Fractional evolution equations with nonlocal conditions in partially ordered Banach space

2018 ◽  
Vol 34 (3) ◽  
pp. 379-390
Author(s):  
HEMANT KUMAR NASHINE ◽  
◽  
HE YANG ◽  
RAVI P. AGARWAL ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Coupled Fixed Point ◽  
Positive Cone ◽  
Bounded Linear ◽  
Partially Ordered ◽  
C0 Semigroup ◽  
Uniformly Bounded

In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form where C Dσ t denotes the Caputo fractional derivative of order σ ∈ (0, 1), −A : D(A) ⊂ X → X generates a positive C0-semigroup T(t)(t ≥ 0) of uniformly bounded linear operator in X, b > 0 is a constant, f is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone K is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.

Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Min Yang ◽  
Qiru Wang
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Hilfer Fractional Derivative ◽  
Fractional Evolution Equations ◽  
Noncompact Measure ◽  
The Fixed Point Theorem ◽  
New Criteria

AbstractIn this paper, we consider a class of evolution equations with Hilfer fractional derivative. By employing the fixed point theorem and the noncompact measure method, we establish a number of new criteria to guarantee the existence and uniqueness of mild solutions when the associated semigroup is compact or not.

scholarly journals Existence of solutions for delay evolution equations with nonlocal conditions

2017 ◽  
Vol 15 (1) ◽  
pp. 616-627 ◽  
Author(s):  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Fixed Point ◽  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Existence Of Solutions ◽  
Fixed Point Theory ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Point Theory ◽  
Existence Results ◽  
Kuratowski Measure Of Noncompactness

Abstract In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

Existence Results of Mild Solutions for Impulsive Fractional Integrodifferential Evolution Equations With Nonlocal Conditions

2019 ◽  
Vol 20 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Xuping Zhang ◽  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Existence Results ◽  
Cosine Family

AbstractIn this paper, we investigate the existence of mild solutions of impulsive fractional integrodifferential evolution equations with nonlocal conditions via the fixed point theorems and fractional cosine family combined with solutions operator theorems. Our results improve and generalize some classical results. Finally, an example is given to illustrate the main results.

Fractional Retarded Evolution Equations with Measure of Noncompactness Subjected to Mixed Nonlocal Plus Local Initial Conditions

2018 ◽  
Vol 19 (1) ◽  
pp. 69-81
Author(s):  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Measure Of Noncompactness ◽  
Evolution Equations ◽  
Fixed Point Theory ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Point Theory ◽  
Linear Operators ◽  
Nonlinear Operators ◽  
Bounded Linear ◽  
Uniformly Bounded

AbstractWe consider the fractional retarded evolution equations $$^{C}D_{t}^{q}u(t)+Au(t)=f\Big(t,u_t,\int_{0}^{t}w(t,s,u_s)ds\Big),\quad t\in[0,a],$$where $^{C}D_{t}^{q}$, $q\in(0,1]$, is the fractional derivative in the Caputo sense, $-A$ is the infinitesimal generator of a $C_0$-semigroup of uniformly bounded linear operators $T(t)$$(t\geq0)$ on a Banach space $X$ and the nonlinear operators $f$ and $w$ are given functions satisfying some assumptions, subjected to a general mixed nonlocal plus local initial condition of the form $u(t)=g(u)(t)+\phi(t)$, $t\in[-h,0]$. Under more general conditions, the existence of mild solutions and positive mild solutions are obtained by means of fractional calculus and fixed point theory for condensing maps. Moreover, we present an example to illustrate the application of abstract results.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

scholarly journals Some New Coupled Coincidence Point and Coupled Fixed Point Results in Partially Ordered Metric-Like Spaces and an Application

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Min Liang ◽  
Chuanxi Zhu ◽  
Zhaoqi Wu ◽  
Chunfang Chen
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Fixed Point Theorems ◽  
Coincidence Point ◽  
Coupled Fixed Point ◽  
Coupled Coincidence Point ◽  
Nonlinear Integral Equations ◽  
Partially Ordered ◽  
Coupled Coincidence

Some new coupled coincidence point and coupled fixed point theorems are established in partially ordered metric-like spaces, which generalize many results in corresponding literatures. An example is given to support our main results. As an application, we discuss the existence of the solutions for a class of nonlinear integral equations.

Sign in / Sign up

Export Citation Format

Share Document