Now showing 1 - 5 of 5
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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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    Mild Solutions of Time Fractional Navier-Stokes Equations Driven by Finite Delayed External Forces
    (01-04-2022)
    Alam, Md Mansur
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    In this work, we consider time-fractional Navier-Stokes equations (NSE) with the external forces involving finite delay. Equations are considered on a bounded domain Ω ⊂ R3 having sufficiently smooth boundary. We transform the system of equations (NSE) to an abstract Cauchy problem and then investigate local existence and uniqueness of the mild solutions for the initial datum (Formula Presented), where r > 0 and A is the Stokes operator. With some suitable condition on initial datum we establish the global continuation and regularity of the mild solutions.We use semigroup theory, tools of fractional calculus and Banach contraction mapping principle to establish our results
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    On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay
    (01-01-2022)
    Alam, Md Mansur
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    In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions, and enlarge the existence theory with two different sets of assumptions. Under the first set of assumptions, we establish the existence of Hölder classical solution. Since the Hölder exponent appears as an exponent on the metric function in contraction inequality, it is not suitable to use Banach contraction mapping principle. Krasnoselskii’s fixed point theorem becomes effective to overcome this situation. Under the second set of assumptions, we obtain only the existence of mild solution using Schauder’s fixed point theorem. Few examples have been provided to illustrate our results.
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    Publication
    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.