Existence of Positive Solutions for a Class of Conformable Fractional Differential Equations with Parameterized Integral Boundary Conditions

Abstract

Fractional calculus and fractional differential equations are recently experiencing rapid development. There are several notions of fractional derivatives, some classical, such as the Riemann-Liouville or Caputo definitions, and some novel, such as conformable fractional derivatives [18], β-derivatives [9], or others [12, 20]. Recently, the new definition of a conformable fractional derivative, given by [1, 2, 18], has drawn much interest from many researchers [6, 7, 17, 22, 23, 24, 26]. Recent results on conformable fractional differential equations can also be found in [3, 8, 11]. In 2017, X. Dong et al.[15] studied the existence and multiplicity of positive solutions for the following conformable fractional differential equation with p-Laplacian operator D(φp(D u(t))) = f(t, u(t)), 0 < t < 1, u(0) = u(1) = Du(0) = Du(1) = 0. Here, 1 < α ≤ 2 is a real number, D is the conformable fractional derivative, φp(s) = |s|p−2s, p > 1, φ−1 p = φq, 1/p+ 1/q = 1, and f : [0, 1]× [0,+∞)→ [0,+∞)