Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions
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RESEARCH
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Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions Jia Mu* *
Correspondence: [email protected] School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Lanzhou, Gansu, People’s Republic of China
Abstract In this article, the theory of positive semigroup of operators and the monotone iterative technique are extended for the impulsive fractional evolution equations with nonlocal initial conditions. The existence results of extremal mild solutions are obtained. As an application that illustrates the abstract results, an example is given. Keywords: impulsive fractional evolution equations; nonlocal initial conditions; extremal mild solutions; monotone iterative technique
1 Introduction In this article, we use the monotone iterative technique to investigate the existence of extremal mild solutions of the impulsive fractional evolution equation with nonlocal initial conditions in an ordered Banach space X ⎧ α ⎪ ⎪ ⎨D u(t) + Au(t) = f (t, u(t)), t ∈ I, t = tk , u|t=tk = Ik (u(tk )), k = , , . . . , m, ⎪ ⎪ ⎩ u() + g(u) = x ∈ X,
(.)
where Dα is the Caputo fractional derivative of order < α < , A : D(A) ⊂ X → X is a linear closed densely defined operator, –A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators T(t) (t ≥ ), I = [, T], T > , = t < t < t < · · · < tm < tm+ = T, f : I × X → X is continuous, g : PC(I, X) → X is continuous (PC(I, X) will be defined in Section ), the impulsive function Ik : X → X is continuous, u|t=tk = u(tk+ ) – u(tk– ), where u(tk+ ) and u(tk– ) represent the right and left limits of u(t) at t = tk , respectively. Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary real or complex order. The subject is as old as differential calculus, and goes back to the time when Leibnitz and Newton invented differential calculus. Fractional derivatives have been extensively applied in many fields which have been seen an overwhelming growth in the last three decades. Examples abound: models admitting backgrounds of heat transfer, viscoelasticity, electrical circuits, electro-chemistry, economics, polymer physics, and even biology are always concerned with fractional derivative [–]. Fractional evolution equations have attracted many researchers in recent years, for example, see [–]. © 2012 Mu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Mu Boundary Value Problems 2012, 2012:71 http://www.boundaryvalueproblems.com/content/2012/1/71
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A strong motivation for investigating the problem (.) comes form physics. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and
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