ON EVOLUTION EQUATIONS IN BANACH SPACES AND COMMUTING SEMIGROUPS A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Saud M. A. Alsulami June 2005 This dissertation entitled ON EVOLUTION EQUATIONS IN BANACH SPACES AND COMMUTING SEMIGROUPS by SAUD M. A. ALSULAMI has been approved for the Department of Mathematics and the College of Arts and Sciences by Quoc-Phong Vu Professor of Mathematics Leslie A. Flemming Dean, College of Arts and Sciences ALSULAMI, SAUD M. A. Ph.D. June 2005. Mathematics On Evolution Equations In Banach Spaces And Commuting Semigroups (102 pp.) Director of Dissertation: Quoc-Phong Vu This dissertation is concerned with several questions about the qualitative behavior of mild solutions of differential equations with multi-dimensional time on Banach spaces. For the differential equations ⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t) where A and B are linear (in general, unbounded) operators defined on a Banach space E, we give a definition of the mild solution of (∗). In order for Eq.(∗)tohave a (mild) solution, we introduce the condition(Ds − A)f2 =(Dt − B)f1 (where Ds and Dt are the partial differential operators with respect to s and t, respectively) which is understood in a generalized (mild) sense. We extend the notion of ad- missible subspaces, for closed translation-invariant subspaces M of BUC(R2,E) with respect to Eq.(∗), and we give characterization of the admissibility in terms M M of solvability of the operator equations AX − XDs = C and BX − XDt = CT . As a tool to investigate almost periodic functions of several variables which is important for the study of asymptotic behavior of Eq.(∗), we give answer to a question raised by Basit in 1971, which is an extension of the classical Bohl-Bohr theorem (to almost periodic functions with two or more variables). We also extend a theorem due to Loomis (who obtained it for functions with values in a finite dimensional space) to functions with values in Banach space E, under conditions introduced by Kadets. As an application, we obtain a result on the almost peri- odicity of a double integral of an almost periodic function defined on the plane. Applying to Eq.(∗), we show that the boundedness implies the almost periodicity for solutions of (∗) in the finite dimensional case. For the infinite dimensional case, and under the conditions that A and B are bounded linear operators, we reduce the question of the almost periodicity (resp, almost automorphicity) of the differential equations (∗) for given almost periodic (resp, almost automorphic) functions f1 and f2 to the question of the almost periodicity(resp, almost auto- morphicity) of the homogeneous system (i.e., f1 = f2 = 0). Finally, in chapter 6, we introduce the notion of C-admissible subspaces and obtain various conditions of C-admissibilities, generalizing well known results of Schuler-Vu and others. Approved: Quoc-Phong Vu Professor of Mathematics DEDICATION I would like to dedicate this work to: My parents, My grandmother, My brothers and sisters, My wife, and My kids. Acknowledgments In the Name of ALLAH[God], the Most Gracious and the Most Merciful. All praises and thanks are due to ALLAH, the Lord of all that exists. May the peace and blessings of ALLAH be upon Muhammad, the messenger of ALLAH, and his family and companions and all who follow them in righteousness until the day of judgment. I would like to express my sincere gratitude to Prof. Quoc- Phong Vu, my advisor, for his guidance and his friendship. Also, I would like to thank the members of my dissertation committee: Prof. A. Gulisashvili, Prof. H. Pasic and Prof. N. Pavel for taking their time to go through my manuscript . There is no doubt that none of this work would have been possible if it was not for the guidance of ALLAH and then for the love, patience and sacrifice of my beloved parents, grandmother and grandfather (who died while I am working on my dissertation) whose sacrifice and prayers have been a major source of strength for me all throughout my life. May ALLAH reward them with the best of reward and grant them mercy, success and happiness as long as they live and when they meet Him on the day of judgment. Also, special thanks go to each one of my brothers and my sisters whose spiritual support greatly enhanced my academic accomplishment. I am also greatly indebted to my beloved wife Um Omar for her patience and sacrifice. A special word of appreciation goes to my beloved kids. Moreover, I would like to thank all of my relatives and all of my friends. Also, I would like to thank King Abdul Aziz University for the scholarship and Ohio University for giving my the chance to continue my higher education. Last but not least, I thank those who help, encourage or pray for me during my stay in the United States. 