Now showing 1 - 6 of 6
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    Solutions to fractional functional differential equations with nonlocal conditions
    (01-01-2014) ;
    Sharma, Madhukant
    In this paper, we discuss the solutions to nonlocal initial value problems of fractional order functional differential equations in a Banach space. In particular, we prove the existence and uniqueness of mild and classical solutions assuming that -A generates a resolvent operator family and nonlinear part is a Lipschitz continuous function. We also investigate the global existence of the solution. At the end, a fractional order partial differential equation is given to illustrate the obtained abstract results. © 2014 Versita Warsaw and Springer-Verlag Wien.
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    Controllability of Sobolev type nonlinear nonlocal fractional functional integrodifferential equations
    (01-10-2015)
    Sharma, Madhukant
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    This paper deals with the controllability of mild solution for a class of Sobolev type nonlinear nonlocal fractional order functional integro-differential equations in a general Banach space X. We use fractional calculus, Krasnoselskii's fixed point theorem and semigroup theory for the main results and render the criteria for the complete controllability of considered problem. We also investigate the null controllability. An application is given to illustrate the abstract results.
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    Analysis of Fractional Functional Differential Equations of Neutral Type with Nonlocal Conditions
    (01-10-2017)
    Sharma, Madhukant
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    This work deals with the existence of solutions for a class of nonlinear nonlocal fractional functional differential equations of neutral type in Banach spaces. In particular, we prove the existence of solutions with the assumptions that the nonlinear parts satisfy locally Lipschitz like conditions and closed linear operator - A(t) generates analytic semigroup for each t≥ 0. We also investigate global existence of solution and study the continuous dependence of solution on initial data. We conclude the article with an application to the developed results.
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    Solvability and Controllability of a Retarded-Type Nonlocal Non-Autonomous Fractional Differential Equation
    (01-01-2023)
    Sharma, Madhukant
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    This paper considers a non-autonomous retarded-type fractional differential equation involving Caputo derivative along with a nonlocal condition in a general Banach space. We present a novel approach to determine the existence-uniqueness and controllability of mild solution to the considered problem using the fixed-point technique, classical semigroup theory, and tools of fractional calculus. It is imperative to mention that the main results are established without assuming the continuity of linear operator −A(t) and compactness condition on semigroup. At the end, the developed theoretical results have been applied to a nonlocal fractional order retarded elliptic evolution equation.
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    Existence of Solutions to Sobolev Type Nonlocal Nonlinear Functional Integrodifferential Equations Involving Caputo Derivative
    (01-10-2022)
    Sharma, Madhukant
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    In this article, we consider a nonlinear Sobolev type fractional functional integrodifferential equations in a Banach space along with a nonlocal condition. Sufficient conditions for existence, uniqueness and dependence on initial data of local solutions of considered problem are derived by employing fixed point techniques and theory of classical semigroup. Further, we also render the criteria for existence of global solution. At the end, we provide an application to elaborate the obtained results.
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    Controllability of nonlocal fractional functional differential equations of neutral type in a Banach space
    (01-01-2015)
    Sharma, Madhukant
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    This paper is concerned with the controllability of nonlinear nonlocal fractional neutral functional evolution system in a Banach space. Sufficient conditions are obtained by using Krasnoselskii's fixed point theorem and semigroup theory. In particular, we assume that the nonlinear parts satisfy locally Lipschitz like conditions and closed linear (not necessarily bounded) operator -A(t) generates analytic semigroup for each t ≥ 0. We also investigate null controllability of the considered system. An example is given to illustrate the effectiveness of our results.