J. Fixed Point Theory Appl. 52:0)81( https://doi.org/10.1007/s11784-018-0510-0 c Springer International Publishing AG,  part of Springer Nature 2018

Journal of Fixed Point Theory and Applications

Fixed point theorems in b-metric spaces and their applications to non-linear fractional differential and integral equations Mian Bahadur Zada, Muhammad Sarwar and Cemil Tunc Abstract. In this paper, we modify L-cyclic (α, β)s -contractions and using this contraction, we prove fixed point theorems in the setting of b-metric spaces. As an application, we discuss the existence of a unique solution to non-linear fractional differential equation, c

Dσ (x(t)) = f (t, x(t)), for all t ∈ (0, 1),

(1)

with the integral boundary conditions,  ρ x(0) = 0, x(1) = x(r)dr, for all ρ ∈ (0, 1), 0

where x ∈ C([0, 1] , R), c Dα denotes the Caputo fractional derivative of order σ ∈ (1, 2], f : [0, 1] × R → R is a continuous function. Furthermore, we established existence result of a unique common solution to the system of non-linear quadratic integral equations,  1 x(t) = 0 H(t, τ )f1 (τ, x(τ ))dτ, for all t ∈ [0, 1]; 1 x(t) = 0 H(t, τ )f2 (τ, x(τ ))dτ, for all t ∈ [0, 1], where H : [0, 1] × [0, 1] → [0, ∞) is continuous at t ∈ [0, 1] for every τ ∈ [0, 1] and measurable at τ ∈ [0, 1] for every t ∈ [0, 1] and f1 , f2 : [0, 1] × R → [0, ∞) are continuous functions. Mathematics Subject Classification. 47H09, 54H25. Keywords. b-Metric spaces, common fixed points, weakly compatible maps, admissible mapping, non-linear quadratic integral equations, nonlinear fractional differential equation.

1. Introduction and preliminaries The theory of integral and differential equations creates an extremely important involvement in mathematical analysis. Many real world problems that come from physical sciences, engineering, biology, geology, economics and applied mathematics, give rise to mathematical models reported by non-linear

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fractional differential and integral equations. There are various advanced methods, that test the existence of a solution to such non-linear mathematical models. One of these powerful tools is the method of fixed point theorems. Several authors apply fixed point results to prove the existence and uniqueness of solution of the non-linear fractional differential equations [2,5– 8,19] and non-linear integral equations [3,13,23,25]. In metric fixed point theory the first and the most widely applied fixed point result is the Banach contraction principle. Many mathematicians generalized this principle either by awakening the contraction conditions or by imposing various additional conditions on metric space or in terms of number of contraction mappings. Samet et al. [20] introduced the notions of α-admissible mapping and studied fixed point results under α − ψ-contraction which extend and generalize many known fixed point theorems, notably the Banach principle. Following this trend several authors gave the generalizations of this innovative approach [4,15,21,23,24]. Alizadeh et al. [4] initiated the concept of a cyclic (α − β)admissible mapping and demonstrated fixed point theorems which generalize and modify some recently established results. In recent times, Isik et al. [15] generalized the notion of a cyclic (α − β)-admissible mapping for two mappings as follows: Definition 1.1. [15] Let K, L : X → X and α, β : X → [0, ∞). Then K is L-cyclic (α, β)-admissible mapping if (i) α(Lu) ≥ 1 for some u ∈ X implies β(Ku) ≥ 1; (ii) β(Lu) ≥ 1 for some u ∈ X implies α(Ku) ≥ 1. In view of (α − β)-admissible mapping Isik et al. [15] presented some common fixed point theorems. In addition, Czerwik [11] proposed the idea of b-metric space, essentially the generalization of ordinary metric space and established fixed point results which generalize the Banach contraction theorem. Definition 1.2. [9,11] Let X be a nonempty set. A function d : X × X → R+ is called b-metric with coefficient s ≥ 1, if for all y1 , y2 , y3 ∈ X, the following assumptions holds: (1) d(y1 , y2 ) = 0 ⇔ y1 = y2 ; (2) d(y1 , y2 ) = d(y2 , y1 ) ∀ y1 , y2 ∈ X; (3) d(y1 , y2 ) ≤ s[d(y1 , y2 ) + d(y3 , y2 )]; and the pair (X, d) is called b-metric space. Every metric space is b-metric space with s = 1, but not conversely, see examples in [12]. Furthermore, Boriceanu [10] interpreted the definitions of a b-convergent sequence and a b-Cauchy sequence in b-metric spaces. In addition, a b-metric space X is complete if and only if each b-Cauchy sequence is convergent in X. In general, b-metric is not continuous, see examples in [14,18]. Thus to study fixed point results, one requires the following lemma. Lemma 1.1. [1] Let (X, d) be a b-metric space with coefficient s ≥ 1 and let {un } and {vn } be b-convergent sequences to the points u, v ∈ X, respectively.

