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A problem with conditions given on inner characteristics and on the line of degeneracy for a mixed-type equation with singular coefficients

Menglibay Kholtojibaevich Ruziev

Author Affiliations

Institute of Mathematics, National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan

Boundary Value Problems 2013, 2013:210  doi:10.1186/1687-2770-2013-210

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/210


Received:11 March 2013
Accepted:12 August 2013
Published:12 September 2013

© 2013 Ruziev; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In an unbounded domain, we consider a problem with conditions given on inner characteristics in a hyperbolic part of the considered domain and on some parts of the line of parabolic degeneracy. We prove the unique solvability of the mentioned problem with the help of the extremum principle. The proof of solvability is based on the theory of singular integral equations, Wiener-Hopf equations and Fredholm integral equations.

Introduction and formulation of a problem

Swedish mathematician Gellerstedt [1] investigated a boundary value problem for the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M1">View MathML</a> (m is an odd number), in which the values of a sought function are given on two pieces of characteristics, and on curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M3">View MathML</a>), the value of its derivative is given. This problem has applications in transonic gas dynamics [2].

The Gellerstedt problem and related problems for mixed elliptic-hyperbolic equations were studied in the works [3-5]. The work [6] is devoted to studying the Gellerstedt problem with data on one family of characteristics and with nonlocal gluing conditions. In the work [7] the unique solvability of the Gellerstedt problem for a parabolic-hyperbolic equation of the second kind was studied. The Cauchy problem was investigated by Jachmann and Reissig [8]. Flaisher [9] studied a problem with data on characteristics, outgoing from the origin.

In an unbounded domain, Wolfersdorf [10] investigated the Tricomi problem for the Gellerstedt equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M5">View MathML</a>.

Boundary value problems for the wave equation and equations of mixed type were investigated in [11]. In the work [12] the general Tricomi-Rassias problem was investigated for the generalized Chaplygin equation. In the paper, the representation of a solution of the general Tricomi-Rassias problem was given for the first time; moreover, the uniqueness and existence of a solution for the problem were proved by a new method. In the works [13,14], fundamental solutions were found and boundary value problems for degenerate elliptic equations were solved.

Due to applications in gas dynamics, the interest in studying boundary value problems for degenerate elliptic and mixed-type equations with singular coefficients has been growing. Note the latest work [15] on this topic, where the Dirichlet problem for a three-dimensional elliptic equation with singular coefficients was investigated. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M6">View MathML</a> be a domain of the complex plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M8">View MathML</a> is a half-plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10">View MathML</a> is a finite domain of the half-plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M11">View MathML</a>, bounded by characteristics AC and BC of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M12">View MathML</a>

(1)

outgoing from the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M14">View MathML</a>, and by the segment AB of the straight line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M16">View MathML</a>. In equation (1) assume that m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M18">View MathML</a> are some real numbers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M21">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M22">View MathML</a> be a finite domain separated from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M8">View MathML</a> by the arc <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M24">View MathML</a> of the normal curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M29">View MathML</a>.

We introduce the following denotations: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M32">View MathML</a> are points of intersection of the characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M33">View MathML</a> with that outgoing from the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M34">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M35">View MathML</a> is an arbitrary fixed number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M37">View MathML</a> is a subdomain in the unbounded domain D.

Consider the diffeomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M38">View MathML</a> mapping the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M39">View MathML</a> into the segment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M40">View MathML</a>; moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M43">View MathML</a>. As an example, we take a linear function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M44">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M48">View MathML</a>.

Note that in the Gellerstedt problem the values of a sought function in the hyperbolic part of the mixed domain D are given on the characteristics <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M50">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M52">View MathML</a>.

Boundary value problem for equation (1) in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M53">View MathML</a>, with data on the piece of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M54">View MathML</a> of the characteristic AC and with inner boundary local shifting condition on AB of the line of degeneracy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15">View MathML</a>, was studied in the work [16], and with data on pieces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M57">View MathML</a> in the work [17].

In the present work, we study a new boundary problem, where characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M49">View MathML</a> is free from the conditions, and the needed condition of Gellerstedt is replaced by an inner boundary condition with local shifting on the parabolic line of degeneracy.

