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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

On estimates of solutions of the periodic boundary value problem for first-order functional differential equations

Eugene Bravyi

Author Affiliations

Scientific center ‘Functional Differential Equations’, Perm National Research Polytechnical University, Komsomol’sky pr. 29, Perm, 614990, Russia

Boundary Value Problems 2014, 2014:119  doi:10.1186/1687-2770-2014-119


Dedicated to Professor Ivan Kiguradze


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/119


Received:29 January 2014
Accepted:6 May 2014
Published:15 May 2014

© 2014 Bravyi; 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 credited.

Abstract

Inequalities for periodic solutions of first-order functional differential equations are obtained. These inequalities are best possible in a certain sense.

MSC: 34K06, 34K10, 34K13.

Keywords:
functional differential equations; periodic solutions; periodic boundary value problem; estimates of solutions

1 Introduction

Periodic solutions of functional differential equations are important in different applications (see, for example, [1-4] and the references therein, and also works on the general theory of boundary value problems for functional differential equations [5-11]). Conditions for the solvability of first-order periodic problems are found in [12-23]. In [15,16] the linear case is considered, and unimprovable sufficient conditions for the solvability of the periodic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M1">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M2">View MathML</a>

(2)

are found in terms of the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a> of linear positive functional operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M6">View MathML</a>

(3)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M7">View MathML</a>

(4)

If both of these conditions are not satisfied for some norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, there exist linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a> with these norms such that problem (1)-(2) has no solution. As to our knowledge, similar unimprovable estimates for solutions of (1)-(2) in terms of norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a> are yet unknown. Here we will fill this gap. Moreover, the estimates obtained here (in Theorems 1, 2, 3) can be expanded to some non-linear functional differential equations (see Remark 1). Theorem 1 gives the best possible estimates of the norm of the Green operator for the periodic boundary value problem. In Theorem 2, we obtain unimprovable estimates of the solutions of (1)-(2) for non-negative f. In Theorem 3, unimprovable bounds of the difference between the maximum and the minimum of a solution are established.

We use the following notation: ℝ is the space of real numbers, C is the space of continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M14">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M15">View MathML</a>; L is the space of integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M16">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M17">View MathML</a>; a linear bounded operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M18">View MathML</a> is called positive if it maps non-negative functions from C into almost everywhere non-negative functions from L.

Consider the periodic boundary value problem (1)-(2), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a> are linear positive operators with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M22">View MathML</a>, 1 is the unit function. An absolutely continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M14">View MathML</a> is called a solution of the problem if it satisfies the periodic boundary condition (2) and equation (1) for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M24">View MathML</a>. We have to solve problem (1)-(2) if, for example, we search for periodic solutions of the equation with delay

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M25">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M26">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M27">View MathML</a>-periodic locally integrable functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M28">View MathML</a> is a measurable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M27">View MathML</a>-periodic non-negative delay. Indeed, suppose that linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a> are defined by the equalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M32">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M33">View MathML</a> and the integer numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M34">View MathML</a> are such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M35">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M36">View MathML</a>. It is easy to show that problem (1)-(2) has a solution if and only if equation (5) has a periodic solution with the period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M37">View MathML</a>.

The conditions (3), (4) for the norms of the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a> are well known [15]. They guarantee the existence and uniqueness of solutions of problem (1)-(2). Note that these conditions are unimprovable in the following sense: if non-negative numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a> satisfy neither (3) nor (4), then problem (1)-(2) has no solution for some linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M43">View MathML</a> and for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a>.

2 The main results

In what follows, we suppose that one of conditions (3), (4) is fulfilled. First, we formulate the results only for the simplest problem (1)-(2) with the null operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M46">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M47">View MathML</a> is a linear positive operator with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a>. The assertions of the following Theorems 1, 2, 3 for problem (6) are as follows.

The solution x of (6) satisfies the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M50">View MathML</a>

(7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M51">View MathML</a>

(8)

If a function f is non-negative, the solution x of (6) satisfies the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M52">View MathML</a>

(9)

All estimates (7), (8) and (9), which are proved in Theorems 1, 2, 3 in the general case, are best possible (see Remarks 3, 5, 6).

