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ISSN 2074-1871 Уфимский математический журнал. Том 12. № 1 (2020). С. 92-103.
EXISTENCE OF SOLUTIONS FOR NONLINEAR SINGULAR g-STURM-LIOUVILLE PROBLEMS
B.P. ALLAHVERDIEV, H. TUNA
Abstract. In this paper, we study a nonlinear q- Sturm-Liouville problem on the semiinfinite interval, in which the limit-circle case holds at infinity for the g-Sturm-Liouville expression. This problem is considered in the Hilbert space L^ (0, ж). We study this problem by using a special way of imposing boundary conditions at infinity. In the work, we recall some necessary fundamental concepts of quantum calculus such as ^-derivative, the Jackson g-integration, the g-Wronskian, the maximal operator, etc. We construct the Green function associated with the problem and reduce it to a fixed point problem. Applying the classical Banach fixed point theorem, we prove the existence and uniqueness of the solutions for this problem. We obtain an existence theorem without the uniqueness of the solution. In order to get this result, we use the well-known Schauder fixed point theorem.
Keywords: Nonlinear g-Sturm-Liouville problem, singular point, Wevl limit-circle case, completely continuous operator, fixed point, theorems.
Mathematics Subject Classification: 39A13, 34B15, 34B16, 34B40
1. Introduction
Nowadays, quantum calculus, or g-ealeulus, attracts a lot of attention because it differs from the classical calculus in the sense that it does not require the concept of limit. It plays an important role in different mathematical areas, such as number theory, orthogonal polynomials, fractal geometry, combinatorics, calculus of variations, mechanics, orthogonal polynomials, as well as in statistic physics, nuclear and high energy physics, conformal quantum mechanics, and theory of relativity. For a general introduction to the quantum calculus, we refer the reader to the references [1]—[3].
So-called ^-difference equations are important in quantum calculus. Recently, much efforts were made in to study the existence of solutions to ^-difference equations, see [4]—[18]. However, there is no results on the existence of solutions to a singular impulsive nonlinear g-Sturm-Liouville problems as the limit-circle case holds at infinity. In this paper, we fill the gap in this area by using a special way of imposing boundary conditions at infinity. While proving our results, we use the machinery and methods of [19], [20].
In the following section, we recall some necessary fundamental concepts of the quantum calculus.
2. Preliminaries
Following the standard notations in [l]-[3], let q be a positive number obeying the inequality 0 < q < 1, А с R and a g A A ^-difference equation is an equation that contains ^-derivatives
B.P. Allahverdiev, H. Tuna, Existence of solutions for nonlinear singular g-Sturm-Liouville problems.
© Allahverdiev B.P., Tuna H. 2020.
Поступила 24 апреля 2019 г.
of a function defined on A Let y be a complex-valued function on A The g-differenee operator Dq, the Jackson ^-derivative is defined by
^ , . y (qx) — y(x) Dqy(x) = —^ for ai 1 x e A. qx — x
Note that there is a connection between g-deformed Heisenberg uncertainty relation and the Jackson derivative on g-basic numbers, see [21], As q ^ 1, the ^-derivative is reduced to the classical derivative. The ^-derivative at zero is defined by
DqV(0) = ^ * (f*\— № (X e A),
n^tt qnx
if this limit exists and is independent of x. The formulation of the extension problems requires
f(x) — f(q-1x)
the definition of Dq-i, which reads as follows:
Dg-i f(x) := < x -q 1X
x e A \ {0},
Dqf (0), X = 0,
provided Dqf (0) exists. Associated with this operator, there is a non-svmmetric formula for
Dq [f(x)9(x)] = g(x)Dqf ^ + f(qx)Dqg (x). A right-inverse to Dqj the Jackson ^-integration is defined as
<x
f(t)dqt = x (1 - q) ^ qnf (qnx) (x e A),
x (1 — ^ 0 n=0 provided the series converges, and
b b a
j № dqt = J f(t)dqt —J f(t) dqt (d ,b E A).
a 0 0
The ^-integration for a function over [0, ro) was defined in [22] by the formula
tt _
f. tt
nn
f(t)dqt =¿2 Qnf (qn).
n=—oo
A function f defined on A, 0 e A, is said to be ^-regular at zero if
lim f(xqn) = f(0),
n^tt
for each x e A In the rest of the paper, we deal only with functions ^-regular at zero,
a a
j g(t)Dq f (t)dqt — J f(qt)Dqg(t) dqt = f(a)g(a) — f(0)g(0).
