Signaling pathways controlling biotic and abiotic stress responses may interact synergistically or antagonistically. To identify the similarities and differences among responses to diverse stresses, we analyzed previously published microarray data on the transcriptomic responses of Arabidopsis to infection with Botrytis cinerea (a biotic stress), and to cold, drought, and oxidative stresses (abiotic stresses). Our analyses showed that at early stages after B. cinerea inoculation, 1498 genes were up-regulated (B. cinerea up-regulated genes; BUGs) and 1138 genes were down-regulated (B. cinerea down-regulated genes; BDGs). We showed a unique program of gene expression was activated in response each biotic and abiotic stress, but that some genes were similarly induced or repressed by all of the tested stresses. Of the identified BUGs, 25%, 6% and 12% were also induced by cold, drought and oxidative stress, respectively; whereas 33%, 7% and 5.5% of the BDGs were also down-regulated by the same abiotic stresses. Coexpression and protein-protein interaction network analyses revealed a dynamic range in the expression levels of genes encoding regulatory proteins. Analysis of gene expression in response to electrophilic oxylipins suggested that these compounds are involved in mediating responses to B. cinerea infection and abiotic stress through TGA transcription factors. Our results suggest an overlap among genes involved in the responses to biotic and abiotic stresses in Arabidopsis. Changes in the transcript levels of genes encoding components of the cyclopentenone signaling pathway in response to biotic and abiotic stresses suggest that the oxylipin signal transduction pathway plays a role in plant defense. Identifying genes that are commonly expressed in response to environmental stresses, and further analyzing the functions of their encoded products, will increase our understanding of the plant stress response. This information could identify targets for genetic modification to improve plant resistance to multiple stresses.

In this paper, owing to the concept of F-contraction, we define two new classes of functions M (S, T) and N(S, T), and we prove some new fixed point results for single-valued and multivalued mappings in complete metric spaces. Our results extend, generalize and unify several known results in the literature. We include an example to show that the generalization is proper.MSC: Primary 30C45; 30C10; secondary 47B38

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In our earlier paper, a generalized Dobrushin ergodicity coefficient of Markov operators (acting on abstract state spaces) with respect to a projection P , has been introduced and studied. It turned out that the introduced coefficient was more effective than the usual ergodicity coefficient. In the present work, by means of a left consistent Markov projections and the generalized Dobrushin's ergodicity coefficient, we investigate uniform and weak P -ergodicities of non-homogeneous discrete Markov chains (NDMC) on abstract state spaces. It is easy to show that uniform P -ergodicity implies a weak one, but in general the reverse is not true. Therefore, some conditions are provided together with weak P -ergodicity of NDMC which imply its uniform P -ergodicity. Furthermore, necessary and sufficient conditions are found by means of the Doeblin's condition for the weak P -ergodicity of NDMC. The weak P -ergodicity is also investigated in terms of perturbations. Several perturbative results are obtained which allow us to produce nontrivial examples of uniform and weak P -ergodic NDMC. Moreover, some category results are also obtained. We stress that all obtained results have potential applications in the classical and non-commutative probabilities.MSC : 47A35; 60J10, 28D05

In this paper, we introduce the concept of modified F -contractions via α-admissible pair of mappings. We also provide several common fixed point results in the setting of metric spaces. Moreover, we present some illustrated examples and we give two applications on a dynamic programming and an integral equation.

Lakzian and Samet 2010 studied some fixed-point results in generalized metric spaces in the sense of Branciari. In this paper, we study the existence of fixed-point results of mappings satisfying generalized weak contractive conditions in the framework of a generalized metric space in sense of Branciari. Our results modify and generalize the results of Laksian and Samet, as well as, our results generalize several well-known comparable results in the literature. Definition 1.1 see 1 . Let X be a nonempty set and d : X × X → 0, ∞ such that for all x, y ∈ X and for all distinct points u, v ∈ X each of them different from x and y, one has 2 Abstract and Applied Analysis p1 :Then, X, d is called a generalized metric space or shortly g.m.s . Any metric space is a generalized metric space, but the converse is not true 1 .Definition 1.2 see 1 . Let X, d be a g.m.s, {x n } a sequence in X, and x ∈ X. We say that {x n } is g.m.s convergent to x if and only if d x n , x → 0 as n → ∞. We denote this by x n → x. Definition 1.3 see 1 . Let X, d be a g.m.s and {x n } a sequence in X. We say that {x n } is a g.m.s Cauchy sequence if and only if for each ε > 0 there exists a natural number N such that d x n , x m < ε for all n > m > N. Definition 1.4 see 1 . Let X, d be a g.m.s. Then, X, d is called a complete g.m.s if every g.m.s Cauchy sequence is g.m.s convergent in X. Very recently, Lakzian and Samet 9 proved the following nice result. Theorem 1.5. Let X, d be a Hausdorff and complete generalized metric space. Suppose that T : X → X is such that for all x, y ∈ X ψ d Tx, Ty ≤ ψ d x, y − φ d x, y , 1.1where ψ : 0, ∞ → 0, ∞ is continuous and nondecreasing with ψ t 0 if and only if t 0, and φ : 0, ∞ → 0, ∞ is continuous and φ t 0 if and only if t 0. Then, there exists a unique point u ∈ X such that u Tu.

In this paper, the concepts of a set-valued contraction of Mizoguchi-Takahashi type in the context of topological vector space (tvs)-cone metric spaces are introduced and a fixed point theorem in the context of tvs-cone metric spaces with respect to a solid cone is proved. We obtained results which extend and generalize the main results of S. H. Cho with J. S. Bae, Mizoguchi with Takahashi and S. B. Nadler Jr. Two examples are given to illustrate the usability of our results. 2010 MSC: 47H10, 54H25.

In this article, we introduce the notion of almost generalized (ψ, j)-contractive mappings in ordered metric spaces and we establish some fixed and common fixed point results in ordered complete metric spaces. Our results generalize several wellknown comparable results in the literature. Finally, an example and an application are given in order to support the useability of our results.