arXiv subjects

Kishor D. Kucche

Publications and source records attributed to Kishor D. Kucche.

At least 19 recordsLinked to original sources

arXiv2023

Existence and multiplicity of solutions for fractional $κ(ξ)$-Kirchhoff-type equation

In this paper, we aim to tackle the questions of existence and multiplicity of solutions to a new class of $κ(ξ)$-Kirchhoff-type equation utilizing a variational approach. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of space $ψ$-fractional $\mathcal{H}^{μ,ν;\,ψ}_{κ(ξ)}(Λ)$. In this sense, we present a few special cases and remark on the outcomes explored.

math.GM↗

arXiv2022

On tempered fractional calculus with respect to functions and the associated fractional differential equations

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the $Ψ$-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the $Ψ$-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann--Liouville type and Caputo type cases of these operators. Furthermore, we study some nonlinear fractional differential equations involving $Ψ$-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle, and proving stability results by using Grönwall type inequalities.

math.CA↗

arXiv2020

On Coupled System of Nonlinear $Ψ$-Hilfer Hybrid Fractional Differential Equations

This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of $Ψ$-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of $Ψ$-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided concrete examples to exhibit the effectiveness of our achieved results.

math.DS↗

arXiv2020

On the Boundary Value Problems of Ψ -Hilfer Fractional Differential Equations

In the current paper, we derive the comparison results for the homogeneous and non-homogeneous linear initial value problem (IVP) for $Ψ$-Hilfer fractional differential equations. In the presence of upper and lower solutions, the obtained comparison results and the location of roots theorem utilized to prove the existence and uniqueness of the solution for the linear $Ψ$-Hilfer boundary value problem (BVP) through the linear non-homogeneous $Ψ$-Hilfer IVP. Assuming the existence of lower solution $w_0 $ and upper solution $z_0 $, we establish the existence of minimal and maximal solutions for the nonlinear $Ψ$-Hilfer BVP in the line segment $[w_0,\,z_0]$ of the weighted space $C_{1-\,γ;\, Ψ}\left( J,\,\R\right)$. Further, it demonstrated that the iterative Picard type sequences that began with lower and upper solutions respectively converges to a minimal and maximal solutions, and that started with any point on a line segment converge to the exact solution of nonlinear $Ψ$-Hilfer BVP. Finally, an example is provided in support of the main results we acquired.

math.AP↗

arXiv2020

On the Nonlinear $Ψ$-Hilfer Hybrid Fractional Differential Equations

In this paper, we initially derive the equivalent fractional integral equation to $Ψ$-Hilfer hybrid fractional differential equations and through it, we prove the existence of a solution in the weighted space. The primary objective of the paper is to obtain estimates on $Ψ$-Hilfer derivative and utilize it to derive the hybrid fractional differential inequalities involving $Ψ$-Hilfer derivative. With the assistance of these fractional differential inequalities, we determine the existence of extremal solutions, comparison theorems and uniqueness of the solution.

math.DS↗

arXiv2020

On Nonlinear Hybrid Fractional Differential Equations with Atangana-Baleanu-Caputo Derivative

In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving Atangana--Baleanu--Caputo (ABC) fractional derivative. We construct the equivalent fractional integral equation and establish the existence results through it. Further, we build up the theory of inequalities for ABC--hybrid fractional differential equations and use it to examine the uniqueness, existence of a maximal and minimal solution and the comparison results.

math.DS↗

arXiv2020

Implicit Fractional Differential Equations in Banach Spaces via Picard and Weakly Picard Operator Theory

In this paper, by employing fixed-point methods, we obtain the existence and uniqueness results for the nonlinear implicit fractional differential equations in Banach spaces. Further, we obtain the uniqueness, dependence of the solution on the initial condition as well as on the functions involved on the right-hand side by means of Picard and weakly Picard operator theory and Pompeiu--Hausdorff functional.

math.DS↗

arXiv2020

Analysis of Nonlinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative

In the present paper, we determine the estimations on Atangana-Baleanu-Caputo fractional derivative at extreme points. With the assistance of the estimations obtained, we derive the comparison results. Peano's type existence results established for nonlinear fractional differential equations involving Atangana-Baleanu-Caputo fractional derivative. The acquired comparison results are then utilized to deal with the existence of local, extremal and global solution.

math.AP↗

arXiv2020

Nonlocal Boundary Value Problem for Generalized Hilfer Implicit Fractional Differential Equations

