For suitable Banach spaces X and Y with Schauder decompositions and a suitable closed subspace M of some compact operator space from X to Y, it is shown that the strong Banach-Saks-ness of all evaluation operators on M is a sufficient condition for the weak Banach-Saks property of M, where for each xÎX and y*ÎY*, the evaluation operators on M are defined by fx (T)= Tx and yy* (T)= T*y*.

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In this paper, we continue the study of A^{**}-biprojectivity of Banach algebras, and the relation between this new notion and phi-amenability of Banach algebras is investigated. A^{**}-biprojectivity of Segal algebras and lower triangular matrix algebras is studied. Also, we introduce the notion of phi-A^{**}-biprojectivity of Banach algebras. Some examples indicate that this notion is weaker than A^{**}-biprojectivity. We obtain the relation between this notion and phi-amenability and phi-inner amenability. Finally, we investigate this new notion on certain Banach algebras such as group algebras, measure algebras, and lower triangular Banach algebras.

We present an isometric version of the complementably universal Banach space P with a Schauder decomposition. The space P is isomorphic to Pe lczynski's space with a universal basis as well as to Kadec’ complementably universal space with the bounded approximation property.

In this paper, we consider a class of nonlinear fractional integro-diff erential equations with fractional derivative of Caputo sense. We shall rely on the Krasnoselskii fi xed point theorem to obtain the existence result in Banach spaces. Moreover, one of Krasnoselskii-Krein conditions is applied to establish the result. Finally, an illustrative example is also presented.