Biharmonic heat equation with gradient non-linearity on Lp space

In this paper, we deal with the biharmonic heat equation with gradient non-linearity. Under the suitable condition of the initial datum, we show that the global unique existence of the mild solution. The main technique in the paper is to use Banach?s fixed point theorem in combination with the Lp-Lq evaluation of biharmonic operator.

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Topological structure of solution sets for control problems governed by semilinear fractional impulsive evolution equations with nonlocal conditions

IMA Journal of Mathematical Control and Information ◽

10.1093/imamci/dnz038 ◽

2020 ◽

Vol 37 (4) ◽

pp. 1089-1113

Author(s):

Yi-rong Jiang ◽

Qiong-fen Zhang ◽

Qi-qing Song

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Control Problem ◽

Mild Solution ◽

Point Theorem ◽

Measure Of Noncompactness ◽

Evolution Equations ◽

Nonlocal Conditions ◽

Solution Set ◽

Impulsive Evolution Equations

Abstract This article investigates the topological structural of the mild solution set for a control problem monitored by semilinear fractional impulsive evolution equations with nonlocal conditions. The $R_{\delta }$-property of the mild solution set is obtained by applying the measure of noncompactness and a fixed point theorem of condensing maps and a fixed point theorem of nonconvex valued maps. Then this result is applied to prove that the presented control problem has a reachable invariant set under nonlinear perturbations. The obtained results are also applied to characterize the approximate controllability of the presented control problem.

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Existence of mild solution of a class of nonlocal fractional order differential equation with not instantaneous impulses

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0029 ◽

2019 ◽

Vol 22 (2) ◽

pp. 495-508 ◽

Cited By ~ 1

Author(s):

Jayanta Borah ◽

Swaroop Nandan Bora

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Mild Solution ◽

Point Theorem ◽

Order Differential Equation ◽

Sufficient Conditions ◽

Banach Fixed Point Theorem ◽

Fractional Order Differential Equation ◽

Krasnoselskii’S Fixed Point Theorem ◽

Fractional Differential

Abstract In this article, we establish a set of sufficient conditions for the existence of mild solution of a class of fractional differential equations with not instantaneous impulses. The results are obtained by using Banach fixed point theorem and Krasnoselskii’s fixed point theorem. An example is presented for validation of result.

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Existence and uniqueness of mild solution for fractional integrodifferential equations of neutral type with nonlocal conditions

Mathematica Slovaca ◽

10.2478/s12175-012-0055-4 ◽

2012 ◽

Vol 62 (5) ◽

Cited By ~ 1

Author(s):

Fang Li

Keyword(s):

Banach Space ◽

Fixed Point ◽

Fixed Point Theorem ◽

Mild Solution ◽

Point Theorem ◽

Existence And Uniqueness ◽

Neutral Type ◽

Nonlocal Conditions ◽

Integrodifferential Equations

AbstractIn this paper, we prove the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations of neutral type with nonlocal conditions in a Banach space. New results are obtained by fixed point theorem.

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EXISTENCE OF SOLUTIONS FOR IMPULSIVE NEUTRAL FUNCTIONAL INTEGRO DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

Kongunadu Research Journal ◽

10.26524/krj223 ◽

2017 ◽

Vol 4 (3) ◽

pp. 1-6

Author(s):

Valliammal N ◽

Ravichandran C

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Differential Equations ◽

Mild Solution ◽

Point Theorem ◽

Existence Of Solutions ◽

Nonlocal Conditions ◽

Fractional Power ◽

Integrodifferential Equations ◽

Fractional Power Of Operators

In this paper, by using fractional power of operators and Sadovskii’s fixed point theorem, we study the existence of mild solution for a certain class of impulsive neutral functional integrodifferential equations with nonlocal conditions. The results we obtained are a generalization and continuation of the recent resultson this issue.

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Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integrodifferential Equations in Hilbert Spaces

Abstract and Applied Analysis ◽

10.1155/2014/480241 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Lijie Li

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Mild Solution ◽

Point Theorem ◽

Hilbert Spaces ◽

Existence And Uniqueness ◽

Almost Automorphic ◽

Banach’S Fixed Point Theorem ◽

Lipschitz Conditions ◽

Stochastic Functional

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

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Controllability of Impulsive ψ-Caputo Fractional Evolution Equations with Nonlocal Conditions

Mathematics ◽

10.3390/math9121358 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1358

Author(s):

Longfei Lin ◽

Yansheng Liu ◽

Daliang Zhao

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Mild Solution ◽

Point Theorem ◽

Evolution Equations ◽

Nonlocal Conditions ◽

Exact Controllability ◽

Fractional Evolution Equations ◽

Mönch Fixed Point Theorem ◽

Controllability Result

This paper is mainly concerned with the exact controllability for a class of impulsive ψ-Caputo fractional evolution equations with nonlocal conditions. First, by generalized Laplace transforms, a mild solution for considered problems is introduced. Next, by the Mönch fixed point theorem, the exact controllability result for the considered systems is obtained under some suitable assumptions. Finally, an example is given to support the validity of the main results.

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Mild Solution for a Stochastic Partial Differential Equation with Noise

WSEAS TRANSACTIONS ON SYSTEMS ◽

10.37394/23202.2020.19.29 ◽

2020 ◽

Vol 19 ◽

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Fixed Point ◽

Fixed Point Theorem ◽

White Noise ◽

Mild Solution ◽

Point Theorem ◽

Stochastic Partial Differential Equation ◽

Stochastic Equation ◽

Multiplicative White Noise

This paper focuses on the study of the existence of a mild solution to time and space-fractional stochastic equation perturbed by multiplicative white noise. The required results are obtained by means of Sadovskii’s fixed point theorem.

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