Abstract Cauchy problem for fractional differential equations

In this paper, a generalized Darbo’s fixed-point theorem associated with Hausdorff measure of noncompactness is established. Then we apply this new variant fixed-point theorem to study some fractional differential equations in Banach spaces via the technique of measure of noncompactness. Many novel existence and uniqueness results for solutions are obtained under the more general conditions.

[1]  Yong Zhou,et al.  Analysis of nonlinear fractional control systems in Banach spaces , 2011 .

[2]  Lishan Liu,et al.  Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces , 2005 .

[3]  Yong-Kui Chang,et al.  Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order , 2009 .

[4]  V. E. Tarasov Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[5]  I. Podlubny Fractional differential equations , 1998 .

[6]  Juan J. Nieto,et al.  Global attractivity for nonlinear fractional differential equations , 2012 .

[7]  Yong Zhou,et al.  A class of fractional evolution equations and optimal controls , 2011 .

[8]  Johnny Henderson,et al.  Existence results for fractional order functional differential equations with infinite delay , 2008 .

[9]  Yong Zhou,et al.  On the concept and existence of solution for impulsive fractional differential equations , 2012 .

[10]  Bashir Ahmad,et al.  Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory , 2010 .

[11]  Johnny Henderson,et al.  Fractional functional differential inclusions with finite delay , 2009 .

[12]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[13]  V. Lakshmikantham,et al.  Nonlinear differential equations in abstract spaces , 1981 .

[14]  K. Deimling Nonlinear functional analysis , 1985 .

[15]  Shuqin Zhang,et al.  Existence of positive solution for some class of nonlinear fractional differential equations , 2003 .

[16]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[17]  Hans-Peter Heinz,et al.  On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions , 1983 .

[18]  Yong Zhou,et al.  Nonlocal Cauchy problem for fractional evolution equations , 2010 .

[19]  Yong Zhou,et al.  EXISTENCE AND UNIQUENESS FOR FRACTIONAL NEUTRAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY , 2009 .

[20]  A. Luo,et al.  Fractional Dynamics and Control , 2011 .

[21]  Chuanzhi Bai,et al.  Impulsive periodic boundary value problems for fractional differential equation involving Riemann–Liouville sequential fractional derivative , 2011 .

[22]  Rong-Nian Wang,et al.  Abstract fractional Cauchy problems with almost sectorial operators , 2012 .

[23]  R. Agarwal,et al.  A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions , 2010 .

[24]  Johnny Henderson,et al.  Corrigendum to: “Fractional Functional Differential Inclusions with Finite Delay” [Nonlinear Anal. 70(5) (2009) 2091–2105] , 2010 .

[25]  Yong Zhou,et al.  Existence and controllability results for fractional semilinear differential inclusions , 2011 .

[26]  Yong Zhou,et al.  On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay , 2011, Journal of Optimization Theory and Applications.

[27]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[28]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .