A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions
Abstract
:1. Introduction
2. Preliminaries
2.1. Fractional Calculus
2.2. Multivalued Analysis
- (i)
- is measurable for each;
- (ii)
- is upper semicontinuous for almost allFurthermore, a Carathéodory function F is calledCarathéodory if:
- (iii)
- for each, there existssuch that
3. Boundary Value Problems for Hilfer Fractional Differential Equations and Inclusions
3.1. Nonlocal Fractional Integral Boundary Conditions
3.1.1. Existence and Uniqueness Results for the Problem (3)
- (1.1)
- There exists a constant such that for each and .
- (2.1)
- , for and .
- (3.1)
- , for , where is continuous and the constant defined by
- (4.1)
- , and .
- (5.1)
- there exists a continuous nondecreasing function and a function such that
- (5.2)
- there exists a constant such that
3.1.2. Existence Results for the Inclusion Problem
- (6.1)
- is -Carathéodory;
- (6.2)
- there exists a continuous nondecreasing function and a function such that
- (7.1)
- is such that is measurable for each
- (7.2)
- for almost all and with and for almost all .
3.2. Pantograph Fractional Differential Equations and Inclusions with Nonlocal Fractional Integral Boundary Conditions
3.2.1. Existence and Uniqueness Results for the Problem (5)
- (8.1)
- there exists a constant such that
- (9.1)
- , and .
- (10.1)
- there exist a continuous nondecreasing function and a function such that
- (10.2)
- there exists a constant such that
3.2.2. Existence Results for the Inclusion Problem
- (11.1)
- is -Carathéodory;
- (11.2)
- where is the function that appears in Definition 11.
- (12.1)
- is a -Carathéodory multivalued map;
- (12.2)
- there exists a function such that
- (13.1)
- there exists a continuous nondecreasing function and a function such that
- (14.1)
- is such that is measurable for each
- (14.2)
- for almost all and with and for almost all .
3.3. Nonlocal Integro-Multipoint Boundary Conditions
3.3.1. Existence and Uniqueness Results for the Problem (7)
- (15.1)
- there exists a constant such that
- (16.1)
- , and .
- (17.1)
- there exist a continuous, nondecreasing, subhomogeneous (that is, for all and ) function and a function such that
- (17.2)
- there exists a constant such that
3.3.2. Existence Results for the Inclusion Problem
- (18.1)
- is -Carathéodory;
- (18.2)
- there exists a function such that
- (19.1)
- there exists a continuous, nondecreasing, subhomogeneous function and a function such that
- (20.1)
- is such that is measurable for each
- (20.2)
- for almost all and with , and for almost all .
4. Boundary Value Problems for Sequential Hilfer Fractional Differential Equations and Inclusions
4.1. Nonlocal Integro-Multipoint Boundary Conditions
4.1.1. Existence and Uniqueness Results for the Problem (9)
- (21.1)
- there exists a constant such that
- (22.1)
- , and .
- (23.1)
- there exists a continuous, nondecreasing, subhomogeneous (that is, for all and ) function and a function such that
- (23.2)
4.1.2. Existence Results for the Inclusion Problem
- (24.1)
- is -Carathéodory;
- (24.2)
- there exists a function such that
- (25.1)
- there exists a continuous, nondecreasing, subhomogeneous function and a function such that
- (26.1)
- is such that is measurable for each
- (26.2)
- for almost all and with and for almost all .
4.2. Nonlocal Integro-Multistrip-Multipoint Boundary Conditions
4.2.1. Existence and Uniqueness Results for the Problem (13)
- (27.1)
- , for all ,
- (27.2)
- for all ,
4.2.2. Existence Results for the Inclusion Problem
- (29.1)
- is -Carathéodory;
- (29.2)
- there exists a continuous, nondecreasing function and a function such that
- (29.3)
- (30.1)
- is such that is measurable for each ;
- (30.2)
- for almost all and with and for almost all .
4.3. Riemann–Stieltjes Integral Multistrip Boundary Conditions
4.3.1. Existence and Uniqueness Results for the Problem (17)
- (31.1)
- and
- (32.1)
- , and .
- (33.1)
- , where is a nondecreasing and continuous function and
- (33.2)
- there exists a constant such that
4.3.2. Existence Results for the Inclusion Problem
- (34.1)
- is -Carathéodory multivalued map;
- (34.2)
- where is a nondecreasing continuous function and
- (34.3)
5. Boundary Value Problems for -Hilfer Fractional Differential Equations
5.1. Existence Results for a -Hilfer Nonlocal Fractional Boundary Value Problem via Topological Degree Theory
- (35.1)
- there exists a constant such that: for all
- (35.2)
- there exists a constant such that: for all
- (36.1)
- for an arbitrary , there exist such that
- (36.2)
- for an arbitrary , there exist such that
5.2. Mixed Nonlocal Boundary Conditions
- (37.1)
- there exists a constant such that
- (38.1)
- , , and
6. Boundary Value Problems for -Hilfer-Type Sequential Fractional Differential Equations and Inclusions
6.1. Multipoint Boundary Conditions
- (40.1)
- there exists a constant such that
- (41.1)
- , and .
- (42.1)
- there exists a continuous, nondecreasing function and a function such that
- (42.2)
- there exists a constant such that
6.2. Integral Multipoint Boundary Conditions
6.2.1. Existence and Uniqueness Results for the Problem (36)
- (43.1)
- there exists a finite number such that, for all and for all the following inequality is valid:
6.2.2. Existence Results for the Inclusion Problem
- (45.1)
- is -Carathéodory multivalued map;
- (45.2)
- there exists a nondecreasing and continuous function and a function such that
- (45.3)
7. Coupled Systems of Hilfer Fractional Differential Equations with Nonlocal Fractional Integral Boundary Conditions
- (46.1)
- there exist constants such that for all and ,
8. Coupled Systems of -Hilfer Sequential Fractional Differential Equations with Nonlocal Boundary Conditions
- (49.1)
- are continuous functions and there exist real constants and such that,
9. Existence and Uniqueness of Solutions for System of Hilfer–Hadamard Sequential Fractional Differential Equations with Two Point Boundary Conditions
- (51.1)
- There exist real constants and , , such that for all , , ,
- (52.1)
- there exist positive constants such that for all , , ,
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Ntouyas, S.K. A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions. Foundations 2021, 1, 63-98. https://0-doi-org.brum.beds.ac.uk/10.3390/foundations1010007
Ntouyas SK. A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions. Foundations. 2021; 1(1):63-98. https://0-doi-org.brum.beds.ac.uk/10.3390/foundations1010007
Chicago/Turabian StyleNtouyas, Sotiris K. 2021. "A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions" Foundations 1, no. 1: 63-98. https://0-doi-org.brum.beds.ac.uk/10.3390/foundations1010007