EXISTENCE OF INFINITELY MANY SOLUTIONS FOR FOURTH ...

Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 117, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF INFINITELY MANY SOLUTIONS FOR FOURTH-ORDER EQUATIONS DEPENDING ON TWO PARAMETERS ARMIN HADJIAN, MARYAM RAMEZANI Communicated by Vicentiu Radulescu

Abstract. By using variational methods and critical point theory, we establish the existence of infinitely many classical solutions for a fourth-order differential equation. This equation has nonlinear boundary conditions and depends on two real parameters.

1. Introduction The aim of this article is to study the fourth-order problem u(iv) (x) = λf (x, u(x))

in [0, 1],

0

(1.1)

u(0) = u (0) = 0, 00

000

u (1) = 0,

u (1) = µg(u(1)),

1

where f : [0, 1] × R → R is an L -Caratéodory function, g : R → R is a continuous function and λ, µ are positive parameters. Problem (1.1) describes the static equilibrium of a flexible elastic beam of length 1 when, along its length, a load f is added to cause deformation. Precisely, conditions u(0) = u0 (0) = 0 mean that the left end of the beam is fixed and conditions u00 (1) = 0, u000 (1) = µg(u(1)) mean that the right end of the beam is attached to a bearing device, given by the function g. Existence and multiplicity of solutions for fourth-order boundary value problems has been discussed by several authors in the last decades; see for example [1, 3, 4, 8, 9, 11, 12, 20, 21, 22] and references therein. Yang et al. [22], used Ricceri’s variational principle [19] to establish the existence of at least two classical solutions generated from g for problem (1.1), with µ = 1. The authors in [8], using a multiplicity result by Cabada and Iannizzotto [7], ensured the existence of at least two nontrivial classical solutions for the problem u(4) (x) + λf (x, u(x)) = 0, 0

0 < x < 1,

00

u(0) = u (0) = u (1) = 0, u000 (1) = λg(u(1)), 2010 Mathematics Subject Classification. 34B15, 58E05. Key words and phrases. Fourth-order equations; variational methods; infinitely many solutions. c

2017 Texas State University. Submitted December 21, 2016. Published May 3, 2017. 1

2

A. HADJIAN, M. RAMEZANI

EJDE-2017/117

where the functions f : [0, 1] × R → R and g : R → R are continuous and λ ≥ 0 is a real parameter. More recently, Bonanno et al. [3], by means of an abstract critical points result of Bonanno [2], studied the existence of at least one non-zero classical solution for problem (1.1). Our goal in this article is to obtain sufficient conditions to guarantee that problem (1.1) has infinitely many classical solutions. To this end, we require that the primitive F of f satisfy a suitable oscillatory behavior either at infinity (for obtaining unbounded solutions) or at the origin (for finding arbitrarily small solutions), while G, the primitive of g, have an appropriate growth (see Theorems 3.1 and 3.6). Our analysis is mainly based on a general critical point theorem (see Lemma 2.1 below) contained in [5]; see also [18]. We just point out that Song [20, Theorem 3.1], using the same variational setting but different technical arguments, ensured the existence of infinitely many classical solutions for the problem u(4) = λf (x, u) + µh(x, u),

0 < x < 1,

0

u(0) = u (0) = 0, 00

u (1) = 0,

u000 (1) = g(u(1)),

where λ, µ are two positive parameters, f, h are two L1 -Carathéodory functions, and g ∈ C(R) is a real function. A special case of our main result reads as follows. Theorem 1.1. Let f : R → R be a nonnegative continuous function. Put F (ξ) = Rξ f (t)dt for all ξ ∈ R and assume that 0 lim inf ξ→+∞

F (ξ) =0 ξ2 7

and

0 < B ? := lim sup ξ→+∞

F (ξ) ≤ +∞. ξ2

4

2 π Then, for each λ > 27B ? , for every nonpositive continuous function g : R → R satisfying the condition Rξ − 0 g(t)dt < +∞, g∞ := lim sup ξ2 ξ→+∞ and for each µ ∈ 0, 2g1∞ , the problem

u(iv) (x) = λf (u(x))

in [0, 1],

0

u(0) = u (0) = 0, 00

u (1) = 0,

(1.2)

000

u (1) = µg(u(1)),

admits infinitely many classical solutions. The plan of the article is as follows. In Section 2 we introduce our notation and a suitable abstract setting (see Lemma 2.1). In Section 3 we present our main result (see Theorems 3.1 and 3.6) and some significative consequences (see Theorem 3.8 as well as Corollaries 3.4, 3.5 and 3.9). A concrete example of an application is exhibited in Example 3.3. In the conclusion, we cite a recent monograph by Kristály, R˘adulescu and Varga [10] as a general reference on variational methods adopted here.

EJDE-2017/117

EXISTENCE OF INFINITELY MANY SOLUTIONS

3

2. Preliminaries We shall prove our results applying the following smooth version of [5, Theorem 2.1], which is a more precise version of Ricceri’s variational principle [18, Theorem 2.5]. We point out that Ricceri’s variational principle generalizes the celebrated three critical point theorem of Pucci and Serrin [16, 17] and is an useful result that gives alternatives for the multiplicity of critical points of certain functions depending on a parameter. Lemma 2.1. Let X be a reflexive real Banach space, let Φ, Ψ : X → R be two Gˆ ateaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, strongly continuous and coercive, and Ψ is sequentially weakly upper semicontinuous. For every r > inf X Φ, let supv∈Φ−1 (−∞,r) Ψ(v) − Ψ(u) , ϕ(r) := inf r − Φ(u) u∈Φ−1 (−∞,r) γ := lim inf ϕ(r), δ := lim inf + ϕ(r). r→+∞

r→(inf X Φ)

Then the following properties hold: (a) If γ < +∞, then for each λ ∈]0, 1/γ[, the following alternative holds: either (a1) Iλ := Φ − λΨ possesses a global minimum, or (a2) there is a sequence {un } of critical points (local minima) of Iλ such that lim Φ(un ) = +∞. n→+∞

(b) If δ < +∞, then for each λ ∈]0, 1/δ[, the following alternative holds: either (b1) there is a global minimum of Φ which is a local minimum of Iλ , or (b2) there is a sequence {un } of pairwise distinct critical points (local minima) of Iλ that converges weakly to a global minimum of Φ. We also refer the interested reader to [6, 13, 14, 15] and the references therein, in which Ricceri’s variational principle and its variants have been successfully used to obtain the existence of multiple solutions for different boundary value problems. Let f : [0, 1] × R → R be an L1 -Caratéodory function and g : R → R be a continuous function. We recall that f : [0, 1] × R → R is an L1 -Carathéodory function if (a) the mapping x 7→ f (x, ξ) is measurable for every ξ ∈ R; (b) the mapping ξ 7→ f (x, ξ) is continuous for almost every x ∈ [0, 1]; (c) for every ρ > 0 there exists a function lρ ∈ L1 ([0, 1]) such that sup |f (x, ξ)| ≤ lρ (x) |ξ|≤ρ

for almost every x ∈ [0, 1]. Corresponding to f, g we introduce the functions F, G as follows Z ξ Z ξ F (x, ξ) := f (x, t)dt, G(ξ) := − g(t)dt, 0

0

for all x ∈ [0, 1] and ξ ∈ R. We consider the space X := {u ∈ H 2 ([0, 1]) : u(0) = u0 (0) = 0},

4


EJDE-2017/117

where H 2 ([0, 1]) is the Sobolev space of all function u : [0, 1] → R such that u and its distributional derivative u0 are absolutely continuous and u00 belongs to L2 ([0, 1]). X is a Hilbert space with the inner product Z 1 hu, vi := u00 (t)v 00 (t)dt 0

and the corresponding norm kuk :=

1

Z

1/2 (u00 (t))2 dt .

0

It is easy to see that the norm k · k on X is equivalent to the usual norm Z 1 |u(t)|2 + |u0 (t)|2 + |u00 (t)|2 dt. 0

It is well known that the embedding X ,→ C 1 ([0, 1]) is compact and kukC 1 ([0,1]) := max{kuk∞ , ku0 k∞ } ≤ kuk

(2.1)

for all u ∈ X (see [22]). We say that u ∈ X is a weak solution of problem (1.1) if Z 1 Z 1 00 00 u (x)v (x)dt − λ f (x, u(x))v(x)dx + µg(u(1))v(1) = 0 0

0

for all v ∈ X. By a classical solution of problem (1.1) we mean a function u ∈ C 1 ([0, 1]) such that u(iv) (x) ∈ C([0, 1]) and the boundary conditions and the equation are satisfied in [0, 1]. In [22, Lemma 2.1] it has been shown that the weak solutions are classical solutions of problem (1.1). 3. Main results Before introducing the main result, we define some notation. We put R1 R1 F (x, ξ)dx max F (x, t)dx |t|≤ξ 3/4 , B := lim sup . A := lim inf 0 ξ→+∞ ξ2 ξ2 ξ→+∞ Theorem 3.1. Let f : [0, 1] × R → R be an L1 -Caratéodory function. Assume that (A1) F (x, ξ) ≥ 0 for all (x, ξ) ∈ [0, 3/4] × R; 27 (A2) A < 64π 4 B. 4

1 Then, for every λ ∈ Λ :=] 32π 27B , 2A [ and for every continuous function g : R → R, whose potential G satisfying the conditions inf ξ>0 G(ξ) = 0 and

G∞ := lim sup ξ→+∞

max|t|≤ξ G(t) < +∞, ξ2

(3.1)

if we put 1 (1 − 2Aλ), 2G∞ where µG,λ = +∞ when G∞ = 0, problem (1.1) has an unbounded sequence of classical solutions for every µ ∈]0, µG,λ [ in X. µG,λ :=

EJDE-2017/117


5

¯∈Λ Proof. Our aim is to apply Lemma 2.1(a) to problem (1.1). To this end, fix λ 1 ¯ and g satisfying our assumptions. Since λ < 2A , we have µG,λ¯ :=

1 ¯ > 0. 1 − 2Aλ 2G∞

Now fix µ ¯ ∈]0, µG,λ¯ [. For each u ∈ X, let the functionals Φ, Ψλ,¯ ¯ µ : X → R be defined by 1 Φ(u) := kuk2 , 2 Z 1 µ ¯ Ψλ,¯ F (x, u(x))dx + ¯ G(u(1)) ¯ µ (u) := λ 0 and put ¯ Iλ,¯ ¯ µ (u) := Φ(u) − λΨ(u),

u ∈ X.

By standard arguments, it follows that Φ is sequentially weakly lower semicontin1 uous, strongly continuous and coercive. Moreover, Φ, Ψλ,¯ ¯ µ ∈ C (X, R) and for any u, v ∈ X, we have Z 1 Φ0 (u)(v) = u00 (x)v 00 (x)dx, 0

Ψ0λ,¯ ¯ µ (u)(v) =

Z 0

1

µ ¯ f (x, u(x))v(x)dx − ¯ g(u(1))v(1). λ

In [22] the authors proved that Ψ0λ,¯ ¯ µ is sequentially weakly ¯ µ is compact. Hence Ψλ,¯ (upper) continuous (see [23, Corollary 41.9]). ¯ < 1/γ. Hence, let {ξn } be a sequence of positive First of all, we show that λ numbers such that limn→+∞ ξn = +∞ and R1 max|t|≤ξn F (x, t)dx = A. lim 0 n→+∞ ξn2 Put rn := ξn2 /2 for all n ∈ N. Then, for all v ∈ X with Φ(v) < rn , taking (2.1) into account, one has kvk∞ < ξn . Note that Φ(0) = Ψλ,¯ ¯ µ (0) = 0. Then, for all n ∈ N, supv∈Φ−1 (−∞,rn ) Ψλ,¯ ¯ µ (v) − Ψλ,¯ ¯ µ (u) ϕ(rn ) = inf −1 rn − Φ(u) u∈Φ (−∞,rn ) supv∈Φ−1 (−∞,rn ) Ψλ,¯ (v) ¯µ ≤ rn h R 1 max µ ¯ max|t|≤ξn G(t) i |t|≤ξn F (x, t)dx ≤2 0 + . ¯ ξn2 ξn2 λ Therefore, from assumption (A2) and condition (3.1), we obtain µ ¯ γ ≤ lim inf ϕ(rn ) ≤ 2 A + ¯ G∞ < +∞. n→+∞ λ ¯

