Abstract
We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.
Highlights
We consider the nonlinear fractional differential equationD0α+u(t) + λh(t)f t, u(t) = 0, t ∈ (0, 1), (1)with the nonlocal boundary conditions u(0) = u (0) = · · · = u(n−2)(0) = 0, m1 D0β+0 u(1) =D0β+i u(t) dHi(t), (2)i=1 0 where α ∈ R, α ∈
We mention that condition (I3) used in our results was first introduced in the paper [18], where the authors proved the existence of at least one positive solution for a fourth-order nonlinear singular Sturm–Liouville eigenvalue problem
We study the existence of positive solutions for the nonlinear Riemann– Liouville fractional boundary value problem (1), (2), where λ is a positive parameter
Summary
In the paper [20], the author presents some conditions for f , which contain height functions defined on special bounded sets under which he proves the existence and multiplicity of positive solutions. The authors use in [1] various height functions of the nonlinearity defined on special bounded sets and two theorems from the fixed point index theory. We mention the paper [33], where the authors prove the existence of positive solutions of fractional differential equation (1) supplemented with the boundary conditions u(0) = u (0) = · · · = u(n−2)(0) = 0,. We mention that condition (I3) (see below, in Section 3) used in our results was first introduced in the paper [18], where the authors proved the existence of at least one positive solution for a fourth-order nonlinear singular Sturm–Liouville eigenvalue problem.
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