In this paper, we investigate the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equationsdenotes left Riemann-Liouville fractional integrals of order 1− α. We prove that if 1 < p < 1 + 2/N , then every nontrivial nonnegative solution blow-up in finite time, and if p ≥ 1 + 2/N and u 0 L qc (R N ) , qc = N (p − 1)/2 is sufficiently small, then the problem has global solution.

Let T be a time scale such that 0, T ∈ T. Consider the following three-point boundary value problem on time scales:where β, γ 0, β + γ > 0, η ∈ (0, ρ(T )), 0 < α < T /η, and d = β(T − αη) + γ (1 − α) > 0. By using fixed point theorems in cones, some new and general results are obtained for the existence of single and multiple positive solutions of the above problem. In particular, our criteria generalize and improve some known results.  2004 Published by Elsevier Inc.

In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scaleswhere ϕ p (u) is p-Laplacian operator, i.e., ϕ p (u) = |u| p−2 u, p > 1. Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel'skii's fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett-Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied.

In this paper, we deal with a class of fractional abstract Cauchy problems of order α ∈ (1, 2) by introducing an operator Sα which is defined in terms of the Mittag-Leffler function and the curve integral. Some nice properties of the operator Sα are presented. Based on these properties, the existence and uniqueness of mild solution and classical solution to the inhomogeneous linear and semilinear fractional abstract Cauchy problems is established accordingly. The regularity of mild solution of the semilinear fractional Cauchy problem is also discussed. Contents 157 3. Properties of Operator S α (t) 159 4. Linear Problem 165 5. Nonlinear Problem 169 6. An Example 175 Acknowledgments 176 References 176

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