On positive solutions of the homogeneous Hammerstein integral equation Текст научной статьи по специальности «Математика»

On positive solutions of the homogeneous Hammerstein integral equation

Yu. Kh. Eshkabilov, F. H. Haydarov National University of Uzbekistan, Tashkent, Uzbekistan yusup62@mail.ru, haydarov_imc@mail.ru

PACS 02.30.Rz DOI 10.17586/2220-8054-2015-6-5-618-627

In this paper the existence and uniqueness of positive fixed points operator for a nonlinear integral operator are discussed. We prove the existence of a finite number of positive solutions for the Hammerstein type of integral equation. Obtained results are applied to the study of Gibbs measures for models on a Cayley tree.

Keywords: integral equation of Hammerstein type, fixed point of operator, Gibbs measure, Cayley tree.

Received: 2 April 2015

Revised: 18 July 2015

1. Introduction

It is well known that integral equations have wide application in engineering, mechanics, physics, economics, optimization, vehicular traffic, biology, queuing theory and so on (see [1-5]). The theory of integral equations is rapidly developing with the help of tools in functional analysis, topology and fixed point theory. Therefore, many different methods are used to obtain the solution of the nonlinear integral equation. Moreover, some methods can be found in Refs. [6-13], to discuss and obtain a solution for the Hammerstein integral equation. In [11], J.Appell and A.S. Kalitvin used fixed point methods and methods of nonlinear spectral theory to obtain a solution for integral equations of the Hammerstein or Uryson type. The existence of positive solutions of abstract integral equations of Hammerstein type is discussed in [9]. In [7], M.A. Abdou, M.M. El-Borai and M.M. El-Kojok discuss the existence and uniqueness of a solution for the nonlinear integral equation of the Hammerstein type with a discontinuous kernel.

In this present paper, we study the solvability of an homogeneous integral equation of the Hammerstein type. An integral equation of the form:

J K(t,u)tf (t,f (u)) du = f (t), (1.1)

0

is called the homogeneous Hammerstein integral equation, where K(t,u) is continuous real-valued function defined on 0 < t < 0, 0 < u < 1, ^ : [0,1] x R ^ R is a continuous function and f (t) is unknown function from C[0,1].

d

Let ^(t, z), — ^(t, z) be continuous and bounded for t e [0,1] and for all z. Then [14],

dz

the Hammerstein integral equation (1.1) has a solution, assuming that ^(t,z) is a bounded continuous function for t e [0,1] and z e R. In this case, the Hammerstein integral equation (1.1) also has a solution [15]. For the necessary details of this theorem and for more results on the Hammerstein integral equation, we refer to Petryshyn and Fitzpatrik [16], Browder [17], Brezis and Browder [18].

i

Recently, the case ^(t, z) = ^(z) was considered [19]. Let ^(z) be a monotonous left-continuous function on [0, and lim-= lim -= 0. Then, the integral

x^ö z x^+TO z

equation of Hammerstein type (1.1) has a solution [19].

In this work, we will consider the following integral equation of Hammerstein type (i.e. in (1.1) tf(t,z) = ) = z^):

JK(t,u)f*(u)du = f(t), 1, (1.2)

0

on the C[0,1], where K(t,u) is a strictly positive continuous function.

By Theorem 44.8 from [4], the existence of a nontrivial positive solution for the Hammerstein equation (1.2) follows. We study the problem of the existence of a finite number of positive solutions for the integral equation of the Hammerstein type (1.2).

Consider the nonlinear operator Ra on the cone of positive continuous functions on [0,1] :

f=(§f)' ■ (13)

where K(t,u) is given in the integral equation of Hammerstein type (1.2) and a > 0. An operator of the form (1.3) arises in the theory of Gibbs measures (see [20-22]). Positive fixed points of the operator Rk, k e N and their numbers are very important to study Gibbs measures for models on a Cayley tree.

In [21], for the case a =1, the uniqueness of positive fixed points of the nonlinear operator Ra (1.3) was proved. In [20], in the case a = k e N, k > 1, for the nonlinear operator Ra, the existence of positive fixed point and the existence Gibbs measure for some mathematical models on a Cayley tree were proved.