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Then 1 d(u, v) ≤ lim inf d(un , vn ) ≤ lim sup d(un , vn ) ≤ s2 d(u, v). n→∞ s2 n→∞ In particular, if u = v, then, lim inf d(un , vn ) = 0. Moreover, for each t ∈ X, n→∞ we write 1 d(u, t) ≤ lim inf d(un , t) ≤ lim sup d(un , t) ≤ sd(u, t). n→∞ s n→∞ Khan et al. [16] through the idea of a control function called altering distance function and define it as “A continuous and non-decreasing function θ : [0, ∞) → [0, ∞) such that θ(t) = 0 if and only if t = 0”. Let K and L be any two self-maps on a nonempty set X, then w ∈ X is a point of coincident of K and L if Ku = Lv = w. and t ∈ X is a common fixed point of K and L if Kt = Lt = t. Recently, Lakzian et al. [19] established fixed point results under α − ψcontraction mappings in the setting of w-distance and studied the existence of unique solution to the following non-linear fractional differential equation: c

Dσ (x(t)) = f (t, x(t)), for all t ∈ (0, 1),

with the integral boundary conditions  ρ x(r)dr, for all ρ ∈ (0, 1). x(0) = 0, x(1) = 0

On the other hand, Allahyari et al. [3] proved some fixed point results for almost generalized (α − ψ − φ − θ)-contraction mappings in partially ordered b-metric spaces and applied their results for unique solution to the following non-linear quadratic integral equation:  1 (2) x(t) = h(t) + λ H(t, τ )f (τ, x(τ ))dτ, t ∈ [0, 1] , λ ≥ 0. 0

The intention of our contribution is to study fixed point results using Lcyclic (α, β)s -contraction in b-metric spaces. Moreover, motivated from the applications presented in [3,19], we establish the existence results for the unique: (a) solution of the non-linear fractional differential equation and (b) common solution to the system of non-linear quadratic integral equations.

2. Main results Throughout this work, R and R+ denotes the sets of real numbers and nonnegative real numbers respectively, γ : [0, ∞) → [0, ∞) is a non-decreasing function and continuous from the right such that ϕ(r) > γ(r) for all r > 0, ϕ is an altering distance function and Ψ denote the collection of all those functions ψ : R5+ → R+ for which the following assumptions holds: ψ1 : ψ is continuous and non-decreasing in each coordinate ; ψ2 : ψ(r, r, r, r, r) ≤ r, ψ(r, r, r, 0, 0) ≤ r, ψ(r, 0, 0, r, 0) ≤ r, ψ(0, r, 0, 0, 0) ≤ r, for all r > 0; ψ3 : ψ(r1 , r2 , r3 , r4 , r5 ) = 0 iff r1 = r2 = r3 = r4 = r5 = 0.

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To obtain common fixed point results, we modify the definition of cyclic (α, β)-contractive mapping in b-metric space as follows: Definition 2.1. Let (X, d) be a b-metric space with coefficient s ≥ 1 and K be L-cyclic (α, β)-admissible mapping. Then K is L-cyclic (α, β)s -contraction mapping of type s if for all x, y ∈ X, α(Lx)β(Ly) ≥ 1 ⇒ ϕ(s3 d(Kx, Ky)) ≤ γ(Ms (x, y)), where ψ ∈ Ψ and

(3)



d(Kx, Ly)d(Lx, Ky) 1 , Ms (x, y) = ψ d(Lx, Ly), d(Ly, Kx), s 2s3 [1 + d(Lx, Ly)]  d(Kx, Lx)d(Ky, Lx) d(Ky, Ly)d(Kx, Ly) , × . 2s[1 + d(Lx, Ly)] 2s[1 + d(Lx, Ly)] Theorem 2.1. Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1. If α, β : X → [0, ∞) and K is L-cyclic (α, β)s −contraction mapping of type s on X satisfying the following conditions: (i) KX ⊆ LX such that LX is closed subspace of X; (ii) there exists x0 ∈ X such that α(Lx0 ) ≥ 1 and β(Lx0 ) ≥ 1; (iii) if {xn } is a sequence in X such that xn → t and β(xn ) ≥ 1 for all n, then β(t) ≥ 1; (iv) α(Lu) ≥ 1 and β(Lv) ≥ 1 whenever Ku = Lu and Kv = Lv, then K and L have a unique point of coincident in X. Moreover, if the pair (K, L) is weakly compatible, then K and L have a unique common fixed point in X. Proof. Starting from x0 ∈ X in the condition (ii) and by using the fact that KX ⊆ LX, we can define two sequences {xn } and {yn } in X by, yn = Kxn = Lxn+1 ,

