Monotone iterative techniques for discontinuous nonlinear differential equations.

*(English)*Zbl 0804.34001The main purpose of this monograph is to present a systematic presentation of a generalized monotone iterative method which is appropriate for the study of upper and lower solutions of discontinuous nonlinear problems.

Dealing with several problems in natural sciences the underlying operator describing the evolution is compatible with the natural ordering structure of a suitable space. It is therefore justified the fact that such problems are usually investigated in the framework of ordered function spaces. The abstract setting of such a theory has now become an important branch in nonlinear functional analysis. Most of the first chapter of this book (sections 1.1 to 1.4) is devoted to those facts from this abstract theory needed in the sequel. Indeed, after introducing the basic concepts related to partially ordered sets, an abstract recursion principle is presented which forms a basis to the generalized monotone iteration method and its application to prove a fixed point theorem for a mapping \(G: P\to P\) where \(P\) is a poset. Such an example is given in Theorem 1.2.1.: Assume that \(P\) is a poset, \(G: P\to P\) a nondecreasing mapping and \(a\in P\) being such that \(a\leq Ga\) and \(G\) is nondecreasing on \([a):= \{u\in P: a\leq u\}\). If \(x_ *= \sup G[C]\), where \(C\) is the well-ordered chain of \(G\)-iterations of \(a\), then \(x_ *\) is the least fixed point of \(G\) in \([a)\) and \(x_ *= \max C=\min \{y\in [a): Gy\leq y\}\). But the central role in developing the theory of discontinuous differential equations is played by some results given for instance in Theorem 1.2.3.: Let \(Y\) be a subset of an ordered metric space \(X\), \([a,b]\) a nonempty order interval of \(Y\), and \(G: [a,b]\to [a,b]\) a nondecreasing mapping. If \(\sup G[C]\) and \(\inf G[\overline C]\) exist in \(Y\) where \(C\) is the well ordered chain of \(G\)-iterations of \(a\) and \(\overline C\) is the inversely well-ordered chain of \(G\)-iterations of \(b\), then \(x_ *= \max C\) is the least fixed point of \(G\), and \(x^*= \min\overline C\) is the greatest fixed point of \(G\) and \(x_ *= \min\{y\mid Gy\leq y\}\), \(x^*= \max\{y\mid y\leq Gy\}\) hold. What is most important in the applications of the method of upper and lower solutions is the meaning of mixed monotone operators. Let \(P\) be a poset; a mapping \(A: p\times p\to p\) is called mixed monotone if \(A(\cdot, z)\) is nondecreasing and \(A(z,\cdot)\) is nonincreasing for each \(z\in P\). A point \((v,w)\) is a coupled fixed point of \(A\) if \(v= A(v,w)\) and \(w= A(w,v)\). Proposition 1.2.3.: Let \(P\) be a poset. A mixed monotone and order bounded mapping \(A: P\times P\to P\) has an extremal coupled fixed point if \(A[C]\) has supremum and infimum in \(P\) whenever \(C\) is a well- ordered or an inversely well-ordered chain in \(P\times P\) with the componentwise ordering. From Section 1.3 and on several function spaces are considered where some properties like ordering, Bochner integrability and a.e. differentiability are studied as well as some fixed point theorems are given. Finally, Section 1.5 is devoted to the study of the method of upper and lower solutions of Carathéodory systems.

Chapter 2 is devoted to applications of the generalized monotone iterative technique to discontinuous ordinary differential equations. First the authors investigate the initial value problem \((*)\) \(x'= f(t,x,x)\), \(x(0)= x_ 0\) and some special cases of it. Existence of extremal solutions (subsection 2.1.1) and existence of extremals among all solutions (subsection 2.1.2) are discussed when \(f(t,x,y)\) can be discontinuous even in all of its arguments (subsection 2.1.3). The dependence of the extremal solutions on the data as well as the existence of extremal solutions of the periodic boundary value problem \(x'= f(t,x,x)\), \(x(0)= x(T)\) are investigated in subsections 2.1.4 and 2.3.1. Almost the same problems are investigated when the previous equation is replaced with \((**)\) \(x'= f(t,x,x,x)\), when \(f(t,x,y,z)\) is nondecreasing in \(y\) and nonincreasing in \(z\). The new concept here is the coupled quasisolutions, namely a pair of absolutely continuous functions \(y\), \(z\) satisfying \(y'(t)= f(t,y(t),y(t),z(t))\) a.e., \(z'(t)= f(t,z(t),z(t),y(t))\), a.e. and \(y(0)= z(0)= x_ 0\). This is the so- called mixed monotone IVP. Sections 2.4, 2.5, 2.6 and 2.7, 2.8, 2.9 are respectively devoted to finite and infinite differential systems where problems as for \((*)\) and \((**)\) are investigated. Notice that many of the results of this chapter are new.

