TY - JOUR

T1 - Vallée-Poussin theorem for fractional functional differential equations

AU - Domoshnitsky, Alexander

AU - Padhi, Seshadev

AU - Srivastava, Satyam Narayan

N1 - Publisher Copyright:
© 2022, Diogenes Co.Ltd.

PY - 2022/8

Y1 - 2022/8

N2 - An analog of the classical Vallée-Poussin theorem about differential inequality in the theory of ordinary differential equations is developed for fractional functional differential equations. The main results are obtained in a form of a theorem about several equivalent assertions. Among them solvability of two-point boundary value problems with fractional functional differential equation, negativity of Green’s function, and its derivatives and existence of a function v(t) satisfying a corresponding differential inequality. Thus the Vallée-Poussin theorem presents one of the possible “entrances” to assertions on nonoscillating properties and assertions about the negativity of Green’s functions and their derivatives for various two-point problems. Choosing the function v(t) in the condition, we obtain explicit tests of sign-constancy of Green’s functions and their derivatives. It can be stressed that a choice of a corresponding function in the Vallée-Poussin theorem leads to explicit criteria in the form of algebraic inequalities, which, as we demonstrate with examples, cannot be improved. Replacing strict inequalities with non-strict ones will already lead to incorrect statements. In some cases, the well-known results obtained in the form of the Lyapunov inequalities for fractional differential equations can be improved based on ours. Another development, we propose, is a concept to consider fractional functional differential equations. The basis of this concept is to reduce boundary value problems for fractional functional differential equations to operator equations in the space of essentially bounded functions. Fractional functional differential equations can appear in various applications and in the process of construction of equations for a corresponding component of a solution-vector of a system of fractional equations.

AB - An analog of the classical Vallée-Poussin theorem about differential inequality in the theory of ordinary differential equations is developed for fractional functional differential equations. The main results are obtained in a form of a theorem about several equivalent assertions. Among them solvability of two-point boundary value problems with fractional functional differential equation, negativity of Green’s function, and its derivatives and existence of a function v(t) satisfying a corresponding differential inequality. Thus the Vallée-Poussin theorem presents one of the possible “entrances” to assertions on nonoscillating properties and assertions about the negativity of Green’s functions and their derivatives for various two-point problems. Choosing the function v(t) in the condition, we obtain explicit tests of sign-constancy of Green’s functions and their derivatives. It can be stressed that a choice of a corresponding function in the Vallée-Poussin theorem leads to explicit criteria in the form of algebraic inequalities, which, as we demonstrate with examples, cannot be improved. Replacing strict inequalities with non-strict ones will already lead to incorrect statements. In some cases, the well-known results obtained in the form of the Lyapunov inequalities for fractional differential equations can be improved based on ours. Another development, we propose, is a concept to consider fractional functional differential equations. The basis of this concept is to reduce boundary value problems for fractional functional differential equations to operator equations in the space of essentially bounded functions. Fractional functional differential equations can appear in various applications and in the process of construction of equations for a corresponding component of a solution-vector of a system of fractional equations.

KW - Boundary value problems

KW - Comparison of solution

KW - Differential inequality

KW - Fractional differential equations

KW - Lyapunov inequality

KW - Positive solutions

KW - Riemann–Liouville derivative

KW - Sign constancy of Green’s function

UR - http://www.scopus.com/inward/record.url?scp=85133585397&partnerID=8YFLogxK

U2 - 10.1007/s13540-022-00061-z

DO - 10.1007/s13540-022-00061-z

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AN - SCOPUS:85133585397

SN - 1311-0454

VL - 25

SP - 1630

EP - 1650

JO - Fractional Calculus and Applied Analysis

JF - Fractional Calculus and Applied Analysis

IS - 4

ER -