##
**Focal boundary value problems for differential and difference equations.**
*(English)*
Zbl 0914.34001

Mathematics and its Applications (Dordrecht). 436. Dordrecht: Kluwer Academic Publishers. viii, 289 p. (1998).

The book is divided into two chapters. The first one is concerned with the \(n\)th-order differential equation \(x^{(n)}(t)=f(t,x(t), \cdots,x^{(q)}(t))\), \(t \in [a,b]\); \(0 \leq q \leq n-1\) fixed, together with the right focal point boundary conditions \(x^{(i)}(a_{i+1}) = A_{i+1}\), \(0 \leq i \leq n-1\), with \(- \infty < a \leq a_1 \leq a_2 \leq \cdots \leq a_n \leq b < \infty\).

Existence and uniqueness results together with approximate solutions via Picard’s method are obtained. Under certain regularity conditions on the function \(f\), a sequence is constructed which converges quadratically to the unique solution. Numerical methods are presented.

Analogous existence, uniqueness and approximation results are given for integro-differential and delay-differential equations.

Uniqueness of some boundary value problems is used to derive existence and uniqueness of different boundary value problems.

For the linear operator \(Lx(t) = x^{(n)}(t) + \sum _{i=0} ^{q} {b_i(t) x^{(i)}(t)}\) necessary and sufficient conditions for right disfocality are proved. A Green function for the equation \(Lx = \phi\) and \(x\) verifying convenient focal boundary value conditions is constructed. The sign of each derivative of such a function is determined.

Continuous dependence and differentiation with respect to boundary values are studied. The monotone convergence is treated in an abstract formulation under the hypothesis of existence of lower and upper solutions. Furthermore, particular cases as unbounded intervals of definition, unordered \(a_i\)’s, singular problems or impulse effects are studied.

In Chapter 2 the \(n\)th-order difference equation \[ \Delta^n u(k) = f(k,u(k), \Delta u(k), \dots, \Delta ^{n-1}u(k)), \] \(k \in \{a,a+1, \ldots, b-1\}\); \(\Delta^iu(a_{i+1})=A_{i+1}\), \(0 \leq i \leq n-1\), is considered. Here, \(a\), \(b \in \mathbb{N}\), \(a \leq a_1 \leq \cdots \leq a_n \leq b\), \(\Delta u(k) = u(k+1) - u(k)\) and \(\Delta^mu(k) = \Delta(\Delta ^{m-1}u(k)) \) for all \(m \in \mathbb{N}\).

The majority of the studied cases in Chapter 1 are treated here.

Since the author has a high knowledge in this area, the book is a very good tool for mathematicians interested in differential and difference equations.

Existence and uniqueness results together with approximate solutions via Picard’s method are obtained. Under certain regularity conditions on the function \(f\), a sequence is constructed which converges quadratically to the unique solution. Numerical methods are presented.

Analogous existence, uniqueness and approximation results are given for integro-differential and delay-differential equations.

Uniqueness of some boundary value problems is used to derive existence and uniqueness of different boundary value problems.

For the linear operator \(Lx(t) = x^{(n)}(t) + \sum _{i=0} ^{q} {b_i(t) x^{(i)}(t)}\) necessary and sufficient conditions for right disfocality are proved. A Green function for the equation \(Lx = \phi\) and \(x\) verifying convenient focal boundary value conditions is constructed. The sign of each derivative of such a function is determined.

Continuous dependence and differentiation with respect to boundary values are studied. The monotone convergence is treated in an abstract formulation under the hypothesis of existence of lower and upper solutions. Furthermore, particular cases as unbounded intervals of definition, unordered \(a_i\)’s, singular problems or impulse effects are studied.

In Chapter 2 the \(n\)th-order difference equation \[ \Delta^n u(k) = f(k,u(k), \Delta u(k), \dots, \Delta ^{n-1}u(k)), \] \(k \in \{a,a+1, \ldots, b-1\}\); \(\Delta^iu(a_{i+1})=A_{i+1}\), \(0 \leq i \leq n-1\), is considered. Here, \(a\), \(b \in \mathbb{N}\), \(a \leq a_1 \leq \cdots \leq a_n \leq b\), \(\Delta u(k) = u(k+1) - u(k)\) and \(\Delta^mu(k) = \Delta(\Delta ^{m-1}u(k)) \) for all \(m \in \mathbb{N}\).

The majority of the studied cases in Chapter 1 are treated here.

Since the author has a high knowledge in this area, the book is a very good tool for mathematicians interested in differential and difference equations.

Reviewer: Alberto Cabada (Santiago)

### MSC:

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

39A11 | Stability of difference equations (MSC2000) |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |