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Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions

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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 21 Issue 3

Abstract

In this paper, we study the existence of positive solutions to the fractional boundary value problem

D0+αx(t)+q(t)f(t,x(t))=0,0<t<1,

together with the boundary conditions

x(0)=x′(0)=⋯=x(n−2)(0)=0,D0+βx(1)=∫01h(s,x(s))dA(s),

where n > 2, n – 1 < α ≤ n, β ∈ [1,α – 1], and D0+α and D0+β 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.

MSC 2010: Primary 34B08, 34B18, 34B15, 34B10
Key Words and Phrases: fractional differential equations; fixed point theory; positive solutions; Riemann-Liouville derivative; boundary value problems

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Received: 2017-11-15
Published Online: 2018-07-12
Published in Print: 2018-06-26

© 2018 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–Volume 21–3–2018)
  4. Research Paper
  5. Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces
  6. Novel polarization index evaluation formula and fractional-order dynamics in electric motor insulation resistance
  7. The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces
  8. Parallel algorithms for modelling two-dimensional non-equilibrium salt transfer processes on the base of fractional derivative model
  9. Some iterated fractional q-integrals and their applications
  10. Lebesgue regularity for nonlocal time-discrete equations with delays
  11. Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions
  12. Error estimates of high-order numerical methods for solving time fractional partial differential equations
  13. The Laplace transform induced by the deformed exponential function of two variables
  14. Attractivity for fractional evolution equations with almost sectorial operators
  15. The multiplicity solutions for nonlinear fractional differential equations of Riemann-Liouville type
  16. Well-posedness of general Caputo-type fractional differential equations
  17. Lyapunov-type inequalities for nonlinear fractional differential equation with Hilfer fractional derivative under multi-point boundary conditions
  18. Inverse source problem for a space-time fractional diffusion equation
  19. Erratum
  20. Erratum to: Invariant subspace method: A tool for solving fractional partial differential equations, in: FCAA-20-2-2017
https://doi.org/10.1515/fca-2018-0038
Keywords for this article
fractional differential equations; fixed point theory; positive solutions; Riemann-Liouville derivative; boundary value problems

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–Volume 21–3–2018)
  4. Research Paper
  5. Hardy-Littlewood, Bessel-Riesz, and fractional integral operators in anisotropic Morrey and Campanato spaces
  6. Novel polarization index evaluation formula and fractional-order dynamics in electric motor insulation resistance
  7. The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces
  8. Parallel algorithms for modelling two-dimensional non-equilibrium salt transfer processes on the base of fractional derivative model
  9. Some iterated fractional q-integrals and their applications
  10. Lebesgue regularity for nonlocal time-discrete equations with delays
  11. Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann-Stieltjes integral boundary conditions
  12. Error estimates of high-order numerical methods for solving time fractional partial differential equations
  13. The Laplace transform induced by the deformed exponential function of two variables
  14. Attractivity for fractional evolution equations with almost sectorial operators
  15. The multiplicity solutions for nonlinear fractional differential equations of Riemann-Liouville type
  16. Well-posedness of general Caputo-type fractional differential equations
  17. Lyapunov-type inequalities for nonlinear fractional differential equation with Hilfer fractional derivative under multi-point boundary conditions
  18. Inverse source problem for a space-time fractional diffusion equation
  19. Erratum
  20. Erratum to: Invariant subspace method: A tool for solving fractional partial differential equations, in: FCAA-20-2-2017
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