If $Y$ is a subset of the space $\mathbb{R}^{n}\times {\mathbb{R}^{n}}$, we call a pair of continuous functions $U$, $V$ $Y$-compatible, if they map the space $\mathbb{R}^{n}$ into itself and satisfy $Ux\cdot Vy\ge 0$, for all $(x,y)\in Y$ with $x\cdot y\ge {0}$. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer’s fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points $P_{\delta }$ is obtained. Then passing to the limits as $\delta $ tends to zero the so-obtained accumulation points are solutions of the problem.
The ability of plants to increase their net CO2 assimilation rate in response to increased irradiance is due to morphological and physiological changes, which might be related to their shade tolerance and leaf ontogeny, but few studies have considered morphology and physiology. Two sympatric oak species (the shade-tolerant Q. petraea and the comparatively shade-intolerant Q. pyrenaica) were grown in hydroponic solution in low-light (LL) and high-light (HL) conditions. 5 months after leaf expansion under these conditions, half of the LL plants were transferred to high light (TLH). Transfer of Q. pyrenaica, from low- to high light led to photoinhibition and after 21 days in higher light there was little acclimation of the maximum rate of carboxylation (VCmax) or the maximum rate of electron transport (Jmax). Q. pyrenaica TLH plants showed lower stomatal conductance at all times compared to plants growing in LL. Stomatal closure was the main limitation to photosynthesis after transfer in Q. pyrenaica. The increase in evaporative demand upon TLH did not affect hydraulic conductivity of Q. pyrenaica. In contrast, the more shade-tolerant Q. petraea showed a greater degree of acclimation of gas exchange in TLH than Q. pyrenaica and two weeks after transfer gas-exchange rates were as high as in LL plants. In Q. petraea, the most important changes occurred at the level of leaf biochemistry with significant increase in VCmax that decreased the Jmax/VCmax ratio below values recorded in HL plants. However, this potential increase in photosynthesis was at least partially hamstrung by a decrease in internal conductance, which highlights the importance of internal conductance in acclimation to higher light in mature leaves. Neither oak species reached the photosynthetic rates of HL plants; however a trend towards leaf acclimation was observed in Q. petraea while the transfer was harmful to the leaves of Q. pyrenaica developed in the shade. and F. J. Cano ... [et al.].