We prove a best proximity point version of Krasnoselskii’s fixed point theorem. As a consequence we obtain the existence of best proximity points of relatively u-continuous maps for a pair of compact convex sets.

Using the concept of asymptotic center we obtain the existence of fixed points having preassigned location for a wider class of asymptotic nonexpansive mappings in a uniformly convex Banach space. This generalization leads us to get a recent result of Alfuraidan and Khamsi for continuous monotone asymptotic nonexpansive mappings as well as the classical fixed-point result of Geobel and Kirk for asymptotic nonexpansive mappings in a uniformly convex Banach space. Also we prove a fixed-point theorem for order preserving continuous maps on a quasiordered closed convex subset of a uniformly convex Banach sapce having monotone norm.

In this survey, we discuss some open problems in metric fixed point theory and some related results for nonexpansive mappings.

In 2013, Jiménez–Melado and Llorens–Fuster proved that the renorming of ℓ ^2, x = max{‖ x ‖_2, p ( x )}, where p is a seminorm on ℓ ^2 satisfying certain conditions, has the weak fixed point property. In this paper, we generalize this result for a Banach space having normal structure and Schauder basis. From this, we derive that every Banach space having normal structureAMS Subject Classification and Schauder basis has an equivalent renorming that lacks asymptotic normal structure but has the weak fixed point property.

In this article, we prove that if K is a nonempty weakly compact convex set in a Banach space such that K has the hereditary fixed-point property (FPP) and f is a commuting family of isometry mappings on K, then there exists a point in C(K) which is fixed by every member in f whenever C(K) is a compact set. Also, we give an example to show that C(K), the Chebyshev center of K, need not be invariant under isometry maps. This example answers the question as to whether the Chebyshev center is invariant under isometry maps. Furthermore, we give a simple example to illustrate that Lim's center, as introduced by Lim, is different from the Chebyshev center.

In a recent paper Shunmugaraj and Thota characterized uniform convexity in terms of property UC. In this paper we introduce a more general notion called property k-UC and characterize k-uniform rotundity in terms of property k-UC. We also obtain some results related to k-uniform rotundity and property k-UC. Some results obtained in this paper generalize some results recently obtained by Shunmugaraj and Thota as well as a result of Suzuki et al.

We introduce the notion of uniform rotundity of a normed space with respect to a finite dimensional subspace as a generalization of uniform rotundity in a direction. We discuss several characterizations of this property and obtain a series of new classes of normed spaces which in a natural way generalize normed spaces that are uniformly rotund in every direction. Indeed for each positive integer k we get normed spaces that are uniformly rotund with respect to every k-dimensional subspace (UREk) with k = 1 reducing to uniform rotundity in every direction. Also UREk implies UREk+1 but not conversely. We show that UREk spaces turn out to be exactly those in which the Chebyshev center of a nonempty bounded set is either empty or is of dimension at most k - 1 thus extending a well known result of Garkavi. These spaces have normal structure which is a sufficient condition for fixed property for nonexpansive maps on weakly compact convex sets. In addition, there is a common fixed point in the self-Chebyshev center of a weakly compact convex set for the collection of all isometric selfmaps on the set. Uniform rotundity with respect to a finite dimensional subspace is defined based on Sullivan's notion of k-uniform rotundity in the same fashion as uniform rotundity in a direction is based on Clarkson's uniform rotundity. But a characterization of the same in terms of Milman's modulus of k-uniform rotundity is also discussed.

In 2005 Eldred et al. introduced a notion called 'proximal normal structure' to study the existence of best proximity points for relatively nonexpansive maps. In this paper we will give a characterization for proximal normal structure. Using this characterization we will prove that if A or B is compact, then the convex pair (A, B) has proximal normal structure. We will also show that in general if (A, B) is a closed bounded convex proximal pair, then the compactness of A need not imply the compactness of B. But in k-strictly convex Banach spaces this is not the case. In addition we will use our characterization to prove the following: (i) If X is a Banach space and the set of all directions in which X is not uniformly convex is contained in a countable union of k-dimensional subspaces of X for some k is an element of N or the set of all directions in which X is not uniformly convex is contained in a subspace of X with countable Hamel basis, then X has proximal normal structure. (ii) Every nearly uniformly convex Banach space has proximal normal structure. (iii) If (X, parallel to-parallel to) is a Hilbert space and vertical bar center dot vertical bar a norm on X and 1 <= beta < root 2 such that (1/beta)vertical bar x vertical bar <= parallel to x parallel to <= vertical bar x vertical bar for all x is an element of X, then (X, vertical bar center dot vertical bar) has proximal normal structure.