Chapter 3 deals with the applicability of the method of upper and lower solutions coupled with a generalized monotone iterative technique to second order discontinuous nonlinear boundary value problems. First the authors consider equations of Carathéodory type and existence results are given. Then they discuss existence of extremal solutions for the \(\text{BVP}- x''= f(t,x,x,x')\), \(B_ j x(t_ j)= a_ j x(t_ j)- (- 1)^ j b_ j x'(t_ j)= c_ j\). Section 3.2 is devoted to second order mixed monotone BVPs involving both continuous and mixed monotone type dependencies on the dependent variable, while in Section 3.3 and 3.4 several results derived for scalar BVPs are extended to finite (subsections 3.3.1, 3.4.1) and infinite (subsections 3.3.2, 3.4.2) discontinuous differential systems.

Chapter 4 is devoted to the study of existence of weak extremal solutions of second order partial differential equations and systems of elliptic and parabolic type, where the dependence contains discontinuous nonlinearities. The first main result is contained in subsection 4.1.3 where an existence result for extremal solutions to a quasilinear elliptic BVP is provided in case both continuous and discontinuous dependence in the equation and in the boundary condition is allowed.

Elliptic partial differential equations of the form \(-Lu= f(x,u,u)\), \(u=0\) on \(\partial\Omega\), where \(L\) is a uniformly elliptic linear operator with \(L^ \infty(\Omega)\) coefficients, are treated in Section 4.2, while in Section 4.3 systems of such equations (i.e. elliptic systems) are investigated. Sections 4.4 and 4.5 are devoted to parabolic equations and systems.

The last chapter deals with the study of discontinuous differential equations in abstract ordered Banach space by using comparison methods, the method of upper and lower solutions as well as the iteration method. Indeed, Section 5.1 contains results on existence, uniqueness and successive approximation for first order Carathéodory differential equations in Banach spaces. The existence and dependence on the data of strong extremal solutions to first order semilinear initial value problems with discontinuous nonlinearities in ordered Banach spaces are studied in Section 5.2. Higher order semilinear initial value problems in an ordered Banach space are discussed in Section 5.3 and the results are new. Here it is assumed that the order cone of the space is regular or fully regular. In Section 5.4 first and second order periodic BVPs are invstigated where the authors seek for extremal solutions. Similar problems for mixed monotone equations are discussed in Section 5.5. Section 5.6 is devoted to mild solutions of an IVP of the form \(x'= A(t)x+ g(t,x)\), \(x(0)= x_ 0\), where \(A(t)\) is the infinitesimal generator of a strongly continuous semigroup on a Banach space. An application to PDEs is given in subsection 5.6.5. The case of second order semilinear IVPs of the form \(x''= Ax+ g(t,x,x')\), \(x(0)= x_ 0\), \(x'(0)= x_ 1\), where \(A\) is again the infinitesimal generator of a strongly continuous cosine family in a Banach space is discussed in Section 5.7 and no compactness assumptions are imposed on the function \(g\). The results obtained here are applied in subsection 5.7.6 to a second order hyperbolic partial differential equation. The last Section 5.8 contains a variety of examples of ordered Banach spaces (most of which are function spaces) which have a regular or full regular order cone, or they are reflexive and their order is introduced by a closed cone or via continuous embedding in an ordered Banach space with fully regular order cone.

Although the title of the book resembles the title of the well-known book “Monotone iterative techniques for nonlinear differential equations” by G. S. Ladde, the second author and A. S. Vatsala [Pitman, Boston (1985; Zbl 0658.35003)] after a detailed discussion one can see that the present book extends the results in that book so that it is a unique reference to the subject. Indeed, it contains a systematic presentation of the complete abstract theory of monotone iterative techniques and applications. That is why in the reviewer’s opinion this self-consistent monograph, based mainly on research work of the authors for a great part, is welcome and strongly recommended to the teachers and researchers on the subject.