(4)

for all n ∈ N ∪ {0}. If yn = yn +1 , then yn +1 is a point of coincidence of K and L and we have nothing to prove. So we may assume that yn = yn+1 for all n ∈ N ∪ {0}. Since K is an L-cyclic (α, β)-admissible mapping and α(Lx0 ) ≥ 1, we get β(Lx1 ) = β(Kx0 ) ≥ 1 ⇒ α(Lx2 ) = α(Kx1 ) ≥ 1 and β(Lx3 ) = β(Kx2 ) ≥ 1 ⇒ α(Lx4 ) = α(Kx3 ) ≥ 1. By continuing this process, one can obtain that, α(Lx2k ) ≥ 1 and β(Lx2k+1 ) ≥ 1, for all k ∈ N ∪ {0}.

(5)

Similarly, since K is a L-cyclic (α, β)-admissible mapping and β(Lx0 ) ≥ 1, we have, α(Lx1 ) = α(Kx0 ) ≥ 1 ⇒ β(Lx2 ) = β(Kx1 ) ≥ 1 and α(Lx3 ) = α(Kx2 ) ≥ 1 ⇒ β(Lx4 ) = β(Kx3 ) ≥ 1. By continuing this process, one can obtain that, β(Lx2k ) ≥ 1 and α(Lx2k+1 ) ≥ 1, for all k ∈ N ∪ {0}.

(6)

From (5) and (6), we can write, α(Lxn ) ≥ 1 and β(Lxn+1 ) ≥ 1, for all n ∈ N ∪ {0},

(7)

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which implies that, α(Lxn )β(Lxn+1 ) ≥ 1, for all n ∈ N ∪ {0}.

(8)

Therefore, by (4), we have, ϕ(d(yn , yn+1 )) = ϕ(d(Kxn , Kxn+1 )) ≤ ϕ(s3 d(Kxn , Kxn+1 )),

(9)

and from (3), we write, ϕ(d(yn , yn+1 )) ≤ γ(Ms (xn , xn+1 )) < ϕ(Ms (xn , xn+1 )).

(10)

Since ϕ is non-decreasing, so, d(yn , yn+1 ) < Ms (xn , xn+1 ),

(11)

where

 Ms (xn , xn+1 ) = ψ d(Lxn , Lxn+1 ), 1s d(Lxn+1 , Kxn ),

d(Kxn ,Lxn+1 )d(Lxn ,Kxn+1 ) , 2s3 [1+d(Lxn ,Lxn+1 )]

d(Kxn ,Lxn )d(Kxn+1 ,Lxn ) d(Kxn+1 ,Lxn+1 )d(Kxn ,Lxn+1 ) , 2s[1+d(Lxn ,Lxn+1 )] 2s[1+d(Lxn ,Lxn+1 )]

 = ψ d(yn−1 , yn ), 1s d(yn , yn ),

d(yn ,yn )d(yn−1 ,yn+1 ) , 2s3 [1+d(yn−1 ,yn )]

d(yn ,yn−1 )d(yn+1 ,yn−1 ) d(yn+1 ,yn )d(yn ,yn ) , 2s[1+d(y ,y )] 2s[1+d(yn−1 ,yn )] n−1 n





  ≤ ψ d(yn−1 , yn ), 0, 0, 12 [d(yn+1 , yn ) + d(yn , yn−1 )], 0 .

(12) Therefore, inequality (11) becomes,   d(yn , yn+1 ) < ψ d(yn−1 , yn ), 0, 0, 12 [d(yn+1 , yn ) + d(yn , yn−1 )], 0 . If d(yj−1 , yj ) ≤ d(yj , yj+1 ) for some j ∈ N, then,   d(yj , yj+1 ) < ψ d(yj−1 , yj ), 0, 0, 12 [d(yj+1 , yj ) + d(yj , yj−1 )], 0   ≤ ψ d(yj , yj+1 ), 0, 0, d(yj , yj+1 ), 0 ≤ d(yj , yj+1 ), which is not possible, and hence d(yn , yn+1 ) < d(yn−1 , yn ), for all n ∈ N. Therefore the sequence {d(yn , yn+1 )} is decreasing and bounded below in X. Thus lim d(yn , yn+1 ) = λ, for some λ ≥ 0. Now, we show that λ = 0. For n→∞

this, assume that λ > 0. Then from (10) and (12), we have, ϕ(d(yn , yn+1 )) ≤ γ(M   s(xn , xn+1 )) ≤ γ ψ d(yn−1 , yn ), 0, 0, 12 [d(yn+1 , yn ) + d(yn , yn−1 )], 0 ≤ γ(ψ(d(yn−1 , yn ), 0, 0, d(yn−1 , yn ), 0)), by using ψ2 , we get, ϕ(d(yn , yn+1 )) ≤ γ(d(yn−1 , yn )).

