scholarly journals Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1165
Author(s):  
Shahram Rezapour ◽  
Sotiris K. Ntouyas ◽  
Abdelkader Amara ◽  
Sina Etemad ◽  
Jessada Tariboon
Keyword(s):  
Differential Equation ◽  
Fixed Point Theorems ◽  
Research Study ◽  
Existence Results ◽  
Gronwall Inequality ◽  
Grönwall Inequality ◽  
Dependence Criterion ◽  
Uniqueness And Existence ◽  
The Right ◽  
Uniqueness And Existence Results

The main intention of the present research study is focused on the analysis of a Caputo fractional integro-differential boundary problem (CFBVP) in which the right-hand side of supposed differential equation is represented as a sum of two nonlinear terms. Under the integro-derivative boundary conditions, we extract an equivalent integral equation and then define new operators based on it. With the help of three distinct fixed-point theorems attributed to Krasnosel’skiĭ, Leray–Schauder, and Banach, we investigate desired uniqueness and existence results. Additionally, the dependence criterion of solutions for this CFBVP is checked via the generalized version of the Gronwall inequality. Next, three simulative examples are designed to examine our findings based on the procedures applied in the theorems.

Uniqueness Implies Existence and Uniqueness Conditions for a Class of (k + j)-Point Boundary Value Problems for n-th Order Differential Equations

2012 ◽  
Vol 55 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Paul W. Eloe ◽  
Johnny Henderson ◽  
Rahmat Ali Khan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Differential Equation ◽  
Unique Solvability ◽  
Existence And Uniqueness ◽  
Existence Results ◽  
Point Boundary ◽  
Uniqueness And Existence ◽  
Uniqueness And Existence Results

AbstractFor the n-th order nonlinear differential equation, y(n) = f (x, y, y′, … , y(n–1)), we consider uniqueness implies uniqueness and existence results for solutions satisfying certain (k + j)-point boundary conditions for 1 ≤ j ≤ n – 1 and 1 ≤ k ≤ n – j. We define (k; j)-point unique solvability in analogy to k-point disconjugacy and we show that (n – j0; j0)-point unique solvability implies (k; j)-point unique solvability for 1 ≤ j ≤ j0, and 1 ≤ k ≤ n – j. This result is analogous to n-point disconjugacy implies k-point disconjugacy for 2 ≤ k ≤ n – 1.

On a discrete fractional stochastic Grönwall inequality and its application in the numerical analysis of stochastic FDEs involving a martingale

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ahmed S. Hendy ◽  
Mahmoud A. Zaky ◽  
Eid H. Doha
Keyword(s):  
Numerical Analysis ◽  
A Priori ◽  
Gronwall Inequality ◽  
Gronwall Lemma ◽  
Discrete Form ◽  
Left Hand ◽  
Grönwall Inequality ◽  
Difference Formula ◽  
The Right ◽  
The Given

Abstract The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.

On a fractional differential inclusion with “maxima”

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Aurelian Cernea
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Differential Inclusion ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Existence Results ◽  
Fractional Differential Inclusion ◽  
Right Hand ◽  
Fractional Differential ◽  
The Right

Abstract We study a boundary value problem associated to a fractional differential inclusion with “maxima”. Several existence results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

scholarly journals Non Uniqueness of Power-Law Flows

2021 ◽  
Author(s):  
Jan Burczak ◽  
Stefano Modena ◽  
László Székelyhidi
Keyword(s):  
Power Law ◽  
Power Index ◽  
Existence Results ◽  
Seminal Paper ◽  
Convex Integration ◽  
Power Law Fluids ◽  
Uniqueness And Existence ◽  
Uniqueness And Existence Results

AbstractWe apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $$d\ge 3$$ d ≥ 3 . For the power index q below the compactness threshold, i.e. $$q \in (1, \frac{2d}{d+2})$$ q ∈ ( 1 , 2 d d + 2 ) , we show ill-posedness of Leray–Hopf solutions. For a wider class of indices $$q \in (1, \frac{3d+2}{d+2})$$ q ∈ ( 1 , 3 d + 2 d + 2 ) we show ill-posedness of distributional (non-Leray–Hopf) solutions, extending the seminal paper of Buckmaster & Vicol [10]. In this wider class we also construct non-unique solutions for every datum in $$L^2$$ L 2 .

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