Abstract
In this paper, we use fixed-point index to study the existence of positive solutions for a system of Hadamard fractional integral boundary value problems involving nonnegative nonlinearities. By virtue of integral-type Jensen inequalities, some appropriate concave and convex functions are used to depict the coupling behaviors for our nonlinearities fii=1, 2.
Highlights
In this paper, we study the existence of positive solutions for the system of Hadamard fractional integral boundary value problems:− HDαu(t) f1(t, − HDαv(t) f2(t, u(j)(1) v(j)(1)u(t), u(t), 0, v(t)), v(t)), t ∈ (1, e), t ∈ (1, e), u(e) v(e)e dt h(t)u(t), t h(t)v(t), (1)t where α ∈ (n − 1, n] is a real number with n ≥ 3, j 0, 1, 2, . . . , n − 2, and HDα is the Hadamard fractional derivative. e nonlinearities fi ∈ C([1, e] × R+ × R+, R+), R+ [0, +∞)
The fractional calculus and fractional differential equations are of importance in mathematics, physics, electroanalytical chemistry, capacitor theory, electrical circuits, biology, control theory, and uid dynamics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]
In [21], the authors used a xed-point theorem of increasing φ-(h, r)-concave operators to establish the existence and uniqueness of solutions for a system of four-point boundary value problems involving Hadamard fractional derivatives:
Summary
We study the existence of positive solutions for the system of Hadamard fractional integral boundary value problems:− HDαu(t) f1(t, − HDαv(t) f2(t, u(j)(1) v(j)(1)u(t), u(t), 0, v(t)), v(t)), t ∈ (1, e), t ∈ (1, e), u(e) v(e)e dt h(t)u(t) , t h(t)v(t) , (1)t where α ∈ (n − 1, n] is a real number with n ≥ 3, j 0, 1, 2, . . . , n − 2, and HDα is the Hadamard fractional derivative. e nonlinearities fi ∈ C([1, e] × R+ × R+, R+), R+ [0, +∞). In [21], the authors used a xed-point theorem of increasing φ-(h, r)-concave operators to establish the existence and uniqueness of solutions for a system of four-point boundary value problems involving Hadamard fractional derivatives: Complexity solutions for the coupled Hadamard fractional integral boundary value problems: U(e) v(e) v(s) u(s) where the nonlinearities f and g satisfy either of the following conditions: (H)Yang1: there exists [θ1, θ2] ⊂ (1, e) such that lim inf u⟶+∞ mint∈[θ1,θ2](f(t, u, v)/u) +∞
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