¯ < 1/γ. It follows from µ ¯ ∈]0, µG,λ¯ [ that γ < 2A + 1−2λ¯λA . Hence λ ¯ Let λ be fixed. We claim that the functional Iλ,¯ ¯ µ is unbounded from below. 27 Since λ1¯ < 32π 4 B, there exist a sequence {dn } of positive numbers and τ > 0 such

6


EJDE-2017/117

that limn→+∞ dn = +∞ and R1 1 27 3/4 F (x, dn )dx ¯ < τ < 32π 4 d2n λ for all n ∈ N large enough. For n ∈ N we define   x ∈ [0, 3/8], 0, wn (x) := dn cos2 (4πx/3), x ∈]3/8, 3/4[,   dn , x ∈ [3/4, 1].

(3.2)

(3.3)

For any fixed n ∈ N, it is easy to see that wn ∈ X and, in particular, one has 32 4 2 Φ(wn ) = π dn . (3.4) 27 On the other hand, bearing (A1) and inf ξ>0 G(ξ) = 0 in mind, from the definition of Ψλ,¯ ¯ µ and (3.2), we infer that Z 1 µ ¯ 32 4 2 Ψλ,¯ π τ dn . (3.5) F (x, dn )dx + ¯ G(dn ) ≥ ¯ µ (wn ) ≥ 27 λ 3/4 It follows from (3.4) and (3.5) that 32 4 2 32 4 ¯ 2 32 4 ¯ )d2 Iλ,¯ π dn − π λτ dn = π (1 − λτ ¯ µ (wn ) ≤ n 27 27 27 ¯ > 1 and dn → +∞ as n → +∞, we have for all n ∈ N large enough. Since λτ lim Iλ,¯ ¯ µ (wn ) = −∞.

n→+∞

Hence, our claim is proved. It follows that Iλ,¯ ¯ µ has no global minimum. Therefore, by Lemma 2.1(a), there exists a sequence {un } of critical points of Iλ,¯ ¯ µ such that limn→+∞ kun k = +∞, and the proof is complete. Remark 3.2. Under the conditions A = 0 and B = +∞, from Theorem 3.1 we see that for every λ > 0 and for each µ ∈]0, 2G1∞ [, problem (1.1) admits a sequence of classical solutions which is unbounded in X. Moreover, if G∞ = 0, the result holds for every λ > 0 and µ > 0. Here, we present a concrete application of Theorem 3.1. Example 3.3. Let f : [0, 1] × R → R be a function defined by ( 0, (x, t) ∈ [0, 1] × {0}, f (x, t) = 2 x t (2 − 2 sin(ln |t|) − cos(ln |t|)) , (x, t) ∈ [0, 1] × (R − {0}). A direct calculation yields ( 0, (x, t) ∈ [0, 1] × {0}, F (x, t) = 2 2 x t 1 − sin(ln |t|) , (x, t) ∈ [0, 1] × (R − {0}). It is easy to see that A = 0 and B = 37/96. Hence, denoting u+ := max{u, 0}, by 10 4 taking Remark 3.2 into account, we have that for every (λ, µ) ∈]0, 2333π [×]0, 1[ the problem u(iv) (x) = λf (x, u(x)) 0

in [0, 1],

u(0) = u (0) = 0,

EJDE-2017/117


u00 (1) = 0,

u000 (1) = µ u+ (1) + 1

7

has a sequence of classical solutions which is unbounded in X. The following corollary is a direct consequence of Theorem 3.1. Corollary 3.4. Let f : [0, 1] × R → R be an L1 -Caratéodory function. Assume that hypothesis (A1) holds, and 32π 4 1 , B> . 2 27 Then, for every continuous function g : R → R, whose potential G satisfying the conditions inf ξ>0 G(ξ) = 0 and (3.1), if we put A
0 G(ξ) = 0 and (3.1), if we put 1 K(ξ) 1 − (2λkhk1 ) lim inf 2 , µ0G,λ := ξ→+∞ ξ 2G∞ where µ0G,λ = +∞ when G∞ = 0, problem (3.6) has an unbounded sequence of classical solutions for every µ ∈]0, µ0G,λ [ in X.

8


EJDE-2017/117

Put R1 0

A := lim inf

0

ξ→0+

max|t|≤ξ F (x, t)dx , ξ2

R1 0

B := lim sup ξ→0+

3/4

F (x, ξ)dx .

ξ2

Using Lemma 2.1(b) and arguing as in the proof of Theorem 3.1, we can obtain the following multiplicity result. Theorem 3.6. Let f : [0, 1] × R → R be an L1 -Caratéodory function. Assume that (A1) and 27 0 (A3) A0 < 64π 4B , 4

1 32π are satisfied. Then, for every λ ∈ Λ :=] 27B 0 , 2A0 [ and for every continuous function g : R → R, whose potential G satisfying the conditions inf ξ>0 G(ξ) = 0 and

G0 := lim sup ξ→0+

max|t|≤ξ G(t) < +∞, ξ2

if we put 1 (1 − 2A0 λ), 2G0 where µ eG,λ = +∞ when G0 = 0, for every µ ∈ [0, µ eG,λ ) problem (1.1) has a sequence of classical solutions, which converges strongly to zero in X. µ eG,λ :=

Remark 3.7. Applying Theorem 3.6, results similar to Theorem 1.1 and Corollaries 3.4 and 3.5 can be obtained. We omit the discussions here. Now, we put A00 := lim inf ξ→+∞

max|t|≤ξ G(t) , ξ2

B 00 := lim sup ξ→+∞

G(ξ) . ξ2

By reversing the roles of λ and µ, we can obtain the following result. Theorem 3.8. Let g : R → R be a continuous function. Assume that 27 00 (A4) A00 < 64π 4 B . 4

32π 1 1 Then, for every µ ∈ Γ :=] 27B eodory function f : 00 , 2A00 [ and for every L -Carat´ [0, 1] × R → R, whose potential F is a nonnegative function satisfying the condition R1 max|t|≤ξ F (x, t)dx < +∞, (3.7) F∞ := lim sup 0 ξ2 ξ→+∞

there exists λF,µ , where 1 (1 − 2A00 µ), 2F∞ such that for every λ ∈]0, λF,µ [, problem (1.1) has an unbounded sequence of classical solutions in X. λF,µ :=

Proof. Fix µ ¯ ∈ Γ and f satisfying our assumptions. Since µ ¯ < ¯ λF,¯µ > 0. Now fix λ ∈ ]0, λF,¯µ [. Set ¯Z 1 λ e F (x, u(x))dx + G(u(1)), Ψλ,¯ ¯ µ (u) := µ ¯ 0 e λ,¯ Ieλ,¯ ¯Ψ ¯ µ (u) := Φ(u) − µ ¯ µ (u), for all u ∈ X. Clearly, Ieλ,¯ ¯ µ = Iλ,¯ ¯ µ.

1 2A00 ,

we have

EJDE-2017/117


9

Let ξn be a sequence of positive numbers such that limn→+∞ ξn = +∞ and lim

n→+∞

max|t|≤ξn G(t) = A00 . ξn2

Let rn = ξn2 /2 for all n ∈ N. Arguing as in the proof of Theorem 3.1 and from the conditions (A4) and (3.7) we obtain ¯ λ γ ≤ lim inf ϕ(rn ) ≤ 2 F∞ + 2A00 < +∞. n→+∞ µ ¯ ¯ Therefore, from λ ∈]0, λF,¯µ [ we obtain µ ¯ < 1/γ. Let µ ¯ be fixed. We claim that the functional Ieλ,¯ ¯ µ is unbounded from below. Since 1/¯ µ< and

27 00 32π 4 B ,

there exist a sequence {dn } and θ > 0 such that limn→+∞ dn = +∞

1 27 G(dn ) 27 It follows that 32 4 e λ,¯ ¯Ψ Ieλ,¯ π (1 − µ ¯θ)d2n < 0 ¯ µ (wn ) ≤ ¯ µ (wn ) = Φ(wn ) − µ 27 for all n ∈ N large enough. Therefore, limn→+∞ Ieλ,¯ ¯ µ (wn ) = −∞, and the proof is complete. Corollary 3.9. Assume that g : R → R be a nonpositive continuous function such that Rξ Rξ − 0 g(t)dt − 0 g(t)dt lim inf = 0, lim sup = +∞. ξ→+∞ ξ2 ξ2 ξ→+∞ Then, for each µ > 0 and for every nonnegative continuous function f : R → R satisfying the condition Rξ f (t)dt f∞ := lim sup 0 2 < +∞, ξ ξ→+∞ and for each λ ∈ 0, 2f1∞ , problem (1.2) admits infinitely many classical solutions. References [1] S. Baraket, V. R˘ adulescu; Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent, Adv. Nonlinear Stud., 16 (2016), 409–419. [2] G. Bonanno; Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205–220. [3] G. Bonanno, A. Chinn`ı, S. Tersian; Existence results for a two point boundary value problem involving a fourth-order equation, Electron. J. Qual. Theory Differ. Equ., 2015 (2015), No. 33, pp. 1–9. [4] G. Bonanno, B. Di Bella; A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166–1176. [5] G. Bonanno G. Molica Bisci; Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009 (2009), 1–20. [6] G. Bonanno, G. Molica Bisci, V. R˘ adulescu; Infinitely many solutions for a class of nonlinear elliptic problems on fractals, C. R. Math. Acad. Sci. Paris, 350 (2012), 187–191.

10


EJDE-2017/117

[7] A. Cabada, A. Iannizzotto; A note on a question of Ricceri, Appl. Math. Lett., 25 (2012), 215–219. [8] A. Cabada, S. Tersian; Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Appl. Math. Comput., 219 (2013), 5261–5267. [9] R. Demarque, O. H. Miyagaki; Multiplicity of solutions of a two point boundary value problem for a fourth-order equation, Adv. Nonlinear Anal., 4 (2015), 135–151. [10] A. Krist´ aly, V. R˘ adulescu. Cs. Varga; Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, no. 136, Cambridge University Press, Cambridge, 2010. [11] F. Li, Q. Zhang, Z. Liang; Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal., 62 (2005), 803–816. [12] T. F. Ma; Positive solutions for a beam equation on a nonlinear elastic foundation, Math. Comput. Modelling, 39 (2004), 1195–1201. [13] G. Molica Bisci, D. Repovˇs; Multiple solutions of p-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ., 59 (2014), no. 2, 271–284. [14] G. Molica Bisci, D. Repovˇs; Multiple solutions for elliptic equations involving a general operator in divergence form, Ann. Acad. Sci. Fenn. Math., 39 (2014), no. 1, 259–273. [15] G. Molica Bisci, D. Repovˇs; Algebraic systems with Lipschitz perturbations, J. Elliptic Parabol. Equ., 1 (2015), 189–199. [16] P. Pucci, J. Serrin; Extensions of the mountain pass theorem J. Funct. Anal., 59 (1984), 185–210. [17] P. Pucci, J. Serrin; A mountain pass theorem, J. Differential Equations 60 (1985), 142–149. [18] B. Ricceri; A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401–410. [19] B. Ricceri; A further three critical points theorem, Nonlinear Anal., 71 (2009), 4151–4157. [20] Y. Song; A nonlinear boundary value problem for fourth-order elastic beam equations, Bound. Value Probl., 2014 (2014), No. 191, pp. 1–11. [21] S. Tersian, J. Chaparova; Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations, J. Math. Anal. Appl., 260 (2001), 490–506. [22] L. Yang, H. Chen, X. Yang; The multiplicity of solutions for fourth-order equations generated from a boundary condition, Appl. Math. Lett., 24 (2011), 1599–1603. [23] E. Zeidler; Nonlinear Functional Analysis and its Applications, vol. II/B and III, SpringerVerlag, Berlin-Heidelberg-New York, 1985. Armin Hadjian (corresponding author) Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran E-mail address: [email protected], [email protected] Maryam Ramezani Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord 94531, Iran E-mail address: [email protected]