(13)

Considering the properties of ϕ, γ and taking limit as n → ∞ in (13), we have, lim ϕ(d(yn , yn+1 )) ≤ lim γ(d(yn−1 , yn )) ⇒ ϕ(λ) ≤ γ(λ),

n→∞

n→∞

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which is not possible. Thus, λ = 0 and hence, lim d(yn , yn+1 ) = 0.

n→∞

(14)

Next, to show that {yn } is a b-Cauchy sequence. Assume that {yn } is not a b-Cauchy sequence, then for subsequences {ym(j) } and {yn(j) } of {yn } along with j ∈ N, m(j) is even number and n(j) is odd number such that n(j) > m(j) ≥ j, there exists > 0 for which, d(ym(j) , yn(j) ) ≥ .

(15)

Corresponding to m(j) we can choose the smallest number n(j) such that n(j) > m(j) ≥ j satisfying (15). Then d(ym(j) , yn(j)−1 ) < .

(16)

Using the triangle inequality and inequalities (15) and (16), we have, ≤ d(y  m(j) , yn(j) ) ≤ s d(ym(j) , yn(j)−1 ) + d(y n(j)−1 , yn(j) ) < s + d(yn(j)−1 , yn(j) ) .

(17)

Taking limit supremum as j → ∞ in (17) and using (14), we get, ≤ lim sup d(ym(j) , yn(j) ) < s .

(18)

j→∞

Now, from triangle inequality, we get,  d(ym(j) , yn(j) ) ≤ s d(ym(j) , yn(j)+1 ) + d(yn(j)+1 , yn(j) ) , and

 d(ym(j) , yn(j)+1 ) ≤ s d(ym(j) , yn(j) ) + d(yn(j) , yn(j)+1 ) .

(19) (20)

Taking limit supremum as j → ∞ in (19), (20) and using (14), (18), we get,   ≤ s lim sup d(ym(j) , yn(j)+1 ) and lim sup d(ym(j) , yn(j)+1 ) ≤ s2 . j→∞

j→∞

Form here we write,

Similarly,

≤ lim sup d(ym(j) , yn(j)+1 ) ≤ s2 . s j→∞

(21)

≤ lim sup d(yn(j) , ym(j)+1 ) ≤ s2 . s j→∞

(22)

Next,

and

 d(ym(j) , yn(j)+1 ) ≤ s d(ym(j) , ym(j)+1 ) + d(ym(j)+1 , yn(j)+1 ) ,

(23)

 d(ym(j)+1 , yn(j)+1 ) ≤ s d(ym(j)+1 , ym(j) ) + d(ym(j) , yn(j)+1 ) .

(24)

Taking limit supremum as j → ∞ in (23), (24) and using (14), (21), we get,   ≤ s lim sup d(ym(j)+1 , yn(j)+1 ) and lim sup d(ym(j)+1 , yn(j)+1 ) ≤ s3 . s j→∞ j→∞

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Which implies that, ≤ lim sup d(ym(j)+1 , yn(j)+1 ) ≤ s3 . s2 j→∞

(25)

From (8), we get α(Lxm(j) )β(Lxn(j) ) ≥ 1 and using (3), we have, ϕ(d(ym(j)+1 , yn(j)+1 )) = ϕ(d(Kxm(j)+1 , Kxn(j)+1 )) ≤ ϕ(s3 d(Kxm(j)+1 , Kxn(j)+1 )) ≤ γ(Ms (xm(j)+1 , xn(j)+1 )),

(26)

where Ms (xm(j)+1 , xn(j)+1 ) = ψ d(Lxm(j)+1 , Lxn(j)+1 ), 1s d(Lxn(j)+1 , Kxm(j)+1 ), d(Kxm(j)+1 ,Lxn(j)+1 )d(Lxm(j)+1 ,Kxn(j)+1 ) , 2s3 [1+d(Lxm(j)+1 ,Lxn(j)+1 )] d(Kxm(j)+1 ,Lxm(j)+1 )d(Kxn(j)+1 ,Lxm(j)+1 ) , 2s[1+d(Lxm(j)+1 ,Lxn(j)+1 )] 



d(Kxn(j)+1 ,Lxn(j)+1 )d(Kxm(j)+1 ,Lxn(j)+1 ) 2s[1+d(Lxm(j)+1 ,Lxn(j)+1 )]