Let (A, B) be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and T: A boolean OR B -> A boolean OR B be a continuous and asymptotically relatively nonexpansive map. We prove that there exists x is an element of A boolean OR B such that parallel to x - Tx parallel to = dist(A, B) whenever T(A) subset of B, T(B) subset of A. Also, we establish that if T(A) subset of A and T(B) subset of B, then there exist x is an element of A and y is an element of B such that Tx = x, Ty = y and parallel to x - y parallel to = dist(A, B). We prove the aforementioned results when the pair (A, B) has the rectangle property and property UC. In the case of A = B, we obtain, as a particular case of our results, the basic fixed point theorem for asymptotically nonexpansive maps by Goebel and Kirk.

In this paper, a notion called proximally complete pair of subsets of a metric space is introduced, which weakens earlier notions in the theory of best-proximity points. By means of this notion, existence and convergence results of best-proximity points are proven for cyclic contraction mappings, which extent other recent results. By observing geometrical properties of Hilbert spaces, the so-called Pythagorean property is introduced. This property is employed to provide sufficient conditions for a cyclic map to be a cyclic contraction.

A normed space is said to have the weak fixed point property (WFPP) if every nonexpansive self map on a weakly compact convex set has a fixed point. Kirk proved that if a normed space has normal structure, then it has WFPP. It is known that if a normed space is uniformly convex in every direction (UCED), then it has normal structure. Also known is that every normed space that is uniformly convex in all but countably many directions has normal structure. We show that a normed space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a countable union of n-dimensional subspaces of X for some positive integer n. We also show that in such a space, the Chebyshev center C(K) of a weakly compact convex set K is a common invariant set for the collection of all isometries that map K into K and also that there is a common fixed point in C(K) for this collection of maps. This was previously known to be true only in the case of a normed space that is UCED. Another observation made in this paper is that a Banach space X has normal structure if the set of all directions in which it is not uniformly convex is contained in a linear subspace with a countable Hamel basis.

CdTe and CZT detectors are considered better choices for high energy γ and X-ray spectroscopy in comparison to Si and HPGe detectors due to their good quantum efficiency and room temperature operation. The performance limitations in CdTe and CZT detectors are mainly associated with poor hole transport and trapping phenomena. Among many techniques that can be used to eliminate the effect of the poor charge transport properties of holes in CdTe and CZT material, the drift ring technique shows promising results. In this work, the performance of a 2.3 mm thick CZT drift ring detector is investigated. Spatially resolved measurements were carried out with an X-ray microbeam (25 and 75 keV) at the Diamond Light Source synchrotron to study the response uniformity and extent of the active area. Higher energy photon irradiation was also carried out at up to 662 keV using different radioisotopes to complement the microbeam data. Different biasing schemes were investigated in terms of biasing the cathode rear electrode (bulk field) and the ring electrodes (lateral fields). The results show that increasing the bulk field with fixed-ratio ring biases and lateral fields with fixed bulk fields increase the active area of the device significantly, which contrasts with previous studies in CdTe, where only an increasing lateral field resulted in an improvement of device performance. This difference is attributed to the larger thickness of the CZT device reported here.

A sufficient condition is given for a non-convex proximal pair to be a proximal parallel pair on Hilbert spaces. Let (A, B) be a nonempty weakly compact non-convex proximal parallel pair in a Hilbert space X over the real field and T : A boolean OR B -> X be a relatively nonexpansive map. We prove that there exists x is an element of A boolean OR B such that parallel to x - Tx parallel to = dist(A, B) whenever A boolean OR B is a cyclic T-regular set. We also establish that there exists (x, y) is an element of A x B such that Tx = x, Ty = y and parallel to x - y parallel to = dist(A, B), if A boolean OR B is a T-regular set, T(A) subset of A and T(B) subset of B. In the above cases, we prove that the Kransnoel'skiis iteration process yields a convergence result under suitable assumption.

