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

Abstract

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.

On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

“…As we know, the existence theory is meat-and-potatoes in every field of science, as it is very applicative to comprehend whether there is a solution to a given differential equation beforehand; otherwise, all the attempts to find a numerical or analytic solution will become valueless. The analysis of fractional differential equations has been carried out by various authors (see, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]).…”

New Hermite–Hadamard and Ostrowski-Type Inequalities for Newly Introduced Co-Ordinated Convexity with Respect to a Pair of Functions

“…Additionally, in recent years, such inequalities have been used in most papers discussing fractional mathematical models, fractional boundary value problems, etc. The application of some of them can be found in numerous articles, including [1][2][3][4][5][6][7][8][9][10][11]. These applications show the importance of study on different generalizations of inequalities.…”

“…2 4(1 − κ)2 (1 − γ) 2 e −2κ−2γ + 4(1 − γ) 2 e −2γ + 2(1 − κ) 2 (1 − γ) 2 e −2κ−2 + (1 − γ) 2 e −2κ−2 + 2(1 − γ) 2 e −2κ−2γ + (1 − κ) 2 (1 − γ) 2 e −2γ +2(1 − κ) 2 (1 − γ) 2 e −2 + 2(1 − γ) 2 e −2 1/2 + (1 − κ) 2 γ 2 4(1 − κ) 2 (1 − γ) 2 e −2κ−2γ + 4(1 − κ) 2 e −2κ + 2(1 − κ) 2 (1 − γ) 2 e −2κ + 2(1 − κ) 2 e −2κ−2γ + (1 − κ) 2 e −2γ−2 + 2(1 − κ) 2 (1 − γ) 2 e −2γ−− κ) 2 (1 − γ) 2 e −2γ−2 + 2(1 − κ) 2 (1 − γ) 2 e −2 + 2(1 − κ) 2 e −2 1/− κ) 2 (1 − γ) 2 4(1 − κ) 2 (1 − γ) 2 e −2κ−2γ + 2(1 − κ) 2 (1 − γ) 2 e −2κ−2 + (1 − γ) 2 e −2κ−− κ) 2 (1 − γ) 2 e −2γ−2 1/2. left hand side The surface of right hand side…”

H-U-Type Stability and Numerical Solutions for a Nonlinear Model of the Coupled Systems of Navier BVPs via the Generalized Differential Transform Method

“…Regarding applications of FDEs, one can observe some fields including aerodynamics, biochemistry, electro-chemistry, bioengineering, physics, viscoelasticity, mathematical biology, and so on (see [1,2]). In the last few years, a large number of studies regarding the existence theory for different FDEs have received much attentions from researchers and some examples include [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Since we can model some of applied phenomena in the framework of the fractional coupled systems, a large number of researchers have conducted many research studies on the existence of solution for such a type of systems (for instances, see [19][20][21][22][23]).…”