An existence result of asymptotically stable solutions for an integral equation of mixed type

In the present Note an existence result of asymptotically stable solutions for the integral equation $$x\left( t\right) =q\left( t\right) +\int_{0}^{t}K\left( t,s,x\left( s\right) \right) ds +\int_{0}^{\infty }G\left( t,s,x\left( s\right) \right) ds$$ is presented

Limits of solutions of a perturbed linear differential equation

Using interesting techniques, an existence result for the problem $\ddot{x}+2f\left( t\right) \dot{x}+x+g\left( t,x\right) =0,$ $\lim\limits_{t\rightarrow +\infty }x\left( t\right) =\lim\limits_{t\rightarrow +\infty }\dot{x}\left( t\right) =0,$ is given in [2]. This note treates the same problem via Schauder-Tychonoff and Banach theorems

Fixed points for some non-obviously contractive operators defined in a space of continuous functions

Let $X$ be an arbitrary (real or complex) Banach space, endowed with the norm $\left| \cdot \right| .$ Consider the space of the continuous functions $C\left( \left[ 0,T\right] ,X\right)$ $\left( T>0\right)$, endowed with the usual topology, and let $M$ be a closed subset of it. One proves that each operator $A:M\rightarrow M$ fulfilling for all $x,y\in M$ and for all $t\in \left[ 0,T\right]$ the condition \begin{eqnarray*} \left| \left( Ax\right) \left( t\right) -\left( Ay\right) \left( t\right) \right| &\leq &\beta \left| x\left( \nu \left( t\right) \right) -y\left( \nu \left( t\right) \right) \right| + \\ &&+\frac{k}{t^{\alpha }}\int_{0}^{t}\left| x\left( \sigma \left( s\right) \right) -y\left( \sigma \left( s\right) \right) \right| ds, \end{eqnarray*} (where $\alpha ,$ $\beta \in \lbrack 0,1)$, $k\geq 0$, and $\nu ,$ $\sigma :\left[ 0,T\right] \rightarrow \left[ 0,T\right]$ are continuous functions such that $\nu \left( t\right) \leq t,$ $\sigma \left( t\right)\leq t,$ $\forall t\in \left[ 0,T\right]$) has exactly one fixed point in $M$. Then the result is extended in $C\left( \mathbb{R}_{+},X\right) ,$ where $\mathbb{R}_{+}:=[0,\infty ).

Transport lattice models of heat transport in skin with spatially heterogeneous, temperature-dependent perfusion

BACKGROUND: Investigation of bioheat transfer problems requires the evaluation of temporal and spatial distributions of temperature. This class of problems has been traditionally addressed using the Pennes bioheat equation. Transport of heat by conduction, and by temperature-dependent, spatially heterogeneous blood perfusion is modeled here using a transport lattice approach. METHODS: We represent heat transport processes by using a lattice that represents the Pennes bioheat equation in perfused tissues, and diffusion in nonperfused regions. The three layer skin model has a nonperfused viable epidermis, and deeper regions of dermis and subcutaneous tissue with perfusion that is constant or temperature-dependent. Two cases are considered: (1) surface contact heating and (2) spatially distributed heating. The model is relevant to the prediction of the transient and steady state temperature rise for different methods of power deposition within the skin. Accumulated thermal damage is estimated by using an Arrhenius type rate equation at locations where viable tissue temperature exceeds 42°C. Prediction of spatial temperature distributions is also illustrated with a two-dimensional model of skin created from a histological image. RESULTS: The transport lattice approach was validated by comparison with an analytical solution for a slab with homogeneous thermal properties and spatially distributed uniform sink held at constant temperatures at the ends. For typical transcutaneous blood gas sensing conditions the estimated damage is small, even with prolonged skin contact to a 45°C surface. Spatial heterogeneity in skin thermal properties leads to a non-uniform temperature distribution during a 10 GHz electromagnetic field exposure. A realistic two-dimensional model of the skin shows that tissue heterogeneity does not lead to a significant local temperature increase when heated by a hot wire tip. CONCLUSIONS: The heat transport system model of the skin was solved by exploiting the mathematical analogy between local thermal models and local electrical (charge transport) models, thereby allowing robust, circuit simulation software to obtain solutions to Kirchhoff's laws for the system model. Transport lattices allow systematic introduction of realistic geometry and spatially heterogeneous heat transport mechanisms. Local representations for both simple, passive functions and more complex local models can be easily and intuitively included into the system model of a tissue

Genetic Diversity in Marginal Populations of <i>Nitraria schoberi</i> L. from Romania

Nitraria schoberi L. (Nitrariaceae) is a halophytic plant with a continuous range in Central Asia and with only two populations in the westernmost distribution limit of species, in Romania. Currently, there is no documented explanation for the species’ presence in Europe, outside the main distribution area. Considering that marginal populations genetics are important in establishing range limits and species adaptative potential, genetic diversity was assessed using Inter-simple sequence repeat markers (ISSR). Both the Shannon’s Information Index (I) and Expected Heterozygosity (He) suggested a relatively low level of genetic diversity within the two populations. However, the Unweighted Pair Group Method with Arithmetic Mean (UPGMA) dendrogram and Principal Coordinates Analysis clearly distinguished the two populations. Our presumptions, based on current results, are that the marginal westernmost population of N. schoberi was established due to the unique conditions from the “islands of desert” developed in a temperate continental climate. The European establishment of this species was likely accidental and probably due to ornithochory. Genetic relatedness between populations could be a consequence of their common origin, presumably from proximal Asian N. schoberi populations, while the separation can be explained by the lack of genetic material exchange between the two populations

Genetic Diversity in Marginal Populations of Nitraria schoberi L. from Romania

Nitraria schoberi L. (Nitrariaceae) is a halophytic plant with a continuous range in Central Asia and with only two populations in the westernmost distribution limit of species, in Romania. Currently, there is no documented explanation for the species&rsquo; presence in Europe, outside the main distribution area. Considering that marginal populations genetics are important in establishing range limits and species adaptative potential, genetic diversity was assessed using Inter-simple sequence repeat markers (ISSR). Both the Shannon&rsquo;s Information Index (I) and Expected Heterozygosity (He) suggested a relatively low level of genetic diversity within the two populations. However, the Unweighted Pair Group Method with Arithmetic Mean (UPGMA) dendrogram and Principal Coordinates Analysis clearly distinguished the two populations. Our presumptions, based on current results, are that the marginal westernmost population of N. schoberi was established due to the unique conditions from the &ldquo;islands of desert&rdquo; developed in a temperate continental climate. The European establishment of this species was likely accidental and probably due to ornithochory. Genetic relatedness between populations could be a consequence of their common origin, presumably from proximal Asian N. schoberi populations, while the separation can be explained by the lack of genetic material exchange between the two populations