The aim of this work is to study the existence of a finite number of positive solutions for the Hammerstein equation (1.2) on the space of continuous functions on [0,1]. The plan of this paper is as follows: in the second section, using properties of Hammerstein equation (1.2), we reduce some statements on the positive fixed point of the operator Ra; in the third section, we construct the strictly positive continuous kernel K(t,u), such that, for given n e N, the corresponding Hammerstein equation (1.2) has n number of positive Solutions; in the fourth section, the obtained results for the operator Ra are applied to study Gibbs measures for models on a Cayley tree.

2. Existence and uniqueness of positive fixed points for the operator Ra

In this section, we study the existence and the uniqueness of positive fixed points for the nonlinear operator Ra (1.3). We set:

C+[0,1] = {f e C[0,1] : f (x) > 0}, C+[0,1] = C+[0,1] \{0 = 0},

where the set C+[0,1] is the cone of positive continuous functions on [0,1]. We define the Hammerstein operator H$ on C[0,1] by the equality:

Hf (t) = J K(t,u)/*(u)du = f (t), tf > 1.

o

Clearly, by Theorem 44.8 from [4], we obtain:

i

i

Theorem 1. Let $ > 1. The equation:

Hf = f (2.1)

has at least one solution in C+[0,1]. We set:

Mo = {f e C +[0,1]: f (0) = 1} .

Lemma 1. Let a > 1. The equation

Raf = f, f e C+[0,1] (2.2)

has a positive solution iff the Hammerstein operator has a positive eigenvalue, i.e. the Hammerstein equation:

Hag = Ag, f e C+[0,1], (2.3)

has a positive solution in M0 for some A > 0.

Proof. We define the linear operator W and the linear functional u on the C[0,1] by the following equalities:

i i (Wf )(t) = y K(t,u)f (u)du, u(f ) = j K(0, u)f (u)du. 00 Necessariness. Let f0 e C+[0,1] be a solution of the equation (2.2). We have:

(Wfo) (t) = u(fo) f).

From this equality, we get:

(H«h) (t) = Aoh(t),

where h(t) = ft) and Ao = u(fo) > 0.

It is easy to see that h e Mo and h(t) is an eigenfunction of the Hammerstein's operator Ha, corresponding the positive eigenvalue Ao.

Sufficiency. Let h e Mo be an eigenfunction of the Hammerstein's operator Ha. Then, there is a number Ao > 0 such that Hah = Aoh. From h(0) = 1, we get Ao = (Hah) (0) = u (ha). Then:

h(t) = (Hi^M.

v u (ha)

From this equality, we get Rafo = fo with fo = ha e C+[0,1]. This completes the proof. □

Theorem 2. The equation (2.2) has at least one solution in C+ [0,1].

Let Ao be a positive eigenvalue of the Hammerstein operator Ha, a > 1. Then, there exists fo e Mo such that Hafo = Aofo. We take A e (0, +rc>), A = Ao. We define Unction

ho (t) e C+[0,1] by

ho(t)= a-1/^fo(t), t e [0,1]. Ao

Then:

Haho = Ha ^ = Ah0,

i.e. the number A is an eigenvalue of Hammerstein operator Ha corresponding the eigenfunction h0(t). This can be easily verified: if the number A0 > 0 is an eigenvalue of the operator

Ha, a > 1, then an arbitrary positive number is an eigenvalue of the operator Ha. Therefore, we have:

Lemma 2. a) Let a > l.The equation Raf = f has a nontrivial positive solution iff the Hammerstein equation Hag = g has a nontrivial positive solution.

Let a > 1. We denote by Nfix.p (Ha) and Nfix.p (Ra) numbers of nontrivial positive solutions of the equations (2.1) and (2.2), respectively.

Theorem 3. Let a > 1. The equality Nfix.p (Ha) = Nfix.p (Ra) is held.