Formulation of the problem

Problem G. In the domain D, find a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59">View MathML</a> satisfying the following conditions:

(1) the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59">View MathML</a> is continuous in any subdomain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M61">View MathML</a> of the unbounded domain D;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59">View MathML</a> belongs to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M63">View MathML</a> and satisfies equation (1) in this domain;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59">View MathML</a> is a generalized solution from the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M66">View MathML</a>) in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10">View MathML</a> [18];

(4) the following equalities are fulfilled:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M68">View MathML</a>

(2)

(5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59">View MathML</a> satisfies the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M70">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M71">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M72">View MathML</a>

(5)

and the conjugation condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M73">View MathML</a>

(6)

Moreover, these limits at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M75">View MathML</a> can have singularity of the order less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M76">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M78">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M81">View MathML</a> are given functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M86">View MathML</a>, μ-const., the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M81">View MathML</a> are expressed as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M88">View MathML</a> in a neighborhood of the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M90">View MathML</a>, and they satisfy Holder’s condition on any intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M93">View MathML</a>. For a sufficiently large absolute value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M94">View MathML</a>, they satisfy the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M95">View MathML</a>, where δ, M are positive constants.

Note that condition (5) is an inner boundary condition with local shifting on the parabolic line of degeneracy [16-18].

The uniqueness of the solution of problem G

Theorem 1Let conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M100">View MathML</a>be fulfilled. Then problemGhas only a trivial solution.

Proof Solution of the modified Cauchy problem for equation (1) in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10">View MathML</a>, satisfying initial conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M105">View MathML</a>, is given by the formula [18]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M106">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M109">View MathML</a> is Euler’s gamma function [19].

Satisfying (7) to condition (4), after some evaluations, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M110">View MathML</a>

(8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M111">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M113">View MathML</a> is a differential operator of fractional order in a sense of Riemann-Liouville [19].

Equality (8) is the first functional relation between functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M115">View MathML</a>, on an interval of the axis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15">View MathML</a> reduced from the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M117">View MathML</a>.

Prove that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M100">View MathML</a>, then the solution of problem G in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M123">View MathML</a> is trivial by virtue of equality (2).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M124">View MathML</a> be a point of positive maximum of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M125">View MathML</a> in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M126">View MathML</a>. By virtue of (2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M127">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M128">View MathML</a> such that at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M129">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M130">View MathML</a>

(9)

Considering designation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M103">View MathML</a>, condition (5) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M133">View MathML</a>

(10)

Hence, at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M75">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M121">View MathML</a>) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M136">View MathML</a>. Then, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M137">View MathML</a> into account, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M138">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M139">View MathML</a>. By the Hopf principle [20], the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59">View MathML</a> cannot reach its positive maximum and negative minimum on inner points of the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M141">View MathML</a>. By virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M100">View MathML</a>, from (10) (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M99">View MathML</a>) it follows that there are no points of extremum in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M144">View MathML</a> of the axis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M145">View MathML</a>.

Assume that the sought function reaches its positive maximum and negative minimum on points of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M146">View MathML</a> of the axis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M15">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M148">View MathML</a> (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M149">View MathML</a>) be a point of positive maximum (negative minimum) of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M150">View MathML</a>. Then [21]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M151">View MathML</a>

(11)

It is well known that on the point of positive maximum (negative minimum) of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M152">View MathML</a> for the differential operators of fractional order, the following inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M153">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M154">View MathML</a>) holds. Hence, considering (8) (where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M155">View MathML</a>), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M156">View MathML</a>

(12)

Inequalities (11) and (12) contradict the gluing condition (6), therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M157">View MathML</a>. Hence, there is no point of positive maximum (negative minimum) of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M158">View MathML</a> in the interval AB. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M159">View MathML</a>. By the Hopf principle and the statements obtained above, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M160">View MathML</a>, and by virtue of (9), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M161">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M162">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M163">View MathML</a>. From here, due to arbitrariness of ε at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M164">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M165">View MathML</a> in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M166">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M167">View MathML</a>

(13)

Considering (13), due to continuity of the solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M126">View MathML</a> and conjugation condition (6), we restore the desired function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M59">View MathML</a> in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M10">View MathML</a> as a solution of the modified Cauchy problem with homogeneous data and get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M165">View MathML</a> in the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M172">View MathML</a>. The proof of Theorem 1 is complete. □

The existence of the solution of problem G

Theorem 2Let the following conditions be fulfilled: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M177">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M182">View MathML</a>. Then problemGhas a solution.

