Positive solutions of conformable fractional differential equations with integral boundary conditions

• Published in: Boundary value problems Vol. 2018; no. 1; pp. 1 - 12
• Main Authors: Zhong, Wenyong, Wang, Lanfang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 12.09.2018; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Boundary conditions; Conformable fractional derivatives; Derivatives; Difference and Functional Equations; Differential equations; Existence theorems; Fixed point theorems; Fixed points (mathematics); Integral boundary value problems; Integrals; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Positive solutions; Fixed point theorems; Positive solutions; Integral boundary value problems; Conformable fractional derivatives
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-018-1056-1

In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation T α x ( t ) + f ( t , x ( t ) ) = 0 , t ∈ [ 0 , 1 ] , subject to the boundary conditions x ( 0 ) = 0 and x ( 1 ) = λ ∫ 0 1 x ( t ) d t , where the order α belongs to ( 1 , 2 ] , T α x ( t ) denotes the conformable fractional derivative of a function x ( t ) of order α , and f : [ 0 , 1 ] × [ 0 , ∞ ) ↦ [ 0 , ∞ ) is continuous. By use of the fixed point theorem in a cone, some criteria for the existence of at least one positive solution are established. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem. A concrete example is given to illustrate the possible application of the obtained results.