= ψ d(ym(j) , yn(j) ), 1s d(yn(j) , ym(j)+1 ),

d(ym(j)+1 ,yn(j) )d(ym(j) ,yn(j)+1 ) , 2s3 [1+d(ym(j) ,yn(j) )]

d(ym(j)+1 ,ym(j) )d(yn(j)+1 ,ym(j) ) d(yn(j)+1 ,yn(j) )d(ym(j)+1 ,yn(j) ) , 2s[1+d(ym(j) ,yn(j) )] 2s[1+d(ym(j) ,yn(j) )]

 .

Taking limit supremum as j → ∞ in above and using (14), (18), (21) and (22), we have,   s 2 , 0, 0 lim sup Ms (xm(j)+1 , xn(j)+1 ) ≤ ψ s , s , 2[1 + ] j→∞ ≤ ψ (s , s , s , , 0, 0) = s .

(27)

Taking limit supremum as j → ∞ in (26) and using (25), (27), we have,



 ϕ (s ) = ϕ s3 s2 ≤ ϕ s3 lim supj→∞ d(ym(j)+1 , yn(j)+1 )  ≤ γ lim supj→∞ Ms (xm(j)+1 , xn(j)+1 ) (28) ≤ γ (s ) , which is not possible. Therefore, {yn } is a b-Cauchy sequence in X. But X is b-complete, so one can find r ∈ X such that lim yn = r and hence from n→∞

(4), we get, lim Kxn = lim Lxn+1 = r.

n→∞

n→∞

(29)

Since LX is closed, so by (29), r ∈ LX and therefore one can find u ∈ X for which Lu = r. Now, we prove that Ku = r. For this, since yn → r and by (7), β(yn ) = β(Lxn+1 ) ≥ 1 for all n ∈ N. Thus by condition (iii) of Theorem 2.1, β(r) = β(Lu) ≥ 1 and hence by (7), α(Lxn )β(Lu) ≥ 1 for all n ∈ N. Now, using (3) with setting x = xn and y = u, we have, ϕ(s3 d(Kxn , Ku)) ≤ γ(Ms (xn , u)),

(30)

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where

 n ,Lu)d(Lxn ,Ku) Ms (xn , u) = ψ d(Lxn , Lu), 1s d(Lu, Kxn ), d(Kx 2s3 [1+d(Lxn ,Lu)] , d(Kxn ,Lxn )d(Ku,Lxn ) d(Ku,Lu)d(Kxn ,Lu) , 2s[1+d(Lxn ,Lu)] 2s[1+d(Lxn ,Lu)]

 .

Taking limit supremum as n → ∞ in above and using (29), we get,  d(r, r)d(r, Ku) 1 lim Ms (xn , u) = ψ d(r, r), d(r, r), , n→∞ s 2s[1 + d(r, r)]  d(r, r)d(Ku, r) d(Ku, r)d(r, r) , 2s[1 + d(r, r)] 2s[1 + d(r, r)] = ψ(0, 0, 0, 0, 0) = 0, and ϕ(s3 d(r, Ku)) ≤ γ(0) < ϕ(0) = 0. Therefore, ϕ(d(r, Ku)) ≤ ϕ(s3 d(r, Ku)) = 0, which is possible only if ψ(d(r, Ku)) = 0. Thus d(r, Ku) = 0 implies that Ku = r and hence, Ku = Lu = r. (31) Thus r is the point of coincident of K and L. For the uniqueness of the point of coincident of K and L. Let r∗ = r be another point of coincident of K and L, then we can find v ∈ X such that, Kv = Lv = r∗ .

(32)

Now, from condition (iv), α(Lu)β(Lv) ≥ 1. Therefore, from (3) with setting x = u, y = v and using (31), (32), we have, ϕ(s3 d(Ku, Kv)) ≤ γ(M (u, v)),

(33)

where

 d(Ku, Lv)d(Lu, Kv) d(Ku, Lu)d(Kv, Lu) , , Ms (u, v) = ψ d(Lu, Lv), 1s d(Lv, Ku), 2s[1 + d(Lu, Lv)] 2s[1  + d(Lu, Lv)] d(Kv, Lv)d(Ku, Lv) 2s[1 + d(Lu, Lv)]   1 d(r, r ∗ )d(r, r ∗ ) d(r, r)d(r∗ , r) d(r∗ , r∗ )d(r, r ∗ ) ∗ ∗ , , = ψ d(r, r ), d(r , r), s 2s[1 + d(r, r ∗ )] 2s[1 + d(r, r ∗ )] 2s[1 + d(r, r ∗ )] ≤ ψ(d(r, r∗ ), d(r∗ , r), d(r, r ∗ ), 0, 0) ≤ d(r, r ∗ ).

