Multiple Positive Solutions for a Boundary Value Problem with Nonlinear Nonlocal Riemann-Stieltjes Integral Boundary Conditions

Abstract

Abstract In this paper, we study the existence of positive solutions to the fractional boundary value problem D 0 + α x ( t ) + q ( t ) f ( t , x ( t ) ) = 0 , 0 < t < 1 , $$\begin{array}{} \displaystyle D^{\alpha }_{0+}x(t)+q(t)f(t,x(t))=0, \,\, 0\lt t \lt1, \end{array}$$ together with the boundary conditions x ( 0 ) = x ′ ( 0 ) = ⋯ = x ( n − 2 ) ( 0 ) = 0 , D 0 + β x ( 1 ) = ∫ 0 1 h ( s , x ( s ) ) d A ( s ) , $$\begin{array}{} \displaystyle x(0)=x^{\prime}(0)= \cdots = x^{(n-2)}(0)=0, D_{0+}^{\beta }x(1)= \int^{1}_{0}h(s,x(s))\,dA(s), \end{array}$$ where n > 2, n – 1 < α ≤ n, β ∈ [1,α – 1], and D 0 + α $\begin{array}{} \displaystyle D^{\alpha }_{0+} \end{array}$ and D 0 + β $\begin{array}{} \displaystyle D^{\beta }_{0+} \end{array}$ are the standard Riemann-Liouville fractional derivatives of order α and β, respectively. We consider two different cases: f, h : [0, 1] × R → R, and f, h : [0, 1] × [0, ∞) → [0, ∞). In the first case, we prove the existence and uniqueness of the solutions of the above problem, and in the second case, we obtain sufficient conditions for the existence of positive solutions of the above problem. We apply a number of different techniques to obtain our results including Schauder’s fixed point theorem, the Leray-Schauder alternative, Krasnosel’skii’s cone expansion and compression theorem, and the Avery-Peterson fixed point theorem. The generality of the Riemann-Stieltjes boundary condition includes many problems studied in the literature. Examples are included to illustrate our findings.