In this paper, we derive the equivalent fractional integral equation to the nonlinear implicit fractional differential equations involving $φ$-Hilfer fractional derivative subject to nonlocal fractional integral boundary conditions. The existence of a solution, Ulam-Hyers, and Ulam-Hyers-Rassias stability has been acquired by means equivalent fractional integral equation. Our investigations depend on the fixed point theorem due to Krasnoselskii and the Gronwall inequality involving $φ$-Riemann--Liouville fractional integral. An example is provided to show the utilization of primary outcomes.

math.DS↗

arXiv2019

On the Nonlinear Impulsive Volterra-Fredholm Integrodifferential Equations

In this paper, we investigate existence and uniqueness of solutions of nonlinear Volterra-Fredholm impulsive integrodifferential equations. Utilizing theory of Picard operators we examine data dependence of solutions on initial conditions and on nonlinear functions involved in integrodifferential equations. Further, we extend the integral inequality for piece-wise continuous functions to mixed case and apply it to investigate the dependence of solution on initial data through $ε$-approximate solutions. It is seen that the uniqueness and dependency results got by means of integral inequity requires less restrictions on the functions involved in the equations than that required through Picard operators theory.

math.CA↗

arXiv2019

Analysis of Volterra Integrodifferential Equations with Nonlocal and Boundary Conditions via Picard Operator

This article investigates the existence and uniqueness of solutions to the second order Volterra integrodifferential equations with nonlocal and boundary conditions through its integral equivalent equations and fixed point of Banach. Further, utilising the Picard operator theory we obtain the dependency of solutions on the initial nonlocal data and on functions involved on the right hand side of the equations.

math.CA↗

arXiv2019

On the Impulsive Implicit $Ψ$--Hilfer Fractional Differential Equations with Delay

In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam--Hyers--Mittag--Leffler stability results for impulsive implicit $Ψ$--Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam--Hyers and generalized Ulam--Hyers stability are the specific cases of Ulam--Hyers--Mittag--Leffler stability. Extended version of Gronwall inequality, abstract Gronwall lemma and Picard operator theory are the primary devices in our investigation. We give an example to illustrate the obtained results.

math.DS↗

arXiv2019

Analysis of Impulsive $φ$--Hilfer Fractional Differential Equations

This paper is concerned with the existence and uniqueness, and Ulam--Hyers stabilities of solutions of nonlinear impulsive $φ$--Hilfer fractional differential equations. Further, we investigate the dependence of the solution on the initial conditions, order of derivative and the functions involved in the equations. The outcomes are acquired in the space of weighted piecewise continuous functions by means of fixed point theorems and the generalized version of Gronwall inequality.

math.DS↗

arXiv2019

Global Existence and Ulam--Hyers Stability of $Ψ$--Hilfer Fractional Differential Equations

In this paper, we consider the Cauchy-type problem for a nonlinear differential equation involving $Ψ$-Hilfer fractional derivative and prove the existence and uniqueness of solutions in the weighted space of functions. The Ulam--Hyers and Ulam--Hyers--Rassias stability of Cauchy--type problem is investigated via successive approximation method. Further, we investigate the dependence of solutions on the initial conditions and uniqueness via $ε$-approximated solution. An example is provided to illustrate the results we obtained.

math.DS↗

arXiv2019

On Impulsive Delay Integrodifferential Equations with Integral Impulses

The present research paper is devoted to investigate the existence, uniqueness of mild solutions for impulsive delay integrodifferential equations with integral impulses in Banach spaces. We also investigate the dependence of solutions on initial conditions, parameters and the functions involved in the equations. Our analysis is based on semigroup theory, fixed point technique and an application of Pachpatte's type integral inequality with integral impulses. An example is provided in support of existence result.

math.DS↗

arXiv2019

On the Nonlinear Impulsive $Ψ$--Hilfer Fractional Differential Equations

In this paper, we consider the nonlinear $Ψ$-Hilfer impulsive fractional differential equation. Our main objective is to derive the formula for the solution and examine the existence and uniqueness of results. The acquired results are extended to the nonlocal $Ψ$-Hilfer impulsive fractional differential equation. We gave an applications to the outcomes we procured. Further, examples are provided in support of the results we got.

math.DS↗

arXiv2018

On Fractional Volterra integrodifferential equations with fractional integrable impulses

We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki's normed Banach spaces. Further, Bielecki--Ulam type stabilities have been demonstrated on a compact interval. A concrete example is provided to illustrate the outcomes we acquired.

math.DS↗

arXiv2018

On the fractional functional differential equation with abstract Volterra operator

The present paper plans to examine the existence, uniqueness and data dependence of the solution of the fractional functional differential equation with the abstract operator of Volterra, in the context of the Picard operators. We present an example, in order to illustrate the results obtained. We present an application, with the end goal to illustrate the results obtained.

math.CA↗