We denote:

m = min K(t,u), M0 = max K(0,u),

i,ue[0,1] m€[0,1]

M = max K(t, u), m0 = min K(0,u). t,ue[o,i] ue[o,i]

Theorem 4. Let a > 1. If the following inequality holds:

f M\a / m\a 1

vmy - \m) <a

then the homogenous Hammerstein equation (2.1) and the equation (2.2) have a unique nontrivial positive solution.

An analogous theorem was proved for a = k e N, k > 2 in [20] and proof of Theorem 4 is analogously obtained.

3. The existence of a finite number of positive solutions for the homogeneous Hammerstein equation

In this section, for a given n e N, we'll show the existence of n number of positive solutions of homogeneous integral equation of Hammerstein type (1.2). For all p, n e N we define following matrices:

f 1 f 1 \ 2(2p+i+j-2) ^ A!P' = 2(2p + i + j) - 3(2 'n'p e N (31)

V J ij = i,n

B [ai,...,an bi, ...bn ] = (—, ai,bj > 0. (3.2)

Vai + b^i,j=^

Cnp) = B [4p, 4(p +1),..., 2(p + n - 1); 1, 5,..., 4n - 3]. (3.3)

Lemma 3. [25] Let n > 2. Then:

ril<i<j<n [(ai - aj)(bi - bj )]

det B [ai,...,an bi,...,bn]

YTi,i=i (ai + bj)

Corollary 1. detA^ = detC^.

2n(2p+n-i)

Proof. Let i,j e 1, 2,...,n. We multiply by 22(p+j i) the j th column of the matrix A „, after that, we multiply by 22(i-i) the i-th row of the matrix obtained. As a result, we get Cnp). □

Lemma 4. Let B 1 [a1, a2,..., an; b^ b2,..., bn] = {,%be an inverse matrix of B [ai, a2,..., a„; bi, 62,..., bn]. Then:

ß

nn=1(as + bj) nn=1,s=i(ai + )

j TT™

m=i,=j (bj- bsrn:=M=i(ai- as) *

Proof. Subtracting the jth column of B [ai, a2,..., an; bi, b2,..., bn] from every other column, we get the following equality:

det B [ai, a2,..., a™; bi, b2,..., bn] = ( 1 1.1

nn=i(a

s=il ^ 1 J

ai + bi 1

a2 + bi 1

1

ai + bj-i ai + bj+i 11 1

a2 + bj-i 1

a2 + bj+i 1

a™ + bi a„ + bj-i a„ + bj+i

1

ai + b„ 1

a2 + b„

+ bn /

Next, we subtract from the j-th row the i-th row for every j e {1,2..., i — 1, i + 1, ...n}. Then,

det B [ai,..., a™; bi, ...,b™]

nn=i,s=j(bj — bs) nn=i,s=i(ai — as) m=i(as + bj) n™=i,s=i(ai + bs)

x det B(i,j) [ai,...,a™; bi,...,b™]

where B(j,j) [ai,..., an; bi,..., bn] is the cofactor of the element

1

ai + a.

in B [ai,..., a™; bi,...,b™ ],

since:

ß

ji

det B(i,j) [ai,..., a™; bi,...,b™] det B [ai,..., a™; bi,...,b™] '

This completes the proof. We let:

□

A

i

{aij }i,7

ij }i,jei,n-

Remark 1. For each a^ element of (A„ ) , the following equality holds:

i

a

ji

42p+i+j-n+i . nn=i(4p + 2s + 2J — 3) nn=i,s=j(4p + 2s + 2J — 3)

nn=i,s=j j — smn=i,s=i(i — s) .

Proof. By Corollary 1 and Lemma 4 we get:

a

= 42p+i+j det B(i,j) [4p, 4p + 2, ...4p + 2(n — 1); 1, 3, ...2n — 1] ' det B [4p, 4p + 2,...4p + 2(n — 1);1, 3,...2n — 1]

42P+i+j nn=i(4P + 2s + 2J — 3) nn=i,s=i(4P + 2s + 2i — 3)

' nn=i,s=j(2j — 2s) nn=i,s=i(2i — 2s) .