From (33), we write, ϕ(d(r, r∗ )) ≤ ϕ(s3 d(r, r∗ )) ≤ γ(d(r, r∗ )) < ϕ(d(r, r∗ )), which is contradiction, thus r = r∗ . Therefore, r is unique point of coincident of K and L. Next, since the pair (K, L) is weak compatible, so that Ku = Lu implies that LKu = KLu, using (31) it follows that Lr = Kr. Also, r is a unique point of coincidence, thus Kr = Lr = r and hence r is a unique common fixed point of K and L. 

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Corollary 2.1. Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1. If α, β : X → [0, ∞) and K, L : X → X be mappings such that, α(Lx)β(Ly)ϕ(s3 d(Kx, Ky)) ≤ γ(Ns (x, y)), for all x, y ∈ X,

(34)

where

 d(Kx, Ly)d(Lx, Ky) 1 , Ns (x, y) = max d(Lx, Ly), d(Ly, Kx), s 2s3 [1 + d(Lx, Ly)]  d(Kx, Lx)d(Ky, Lx) d(Ky, Ly)d(Kx, Ly) , . 2s[1 + d(Lx, Ly)] 2s[1 + d(Lx, Ly)] Assume that the following conditions hold: (i) KX ⊆ LX such that LX is closed subspace of X; (ii) there exists x0 ∈ X such that α(Lx0 ) ≥ 1 and β(Lx0 ) ≥ 1; (iii) if {xn } is a sequence in X such that xn → t and β(xn ) ≥ 1 for all n, then β(t) ≥ 1; (iv) α(Lu) ≥ 1 and β(Lv) ≥ 1 whenever Ku = Lu and Kv = Lv. Then K and L have a unique point of coincident in X. Moreover, if the pair (K, L) is weakly compatible, then K and L have a unique common fixed point in X. Proof. The proof follows from Theorem (2.1) by taking, α(Lx)β(Lx) ≥ 1 for  all x ∈ X and ψ(r1 , r2 , r3 , r4 , r5 ) = max{r1 , r2 , r3 , r4 , r5 }. Taking α(Lx) = β(Lx) = 1, for all x ∈ X in Theorem (2.1), we get the following corollary: Corollary 2.2. Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1 and K, L : X → X be mappings such that, ϕ(s3 d(Kx, Ky)) ≤ γ(Ms∗ (x, y)), for all x, y ∈ X,

(35)

where

 d(Kx, Ly)d(Lx, Ky) 1 , Ns (x, y) = max d(Lx, Ly), d(Ly, Kx), s 2s3 [1 + d(Lx, Ly)]  d(Kx, Lx)d(Ky, Lx) d(Ky, Ly)d(Kx, Ly) , . 2s[1 + d(Lx, Ly)] 2s[1 + d(Lx, Ly)] If KX ⊆ LX such that LX is closed subspace of X, then K and L have a unique point of coincident in X. Moreover, if the pair (K, L) is weakly compatible, then K and L have a unique common fixed point in X. Corollary 2.3. Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1 and K, L : X → X be mappings such that, k (36) d(Kx, Ky) ≤ 3 Ms (x, y)), for all x, y ∈ X. s If KX ⊆ LX such that LX is closed subspace of X, then K and L have a unique point of coincident in X. Moreover, if the pair (K, L) is weakly compatible, then K and L have a unique common fixed point in X. Proof. The proof easily follows from Theorem (2.1) by taking α(Lx) = β(Lx) = 1 for all x ∈ X and ψ(t) = γ(t) = t, for all t ∈ [0, ∞). 

