Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

DOI10.1186/s13662-016-0760-3zbMath1422.39027OpenAlexW2260101277WikidataQ59468200 ScholiaQ59468200MaRDI QIDQ1796468

Baoguo Jia, Christopher S. Goodrich, Allan C. Peterson, Lynn H. Erbe

Publication date: 17 October 2018

Published in: Advances in Difference Equations (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1186/s13662-016-0760-3

zbMATH Keywords

monotonicityconvexityasymptotic behavior of solutionfractional difference calculusfractional initial value problem

Mathematics Subject Classification ID

Fractional derivatives and integrals (26A33) Additive difference equations (39A10) Functional inequalities, including subadditivity, convexity, etc. (39B62) Difference equations, scaling ((q)-differences) (39A13) Convexity of real functions in one variable, generalizations (26A51) Monotonic functions, generalizations (26A48) Growth, boundedness, comparison of solutions to difference equations (39A22) Stability theory for difference equations (39A30) Linear difference equations (39A06)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Generalized Gronwall fractional summation inequalities and their applications
• Discrete Grüss type inequality on fractional calculus
• Gronwall's inequality on discrete fractional calculus
• Positive solutions for a class of higher order boundary value problems with fractional \(q\)-derivatives
• On a fractional boundary value problem with fractional boundary conditions
• Multiplicity and uniqueness for a class of discrete fractional boundary value problems.
• Linear systems of fractional nabla difference equations
• Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions
• Positive solutions for a class of boundary value problems with fractional \(q\)-differences
• Discrete Mittag-Leffler functions in linear fractional difference equations
• Foundations of nabla fractional calculus on time scales and inequalities
• Principles of delta fractional calculus on time scales and inequalities
• Existence of a positive solution to a system of discrete fractional boundary value problems
• On discrete sequential fractional boundary value problems
• The Laplace transform in discrete fractional calculus
• Continuity of solutions to discrete fractional initial value problems
• Constrained local controllability of dynamic systems on time scales
• Modeling with fractional difference equations
• Nabla discrete fractional calculus and nabla inequalities
• On the existence of solutions for a fractional finite difference inclusion via three points boundary conditions
• Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions
• Multiple positive solutions of nonlinear boundary value problem for finite fractional difference
• Existence theory for \(q\)-antiperiodic boundary value problems of sequential \(q\)-fractional integrodifferential equations
• On a first-order semipositone discrete fractional boundary value problem
• Impulsive quantum difference systems with boundary conditions
• On delta and nabla Caputo fractional differences and dual identities
• On positive solutions to fractional difference inclusions
• A convexity result for fractional differences
• Two monotonicity results for nabla and delta fractional differences
• Erratum to: ``A monotonicity result for discrete fractional difference operators
• Dual identities in fractional difference calculus within Riemann
• Nabla discrete fractional Grüss type inequality
• Asymptotic behavior of solutions of fractional nabla \(q\)-difference equations
• Solutions to a discrete, nonlinear, (\(N-1,1\)) fractional boundary value problem
• Discrete fractional logistic map and its chaos
• A monotonicity result for discrete fractional difference operators
• Comparison theorems and asymptotic behavior of solutions of discrete fractional equations
• Fractional h-difference equations arising from the calculus of variations
• On semipositone discrete fractional boundary value problems with non-local boundary conditions
• On a discrete fractional three-point boundary value problem
• Systems of semipositone discrete fractional boundary value problems
• Two-point boundary value problems for finite fractional difference equations
• Nontrivial solutions for fractional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math>-difference boundary value problems
• Sum and Difference Compositions in Discrete Fractional Calculus
• q− Fractional Inequalities
• Discrete Fractional Calculus
• Existence and uniqueness of solutions of sequential nonlinear fractional difference equations with three‐point fractional sum boundary conditions
• Discrete fractional calculus with the nabla operator
• On (q, h)-Analogue of Fractional Calculus
• Initial value problems in discrete fractional calculus
• Convexity for nabla and delta fractional differences
• Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions
• Analysis of discrete fractional operators
• Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one
• Monotonicity results for delta fractional differences revisited
• A discrete fractional Gronwall inequality
• Exponential functions of discrete fractional calculus

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