00
Let L2q(0, ro) be the space of all complex-valued functions defined on (0, ro) such that
If (x)f dq x] < ro.
x
The space L2q(0, œ) is a separable Hilbert space with the inner product
{f,g)'-= J f (x)g {x)dqx, f,g e Lq{0, X)), 0
see [3], [23],
The q-Wronskian of y(x), z(x) is defined as
Wq(y,z)(x): = y(x)Dqz(x) — z(x)Dqy(x), x e [0,a]. (2,1)
We consider the following nonlinear q-Sturm-Liouville equation
l(y) : = — 1 Dq-i (p(x)Dqy(x)) + r(x)y(x) = f (x,y(x)), (2.2)
where p, r are real-valued functions defined on [0, <x) and continuous at zero, 1 ,r e Lqqloc (0, <x>) and y = y(x) is a sought solution.
We denote bv V the linear set of all functions y e Lq(0, <x>) such that ^d pDqy are q-regular at zero and I (y) e L2q (0, x). The operator L defined by Ly = l(y) is called the maximal operator on L2q(0, <x).
For each y,z e V we have q-Green formula (or q-Lagrange identity)
J (Ly)(x)z(x)dqx — J y(x)(Lz)(x)dqx = [y, z]t — [y, z]o, t e (0, x), (2.3) 00
where
[y,z]x := p(x){y(x)Dq-iz(x) — Dq-iy(x)z(x)},
see [3], [23].
In view of (2.3), it is clear that the limit
[y,z}^ = lim [y,z] (q-n)
exists and is finite for all y, z eV.
For each function y e V, the values y(0) and (pDq-iy)(0) can be defined as
y(0) := lim y(qn)
and
(pDq-iy)(0) := lim(pDq-iy)(qn).
n—>oo
These limits exist and are finite since y and (pDq-i )y are q-regular at zero. We assume that the following conditions are satisfied. (Al) The functions p and r are such that all solutions of the equation
l(y) = 0 (2.4)
belong to Lq (0, œ), i.e., the Wevl limit-circle case holds for the g-Sturm-Liouville expression I [23].
(A2) The function f (x, y) is real-valued and continuous in (x,( ) E (0, œ) x R, and, for all (x, () in (0, œ) x R,
If (x,()| ^ g(x)+ $ KI , (2.5)
where g(x) ^ 0, g E Lq (0, œ), and ê is a positive constant.
Denote bv u(x) v(x) the solution of equation (2.4) satisfying the initial conditions
u(0) = 0, (pDq-iu) (0) = 1, v(0) = -1, (pDq-iv))(0) = 0. (2.6)
t
t
Since the Wronskian of any two solutions of equation (2,4) are constant, we have Wq(u, v) = 1, Then, u and v are linearly independent and they form a fundamental system of solutions of equation (2,4), By the condition (Al), we get u, v E L2q (0, œ) and moreover, u, v EV. So, the values [y,u]m and [y, exist and are finite for each y eV. By using Green formula (2,3) and conditions (2,6), we obtain:
[y,u]cx = y(0) + y u (x) l(y(x))dqX, 0
[y,v}^ = (pDq-i y)(0) + v(x)l(y(x))dq x.
(2.7)
We complete problem (2,2) by the boundary conditions
y(0) cos a + (pDq-iy)(0) sin a = d\, [y,u]^ cos P + [y,v]^ sin P = dq,
where a, [3 E R. Our next assumption is as follows, (A3) The inequality holds:
p := cos a sin ¡3 — cos ¡3 sin a = 0,
and d\, dq are arbitrary given real numbers.
Since the function y in (2,8) satisfies equation (2,2), we have
(2.8)
[y,u]^ = y(0)+ / u (x) f (x,y(x))dqx,
[y,v]^ = (pDq-i y)(0) + v(x)f (x,y(x))dq x.
