##
**Nonlinear differential equations in ordered spaces.**
*(English)*
Zbl 0948.34001

Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. 111. Boca Raton, FL: Chapman & Hall/CRC. vi, 323 p. (2000).

In 1994, the monograph [Monotone iterative techniques for discontinuous nonlinear differential equations. New York: Marcel Dekker, Inc. (1994; Zbl 0804.34001)], by the second author of this book and V. Lakshmikantham, showed that the monotone iterative technique coupled with the method of upper and lower solutions could be treated from an abstract point of view, allowing the consideration of discontinuous differential equations.

During the last few years the authors, S. Carl and S. Heikkilä, maintained a very important research activity in this subject, obtaining new generalizations of the method of upper and lower solutions to investigate the existence of extremal solutions of different explicit and implicit problems for ordinary, functional and partial differential equations. The obtained results constitute the basis of the contents of the present monograph. However, the important point to note here is the fact that the authors present again their results from an abstract point of view, which permits them to study many different problems in a systematic way. Indeed, most of the problems considered in the book can be represented as an abstract equation \(Lu=Nu\), where \(L\) and \(N\) are mappings from a partially ordered set to an ordered normed space. Here, the operator \(L\) is not necessarily linear as in previous works that use this abstract scheme [see, for instance, Juan J. Nieto, Nonlinear Anal., Theory Methods Appl. 28, No. 12, 1923-1933 (1997; Zbl 0883.47058)]. Moreover, \(N\) may depend on \(Lu\), which permits the authors to consider also implicit problems, and they also emphasize the generality of the problems under consideration based on the fact that the nonlinearities are allowed to depend discontinuously of the solution to the problem.

Although the book is mainly devoted to derive results on the existence of extremal solutions to different problems between a lower and an upper solution, also uniqueness and well-posedness results, as well as conditions which ensure the compactness of the solution set are given.

The book consists of seven chapters and an appendix which contains basic results on ordered spaces, Sobolev spaces, pseudomonotone and quasilinear elliptic operators, nonlinear first-order evolution equations and nonsmooth analysis. Each chapter contains a useful section with bibliographical notes. Next we describe briefly the contents of the chapters.

In Chapter 1, some existence and comparison results for extremal solutions to the abstract equation \(Lu=Nu\) are derived from a fixed point theorem for an increasing map defined in an ordered normed space. These results are the basis of many applications in the remaining chapters of the book. This applicability is illustrated with several results for explicit and implicit problems for ordinary and partial differential equations. Some numerical examples are provided.

Chapter 2 is devoted to derive existence and comparison results for first-order ordinary differential equations. The results of Chapter 1 permit the authors to consider also implicit differential equations with functional dependence. Sections 2.1 and 2.2 deal with the generalized first-order differential equation \(\frac{d}{dt}\varphi(u(t))=g(t,u(t))\) for a.e. \(t\in J=[t_0,t_1]\), where \(\varphi\) is an increasing homeomorphism. The existence of extremal solutions to this equation is investigated both for initial conditions and functional-boundary conditions of the form \(B(u(t_0),u)=0\), where \(B: \mathbb{R}\times C(J)\to \mathbb{R}\). In Section 2.3 functional dependence on the nonlinearity in the form \(\frac{d}{dt}\varphi(u(t))=g(t,u(t),u)\) is taken up. Finally, using the abstract results of Chapter 1 and the results of section 2.3, the authors investigate in section 2.4 the existence of extremal solutions to an implicit first-order differential equation with implicit boundary conditions.

Chapter 3 presents uniqueness, comparison and well-posedness results to initial and boundary value problems for first- and second-order quasilinear differential equations. Sections 3.1, 3.2 and 3.3 deal with explicit and implicit first-order problems and include some comparison principles which are used to derive some results on comparison, uniqueness and continuous dependence of the solutions. Section 3.4 is devoted to obtain comparison and uniqueness results for the second-order quasilinear equation \(\frac{d}{dt}\varphi(t,u'(t))=g(t,u(t),u'(t))\) with separated, periodic, Neumann or Dirichlet boundary conditions.

In Chapter 4, existence and comparison results for second-order ordinary and functional differential equations are discussed. Sections 4.1 and 4.2 are devoted to the study of explicit and implicit Sturm-Liouville problems. Special cases where the existence of extremal solutions can be proved by the method of successive approximations are considered in section 4.3. Finally, explicit and implicit functional phi-Laplacian differential equations with functional initial conditions are discussed in sections 4.4 and 4.5.

In Chapter 5, the authors present some existence and extremality results for the Dirichlet problem for general quasilinear elliptic and parabolic problems. Compactness results of the solution set among the given upper and lower solutions are also proved.

Chapter 6 is dedicated to develop a method of upper and lower solutions for differential inclusions of hemivariational type in the form \(h\in Au+\partial Ju\) in \(X^*\), where \(X^*\) is the dual space of a reflexive Banach space \(X\), \(A:X\to X^*\) is a pseudomonotone and coercive operator, and \(\partial Ju\) denotes Clarke’s generalized gradient of a locally Lipschitz functional \(J:X\to \mathbb{R}\). The existence of extremal solutions in a sector determined by an upper and a lower solution and the compactness of the solution set are investigated.

Finally, in Chapter 7 some existence results for discontinuous implicit elliptic and parabolic problems with initial-Dirichlet boundary conditions are provided. The existence of extremal solutions is proved using the results given in Chapter 1 for the abstract equation \(Lu=Nu\). Sections 7.2 and 7.3 are devoted to an implicit parabolic equation in the form \(\Lambda u=f(x,t,u,\Lambda u)\), where \(\Lambda\) is a semilinear parabolic differential operator. In section 7.4 the implicit elliptic equation \(\Lambda u=f(x,u,\Lambda u)\), with \(\Lambda\) a semilinear elliptic operator, is discussed. For both equations, the function \(f\) is allowed to be discontinuous in all its arguments.

To end this review, I would like to point out that the book is self-consistent and well organized. In the reviewer’s opinion, it will serve as a basis for many further developments in the study of the extremal solutions to discontinuous implicit and explicit problems for ordinary, functional and partial differential equations.

During the last few years the authors, S. Carl and S. Heikkilä, maintained a very important research activity in this subject, obtaining new generalizations of the method of upper and lower solutions to investigate the existence of extremal solutions of different explicit and implicit problems for ordinary, functional and partial differential equations. The obtained results constitute the basis of the contents of the present monograph. However, the important point to note here is the fact that the authors present again their results from an abstract point of view, which permits them to study many different problems in a systematic way. Indeed, most of the problems considered in the book can be represented as an abstract equation \(Lu=Nu\), where \(L\) and \(N\) are mappings from a partially ordered set to an ordered normed space. Here, the operator \(L\) is not necessarily linear as in previous works that use this abstract scheme [see, for instance, Juan J. Nieto, Nonlinear Anal., Theory Methods Appl. 28, No. 12, 1923-1933 (1997; Zbl 0883.47058)]. Moreover, \(N\) may depend on \(Lu\), which permits the authors to consider also implicit problems, and they also emphasize the generality of the problems under consideration based on the fact that the nonlinearities are allowed to depend discontinuously of the solution to the problem.

Although the book is mainly devoted to derive results on the existence of extremal solutions to different problems between a lower and an upper solution, also uniqueness and well-posedness results, as well as conditions which ensure the compactness of the solution set are given.

The book consists of seven chapters and an appendix which contains basic results on ordered spaces, Sobolev spaces, pseudomonotone and quasilinear elliptic operators, nonlinear first-order evolution equations and nonsmooth analysis. Each chapter contains a useful section with bibliographical notes. Next we describe briefly the contents of the chapters.

In Chapter 1, some existence and comparison results for extremal solutions to the abstract equation \(Lu=Nu\) are derived from a fixed point theorem for an increasing map defined in an ordered normed space. These results are the basis of many applications in the remaining chapters of the book. This applicability is illustrated with several results for explicit and implicit problems for ordinary and partial differential equations. Some numerical examples are provided.

Chapter 2 is devoted to derive existence and comparison results for first-order ordinary differential equations. The results of Chapter 1 permit the authors to consider also implicit differential equations with functional dependence. Sections 2.1 and 2.2 deal with the generalized first-order differential equation \(\frac{d}{dt}\varphi(u(t))=g(t,u(t))\) for a.e. \(t\in J=[t_0,t_1]\), where \(\varphi\) is an increasing homeomorphism. The existence of extremal solutions to this equation is investigated both for initial conditions and functional-boundary conditions of the form \(B(u(t_0),u)=0\), where \(B: \mathbb{R}\times C(J)\to \mathbb{R}\). In Section 2.3 functional dependence on the nonlinearity in the form \(\frac{d}{dt}\varphi(u(t))=g(t,u(t),u)\) is taken up. Finally, using the abstract results of Chapter 1 and the results of section 2.3, the authors investigate in section 2.4 the existence of extremal solutions to an implicit first-order differential equation with implicit boundary conditions.

Chapter 3 presents uniqueness, comparison and well-posedness results to initial and boundary value problems for first- and second-order quasilinear differential equations. Sections 3.1, 3.2 and 3.3 deal with explicit and implicit first-order problems and include some comparison principles which are used to derive some results on comparison, uniqueness and continuous dependence of the solutions. Section 3.4 is devoted to obtain comparison and uniqueness results for the second-order quasilinear equation \(\frac{d}{dt}\varphi(t,u'(t))=g(t,u(t),u'(t))\) with separated, periodic, Neumann or Dirichlet boundary conditions.

In Chapter 4, existence and comparison results for second-order ordinary and functional differential equations are discussed. Sections 4.1 and 4.2 are devoted to the study of explicit and implicit Sturm-Liouville problems. Special cases where the existence of extremal solutions can be proved by the method of successive approximations are considered in section 4.3. Finally, explicit and implicit functional phi-Laplacian differential equations with functional initial conditions are discussed in sections 4.4 and 4.5.

In Chapter 5, the authors present some existence and extremality results for the Dirichlet problem for general quasilinear elliptic and parabolic problems. Compactness results of the solution set among the given upper and lower solutions are also proved.

Chapter 6 is dedicated to develop a method of upper and lower solutions for differential inclusions of hemivariational type in the form \(h\in Au+\partial Ju\) in \(X^*\), where \(X^*\) is the dual space of a reflexive Banach space \(X\), \(A:X\to X^*\) is a pseudomonotone and coercive operator, and \(\partial Ju\) denotes Clarke’s generalized gradient of a locally Lipschitz functional \(J:X\to \mathbb{R}\). The existence of extremal solutions in a sector determined by an upper and a lower solution and the compactness of the solution set are investigated.

Finally, in Chapter 7 some existence results for discontinuous implicit elliptic and parabolic problems with initial-Dirichlet boundary conditions are provided. The existence of extremal solutions is proved using the results given in Chapter 1 for the abstract equation \(Lu=Nu\). Sections 7.2 and 7.3 are devoted to an implicit parabolic equation in the form \(\Lambda u=f(x,t,u,\Lambda u)\), where \(\Lambda\) is a semilinear parabolic differential operator. In section 7.4 the implicit elliptic equation \(\Lambda u=f(x,u,\Lambda u)\), with \(\Lambda\) a semilinear elliptic operator, is discussed. For both equations, the function \(f\) is allowed to be discontinuous in all its arguments.

To end this review, I would like to point out that the book is self-consistent and well organized. In the reviewer’s opinion, it will serve as a basis for many further developments in the study of the extremal solutions to discontinuous implicit and explicit problems for ordinary, functional and partial differential equations.

Reviewer: Eduardo Liz (Vigo)

### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |