الفهرس 
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Abstract
This thesis is concerned with the Whittaker interpolation series as well as Whittaker-
type nonlinear boundary value problems. Like Lidstone series, Whittaker series is
viewed as a two-point Taylor series, which solves the interpolation problem
f
(2i+1)(0) = ai
, f(2i)
(1) = bi
, i ∈ N0.
Whittaker’s series introduced by J. M. Whittaker in 1934, where convergence analysis
is established for entire functions of exponential type less than π/2. In some literature,
Whittaker’s series is called a modied Abel series.
The rst task of this thesis is to investigate the remainder of the Whittaker’s series,
or the Abel-Whittaker series. We introduce sharp error estimates for the remainder
associated with Whittaker’s series. The remainder is established in terms of Euler’s
polynomials and numbers. We give both pointwise and uniform estimates for the
remainder. We carry out several numerical experiments, which assure the accuracy
of the obtained theoretical and analytical results. Dually, we investigate the dual
Abel-Whittaker series, which solves the interpolation problem
f
(2i)
(0) = ai
, f(2i+1)(1) = bi
, i ∈ N0.
This analysis is established in Chapter 2.
Chapter 3 is devoted to study the nonlinear Abel-Whittaker boundary-value prob-
lem
(−1)n
y
(2n)
(x) = λf(x, y), 0 ≤ x ≤ 1,
y
(2i+1)(0) = y
(2i)
(1) = 0, i = 0, 1, ..., n − 1,
where n ∈ N is xed and λ > 0 is the eigenvalue parameter. We investigate the exis-
tence of positive eigenvalues and the existence of corresponding positive eigenfunctions.
We prove under standered conditions on f(x, y), the existence of intervals of positive
eigenvalues with positive eigenfunctions. We also consider the duall problem
(−1)n
y
(2n)
(x) = λf(x, y), 0 ≤ x ≤ 1,
y
(2i+1)(1) = y
(2i)
(0) = 0, i = 0, 1, ..., n − 1,
and establish some existence results. Results are illustrated with examples.
In Chapter 4, a two-parameter nonlinear system of boundary-value problems is
studied. In this setting, we prove the existence of rectangles of positive eigenvalues
iii
with positive corresponding eigenfunctions. We also consider the multiplicity problem,
i.e. when positive eigenpair has more than one positive eigenfunctions. We also work
out some examples of nonlinear systems.
The thesis is also concluded with an introductory chapter, namely Chapter 1,
which involves some basic denitions and tools that will be needed in the sequell. The
rst essential tool that is required in the three chapters in the Green’s function of
self adjoint eigenvalue problems. Then we give a brief account on Bernoulli and Euler
polynomials and numbers and some of their properties. For establishing the results of
Chapters 3 and 4, we state the denition of compact operators in Banach spaces, i.e.
completely continuous operators, and some xed point theorems, involving Krasnosel-
skii’s cone xed point theorem.