Abstract
In this paper, we consider a new class of singular nonlinear higher order fractional boundary value problems supplemented with sum of Riemann–Stieltjes integral type and nonlocal infinite-point discrete type boundary conditions. The fractional derivative of different orders is involved in the nonlinear terms and boundary conditions, and the nonlinear terms are allowed to be singular in regard to not only time variable but also space variables. A unique positive solution is established by using the fixed point theorem of mixed monotone operator. In addition, some significant properties of the unique solution depending on the parameter λ are stated. In the end, two examples are worked out to illustrate our main results.
Highlights
In this paper, we are investigating the following singular nonlinear higher order fractional boundary value problem (BVP for short):⎧ ⎪⎪⎨Dγ0+z(t) + λf (t, z(t), Dν01+z(t), . . . , Dν0n+–3 z(t), Dν0n+–2 z(t)) = 0, 0 < t < 1,⎪⎪⎩Dz(γ000+)z=(1D) q0=+1 z(0) = · p i=1 ai ·· = Dq0+n–2 z(0) = 0, Ii wi(s)Dα0+i z(s) dAi (s) + ∞ j=1 bj Dβ0+j z(ξj ), (1.1)
A function z ∈ C[0, 1] is called a positive solution of BVP (1.1) if it satisfies (1.1) and z(t) > 0 for t ∈ (0, 1)
In this paper, by using the fixed point theorem of mixed monotone operator, we obtain the uniqueness of positive solution under the assumption that f may be singular with respect to both the time and space variables
Summary
A function z ∈ C[0, 1] is called a positive solution of BVP (1.1) if it satisfies (1.1) and z(t) > 0 for t ∈ (0, 1). Lemma 2.2 ([10, 12]) Let P be a normal cone in the Banach space E, and A, B : Pe ×Pe → Pe be two mixed monotone operators which satisfy the following conditions: (i) For any μ ∈ (0, 1), there exists φ(μ) ∈ The equation λA(u, u) + λB(u, u) = u has a unique solution uλ in Pe for all λ > 0, which satisfies: (i) If there exists r ∈ (0, 1) such that φ(μ) ≥ μr – μ + μr, ∀μ ∈ (0, 1), κ uλ is continuous with respect to λ ∈ (0, +∞).
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