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**Boundary value problems for higher order differential equations.**
*(English)*
Zbl 0619.34019

Singapore - Philadelphia, Pa.: World Scientific Publishing Co. (distr. by John Wiley & Sons Ltd., Chichester etc.). X, 307 p. £29.85 (1986).

The monograph is an excellent account of the various techniques available in the literature to prove existence and uniqueness of various boundary value problems which occur in applications. Graduate students and research mathematicians will find it very useful and stimulating. Chapter 1 gives various concrete examples of situations of occurrence of boundary value problems (BVP) to provide motivation for some of the equations considered subsequently. The author considers the nth order linear differential equation (1) \(L[x]=x^{(n)}+\sum^{n}_{i=1}P_ i(t)x^{(n-1)}=f(t)\) where \(P_ i(t)\), \(1\leq i\leq n\) and f(t) are continuous on \(I=[a,b]\) with separated boundary conditions: (2) \(\ell [x]=\sum^{n-1}_{k=0}c_{ik}x^{(k)}(a_ i)=A_ i,\) \(1\leq i\leq n\), where \(a\leq a_ 1\leq a_ 2...\leq a_ n\leq b\). These boundary conditions include as special cases several other boundary conditions. In Chapters 2 to 6 the existence and uniqueness of the solutions of the BVP (1) and (2) are discussed by means of Green function, the method of complementary functions, the method of adjoint equation and the method of chasing. The method of chasing converts the BVP to its equivalent initial value problem. In Chapter 7, the theory of differential inequalities arising from Picard is applied fruitfully to show the existence and uniqueness of solutions of the two point boundary value problem, while in Chapter 8, error estimates which are best possible are established by using polynomial interpolation theory. If f is continuous, satisfies the Lipschitz condition and det \(\ell_ i([t^{j-1}])\neq 0,\) \(1\leq i,j\leq n\); the results of Chapter 9 provide a sufficiently small interval on which the BVP (2) and (3) \(x^{(n)}=f(t,\underline x)\) has a unique solution where \b{x}\(=(x,x',...,x^{(q)})\), \(0\leq q\leq n-1\). Chapter 10 discusses the Picard method for the BVP (3) and (2), where (2) is the Hermite (r point) condition. Chapter 11 provides an upper estimate on the length of the interval so that the quasilinear iterative scheme generated converges to a unique solution of the BVP of Chapter 10. Chapter 12 employs a weight function technique to enlarge the region of existence and uniqueness of solutions of the BVP of Chapter 10, compared with what can be deduced from the results of Chapters 10 and 11.

In the special case where the explicit Green function and some of its properties are known it is possible to find best existence and uniqueness intervals. Chapter 13 provides best possible results by the shooting method while Chapter 14 contains a slightly strengthened version of a fixed point theorem for isotone operators in partially ordered spaces. These are used to provide existence results of the BVP (3) with special boundary conditions of the type (2). In Chapter 15, Kamke’s convergence theorem is used to show uniqueness of solutions of the BVP (3) and (2) implies existence, which generalizes a well-known result for linear BVP and whose proof is based on the linear structure of the fundamental system of solutions. Chapter 16 shows that in the case where (3) is of order 2 or 3, certain special conditions imply compactness condition. Existence of generalized solutions are also obtained. Chapter 17 shows that for the BVP (3) and (2), the uniqueness of m point BVPs on (a,b) implies the uniqueness of r(\(>m^{<m})\) point BVPs on (a,b) while Chapter 18 extends the concept of boundary value function for third order linear equation to equation (3) with certain special conditions of Chapter 15. Chapter 19 uses a newly developed topological transversality method of Gauss which itself generalizes the continuation theorem of Leray and Schauder to study the existence of solutions of the following class of BVP: \[ (4)\quad L[x]=x^{(n)}+\sum^{n}_{i=1}P_ i(t)x^{(n-1)}=f(t,\underline x) \] (5) \(U[x]=\ell\) where \(P_ 1(t)\), \(1\leq i\leq n\) are continuous on \(I=[a,b]\), f is continuous on \(I\times R^{q+1}\), \(U: C^{(n-1)}(I)\to R^ n\) is a continuous linear operator and \(\ell\) is a given vector in \(R^ n\). By considering solutions of an associated family of differential equations satisfying (5), existence of solutions of the BVP (4) and (5) are given. The methods of Chapters 10, 12 and 13 and optimal control theory are used to give the best possible intervals of existence in Chapter 20 while Chapter 21 employs the matching methods to establish the existence of solutions of a given BVP with the help of several other related BVPs. Chapter 22 establishes the existence of a maximal solution for the initial value problem (3) with initial values \(x^{(i)}(t_ 0)=x_ i\), \(i=0,1,...,n-1\), where \(t_ 0\in (a,b)\) and f is assumed to be continuous on \((a,b)\times R^{q+1}\). Chapter 23 gives a generalization of the maximum principle to higher order inequalities with an application to a class of boundary value problems. Chapter 24 provides sufficient conditions for the existence of solutions of BVPs on semi-infinite and infinite intervals while Chapter 25 establishes the existence and uniqueness of solutions of higher order differential equations with deviating arguments with certain boundary conditions.