Remark 1 Consider also the non-linear periodic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M53">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M54">View MathML</a>

(11)

provided there exist non-negative functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55">View MathML</a> with norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M56">View MathML</a>

(12)

such that the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M57">View MathML</a> satisfy the inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M58">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M59">View MathML</a>

(14)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M60">View MathML</a>.

It follows from Lemma 3 and the proofs of Theorems 1, 2, 3 that all statements of these theorems are also valid for solutions of periodic problem (10)-(11) (if the solutions exist).

Theorem 1If the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M61">View MathML</a>of the linear positive operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M63">View MathML</a>

(15)

andxis a solution of (1)-(2), then the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M64">View MathML</a>

(16)

holds.

If the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M65">View MathML</a>of the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M67">View MathML</a>

(17)

andxis a solution of problem (1)-(2), then the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M68">View MathML</a>

(18)

holds.

Remark 2 ([15])

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M69">View MathML</a> and both of the conditions (15), (17) are not fulfilled, then there exist linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a> such that problem (1)-(2) has no solution.

Remark 3 From the proof of Theorem 1 it follows that estimates (16), (18) are best possible: if non-negative numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> satisfy (15) (or (17)), then equality holds in condition (16) (or (18)) for a unique solution x of problem (1)-(2) for some linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> and for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M82">View MathML</a>.

The estimates of solutions (1)-(2) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M83">View MathML</a> can be obtained in the same way.

Theorem 1If the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M84">View MathML</a>of the linear positive operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M86">View MathML</a>

(19)

andxis a solution of (1)-(2), then the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M87">View MathML</a>

(20)

holds.

If the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M88">View MathML</a>of the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M90">View MathML</a>

(21)

andxis a solution of problem (1)-(2), then the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M91">View MathML</a>

(22)

holds.

Remark 2 ([15])

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M69">View MathML</a> and both of conditions (19), (21) are not fulfilled, then there exist linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a> such that problem (1)-(2) has no solution.

Remark 3 From the proof of Theorem 1 it follows that estimates (20), (22) are best possible: if non-negative numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> satisfy (19) (or (21)), then equality holds in condition (20) (or (22)) for a unique solution x of problem (1)-(2) for some linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> and for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M82">View MathML</a>.

In the next statement we get the best possible lower bounds for solutions of problem (1)-(2) for non-negative f.

Theorem 2Letxbe a solution of problem (1)-(2) for some non-negativef.

If the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>of the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M109">View MathML</a>

(23)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M110">View MathML</a>

(24)

if the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>of the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M114">View MathML</a>

(25)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M115">View MathML</a>

(26)

if the norms<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>of the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M62">View MathML</a>satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M119">View MathML</a>

(27)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M120">View MathML</a>

(28)

Remark 4 ([15])

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M121">View MathML</a> and all of conditions (23), (25), (27) are not fulfilled, then there exist linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a> such that problem (1)-(2) has no solution.

Remark 5 From the proof of Theorem 2 it follows that estimates (24), (26), (28) are best possible: if non-negative numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a> satisfy (23) ((25) or (27)), then equality holds in condition (24) ((26) or (28)) for a unique solution x of problem (1)-(2) for some linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a> and for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M82">View MathML</a>.

Now we estimate the difference between the maximum and the minimum of solutions.

Theorem 3Let the solvability conditions (4) be fulfilled andxbe a unique solution of (1)-(2). If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M135">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M136">View MathML</a>

(29)

otherwise

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M137">View MathML</a>

(30)

Remark 6 From the proof of Theorem 3 it follows that inequalities (29) and (30) are unimprovable. It means that for every number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a> satisfying the conditions of the theorem, equality holds in conditions (29) or (30) for the solution x of problem (1)-(2) for some positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M3">View MathML</a>, and for some non-negative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M144">View MathML</a>.

Remark 7 Theorems 2, 3, as Theorem 1, can be easily reformulated for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M88">View MathML</a> when the solvability condition (3) holds.

