Existence and uniqueness results for pantograph equations with generalized fractional derivative

In this manuscript, we establish new existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained results.

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Existence and uniqueness results for Φ-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition

Advances in Difference Equations ◽

10.1186/s13662-020-03008-x ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Idris Ahmed ◽

Poom Kumam ◽

Thabet Abdeljawad ◽

Fahd Jarad ◽

Piyachat Borisut ◽

...

Keyword(s):

Differential Equation ◽

Green’S Function ◽

Green's Function ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Uniqueness Of Solutions ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Special Cases ◽

Generalized Fractional Derivative

Abstract The present paper describes the implicit fractional pantograph differential equation in the context of generalized fractional derivative and anti-periodic conditions. We formulated the Green’s function of the proposed problems. With the aid of a Green’s function, we obtain an analogous integral equation of the proposed problems and demonstrate the existence and uniqueness of solutions using the techniques of the Schaefer and Banach fixed point theorems. Besides, some special cases that show the proposed problems extend the current ones in the literature are presented. Finally, two examples were given as an application to illustrate the results obtained.

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Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

Open Mathematics ◽

10.2478/s11533-012-0141-4 ◽

2013 ◽

Vol 11 (3) ◽

Cited By ~ 2

Author(s):

Svatoslav Staněk

Keyword(s):

Differential Equation ◽

Fractional Derivative ◽

Fractional Differential Equation ◽

Existence And Uniqueness ◽

Existence Results ◽

Point Boundary ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Carathéodory Function ◽

Fractional Differential

AbstractWe investigate the fractional differential equation u″ + A c D α u = f(t, u, c D μ u, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.

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Existence and uniqueness results for Ψ-fractional integro-differential equations with boundary conditions

Publications de l Institut Mathematique ◽

10.2298/pim2021145v ◽

2020 ◽

Vol 107 (121) ◽

pp. 145-155

Author(s):

Devaraj Vivek ◽

E.M. Elsayed ◽

Kuppusamy Kanagarajan

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Differential Equations ◽

Boundary Conditions ◽

Fractional Derivative ◽

Existence And Uniqueness ◽

Contraction Mapping ◽

Contraction Mapping Principle ◽

Existence And Uniqueness Results ◽

Uniqueness Results

We study boundary value problems (BVPs for short) for the integro- differential equations via ?-fractional derivative. The results are obtained by using the contraction mapping principle and Schaefer?s fixed point theorem. In addition, we discuss the Ulam-Hyers stability.

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Analysis of Impulsive Boundary Value Pantograph Problems via Caputo Proportional Fractional Derivative under Mittag–Leffler Functions

Fractal and Fractional ◽

10.3390/fractalfract5040251 ◽

2021 ◽

Vol 5 (4) ◽

pp. 251

Author(s):

Bounmy Khaminsou ◽

Weerawat Sudsutad ◽

Chatthai Thaiprayoon ◽

Jehad Alzabut ◽

Songkran Pleumpreedaporn

Keyword(s):

Differential Equation ◽

Fixed Point ◽

Fractional Derivative ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Boundary Value ◽

Nonlinear Functional ◽

Numerical Examples ◽

Existence And Uniqueness Results ◽

Uniqueness Results

This manuscript investigates an extended boundary value problem for a fractional pantograph differential equation with instantaneous impulses under the Caputo proportional fractional derivative with respect to another function. The solution of the proposed problem is obtained using Mittag–Leffler functions. The existence and uniqueness results of the proposed problem are established by combining the well-known fixed point theorems of Banach and Krasnoselskii with nonlinear functional techniques. In addition, numerical examples are presented to demonstrate our theoretical analysis.

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Existence and uniqueness results for a fractional evolution equation in Hilbert space

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-012-0017-0 ◽

2012 ◽

Vol 15 (2) ◽

Cited By ~ 9

Author(s):

Emilia Bazhlekova

Keyword(s):

Hilbert Space ◽

Evolution Equation ◽

Fractional Derivative ◽

Existence And Uniqueness ◽

Accretive Operators ◽

Existence And Uniqueness Results ◽

Fractional Evolution Equation ◽

Uniqueness Results ◽

Uniqueness Of The Solution ◽

Liouville Fractional Derivative

AbstractThe existence and uniqueness of the solution of a fractional evolution equation with the Riemann-Liouville fractional derivative of order α ∈ (0, 1) is studied in Hilbert space, based on the theory of sums of accretive operators. The results are applied to some subdiffusion problems.

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Existence and uniqueness of some Cauchy type problems in fractional q-difference calculus

Filomat ◽

10.2298/fil2013429s ◽

2020 ◽

Vol 34 (13) ◽

pp. 4429-4444

Author(s):

S. Shaimardan ◽

L.E. Persson ◽

N.S. Tokmagambetov

Keyword(s):

Fractional Derivative ◽

Existence And Uniqueness ◽

Nonlinear Differential Equations ◽

Sufficient Condition ◽

Cauchy Type ◽

Liouville Type ◽

Hilfer Fractional Derivative ◽

Derivative Operator ◽

Existence And Uniqueness Results ◽

Uniqueness Results

In this paper we derive a sufficient condition for the existence of a unique solution of a Cauchy type q-fractional problem (involving the fractional q-derivative of Riemann-Liouville type) for some nonlinear differential equations. The key technique is to first prove that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. Moreover, we define the q-analogue of the Hilfer fractional derivative or composite fractional derivative operator and prove some similar new equivalence, existence and uniqueness results as above. Finally, some examples are presented to illustrate our main results in cases where we can even give concrete formulas for these unique solutions.

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Multi-term fractional differential equations with nonlocal boundary conditions

Open Mathematics ◽

10.1515/math-2018-0127 ◽

2018 ◽

Vol 16 (1) ◽

pp. 1519-1536

Author(s):

Bashir Ahmad ◽

Najla Alghamdi ◽

Ahmed Alsaedi ◽

Sotiris K. Ntouyas

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Fractional Differential ◽

The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

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