3. Green function
In this section, we construct a Green function for boundary value problem (2.2), (2.8), and then, we reduce this problem to a fixed point problem. We consider a linear boundary value problem
-1 Dg-1 (p(x)Dqy(x)) + r(x)y(x) = h (x) , x E (0, to), h E L2q (0, to) (3.1) y(0) cos a + (pDq-iy)(0) sin a = 0,
[y,u]^ cos /3 + [y,v](X sin /3 = 0, a,P E R, ()
where y is a sought solution, u and v are solutions of equation (2.4) satisfying conditions (2.6). We let
p(x) = cos au(x) + sin av (x), ip(x) = cos fiu(x) + sin0v(x), (3.3)
where
Wq (p, ip) = cos a sin ¡3 — cos ¡3 sin a = W.
It is clear that these functions are solutions of equation (2.4) and are in L2(0, œ). Further, we have
[<p,u]x = <(0) = - sin a, [<, v]x = (pDq-i<p) (0) = cos a, (3.4)
[tp, u]x = p(a) = - sin ft, [tp, v]x = (pDq-ip) (0) = cos ft, (3.5)
[tp,u]^ = - sin ft, [tp, = cos ft. (3.6)
We introduce a function
<(x)p(t)
as i ^ x,
G(x, t)=< ^ (3.7)
<(t)t(x) as t>x.
W
The function G (x, t) is the Green function of boundary value problem (3.1)-(3.2). Since p,ip e Lq (0, x), we have
oo oo
j J\G(x, t)\qdqxdqt < X, (3.8)
00
that is, G(x, t) is a q-Hilbert-Schmidt kernel. Theorem 3.1. The function
oo
v(x) = Jg(x, t)h(t)dqt, x e (0, x) , (3.9)
0
is the solution to boundary value problem (3.1)-(3.2).
Proof. By a variation of constants formula, the general solution of equation (3.1) has the form
X
y(x) = h<p(x) + k2p (x) + r^p(x) < (qt) h(t)dgt - ^<p(x) I p(qt)h(t)dgt, (3.10)
where k\ and k2 are arbitrary constants. By (3.10), we get
(PDq-iy) (x) = ki (pDq-i<) (x) + k2 (pDg-ip) (x)
q
X
+ £ (pDg-it) (x) <p(qt)h(t)dgt
W
o
[•X
- W (pDg-i<p)(x) t(qt)h(t)dqt.
Hence, we have
y(0) = kq<^(0) + k2p (0) = — kq sin a — k2 sin ft, (pDq-1 y) (0) = kq (pDq-ip) (0) + k2 (pDq-1^) (0) (3.11)
= kq cos a + k2 cos ft. Substituting (3.11) into (3.2), we get
k2(cosa sin — sin a cos,) = 0, k2W = 0,
X
EXISTENCE OF SOLUTIONS FOR NONLINEAR SINGULAR q-STURM-LIOUVILLE PROBLEMS 97 that is, kq = 0. Further, we have
[y,u]x =p(x){ y(x)Dq-i u(x) — Dq-1 y(x)u(x)}
=ki[<p,u]x + kq[^,u]x + <p(qt)h(t)dqt — [p,u]x j i>(qt)h (t) dqt
0_
J TXI-r- V/"^ ■ w
= — kl sin a — — sin ft j (p(qt)h (t) dqt + — sin a ip(qt)h(t)dqt
0
X
= — kl sin a + I (— sin ftp (qt) + sin a^(qt)) h(t)dqt
0
X
= — kl sin a + I u(qt)h (t) dqt.
Thus,
oo
[y,u]o = — kl sin a + q u(qt)h(t)dqt.
0
Similarly, we get
[y,v]x =p(x){ y(x)Dq-i v(x) — Dg-1 y(x)v(x)}
x
--kl[p,v]x + [^,v]x <p (qt) h(t)dqt — [p,v]x I ^(qt)h (t) dqt
and
oo
[y,v]<x = kl cos a + q J v(qt)h(t)dqt.
0
By conditions (3.2) we obtain
kl (— sin a cos ft + cos a sin ft) + q j [cos ftu(qt) + sin ftv (qt)] h(t)dqt = 0.
Hence,
Mgv
0
kl = — ^ I ^(qt)h(t)dqt.
By (3.10), we get
y(x) = — W ^(^t)^(x)h(t)dqt — ^ I ^(^)^(Qt)h(t)dqt
0 x
that is, (3.7) and (3.9) hold true. □
Our next statement is the following theorem.
X
X
X
X
X
X
Theorem 3.2. The unique solution of boundary value problem (3.1) subject to conditions (2.8) is given by the formula
o
y(x) = u(x) + J G (x,t) h(t)dgt, 0
where
u(x) = V(x) — W ^(X).
Proof. By the conditions (3.4)-(3.6), the function u(x) is a unique solution of the boundary value problem (3.1) satisfying conditions (2.8). This completes the proof. □
From Theorem 3.2, boundary value problem (2.2), (2.8) in Lq (0, to) is equivalent to the nonlinear ^-integral equation
o
y(x) = u(x) + y G (x,t) f (t, y(t)) dq t, (3.12)
0
where the functions u(x) and G(x,t) are defined above. In what follows we study equation (3.12).
By (2.5) and (3.8), we can define the operator T : Lq (0, to) ^ Lq (0, to) by the formula
o
(Ty)(x) = u(x) + J G (x,t) f (t, y(t)) dqt, x E (0, to) , (3.13)
0
where y,u E Lq (0, to). Then equation (3.12) can be written as y = Ty.
Our next step is to find the fixed points of the operator T because it is equivalent to solving the equation (3.12). In the next section we study the operator T by using a Banach fixed point theorem.
4. Fixed points of operator T
Definition 4.1 ([24]). Let A be a mapping of a metric space R into itself. Then x is called a fixed point of A if Ax = x. Suppose there exists a number a < 1 such that
P (Ax,Ay) ^ ap(x,y)
for each pair of points x,y E R. Then A is said to be a contraction mapping.
Theorem 4.2 ([24]). Each contraction mapping A defined on a complete metric space R has a unique fixed point.
Theorem 4.3. Suppose that conditions (A1), (A2) and (A3) are satisfied. Let the function f (x,y) satisfy the following Lipschitz condition: there exist a constant K > 0 such that
o o
J \f (x,y(x)) — f (x,z(x))\2 dq x ^ K2 j \y(x) — z(x)\2 dq x 00 for each y,z E Lq (0, to) . If
o o
K If [\G (x,t)\q dqxdqt I < 1, (4.2)
00
then boundary value problem (2.2), (2.8) has a unique solution in Lq (0, to) .
Proof. It is sufficient to show that the operator T is a contraction operator. For y,z E L2 (0, x), we have
2
12
I(Ty)(x) - (Tz)(x)[
G(x,t) [f (t,y (t)) - f (t,z(t))] dqt
0
oo oo
^ J IG(x,t)l2 dqtj If (t,y(t)) - f (t,z(t))l2 dqt 00
^ K2 \\y - z\\2 IG (x,t)f dqt, x E (0, x).
Thus, we get
where
\\Ty - Tz\\ ^ « \\y - z\\
a = K I IG (x,t)I2 dgxdgt] < 1
00
and hence, T is a contraction mapping.
□
In the next theorem we consider the case, when the function f (x,y) satisfies a Lipschitz condition on a subset of L"^ (0, x); this property is not assumed to hold on the entire space.
Theorem 4.4. Suppose that conditions (A1), (A2) and (A3) are satisfied. In addition, let the function f (x,y) satisfy the following Lipschitz condition: there exist constants M, K > 0 such that
If (x,y(x)) - f (x,z(x))f dg x ^ K2 Iy(x) - z(x)I2 dq x
(4.3)
for all y and z in Sm = {y E L" (0, x) : \\y\\ ^ M}, where K may depend on M. If
2 / OO (X
Iu(x)f dqx I + 1
IG(x,t)I2 dgxdgt I sup [/ If (t,y(t))I2 dgt\ ^ M (4.4)
y£sm
0 0 0
and
K | / / IG (x,t)I2 dgxdgt \ < 1,
(4.5)
00
then boundary value problem (2.2)-(2.8) has a unique solution. This solution satisfies the estimate
(x)I2 dgX ^ M2
2
Proof. It is clear that Sm is a closed set of L"^ (0, to). First, we are going to prove that the operator T maps Sm into itself. For y E Sm we have
\\Ty\\
u (.)+ G (.,t) f (t,y(t)) dqt
< \M\ +
0
oo
G (.,t) f (t,y(t)) dqt
^ \M\ +
IG(x,t)f dqxdqt\ sup \ If (t,y(t))\2 dgt\ ^ M.
y£sm
0 0 0 Thus, T :Sm ^ Sm.