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If we choose L = IX in Corollary (2.2), then we get the following corollary: Corollary 2.4. Let (X, d) be a b-complete b-metric space with coefficient s ≥ 1 and K : X → X be mapping such that, ϕ(s3 d(Kx, Ky)) ≤ γ(Ms∗ (x, y)), for all x, y ∈ X, where

(37)

 d(Kx, y)d(x, Ky) 1 , Ms∗ (x, y) = max d(x, y), d(y, Kx), s 2s3 [1 + d(x, y)]  d(Kx, x)d(Ky, x) d(Ky, y)d(Kx, y) , . 2s[1 + d(x, y)] 2s[1 + d(x, y)]

Then K has a unique fixed point in X. To present our next result, we define the concept of a common limit in the range property in b-metric spaces. Definition 2.2. [22] Let (X, d) be a b-metric space and K, L be self-mappings on X. Then the pair (K, L) satisfies common limit in the range of L (denoted by b − (CLRL )), if we can find a sequence {xn } in X such that, lim Kxn = lim Lxn = Lx, for some x ∈ X.

n→∞

n→∞

Note that CLR property relax the completeness of the space. Theorem 2.2. Let (X, d) be b-metric space with coefficient s ≥ 1. If α, β : X → [0, ∞) and K, L : X → X be mappings such that, α(Lx)β(Ly) ≥ 1 ⇒ ϕ(s3 d(Kx, Ky)) ≤ γ(Ns (x, y)), for all x, y ∈ X, (38) where  d(Kx, Ly)d(Lx, Ky) 1 , Ns (x, y) = ψ d(Lx, Ly), d(Ly, Kx), s 2s[1 + d(Lx, Ly)]  d(Kx, Lx)d(Ky, Lx) d(Ky, Ly)d(Kx, Ly) , . 2s[1 + d(Lx, Ly)] 2s[1 + d(Lx, Ly)] Assume that the following conditions holds: (i) (K, L) satisfies (CLRL ) property; (ii) if {xn } is a sequence in X such that lim Kxn = lim Lxn = x, then n→∞

n→∞

α(x) ≥ 1 and β(x) ≥ 1; (iii) α(Lu) ≥ 1 and β(Lv) ≥ 1 whenever Ku = Lu and Kv = Lv. Then K and L have a unique point of coincident in X. Moreover, if the pair (K, L) is weakly compatible, then K and L have a unique common fixed point in X. Proof. Assume that the pair (K, L) satisfies (CLRL ) property, so we can find a sequence {xn } in X such that, lim Kxn = lim Lxn = Lu, for some u ∈ X.

n→∞

n→∞

(39)

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25

By condition (ii) of Theorem 2.2, we get α(Lu)β(Lu) ≥ 1. We show that Ku = Lu. Assume that Ku = Lu, then on putting x = u and y = xn in (38), we have, (40) ϕ(s3 d(Ku, Kxn )) ≤ γ(Ns (u, xn )), where  1 d(Ku, Lxn )d(Lu, Kxn ) , Ns (u, xn ) = ψ d(Lu, Lxn ), d(Lxn , Ku), s 2s[1 + d(Lu, Lxn )]  d(Ku, Lu)d(Kxn , Lu) d(Kxn , Lxn )d(Ku, Lxn ) , . 2s[1 + d(Lu, Lxn )] 2s[1 + d(Lu, Lxn )] Taking limit supremum as n → ∞ and using (39), we can write,  limn→∞ Ns (u, xn ) = ψ d(Lu, Lu), 1s d(Lu, Ku), d(Ku,Lu)d(Lu,Lu) 2s[1+d(Lu,Lu)] ,



d(Ku,Lu)d(Lu,Lu) d(Lu,Lu)d(Ku,Lu) 2s[1+d(Lu,Lu)] , 2s[1+d(Lu,Lu)]

,

≤ ψ(0, d(Ku, Lu), 0, 0, 0), ≤ d(Lu, Ku). From (40), we have, ϕ(d(Ku, Lu)) ≤ ϕ(s3 d(Ku, Lu)) ≤ γ(d(Ku, Lu)) < ϕ(d(Ku, Lu)), which is a contradiction. Thus, Ku = Lu and hence, Ku = Lu = w(say).

(41)

That is w is the point of coincident of K and L. To show the uniqueness of the point of coincident of K and L. Let w∗ = w be another point of coincident of K and L, then we can find v ∈ X such that, Kv = Lv = w∗ .

(42)

Now, from condition (iv) of Theorem 2.2, α(Lu)β(Lv) ≥ 1. Therefore, from (38) with setting x = u, y = v and using (41), (42), we have ϕ(s3 d(Ku, Kv)) ≤ γ(Ns (u, v)),

(43)

where  1 d(Ku, Lv)d(Lu, Kv) Ns (u, v) = ψ d(Lu, Lv), d(Lv, Ku), , s 2s[1 + d(Lu, Lv)]  d(Ku, Lu)d(Kv, Lu) d(Kv, Lv)d(Ku, Lv) , , 2s[1 + d(Lu, Lv)] 2s[1 + d(Lu, Lv)]  ∗ ∗ 1 d(w, w )d(w, w ) = ψ d(w, w∗ ), d(w∗ , w), , s 2s[1 + d(w, w∗ )]  d(w, w)d(w∗ , w) d(w∗ , w∗ )d(w, w∗ ) , 2s[1 + d(w, w∗ )] 2s[1 + d(w, w∗ )] ≤ ψ(d(w, w∗ ), d(w∗ , w), d(w, w∗ ), 0, 0) ≤ d(w, w∗ ).