In the special case where the explicit Green function and some of its properties are known it is possible to find best existence and uniqueness intervals. Chapter 13 provides best possible results by the shooting method while Chapter 14 contains a slightly strengthened version of a fixed point theorem for isotone operators in partially ordered spaces. These are used to provide existence results of the BVP (3) with special boundary conditions of the type (2). In Chapter 15, Kamke’s convergence theorem is used to show uniqueness of solutions of the BVP (3) and (2) implies existence, which generalizes a well-known result for linear BVP and whose proof is based on the linear structure of the fundamental system of solutions. Chapter 16 shows that in the case where (3) is of order 2 or 3, certain special conditions imply compactness condition. Existence of generalized solutions are also obtained. Chapter 17 shows that for the BVP (3) and (2), the uniqueness of m point BVPs on (a,b) implies the uniqueness of r(\(>m^{<m})\) point BVPs on (a,b) while Chapter 18 extends the concept of boundary value function for third order linear equation to equation (3) with certain special conditions of Chapter 15. Chapter 19 uses a newly developed topological transversality method of Gauss which itself generalizes the continuation theorem of Leray and Schauder to study the existence of solutions of the following class of BVP: \[ (4)\quad L[x]=x^{(n)}+\sum^{n}_{i=1}P_ i(t)x^{(n-1)}=f(t,\underline x) \] (5) \(U[x]=\ell\) where \(P_ 1(t)\), \(1\leq i\leq n\) are continuous on \(I=[a,b]\), f is continuous on \(I\times R^{q+1}\), \(U: C^{(n-1)}(I)\to R^ n\) is a continuous linear operator and \(\ell\) is a given vector in \(R^ n\). By considering solutions of an associated family of differential equations satisfying (5), existence of solutions of the BVP (4) and (5) are given. The methods of Chapters 10, 12 and 13 and optimal control theory are used to give the best possible intervals of existence in Chapter 20 while Chapter 21 employs the matching methods to establish the existence of solutions of a given BVP with the help of several other related BVPs. Chapter 22 establishes the existence of a maximal solution for the initial value problem (3) with initial values \(x^{(i)}(t_ 0)=x_ i\), \(i=0,1,...,n-1\), where \(t_ 0\in (a,b)\) and f is assumed to be continuous on \((a,b)\times R^{q+1}\). Chapter 23 gives a generalization of the maximum principle to higher order inequalities with an application to a class of boundary value problems. Chapter 24 provides sufficient conditions for the existence of solutions of BVPs on semi-infinite and infinite intervals while Chapter 25 establishes the existence and uniqueness of solutions of higher order differential equations with deviating arguments with certain boundary conditions.

Reviewer: O.Akinyele

### MSC:

34B05 | Linear boundary value problems for ordinary differential equations |

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

34A30 | Linear ordinary differential equations and systems |

34B27 | Green’s functions for ordinary differential equations |