We now proceed analogously to the proof of Theorem 4.2 and we get
\\Ty - Tz\\ ^ a \\y - z\\ , y,z E SM.
We apply the Banach fixed point theorem and we obtain a unique solution of boundary value problem (2.2), (2.8) in Sm. The proof is complete. □
5. Existence theorem without uniqueness
In this section, we obtain an existence theorem without the uniqueness of the solution. In order to get this result, we will use the following Schauder fixed point theorem:
Definition 5.1 ([19, 20]). An operator acting in a Banach space is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets.
Theorem 5.2 ([19, 20]). Let B be a Banach space and S be a non-empty bounded, convex, and closed subset of B. Assume that A : B ^ B is a completely continuous operator. If the operator A maps the set S into itself, that is, if A (S) C S, then A has at least one fixed point in S.
Theorem 5.3. The operator T defined by (3.13) is a completely continuous operator under conditions (A1), (A2) and (A3).
Proof. Let y0 E L2 (0, to). Then we obtain:
I(Ty)(x) - (Tyo)(x)\2
G(x,t) [f (t,y(t)) - f (t,yo(t))] dqt
0 oo
oo
^y \G(x,t)\2 dqtj If (t,y(t)) - f (t,yo(t))\2 dqt. 00
Thus,
where
\\Ty - Tyo\\2 ^ KJ \f (t,y(t)) - f (t,yo(t))\2 dqt, 0
(5.1)
K = I I J \G (x,t)\2 dqxdqt ,o o
2
2
2
We know that the operator F defined by Fy(x) = f (x, y(x)) is continuous in L"^ (0, x) under condition (A2), see [25]. Hence, for a given e > 0, we can find a 5 > 0 such that the inequality | | y - yo | | <5 implies
OO
/ If (t, y(t)) - f(t, yo(t))l2 dqt < ^. o
It follows from (5.1) that
| | Ty -Tyoll < t,
that is, T is continuous. We denote
Y = {y EL" (0, x) : || y KG} .
By (3.13) we have
| | Ty | | ^ | | w | | + {k j™ If (t, y(t))I2dqt |2 forall yeY. Furthermore, using (2.5), we get
O OO
J If (t, y(t))I2 dqt ^ J [g(t)+>& Iy (i)|]2 dqt oo
oo
^ 2 J [g2(t)+if Iy(t)I2} dqt o
= 2 {\\g112 + #2M2) ^ 2 (\\g112 + #2C2) .
Thus, for all y EY, we obtain
| | Ty | | w | | + [2 K {\\g 112 + $2C2)] 2 ,
that is, T(y) is a bounded set in L2 (0, x).
For all E Y we have
L L L
J \Ty(x)|2 dqx ^ 2 (1^||2 + $2C2) J J IG(x, t)I2 dqxdqt.
N N 0
Hence, by (3.8), we see that for a given e > 0 there exists a positive number N depending only on such that
L
J \Ty(x)|2 dqx < e2
N
for all y E Y. Thus, T(y) is relatively compact in L2 (0, x), and the operator T is therefore completely continuous. □
Theorem 5.4. Suppose that conditions (A1), (A2) and (A3) are satisfied. In addition, let there exists a constant M > 0 such that
\ 2 / oo oo \ 1 / oo
Mx)|2 dqx\ + If i G)|2 dqxdqt | sup [I If (t, y(t))I2 dqt I ^M, (5.2) loo J yeSM Vo
where
SM = {y E L] (0, to): \\ y\\ ^ M} . Then boundary value problem (2.2), (2.8) has at least one solution with
i \y(x)\2 dqx ^ M2.
Proof. We define an operator T : L" (0, to) ^ L" (0, to) by (3.13). By Theorems 4.4 and 5.3 and inequality (5.2) we conclude that T maps the set Sm into itself. It is clear that the set Sm is bounded, convex and closed. Now theorem follows Theorem 5.2. □
REFERENCES
1. V. Kac, P. Cheung. Quantum calculus. New York, NY, USA, Springer-Verlag (2002).
2. T. Ernst. The History of q-calculus and a new method // U. U. D. M. Report. Uppsala, Sweden: Department of Math., Uppsala Univ (2000).
3. M.H. Annaby, Z.S. Mansour. q-Fractional Ccalculus and equations. Lecture Notes Math. 2056. Heidelberg, Germany, Springer-Verlag (2012).
