UPPER AND LOWER SOLUTION METHOD FOR FRACTIONAL EVOLUTION EQUATIONS WITH ORDER 1 < α < 2

Abstract

Abstract. In this work, we investigate the existence of the extremalsolutions for a class of fractional partial differential equations with or-der 1 < α < 2 by upper and lower solution method. Using the theoryof Hausdorff measure of noncompactness, a series of results about thesolutions to such differential equations is obtained. 1. IntroductionFractionalorderdifferentialequationhasbroadapplicationsin resolvingreal-world problems, and as such it attracted researchers’ attention from differentareas. In orderto evaluate the behaviorsoffractional orderdifferential equationbased models, one need to know the properties of such equation systems, inparticular, the existence of solutions to such equations. Recently, the existenceof solutions to different forms of fractional differential equation systems hasbeen investigated [1, 2, 3, 4, 5, 9, 10, 11, 15, 16, 19, 20, 21, 23, 26, 27, 29, 30,32, 33, 34].Using the upper and lower solution method to study the existence of ex-tremal solutions for fractional differential equations is an interesting topic ofresearch, which has been gaining increasing attention recently [1, 15, 16, 20,23, 27, 30, 32, 33, 34]). Presently, the upper and lower solution methodis widely used to investigate fractional ordinary differential equations (see[1, 15, 20, 23, 30, 32, 33, 34]). However, this method is seldom used to studysemilinear fractional evolution equations. [27] considered the existence of ex-tremal solutions to the following semilinear fractional evolution equation(1.1)(