1

a

n

On positive solutions of the homogeneous Hammerstein integral equation Here, we denote

¥>(w)(u) = asiu2p-1 + ... + «snu2(n+p)-3, s,n,p G N, u G [0,1],

n

K(ra>p)(i,u; k) = 1 + ^ (V1 + t2(p+s)-1 - 1) ^(s,n,P)(u), k G N,k > 2, t,u G [0,1].

s=1

Remark 2. For the given k e N, k > 2. the following inequality holds:

1. kl, (t.u^G[0. 12 2 2

We set: f

64 4n - 1( (4n +1)!!

K(n,P^t - 1 ,u - 1 k^ < K(n,i^t - 1 ,u - 1 k^ , (t,u) G [0,1]2,n,p G N.

Zo(n) — ■

9 4n + 1\(n - 1)!(2n + 1)!!

Lemma 5. Let n e N. If k > (0(n), then the following inequality holds:

KKP) (t - 2,u - 1; ^ > 0, (t,u) e [0,1]2,p e N. Proof. For p =1 from Remark 1, we have:

Then:

and

a

j

4i+j-n+3 nn=l(2i + 2s + 1) nn=1,s=j (2j + 2s + 1)

rc=i,=i(i - n=ls=, j - s)

a

a

»J

4(4 j + 1)(2j + 2n + 3) (2j + 3)(4j + 5) :

1, n, j = 1, n - 1

ai+1,.

a

»J

4(n - i)(2i + 2n + 3)

i = 1, n, j = 1, n - 1.

From the above, one has: maxi |aij | = |ann|. By Remark 1, we can take:

K(n,p) (t - 1 ,u - 1 ^ > 1 - 2 max |aj1 JJ | ^ 1 + 2 1 ann1

2s+1

-1 I >

n / 1 N 2s+1

E

3k ^ V 2

s=1

> 1 - (4n - 1) ■

64(2n + 3)2(2n + 5)2...(4n - 1)2(4n + 1) 9k ((n - 1)!)2 '

Since k > (0(n), one gets K(n,p) ^t - 1 ,u - 1 k^ > 0. This completes the proof.

Proposition 1. Let n e N. If k > (0(n), then the Hammerstein's nonlinear operator with kernel K(n,p) ^ t — 1, u — k^ (p e N) has at least n number of positive fixed points.

Proof Let fj(u) = ^ 1 + u2(p+j)-i, j = 17n and ui = u — 1, ti = t — 1. Put gj(t) = fj ^t — ^. We will show functions gj (t) are fixed points of the Hammerstein operator

with the kernel K(n,p) ^t — 1, u — 1 k

1 1

î -1, u - 2; g?(u)du

i 2

K(n,p) ( t - 1, u - ^ k j /j ( u - 1 j du = K(ra>p) (ti, Ui; k) /jfc(ui)dui =

1 + £ ( V1+ t2(P+S)-i - 0 ^(s-)(Ui) (l + u2(p+j)-i)

2(p+j)-i ^ dUi

1 + £ ( V1 + t2(P+S)-i - 1 ) / (asiu4(p-i)+2s+2j + ... + asnu4p+2(s+j+n)-^ dui =

s=i

1 + it (V1+ t2(p+s)-i - ^ (asjßsi + ... + anjßsn) = yï+ii(p+j)-i.

s=i ^ '

Hence:

K(n,p^î -1 ,u - 2; ^ gk(u)du = gjj G {1,2,...,n}.

□

Theorem 5. For each n G N, there exists ^ > 1 and a positive continuous kernel K(t,u) such that, the number of positive solutions for the Hammerstein integral equation (1.2) is equal to at least n.

i

i

4. Gibbs measures for models on Cayley tree rk

In this section we study Gibbs measures for models on Cayley tree. You may be familiar with the definitions and properties of Gibbs measures in books [22-24]. A Cayley tree (Bethe lattice) rk of order k e N is an infinite homogeneous tree, i.e., a graph without cycles, such that exactly k + 1 edges originate from each vertex. Let rk = (V, L) where V is the set of vertices and L that of edges (arcs). Two vertices x and y are called nearest neighbors if there exists an edge l e L connecting them. We will use the notation l = (x,y). A collection of nearest neighbor pairs (x,xi), (xi,x2), ...(xd-i,y) is called a path from x to y. The distance d(x, y) on the Cayley tree is the number of edges of the shortest path from x to y.