4. B. Ahmad, J.J. Nieto. Basic theory of nonlinear third-order q-difference equations and inclusions // Math. Model. Anal. 18:1, 122-135 (2013).
5. B. Ahmad, S. K. Ntouyas. Boundary value problems for q-difference inclusions // Abstr. Appl. Anal. 2011, 292860, (2011).
6. B. Ahmad, S. K. Ntouyas. Boundary value problems for q-difference equations and inclusions with nonlocal and integral boundary conditions // Math. Model. Anal. 19:5, 647-663 (2014).
7. B. Ahmad, S. K. Ntouyas, I. K. Purnaras. Existence results for nonlinear q-difference equations with nonlocal boundary conditions // Comm. Appl. Nonl. Anal. 19:3, 59-72 (2012).
8. T. Saengngammongkhol, B. Kaewwisetkul, T. Sitthiwirattham. Existence results for nonlinear second-order q-difference equations with q-integral boundary conditions // Diff. Equat. Appl. 7:3, 303-311 (2015).
9. T. Sitthiwirattham, J. Tariboon, S.K. Ntouyas. Three-point boundary value problems of nonlinear second-order q-difference equations involving different numbers of q // J. Appl. Math. 2013, 763786 (2013).
10. W. Sudsutad, S. K. Ntouyas, J. Tariboon. Quantum integral inequalities for convex functions // J. Math. Inequal. 9:3, 781-793 (2015).
11. P. Thiramanus, J. Tariboon. Nonlinear second-order q-difference equations with three-point boundary conditions // Comput. Appl. Math. 33:2, 385-397 (2014).
12. C. Yu, J. Wang. Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives // Adv. Difference Equat. 2013, 124 (2013).
13. M. El-Shahed, H.A. Hassan. Positive solutions of q-difference equation // Proc. Amer. Math. Soc. 138:5, 1733-1738 (2010).
14. F.H. Jackson. On q-difference equations // Amer. J. Math. 32:4, 305-314 (1910).
15. Z. Hefeng, L. Wenjun. Existence results for a second-order q-difference equation with only integral conditions // Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 79:4, 221-234 (2017).
16. Z. Mansour, M. Al-Towailb.q-Lidstone polynomials and existence results for q-boundary value problems // Bound. Value Probl. 2017, 178 (2017).
17. M.J. Mardanov, Y.A. Sharifov. Existence and uniqueness results for q-difference equations with two-point boundary conditions // AIP Conf. Proc. 1676, 020065 (2015).
18. R. P. Agarwal, G. Wang, B. Ahmad, L. Zhang, A. Hobiny, S. Monaquel. On existence of solutions for nonlinear q- difference equations with nonlocal q-integral boundary conditions // Math. Model. Anal. 20:5, 604-618 (2015).
19. G.Sh. Guseinov, I. Yaslan. Boundary value problems for second order nonlinear differential equations on infinite intervals // J. Math. Anal. Appl. 290:2, 620-638 (2004).
20. B.P. Allahverdiev, H. Tuna. Nonlinear singular Sturm-Liouville problems with impulsive conditions // Facta Univ., Ser. Math. Inf. 34:3, 439-457 (2019).
21. P.N. Swamy. Deformed Heisenberg algebra:origin of q-calculus // Physica A: Statist. Mechan. Appl. 328:1-2, 145-153 (2003).
22. W. Hahn. Beitraage zur Theorie der Heineschen Reihen // Math. Nachr. 2, 340-379 (1949) (in German).
23. M.H. Annaby, Z.S. Mansour, I.A. Soliman.q-Titchmarsh-Weyl theory: series expansion // Nagoya Math. J. 205, 67-118 (2012).
24. A.N. Kolmogorov, S.V. Fomin. Introductory Real Analysis. Dover Publ., New York (1970).
25. M.A. Krasnosel'skii. Topological methods in the theory of nonlinear integral equations. Gostekhte-oretizdat, Moscow (1956) [Pergamon Press, New York, (1964)].
Bilender Pa§aoglu Allahverdiev, Süleyman Demirel University, Department of Mathematics 32260, Isparta, Turkey E-mail: bilenderpasaoglu@sdu.edu.tr
Hüseyin Tuna,
Mehmet Akif Ersoy University, Department of Mathematics, 15030, Burdur, Turkey E-mail: hustuna@gmail.com
