Abstract
In this article, we study a class of nonlinear fractional differential equations with mixed-type boundary conditions. The fractional derivatives are involved in the nonlinear term and the boundary conditions. By using the properties of the Green function, the fixed point index theory and the Banach contraction mapping principle based on some available operators, we obtain the existence of positive solutions and a unique positive solution of the problem. Finally, two examples are given to demonstrate the validity of our main results.
Highlights
The operator T has at least one fixed point on (BR \ Br) ∩ P. This implies that boundary value problem (BVP) (1) has at least one positive solution
By Theorem 3, we know that the BVP (1) has a unique positive solution
By Theorem 3, we know that the boundary value problem (BVP) (1) has a unique positive solution
Summary
In [25], Qarout et al studied the following semi-linear Caputo fractional differential equation: cD0q+x(t) = f t, x(t) , 0 < t < 1, n − 1 < q n, x(0) = x (0) = x (0) = · · · = x(n−2)(0) = 0, ξ m−2 x(1) = a x(s) d(s) + b αix(ηi), i=1 where cD0q+ denotes the Caputo fractional derivative of order q, f : [0, 1] × R+ → R+ is a continuous function, a and b are real constants and αi are positive real constants They got the existence of solutions by using some standard tools of fixed point theory.
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