Proof The solution of the Dirichlet problem satisfying condition (3) and the requirement <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M103">View MathML</a>, can be represented in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M185">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M186">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M187">View MathML</a>

Differentiating (14) along y and considering the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M188">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M189">View MathML</a>

Integrating by parts (taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M191">View MathML</a> into account), after some evaluations, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M192">View MathML</a>

(15)

Multiplying both sides of (15) to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M193">View MathML</a>, and passing to the limit at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M194">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M195">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M196">View MathML</a>.

Equation (16) is the second functional relation between unknown functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M197">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M198">View MathML</a> in the interval I of the axis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M199">View MathML</a> deduced from the upper half-plane.

Note that relation (16) is valid for the whole I. Breaking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M200">View MathML</a> into the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M202">View MathML</a>, then into the integral with bounds <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M201">View MathML</a>, we make change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M204">View MathML</a>. By virtue of (10), from (16) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M205">View MathML</a>

(17)

Considering (6), excluding the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M197">View MathML</a> from (8) and (17), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M207">View MathML</a>

(18)

Applying the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M208">View MathML</a> to the both sides of (18), considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M209">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M210">View MathML</a>

(19)

Further, one can easily prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M211">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M212">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M213">View MathML</a>

(22)

Substituting (20)-(22) into (19), after some calculations, we get the singular integral equation regarding the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M198">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M215">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M218">View MathML</a>. The integral operator on the right-hand side of (23) is not regular since expression under the integral has isolated singularity of the first order at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M220">View MathML</a>, and this is why this item is written separately. Setting the right-hand side of (23) temporarily as a known function, we rewrite it as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M221">View MathML</a>

(24)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M222">View MathML</a>

(25)

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M224">View MathML</a>, we rewrite equation (24) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M225">View MathML</a>

(26)

We search for a solution of equation (26) in the class of functions, satisfying Holder’s condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M146">View MathML</a> and bounded at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M227">View MathML</a>, and at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M228">View MathML</a> they can tend to infinity with order less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M76">View MathML</a>. Index χ of this class is equal to zero, and solution can be explicitly found by the method of Carleman-Vekua [22]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M230">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M233">View MathML</a>. From here, according to designation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M234">View MathML</a>

(27)

Substituting (25) into (27), after some evaluations, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M235">View MathML</a>

(28)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M236">View MathML</a>

Equation (28), by virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M173">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M239">View MathML</a>, can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M240">View MathML</a>

(29)

Further, in (29) we calculate the inner integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M241">View MathML</a>

Expanding the rational factor of the integrand in simple fractions and completing simple calculations, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M242">View MathML</a>

(30)

Substituting (30) into (29), considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M244">View MathML</a>, after some calculations, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M245">View MathML</a>

(31)

Considering formulas <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M247">View MathML</a>, we rewrite equality (31) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M248">View MathML</a>

(32)

In (32) allocating a characteristic part, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M249">View MathML</a>

(33)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M250">View MathML</a>

is a regular operator. We rewrite equation (33) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M251">View MathML</a>

(34)

Making change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M253">View MathML</a> and designating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M254">View MathML</a>, we write equation (34) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M255">View MathML</a>

(35)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M256">View MathML</a>

We introduce the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M257">View MathML</a>

(36)

By virtue of (36), equation (35) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M258">View MathML</a>

(37)

Equation (37) is an integral equation of Wiener-Hopf [23]. Functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M260">View MathML</a> are indicative of decrease at infinity; moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M262">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M263">View MathML</a>, and a solution of equation (37) will be sought in the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M264">View MathML</a>[23]. Using Fourier transformation, equation (37) is deduced to the Riemann problem and is solved in quadratures. Fredholm’s theorems are only valid in one particular case, when the index of these equations is equal to zero.

Calculate the index of the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M265">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M266">View MathML</a>

(38)

Using the theory of residues, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M267">View MathML</a>

(39)

Substituting (39) into (38), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M268">View MathML</a>

By virtue of the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M269">View MathML</a>, and since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M270">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M271">View MathML</a>. Therefore, the index of equation (37) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M272">View MathML</a>, i.e., the variation of the argument of the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/210/mathml/M265">View MathML</a> on the real axis expressed in complete revolutions equals zero [23]. Hence, taking into account the fact that the solution to problem G is unique, we deduce the unique solvability of equation (37) and, consequently, that of problem G. Theorem 2 is proved. □

Competing interests

The author declares that he has no competing interests.

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