Brodskii and Milman proved that there exists a point in C ( A ), the set of all Chebyshev centers of A , which is fixed by every surjective isometry from A into A whenever A is a nonempty weakly compact convex set having normal structure in a Banach space. Motivated by this result, Lim et al. proved that every isometry from A into A has a fixed point in C ( A ) whenever A is a nonempty weakly compact convex set having normal structure in a Banach space. In this paper, we prove that every relatively isometry map T : A ∪ B → A ∪ B , satisfying T ( A ) ⊆ B and T ( B ) ⊆ A, has a best proximity point in CA(B), the set of all Chebyshev centers of B relative to A , whenever the nonempty weakly compact convex proximal pair ( A , B ) has proximal normal structure and rectangle property. Also, we prove that, under suitable assumptions, an analogous result of Brodskii and Milman for relatively isometry mappings holds. In case of A = B , we obtain the results of Brodskii and Milman, and Lim et al. as a particular case of our results.

CdTe and CdZnTe material is an excellent candidate for the fabrication of high energy X-ray spectroscopic detectors due to their good quantum efficiency and room temperature operation. The main material limitation is associated with the poor charge transport properties of holes. The motivation of this work is to investigate the performance characteristics of a detector fabricated with a drift ring geometry that is insensitive to the transport of holes. The performance of a prototype Ohmic CdTe drift ring detector fabricated by Acrorad with 3 drift rings is reported; measurements include room temperature current voltage characteristics (IV) and spectroscopic performance. The data shows that the energy resolution of the detector is limited by leakage current which is a combination of bulk and surface leakage currents. The energy resolution was studied as a function of incident X-ray position with an X-ray microbeam at the Diamond Light Source. Different ring biasing schemes were investigated and the results show that by increasing the lateral field (i.e. the bias gradient across the rings) the active area, evaluated by the detected count rate, increased significantly.

Semiconductor InAsxSb(1-x) crystals exhibit superior infrared detection properties but their applications are limited by the lack of production of these stoichiometric single crystal substrates by any melt crystal growth techniques. Indium Arsenic Antimonide (InAsxSb(1-x)) crystals have been grown from melts of Indium(In), Arsenic(As) and Antimony(Sb) by the vertical directional solidification technique using resistive heating furnace and quartz "double ampoule'' after overcoming some problems like ampoule breaking and crystal cracking. A cooling rate of 2 degrees C/h is better for producing homogenous crystals. Cooling rates less than 2 degrees C/h result in breaking of the ampoule. The grown InAsxSb(1-x) crystals are characterized by X-ray diffraction (XRD), electron probe micro analysis (EPMA) and Fourier transform infra red (FT-IR) which contributes to the identification of arsenic (x). Transmission spectra have been taken for the different sections of the crystal and the band gap has been calculated for as grown crystals. Optical transmission analysis indicates the incorporation of arsenic inside the grown crystals of InAsxSb(1-x) . The energy gap decreases with the increase in the arsenic concentration, which implies the incorporation of Arsenic. (C) 2012 Elsevier B.V. All rights reserved.

We obtain a uniform boundedness type theorem in the frame of asymmetric normed spaces. The classical result for normed spaces follows as a particular case.

It is well known that semiconductor detectors operating at room temperature can be read out at high rate, with good noise performance and low sensitivity to ballistic deficit, by using trapezoidal (flat-topped) pulse shaping. Nevertheless, the energy resolution of these detectors is also affected by chargetrapping inside the detector crystal, which can not be compensated by the standard trapezoidal pulse shaping. A new digital algorithm based on trapezoidal pulse shaping, to compensate for the charge-trapping effect while minimizing the electronic noise, has been developed. The application of the pulse processing algorithm to a 5 × 5 × 1 mm3 planar Schottky CdTe detector leads to an energy resolution of 1.15% FWHM at 662 keV at room temperature, which is considerably superior to the results of the standard pulse filters.

Growth by the Multi-tube Physical Vapour Transport technique and characterisation of bulk (Cd,Zn)Te is described. The crystalline perfection and uniformity of zinc content have been mapped by infra-red transmission and microscopy, X-ray diffraction and photoluminescence. X-ray double crystal rocking curve full widths at half maximum as low as 43in have been obtained and a mean zinc mole fraction of 0.03 has been found to vary by less than ±0.003 over the diameter of a 50mm boule. The material exhibits a resistivity in the 2×109 Ωcm range and planar devices fabricated from this material have shown electron mobility lifetime products of 4.07×10−3cm2V−1.