In the article, the existence and uniqueness of positive solutions for a class of fractional differential equation with nonlinear boundary conditions are discussed. By applying some fixed point theorems on cone, we gain a unique positive solution and construct two iterative sequences to approximate the solution. Moreover, as applications of our main results, some examples are also presented.

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This paper develops some new existence and uniqueness theorems of a fixed point for a class of sum operator equations with parameter λ 1 A(x, x) + λ 2 B(x, x) + λ 3 Cx + λ 4 Dx = x, where A, B are two mixed monotone operators, C is an increasing operator, D is a decreasing operator. In the case of positive parameters, the results obtained in this paper extend many existing conclusions in the field of study. Furthermore, by using the properties of Green's function and the above fixed point theorems of sum operator, the unique positive solution a class of fractional differential equations with integro-differential boundary value conditions is given. Application of the results to the study of fractional differential equations is also given in the article.

In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.