Solvability for Discrete Fractional Boundary Value Problems with ap-Laplacian Operator

Abstract

This paper is concerned with the solvability for a discrete fractionalp-Laplacian boundary value problem. Some existence and uniqueness results are obtained by means of the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main results.

Existence and Uniqueness of Solutions for a Discrete Fractional Mixed Type Sum-Difference Equation Boundary Value Problem

“…Then, for any ∈ , by (35), (37), and the fact Γ( ) ∈ (0, 1], we can verify that ‖F ‖ ≤ , which implies F : → .…”

“…which hold for ∈ N 0 . So (35) and (37) imply that F : → is well defined and bounded. Furthermore, from Lemma 11, we can transform problem (1) into an operator equation = F and it is clear to see that is a solution of problem (1) which is equivalent to a fixed point of F. for ( , , V, , ) ∈ N −1 × R × R × R × R, which implies that condition (C 2 ) is stronger than (C 2 ).…”

“…However, compared to the long and rich history of continuous fractional calculus, discrete fractional calculus attracted mathematicians and scientists into its fairly new research area in a short period of time. In this time period, the theory of discrete calculus has been developed in many directions parallel to the theory in continuous fractional calculus such as initial value problems and boundary value problems for fractional difference equations, discrete Mittag-Leffler functions, and inequalities with discrete fractional operators; see Although, among all recently research topics, the branch of discrete finite fractional difference boundary value problems is currently undergoing active investigation [16,[31][32][33][34][35][36][37][38], significantly less is known about discrete infinite fractional difference boundary value problems with the nonlinear term dependent on a fractional difference operator. Here, we should point out that in [39], Lv and Feng, by simple analogy with the ordinary case, introduced some basic definitions of discrete fractional calculus for Banach-valued functions and initially studied a class of discrete infinite fractional mixed type sum-difference equation boundary value problems in abstract spaces by using contracting mapping principle.…”