For a fixed x0 G V, called the root, we set:

n

Wn = {x G V|d(x, x0) = n}, Vn = U Wm

m=0

and denote:

S(x) = {y G Wn+i : d(x,y) = 1},x G Wn,

the set of direct successors of x.

Consider models where the spin takes values in the set [0,1], and is assigned to the vertices of the tree. For A c V a configuration on A is an arbitrary function : A ^ [0,1]. We denote Qa = [0,1]A the set of all configurations on A and Q = [0,1]V. The Hamiltonian on rk of the model is:

H(a) = -J £ £ (a(x),a(y)), a G Q, (4.1)

(x,y>eL

where J G R \ {0} and £ : (u, v) G [0,1]2 M £u,v G R is a given bounded, measurable function.

Let A be the Lebesgue measure on [0,1]. On the set of all configurations on A the a priori measure Aa is introduced as the |A| fold product of the measure A. Here and subsequently, |A| denotes the cardinality of A. We consider a standard sigma-algebra B of subsets of Q = [0,1]V generated by the measurable cylinder subsets.

Let an : x G K, M an(x) be a configuration in Vn and h : x G V M hx = (ht,x,t G [0,1]) G R[0,1] be mapping of x G V \ {x0}. Given n = 1,2,..., consider the probability distribution ^,(n) on QVn defined by:

^,(n)(a„) = Z—1 exp ( -ftH(a„) + ^ h^x ) . (4.2)

V xGW„ J

Here, as before, an : x G K, M a(x) and Zn is the corresponding partition function:

Z„ = e

-- / exp I -ftH(<j„) + haf(x),^ Ayn (<rra), (4.3)

where ft = T-1, T > 0 - temperature. The probability distributions ^(n) are compatible [21] if

for any n > 1 and an-1 G QVn-1 :

J ^(n) (a„_1 V w„) Aw„ №„)) = ^(n-1) (ffn-1). (4.4)

nWn

Here, an-1 V G QVn is the concatenation of an-1 and In this case, there exists [21] a

unique measure ^ on QV such that, for any n and an G QVn, ^ < a

= 0"n

Vn

) =

The measure ^ is called the splitting Gibbs measure, corresponding to Hamiltonian (4.1) and function x M hx, x = x0.

The following statement describes conditions on hx guaranteeing compatibility of the corresponding distributions ^(n)(an).

Proposition 2. [21] The probability distributions ^(n)(an), n = 1, 2,..., in (4.2) are compatible iff for any x G V \ {x0} the following equation holds:

f (t,x)= n Jo exp( J^,u)f (u,y)du. (4.5)

fo exP(J^Co,u)f (u,y)du

Here and below, f (t, x) = exp(ht,x — h0,x), t e [0,1] and du = A(du) is the Lebesgue measure.

We consider as a continuous function and we are going to solve equation (4.5) in the class of translation — invariant functions f (t,x) (i.e. f (t,x) = f (t) for all x e rk \ {x0}). We'll show that there exists a finite number of translation — invariant Gibbs measures for model (4.1).

For translation — invariant functions, equation (4.5) can be written as:

(Rkf)(t) = f (t), k e N, (4.6)

where K(t,u) = Q(t,u) = exp(J0ft„), f (t) e C0+[0,1], t,u e [0,1] (see [20,21]).

Consequently, for each k e N, k > 2, the Hammerstein integral equation corresponding to the equation (4.6) has the following form:

i

J Q(t,u)fk(u)du = f (t). (4.7)

o

By Theorem 3 and Propositions 1and 2 we'll obtain the following Theorem:

Theorem 6. Let n e N. If k > (0(n), then number of translation-invariant Gibbs measures for the model:

H(*) = — 1 E l^K(n,p^a(x) — 2,a(y) — k]) , a e e N),

on the Cayley tree rk is equal to at least n. References

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