7 Contents Abstract 3 Dedication 5 Acknowledgments 6 Introduction 9 1 Preliminaries 18 1.1 Almost periodic and almost automorphic functions from R2 into a Banachspace............................. 18 1.2 Thespectrumofvector-valuedfunctions.............. 22 1.3Semigroupsofoperators....................... 24 1.4 The operator equation AX − XB = C ............... 30 2 On mild solutions of differential equations 35 2.1 On mild solutions of differential equations with multi-time . 35 3 Regular admissibility of subspaces 50 3.1 Operator equations AX − XDs = C and BX − XDt = CT .... 50 3.2 Admissibility of difference equations . 56 4 On the integral of almost periodic functions 64 4.1OnBasit’squestion.......................... 64 4.2 Loomis’ theorem for vector-valued almost periodic functions . 69 8 4.3 An alternative form of the theorem on the integral of almost peri- odicfunctions ............................ 72 5 Almost periodic solutions of differential equations 74 5.1Thefinitedimensionalcase ..................... 74 5.2 The infinite dimensional case . 79 6 C-admissibility and analytic C-semigroups 87 6.1 C-admissibility . 87 6.2 C-semigroups and analytic C-semigroups.............. 90 6.3 Operator equation AX −XB = CD,withA generating an analytic C-semigroup.............................. 92 References 96 9 Introduction Consider the differential equation u(t)=Au(t)+f(t),t∈ R (1) where A is a linear operator on a Banach space E and f is a (continuous) function from R to E.IfA is a bounded operator, then the solution of (1) is defined by the following formula t u(t)=eA(t−s)u(s)+ eA(t−v)f(v)dv ∀t ≥ s (2) s where eA is defined by ∞ Ak eA = . (3) k=0 k! InthecasethatA is unbounded, but is generator of a C0-semigroup T (t), then (2) is replaced by t u(t)=T (t − s)u(s)+ T (t − v)f(v)dv ∀t ≥ s (4) s In general, the formula (4) is used as a definition of mild solution of equation (1). See for example: [5], [20] and [36] . The question of asymptotic behavior of (1) is intensively studied by many authors during the last decades, see e.g. [5], [16] and [54]. The primary objective of this dissertation is to initiate a systematic study of the questions of solvability and asymptotic behavior of differential equations with multi-dimensional time (or, for short, equations with multi-time). We are moti- vated by series of papers by Perov and others. See for examples: [37], [38], [39], 10 [40], [41] and [42] and the references therein. The theory of differential equations with multi-time has applications in different fields. We mention for example the multi-time Hamilton-Jacobi equations which arise from the study of problems in mathematical economics, see for example [7] and [31] and the references therein. The importance of the study of almost periodic solutions of differential equations rely on the applications in mathematical physics and astronomy (where are some objects has the almost periodic behavior). For sake of simplicity we restrict our- selves to the case of time dimension two, but we note that the results remain true for arbitrary finite time dimension. The first question we are concerned about in this dissertation is to find a general formula for mild solutions of the following differential equations ⎫ ∂u(s,t) ⎪ ∂s = Au(s, t)+f1(s, t) ⎬ ∗ ⎪ ( ) ∂u(s,t) ⎭⎪ ∂t = Bu(s, t)+f2(s, t) where A and B are commuting closed linear operators on a Banach space E. In chapter 2, we obtain the following formula for mild solutions of Eq. (∗): in order to be a mild solution of (∗)onR × R, u(s, t) must satisfy simultaneously t u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − s0) T2(t − w) f2(s0,w) dw t0 s + T1(s − v) f1(v,t) dv (s ≥ s0,t≥ t0) s0 and s u(s, t)=T1(s − s0) T2(t − t0) u(s0,t0)+ T1(s − v) T2(t − t0) f1(v,t0) dv s0 11 t + T2(t − w) f2(s, w) dw (s ≥ s0,t≥ t0) t0 Note that mild solutions of ( ∗ )onR+ × R+ are given by t u(s, t)=T1(s) T2(t) u(0, 0) + T1(s) T2(t − w) f2(0,w) dw 0 s + T1(s − v) f1(v,t) dv (s ≥ 0,t≥ 0) 0 which must simultaneously satisfy s u(s, t)=T1(s) T2(t) u(0, 0) + T1(s − v) T2(t) f1(v,0) dv 0 t + T2(t − w) f2(s, w) dw (s ≥ 0,t≥ 0) 0 We show that in order to the system (∗) to have a solution, the following condition must be satisfied: (Ds − A)f2 =(Dt − B)f1 where Ds and Dt are the partial differentiations with respect to s and t, respectively.