A variety of problems in natural sciences, such as mathematical biology, kinetics of chemical reactions, diffusion processes, and the theory of nonlinear heat conduction, lead to semilinear BVPs of the form \(-\Delta u=f(x,u)\) in \(\Omega\), \(u=g\) on \(\partial \Omega\), or to analogous problems for parabolic differential equations and ODEs, respectively. In many important cases the underlying linear differential operators satisfy maximum principles and are thus compatible with the natural order structures of appropriate Banach spaces. It is not surprising therefore that the investigation of equations in ordered Banach spaces has become an important branch in the framework of nonlinear functional analysis. To provide the reader with a first impression we give the following abstract theorem: One studies the fixed point problem (1) \(Tu=u\) and supposes that t: [v\({}_ 0,w_ 0]\subset X\to X\) is a compact monotone increasing operator on a real ordered Banach space X. Here \([v_ 0,w_ 0]=\{u\in X:\quad v_ 0\leq u\leq w_ 0\}\) denotes the order interval (sector) between \(v_ 0\) and \(w_ 0\). If \(v_ 0\) is a subsolution and if \(w_ 0\) is a supersolution of (1) (i.e. \(v_ 0\leq Tv_ 0\), \(w_ 0\geq Tw_ 0)\) then (1) has a solution. Furthermore the sequences defined by \(v_{n+1}=Tv_ n\), \(w_{n+1}=Tw_ n\) converge to the smallest [resp. greatest] fixed point of T in \([v_ 0,w_ 0]\). In this case one has much of what mathematicians like: existence, a constructive method for the determination of a solution, and error estimates \((v_ n\leq u\leq w_ n).\)
The aim of the present book is the unified treatment of a large class of ODE and PDE satisfying maximum and comparison principles. The titles of the chapters are: (1) First-order differential equations, (2) Second- order differential equations, (3) Elliptic differential equations, (4) Parabolic differential equations, (5) Hyperbolic equations. In the appendix some additional material from the theory of ODEs and PDEs is compiled (e.g., Agmon-Douglis-Nirenberg estimates, Leray-Schauder principle).
For convenience we describe the contents of Chapter 3 in detail. In Section 3.1 the authors give three comparison results for the BVP \(f(x,u,Du,D^ 2u)=0\) in \(\Omega \subset {\mathbb{R}}^ n\), \(Bu=g\) on \(\partial \Omega\), where \(Bu=p(x)u+q(x)\partial u/\partial v\) and f is assumed to be elliptic in the sense that \(f(x,u,P,Q)\leq f(x,u,P,R)\) for all \(q,r\in {\mathbb{R}}^{n^ 2}\) that satisfy the inequality \(\sum^{n}_{i,k=1}(q_{ik}-r_{ik})\xi_ i\xi_ j\leq 0\) \((\xi \in {\mathbb{R}}^ n\) arbitrary).
The subject of Section 3.2 is the differential equation \((2)\quad - Lu=f(x,u,Du)\) in \(\Omega\), \(Bu=g\) on \(\partial \Omega\), where \(L=a_{ij}D_ iD_ j+b_ iD_ i+c\) (c\(\leq 0)\) is a uniformly elliptic differential operator with \(a_{ij}\), \(b_ i\), \(c\in C^{\alpha}\). The function \(f=f(x,u,D)\) is assumed to have quadratic growth in D, that is, \(| f(x,u,D)| \leq \psi (| u|)(1+| D|^ 2).\) The existence of a classical solution of (2) in a sector \([v_ 0,w_ 0]\) can be proved provided that both a subsolution \(v_ 0\) and a supersolution \(w_ 0\) exist. If f satisfies a one-sided Lipschitz condition, which can be roughly stated as \((3)\quad f(x,u_ 1,y)-f(x,u_ 2,y)\geq -M(u_ 1-u_ 2)\) whenever \(v_ 0\leq u_ 2\leq u_ 1\leq w_ 0\), then starting with \(v_ 0\) and \(w_ 0\), respectively, two sequences \(\{v_ n\}\) and \(\{w_ n\}\) may be determined by the solution of (nonlinear) elliptic BVPs. It is shown that these sequences are monotone and converge in \(C^ 2\) to the smallest and to the greatest solution of (2), respectively. Things become more complicated in Sections 3.4 and 3.5, where systems of elliptic equations are investigated: (4) \(-L_ ku_ k=f_ k(x,u,Du_ k)\) in \(\Omega\), \(B_ ku_ k=g_ k\) on \(\partial \Omega\) \((k=1,...,N)\). Here the trick is to split \(u\in {\mathbb{R}}^ n\) into \(u=(u_ k,[u]_{b_ k},[u]_{d_ k})\) with \(b_ k+d_ k=N-1\). The functions u and v are said to be coupled quasisolutions of (4) if \(-L_ ku_ k=f_ k(x,u_ k,[u]_{b_ k},[v]_{d_ k},Du_ k),\quad -L_ kv_ k=f_ k(x,v_ k,[v]_{b_ k},[u]_{d_ k},Dv_ k)\) in \(\Omega\), and \(B_ ku_ k=B_ kv_ k=g_ k\) on \(\partial \Omega\). Coupled sub- and supersolutions are defined analogously. Under appropriate monotonicity assumptions on f and a condition similar to (3), some generalizations of the scalar results mentioned above are proven. Chapter 3 is concluded by a detailed discussion of the Belousov-Zhabotinskii reaction and an example describing a predator-prey system.
Let us have a glance at the other chapters: In Chapter 1 the basic ideas are introduced. Here, the first-order ODEs serve as an example to demonstrate the use of sub- and supersolutions and the method of monotone iterations. Besides scalar equations, systems of ODEs are also considered. The results are applied to discuss models in population dynamics. Finally, the monotone iterations technique is shown to be also useful for solving nonlinear equations. The main application in Chapter 2 is a detailed investigation of slider bearing problems. Chapter 4 follows essentially the pattern of Chapter 3. A new feature is the discussion of flow invariance in Sections 4.1 and 4.4. Combustion problems and population dynamics are the interesting practical examples here. The first part of Chapter 5 is devoted to hyperbolic first-order equations of the form \((5)\quad u_ t+\sum^{n}_{i=1}f_ i(t,x)u_{x_ i}=g(t,x,u),\) \(u(0,x)=g(x)\), \(x\in \prod^{n}_{i=1}[a_ i,b_ i]\), \(t\in [0,T]\). For this equation a comparison theorem and an existence result are derived. Under additional hypotheses one may build up monotone sequences and gain error estimates for solutions of (5). The equation (6) \(u_{xy}=f(x,y,u,u_ x,u_ y)\) is the subject of the rest of this chapter. A comparison result is proved, and it is shown that even in this case a monotone iteration may be constructed.
The reviewer is sure that the book, based on research work of the authors for the most part, is welcome since it comprises the results of numerous papers hitherto dispersed in the journal literature. It presents the material in a very systematic way and the treatment is self-consistent and clear.
In conclusion, it should be emphasized that the book neither develops nor uses the theory of operators in ordered Banach spaces explicitly. Thus it is also accessible to mathematically educated nonmathematicians. Readers wanting to see the interplay between the functional analytic theory and the concrete applications are advised additionally to read Chapter 7 in Volume I of E. Zeidler’s book [“Nonlinear functional analysis and its applications. II: Fixed-point theorems.” (1986; Zbl 0583.47050)].