Existence of a positive solution to a system of discrete fractional boundary value problems

• Published in: Applied mathematics and computation Vol. 217; no. 9; pp. 4740 - 4753
• Main Author: Goodrich, Christopher S.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 2011; Elsevier
• Subjects: Boundary conditions; Boundary value problems; Computation; Cone; Coupled system of boundary value problems; Difference equations; Discrete fractional calculus; Eigenvalues; Exact sciences and technology; Functionals; Mathematical analysis; Mathematical models; Mathematics; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations, boundary value problems; Positive solution; Sciences and techniques of general use; Eigenvalues; Coupled system of boundary value problems; Cone; Discrete fractional calculus; Positive solution; Transcendental equation; Existence of solution; Eigenvector; Numerical linear algebra; Eigenvalue; Discrete system; Boundary condition; Partial differential equation; Equation system; Non linear equation; Integer; Numerical analysis; Boundary value problem; Applied mathematics; Algebraic equation
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.11.029

We analyze a system of discrete fractional difference equations subject to nonlocal boundary conditions. We consider the system of equations given by - Δ ν i y i ( t ) = λ i a i ( t + ν i - 1 ) f i ( y 1 ( t + ν 1 - 1 ) , y 2 ( t + ν 2 - 1 ) ) , for t ∈ [ 0 , b ] N 0 , subject to y i ( ν i − 2) = ψ i ( y i ) and y i ( ν i + b) = ϕ i ( y i ), for i = 1, 2, where ψ i , ϕ i : R b + 3 → R are given functionals. We also assume that ν i ∈ (1, 2], for each i. Although we assume that both a i and f i ( y 1, y 2) are nonnegative for each i, we do not necessarily presume that each ψ i ( y i ) and ϕ i ( y i ) is nonnegative for each i and each y i ⩾ 0. This generalizes some recent results both on discrete fractional boundary value problems and on discrete integer-order boundary value problems, and our techniques provide new results in each case.