domains Ω, which are not conformally equivalent to starshaped domains, such that the Dirichlet problem: − Δ u = u n + 2 n − 2 , u > 0 in Ω; u = 0 on ∂ Ω has no solution. Some nonexistence results about the Dirichlet problem: − Δ u = λ u + u 5 , u > 0 in Ω; u = 0 on ∂ Ω for certain λ > 0 over three dimensional non-starshaped domains are also obtained.

Common fixed point and invariant approximation results in certain metrizable topological vector spaces

AbstractWe obtain common fixed point results for generalized "Equation missing"-nonexpansive "Equation missing"-subweakly commuting maps on nonstarshaped domain. As applications, we establish noncommutative versions of various best approximation results for this class of maps in certain metrizable topological vector spaces.

Common Fixed Point and Invariant Approximation Results on Non-Starshaped Domains

Abstract We extend the concept of 𝑅-subweakly commuting maps due to Shahzad [J. Math. Anal. Appl. 257: 39–45, 2001] to the case of non-starshaped domains and obtain common fixed point results for this class of maps on non-starshaped domains in the setup of Fréchet spaces. As applications, we establish Brosowski–Meinardus type approximation theorems. Our results unify and extend the results of Al-Thagafi, Dotson, Habiniak, Jungck and Sessa, Sahab, Khan and Sessa and Shahzad.

Uniqueness for semilinear elliptic problems: supercritical growth and domain geometry

As key problems we consider semilinear elliptic equations $\triangle u + \lambda f(u)=0\, \quad u|_{\partial {\Omega}} =0$ whose nonlinearities have supercritical growth near $u=\infty$ with $f(0)\ge 0$. For examples of these problems on ball domains it is known that there is a $\lambda^*>0$ such that there exists a unique small nonnegative solution for $0 <\lambda <\lambda^*$. This is in contrast to the situation where $f$ has superlinear but subcritical growth , in which case always also a large unstable solution can be found. We show that if $\Omega\subset{{\Bbb R}}^n$ is starshaped, $f(s)>0$ as $s\to\infty$ and $\lim\sup_{s\to\infty} \frac{F(s)}{sf(s)} <\frac{n-2}{2n}$ that a $\lambda^*$ as above always exists. We do not have to assume anything on the behavior of $f$ between zero and infinity. More generally we can assign to any domain $\Omega$ a number $M(\Omega)$ which in the case that $M(\Omega) < \frac12$ allows the same conclusion if in the above growth condition $\frac{n-2}{2n}$ is replaced by $\frac12-M(\Omega)$. There exist nonstarshaped domains in ${{\Bbb R}}^n$, $n \ge 3$ and topologically nontrivial ones in ${{\Bbb R}}^n$, $n\ge 4$ for which $M(\Omega) < \frac 12$. This indicates that for such domains a generalized critical exponent $\le\frac{1+2M(\Omega)}{ 1-2M(\Omega)}$ exists. Moreover, for starshaped domains $M(\Omega)=\frac 1 n$, so in this case the expression gives the usual critical exponent $\frac {n+2}{n-2}$.

Some constancy results for nematic liquid crystals and harmonic maps

Uniqueness results for nematic liquid crystals and harmonic maps with constant boundary data are established. Nonsolvability on non-starshaped domains for a semilinear elliptic equation is also given.

A nonexistence result for a nonlinear equation involving critical Sobolev exponent

Given any constant C > 0, we show that there exists smooth bounded nonstarshaped domains U in ℝN (N ≧ 5), such that the problem [Display omitted] has no solution u, whose energy, ∫U|∇u|2 is less than C.