Abstract
This paper is concerned with the integral boundary value problems of higher-order fractional differential equation with. In the sense of a monotone homomorphism, some sufficient criteria are established to guarantee the existence of at least two monotone positive solutions by employing the fixed point theorem of cone expansion and compression of functional type proposed by Avery, Henderson and O’Regan. As applications, some examples are provided to illustrate the validity of our main results.
Highlights
In recent years, the fractional order differential equation has aroused great attention due to both the further development of fractional order calculus theory and the important applications for the theory of fractional order calculus in the fields of science and engineering such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth
Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes
The boundary value problems with Riemann-Stieltjes integral boundary conditions arise in a variety of different areas of applied mathematics and physics
Summary
The fractional order differential equation has aroused great attention due to both the further development of fractional order calculus theory and the important applications for the theory of fractional order calculus in the fields of science and engineering such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth. There have been many papers focused on boundary value problems of fractional ordinary differential equations (see [ – ]). When φ is p-Laplacian operator, that is, φ(u) = φp(u) = |u|p– u (p > ) and the nonlinear term does not depend on the first-order derivative, the existence prob-. Zhao and Liu Advances in Difference Equations (2016) 2016:20 lems of positive solutions of boundary value problems have attracted much attention. The existence of positive solutions for fractional differential equations with p-Laplacian operator have been studied by several authors (see [ – ] and the references therein). In this article, motivated by the above mentioned discussion, we study the existence of at least two monotone and concave positive solutions of integral boundary value problem for the nonlinear fractional differential equation with sign-changing nonlinearity and delayed or advanced arguments as follows
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