Dealing with several problems in natural sciences the underlying operator describing the evolution is compatible with the natural ordering structure of a suitable space. It is therefore justified the fact that such problems are usually investigated in the framework of ordered function spaces. The abstract setting of such a theory has now become an important branch in nonlinear functional analysis. Most of the first chapter of this book (sections 1.1 to 1.4) is devoted to those facts from this abstract theory needed in the sequel. Indeed, after introducing the basic concepts related to partially ordered sets, an abstract recursion principle is presented which forms a basis to the generalized monotone iteration method and its application to prove a fixed point theorem for a mapping \(G: P\to P\) where \(P\) is a poset. Such an example is given in Theorem 1.2.1.: Assume that \(P\) is a poset, \(G: P\to P\) a nondecreasing mapping and \(a\in P\) being such that \(a\leq Ga\) and \(G\) is nondecreasing on \([a):= \{u\in P: a\leq u\}\). If \(x_ *= \sup G[C]\), where \(C\) is the well-ordered chain of \(G\)-iterations of \(a\), then \(x_ *\) is the least fixed point of \(G\) in \([a)\) and \(x_ *= \max C=\min \{y\in [a): Gy\leq y\}\). But the central role in developing the theory of discontinuous differential equations is played by some results given for instance in Theorem 1.2.3.: Let \(Y\) be a subset of an ordered metric space \(X\), \([a,b]\) a nonempty order interval of \(Y\), and \(G: [a,b]\to [a,b]\) a nondecreasing mapping. If \(\sup G[C]\) and \(\inf G[\overline C]\) exist in \(Y\) where \(C\) is the well ordered chain of \(G\)-iterations of \(a\) and \(\overline C\) is the inversely well-ordered chain of \(G\)-iterations of \(b\), then \(x_ *= \max C\) is the least fixed point of \(G\), and \(x^*= \min\overline C\) is the greatest fixed point of \(G\) and \(x_ *= \min\{y\mid Gy\leq y\}\), \(x^*= \max\{y\mid y\leq Gy\}\) hold. What is most important in the applications of the method of upper and lower solutions is the meaning of mixed monotone operators. Let \(P\) be a poset; a mapping \(A: p\times p\to p\) is called mixed monotone if \(A(\cdot, z)\) is nondecreasing and \(A(z,\cdot)\) is nonincreasing for each \(z\in P\). A point \((v,w)\) is a coupled fixed point of \(A\) if \(v= A(v,w)\) and \(w= A(w,v)\). Proposition 1.2.3.: Let \(P\) be a poset. A mixed monotone and order bounded mapping \(A: P\times P\to P\) has an extremal coupled fixed point if \(A[C]\) has supremum and infimum in \(P\) whenever \(C\) is a well- ordered or an inversely well-ordered chain in \(P\times P\) with the componentwise ordering. From Section 1.3 and on several function spaces are considered where some properties like ordering, Bochner integrability and a.e. differentiability are studied as well as some fixed point theorems are given. Finally, Section 1.5 is devoted to the study of the method of upper and lower solutions of Carathéodory systems.

Chapter 2 is devoted to applications of the generalized monotone iterative technique to discontinuous ordinary differential equations. First the authors investigate the initial value problem \((*)\) \(x'= f(t,x,x)\), \(x(0)= x_ 0\) and some special cases of it. Existence of extremal solutions (subsection 2.1.1) and existence of extremals among all solutions (subsection 2.1.2) are discussed when \(f(t,x,y)\) can be discontinuous even in all of its arguments (subsection 2.1.3). The dependence of the extremal solutions on the data as well as the existence of extremal solutions of the periodic boundary value problem \(x'= f(t,x,x)\), \(x(0)= x(T)\) are investigated in subsections 2.1.4 and 2.3.1. Almost the same problems are investigated when the previous equation is replaced with \((**)\) \(x'= f(t,x,x,x)\), when \(f(t,x,y,z)\) is nondecreasing in \(y\) and nonincreasing in \(z\). The new concept here is the coupled quasisolutions, namely a pair of absolutely continuous functions \(y\), \(z\) satisfying \(y'(t)= f(t,y(t),y(t),z(t))\) a.e., \(z'(t)= f(t,z(t),z(t),y(t))\), a.e. and \(y(0)= z(0)= x_ 0\). This is the so- called mixed monotone IVP. Sections 2.4, 2.5, 2.6 and 2.7, 2.8, 2.9 are respectively devoted to finite and infinite differential systems where problems as for \((*)\) and \((**)\) are investigated. Notice that many of the results of this chapter are new.

