Abstract
This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the α -Riemann-Liouville fractional derivative, with α ∈ ( 1 , 2 ] . To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.
Highlights
Fractional calculus has been applied to a huge number of fields in science, engineering, and mathematics
The existence of solutions of nonlinear boundary value problem coupled with integral boundary conditions in ordinary and fractional cases has been widely studied by many authors, see for example [9,10,11,12,13,14] and the references therein
We use the qualitative properties obtained on those reference and study the parameter relationship between α, λ, μ and η that ensure the constant sign of the Green function related to the linear problem (2)
Summary
Fractional calculus has been applied to a huge number of fields in science, engineering, and mathematics. In 2009, Ahmad and Nieto [15] obtained some existence results for the following nonlinear fractional integrodifferential equations with integral boundary conditions:. The following nonlinear fractional differential equation with non-homogeneous integral boundary conditions is considered:. We use the qualitative properties obtained on those reference and study the parameter relationship between α, λ, μ and η that ensure the constant sign of the Green function related to the linear problem (2). The paper is scheduled as follows: after some introductory results, we study, in Section 3, the related linear equation and deduce suitable properties on the qualitative behavior and constant sign of the related Green function. In last section, some examples are given to point out the applicability of the obtained results
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