Abstract
We study the existence of solutions for the boundary value problem −Δνy1(t) = f(y1(t + ν − 1), y2(t + μ − 1)), −Δμy2(t) = g(y1(t + ν − 1), y2(t + μ − 1)), y1(ν − 2) = Δy1(ν + b) = 0, y2(μ − 2) = Δy2(μ + b) = 0, where 1 < μ, ν ≤ 2, f, g : ℝ × ℝ → ℝ are continuous functions, b ∈ ℕ0. The existence of solutions to this problem is established by the Guo‐Krasnosel′kii theorem and the Schauder fixed‐point theorem, and some examples are given to illustrate the main results.
Highlights
In recent years, fractional differential equations have been of great interest
This paper gives the existence of a positive solution on discrete fractional boundary value problems
Motivated by all the works above, in this paper, we discuss the existence of solutions to a system of discrete fractional boundary value problem
Summary
Fractional differential equations have been of great interest. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. Some of the recent progress in the continuous fractional calculus has included a paper 1 in which the authors explored a continuous fractional boundary value problem of conjugate type. Using cone theory, they deduced the existence of one or more positive solutions. Goodrich studied a two-point fractional boundary value problem in 24 , which gave the existence results for a certain two-point boundary value problem of right-focal type for a fractional difference equation. This paper gives the existence of a positive solution on discrete fractional boundary value problems. Motivated by all the works above, in this paper, we discuss the existence of solutions to a system of discrete fractional boundary value problem.
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