Chapter 3 deals with the applicability of the method of upper and lower solutions coupled with a generalized monotone iterative technique to second order discontinuous nonlinear boundary value problems. First the authors consider equations of Carathéodory type and existence results are given. Then they discuss existence of extremal solutions for the \(\text{BVP}- x''= f(t,x,x,x')\), \(B_ j x(t_ j)= a_ j x(t_ j)- (- 1)^ j b_ j x'(t_ j)= c_ j\). Section 3.2 is devoted to second order mixed monotone BVPs involving both continuous and mixed monotone type dependencies on the dependent variable, while in Section 3.3 and 3.4 several results derived for scalar BVPs are extended to finite (subsections 3.3.1, 3.4.1) and infinite (subsections 3.3.2, 3.4.2) discontinuous differential systems.

Chapter 4 is devoted to the study of existence of weak extremal solutions of second order partial differential equations and systems of elliptic and parabolic type, where the dependence contains discontinuous nonlinearities. The first main result is contained in subsection 4.1.3 where an existence result for extremal solutions to a quasilinear elliptic BVP is provided in case both continuous and discontinuous dependence in the equation and in the boundary condition is allowed.

Elliptic partial differential equations of the form \(-Lu= f(x,u,u)\), \(u=0\) on \(\partial\Omega\), where \(L\) is a uniformly elliptic linear operator with \(L^ \infty(\Omega)\) coefficients, are treated in Section 4.2, while in Section 4.3 systems of such equations (i.e. elliptic systems) are investigated. Sections 4.4 and 4.5 are devoted to parabolic equations and systems.

The last chapter deals with the study of discontinuous differential equations in abstract ordered Banach space by using comparison methods, the method of upper and lower solutions as well as the iteration method. Indeed, Section 5.1 contains results on existence, uniqueness and successive approximation for first order Carathéodory differential equations in Banach spaces. The existence and dependence on the data of strong extremal solutions to first order semilinear initial value problems with discontinuous nonlinearities in ordered Banach spaces are studied in Section 5.2. Higher order semilinear initial value problems in an ordered Banach space are discussed in Section 5.3 and the results are new. Here it is assumed that the order cone of the space is regular or fully regular. In Section 5.4 first and second order periodic BVPs are invstigated where the authors seek for extremal solutions. Similar problems for mixed monotone equations are discussed in Section 5.5. Section 5.6 is devoted to mild solutions of an IVP of the form \(x'= A(t)x+ g(t,x)\), \(x(0)= x_ 0\), where \(A(t)\) is the infinitesimal generator of a strongly continuous semigroup on a Banach space. An application to PDEs is given in subsection 5.6.5. The case of second order semilinear IVPs of the form \(x''= Ax+ g(t,x,x')\), \(x(0)= x_ 0\), \(x'(0)= x_ 1\), where \(A\) is again the infinitesimal generator of a strongly continuous cosine family in a Banach space is discussed in Section 5.7 and no compactness assumptions are imposed on the function \(g\). The results obtained here are applied in subsection 5.7.6 to a second order hyperbolic partial differential equation. The last Section 5.8 contains a variety of examples of ordered Banach spaces (most of which are function spaces) which have a regular or full regular order cone, or they are reflexive and their order is introduced by a closed cone or via continuous embedding in an ordered Banach space with fully regular order cone.

Although the title of the book resembles the title of the well-known book “Monotone iterative techniques for nonlinear differential equations” by G. S. Ladde, the second author and A. S. Vatsala [Pitman, Boston (1985; Zbl 0658.35003)] after a detailed discussion one can see that the present book extends the results in that book so that it is a unique reference to the subject. Indeed, it contains a systematic presentation of the complete abstract theory of monotone iterative techniques and applications. That is why in the reviewer’s opinion this self-consistent monograph, based mainly on research work of the authors for a great part, is welcome and strongly recommended to the teachers and researchers on the subject.

Reviewer: G.Karakostas (Ioannina)

##### MSC:

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

65L99 | Numerical methods for ordinary differential equations |

35A35 | Theoretical approximation in context of PDEs |

34A45 | Theoretical approximation of solutions to ordinary differential equations |

34A37 | Ordinary differential equations with impulses |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

34C25 | Periodic solutions to ordinary differential equations |

34G20 | Nonlinear differential equations in abstract spaces |

35J99 | Elliptic equations and elliptic systems |

35K99 | Parabolic equations and parabolic systems |

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |