Almost Periodic Type Solutions Of Differential Equations With Strictly Monotone Operators

The theory of almost periodic functions was mainly created by the Danish math-ematician H. Bohr during 1925-1926. Almost periodic functions are the class of con-tinuous functions possessing certain structural properties and are a generalization ofpure periodicity. The theory was developed further by some mathematicians. Onedirection is the broader study of functions of almost periodic type, another directionis the application in certain subject, in particular, the existence of the almost periodictype solutions of equations.The paper consists of two parts:one part concerns the existence and uniquenessof the solutions of some one-order almost periodic type differential equations, theother part concerns the existence and uniqueness of the solutions of some second-order almost periodic type differential equations with gradient operators.Some people discussed the almost periodic solutions of differential equationswith strictly monotone operators, and they have given some important results. Sara-son defined slowly oscillating functions. To our knowledge, nobody has applied suchfunctions to the theory of differential equations with strictly monotone operators. In1941, asymptotically almost periodic functions was originally introduced by Fre′chet.Asymptotically almost periodic functions are a generalization of almost periodic func-tions. To our knowledge, nobody has studied the existence and uniqueness of asymp-totically almost periodic solutions of the differential equations with strictly monotoneoperators.Second-order equations with gradient operators is a kind of special nonlineardifferential equations. The equations have applications in certain chemical industryand electron. By means of variational principle, at the beginning of 1990s peopleprovided results about the existence of the solutions of such equations. Because ofthe convexity of functions, the existence and uniqueness of almost periodic solutionsof such equations have been solved. In the following years people deeply study widersecond-order equations with gradient operators and get many perfect results of theexistence and uniqueness of almost periodic solutions of such equations.In recent year, more and more people have given important contributions to the questions of almost periodic solutions of the differential equations with infinitesimalgenerators of semigroups. But slowly oscillating solutions of these differential equa-tions have seldom been discussion.In this paper, we mainly solve the following questions:Firstly, we give the relation of slowly oscillating functions and differential equa-tions with strictly monotone operators. By means of reduction to absurdity, the au-thors discuss a necessary and a sufficient conditions for the existence and uniquenessof slowly oscillating solutions of the equations. Then we give examples to show thatthey can't be necessary and sufficient conditions. Particularly, as a special class, theauthors give necessary and sufficient conditions for the existence and uniqueness ofslowly oscillating solutions of the differential equations with gradient operators.Secondly, By means of the properties of asymptotically almost periodic func-tions, the authors discuss a necessary and a sufficient conditions for the existence anduniqueness of asymptotically almost periodic solutions of the equations with strictlymonotone operators. Then we give examples to show that they can't be necessary andsufficient conditions. Particularly, as a special class, the authors give necessary andsufficient conditions for the existence and uniqueness of asymptotically almost peri-odic solutions for the differential equations with gradient operators. Then we extendthe results to some second-order equations.Thirdly, we solve the problem of existence and uniqueness of asymptoticallyalmost periodic solutions of the second-order equations with gradient operators. Bymeans of the properties of asymptotically almost periodic functions, we get the exis-tence on R+ of asymptotically almost periodic solutions of the equations. On the otherhand, we transform the equations. Thus we present existence and uniqueness theoremfor asymptotically almost periodic solutions of the equations by means of iterationand the conditions for the existence and uniqueness of almost periodic solutions ofcorrelative linear differential equations.Fourthly, we discuss some conditions for the existence and uniqueness of slowlyoscillating (mild) solutions of the differential equations with infinitesimal generatorsof semigroups by Banach fixed point theorem.