3 Proofs

We need three lemmas to prove the main theorems.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a>be linear positive operators, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149">View MathML</a>. Then there exist points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M150">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M152">View MathML</a>

(31)

such that the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M153">View MathML</a>

(32)

holds.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M155">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149">View MathML</a> and the linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a> are positive, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M158">View MathML</a>

Therefore, for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151">View MathML</a> satisfying (31), equality (32) holds. □

Lemma 2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149">View MathML</a>, functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55">View MathML</a>are non-negative, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151">View MathML</a>satisfies (31), then there exist linear positive operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M5">View MathML</a>with the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M164">View MathML</a>

(33)

such that equality (32) holds.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M24">View MathML</a>. Then the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M11">View MathML</a> defined by the equalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M170">View MathML</a>

satisfy the conditions of the lemma. □

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M57">View MathML</a>satisfy (13)-(14), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149">View MathML</a>. Then there exist a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151">View MathML</a>satisfying (31) and points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M150">View MathML</a>such that the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M175">View MathML</a>

(34)

holds.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M155">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M149">View MathML</a> and using (13), (14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M179">View MathML</a>

Therefore, for some function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151">View MathML</a> satisfying (31), equality (34) holds. □

Remark 8 It is obvious that one can choose the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M182">View MathML</a> in Lemmas 1 and 3 in such a way that the solution y takes its maximum and minimum at these points.

Proofs of Theorems 1, 2, 3 If x is a solution of problem (1)-(2) ((10)-(11)), then by Lemma 1 (3) this solution satisfies the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M183">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M184">View MathML</a>

(36)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151">View MathML</a> and non-negative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55">View MathML</a> satisfy (31), (33). If condition (3) or (4) holds, then problem (35)-(36) has a unique solution, which can be easily found explicitly. Since we are only interested in the maximal and minimal values of the solutions, by Remark 8, we have to obtain only representations for values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M188">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M191">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M192">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M188">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M195">View MathML</a>

(37)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M196">View MathML</a>

(38)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M197">View MathML</a>

(39)

Suppose here that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M121">View MathML</a> and condition (4) is fulfilled.

Define by P the set of all solutions of problem (35)-(36) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M189">View MathML</a>, for all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M151">View MathML</a> and non-negative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M55">View MathML</a> such that conditions (12), (31) hold, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M203">View MathML</a>.

Let S be the subset of P corresponding to non-negative functions f.

From Lemmas 1 and 2, it follows that the set P coincides with the set of all solutions of problem (1)-(2) for all linear positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M204">View MathML</a> with norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M206">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M208">View MathML</a>. The subset S consists of all solutions of corresponding problems (1)-(2) with non-negative f.

Define the constants

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M209">View MathML</a>

From representations (37), (38), (39), it easily follows that all the constants are defined correctly and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M210">View MathML</a>

Moreover, for every solution x of (1)-(2), the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M211">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M19">View MathML</a> is non-negative, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M213">View MathML</a>

where the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M217">View MathML</a> are best possible.

It remains to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M216">View MathML</a>.

The numerator and denominator of fractions in (37), (38), (39) are linear with respect to variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M222">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M188">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M225">View MathML</a> take their minimal and maximal values at the bounds of restriction (31) with respect to variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M226">View MathML</a> on each of the sets E and I. Hence we have to consider only the following four different cases:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M227">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M228">View MathML</a>

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M229">View MathML</a>,

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M230">View MathML</a>.

In case (i) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M231">View MathML</a>

In case (ii) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M232">View MathML</a>

In case (iii) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M233">View MathML</a>

In case (iv) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M234">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M238">View MathML</a> be the subsets of S for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M226">View MathML</a> corresponding to cases (i), (ii), (iii), (iv).

We can easily calculate the minimal and maximal values in every case.

In case (iv) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M240">View MathML</a>

In case (iii) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M241">View MathML</a>

Therefore, in cases (iii) and (iv) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M242">View MathML</a>

In case (i) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M243">View MathML</a>

In case (ii) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M244">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M245">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M246">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M247">View MathML</a>

Considering extremal values in all cases (i), (ii), (iii) and (vi), by elementary calculation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M248">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M250">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M252">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M253">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M255">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M257">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/119/mathml/M258">View MathML</a>

This proves all Theorems 1, 2, 3. □

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Acknowledgements

Research was supported by the Russian Foundation for Basic Research (14-01-0033814). The author would like to thank both reviewers for their careful reading of the manuscript and valuable remarks.

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