From (43), we have, ϕ(d(w, w∗ )) ≤ ϕ(s3 d(w, w∗ )) ≤ γ(d(w, w∗ )) < ϕ(d(w, w∗ )), which is a contradiction, thus w = w∗ . Thus w is a unique point of coincident of K and L. Next, since the pair (K, L) is weak compatible, so that Ku = Lu

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implies that LKu = KLu, using (31) it follows that Lw = Kw. Also, w is a unique point of coincidence, thus Kw = Lw = w and hence w is a unique common fixed point of K and L. 

3. Existence result for a solution to non-linear fractional differential equation In this section, we give an existence theorem for a solution of the non-linear fractional differential equation by using Corollary 2.4. First, let us recall the definition of Caputo fractional derivative [17]. The Caputo fractional derivative of order σ > 0 (denoted by c Dσ ) is defined by:  t 1 c σ D g(t) = (t − τ )m−σ−1 g m (τ )dτ, Γ(m − σ) 0 where σ ∈ [m − 1, m) with m = [σ] + 1 ∈ N, [σ] represents the integer part of σ and g : [0, ∞) → R is a continuous function. Throughout this section, X = C([0, 1], R), denotes the set of all continuous functions from [0, 1] into R. Now, we deal with the existence of unique solutions to a non-linear fractional differential equation: c

Dσ (x(t)) = f (t, x(t)),

with the integral boundary conditions:



(44)

ρ

x(τ )dτ,

x(0) = 0, x(1) = 0

where x ∈ X , t, ρ ∈ (0, 1), σ ∈ (1, 2] and f : [0, 1] × R → R is a continuous function. Note that, x ∈ X is a solution of (44) iff x ∈ X is a solution of the integral equation:  t 1 x(t) = (t − τ )σ−1 f (τ, x(τ ))dτ Γ(σ) 0  1 2t − (1 − τ )σ−1 f (τ, x(τ ))dτ (2 − ρ2 )Γ(σ) 0   ρ  τ 2t σ−1 + (τ − r) f (r, x(r))dr dτ. (45) (2 − ρ2 )Γ(σ) 0 0 Define the operator K : X → X by  t 1 Kx(t) = Γ(σ) (t−τ )σ−1 f (τ, x(τ ))dτ 0  1 2t (1−τ )σ−1 f (τ, x(τ ))dτ − (2 − ρ2 )Γ(σ) 0   ρ τ 2t σ−1 + (τ − r) f (r, x(r))dr dτ, (2 − ρ2 )Γ(σ) 0 0

Fixed point theorems in b-metric spaces

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25

where X is a b-metric space with b-metric defined as p

d(u, v) = sup |u(t) − v(t)| , for all u, v ∈ X ,

(46)

t∈[0,1]

with coefficient s = 2p−1 . Theorem 3.1. The non-linear fractional differential equation (44) has a unique solution if the following condition holds: Γ(σ + 1)

|f (a, μ) − f (a, ν)| ≤

1

(Θ(|μ − ν|p )) p ,

(47) 2 where p > 1 and Θ is a continuous and non-decreasing function on [0, ∞) such that Θ(0) = 0 and Θ(a) < a for all a > 0. 5p2 −3p+1 p2

Proof. We have to show that condition (37) of Corollary 2.4 holds. For this, we have,  t 1 σ−1 |Kx(t) − Ky(t)| = Γ(σ) (t−τ ) f (τ, x(τ ))dτ 0  1 2t σ−1 − (1−τ ) f (τ, x(τ ))dτ (2 − ρ2 )Γ(σ) 0   ρ τ 2t σ−1 (τ − r) f (r, x(r))dr dτ + 2 (2 − ρ )Γ(σ) 0 0 t 1 σ−1 − (t − τ ) f (τ, y(τ ))dτ Γ(σ) 0  1 2t σ−1 + (1 − τ ) f (τ, y(τ ))dτ 2 (2 − ρ )Γ(σ) 0   ρ τ 2t σ−1 (τ − r) f (r, y(r))dr dτ − (2 − ρ2 )Γ(σ) 0 0 t 1 σ−1 ≤ (t − τ ) |f (τ, x(τ ))dτ − f (τ, y(τ ))| dτ Γ(σ) 0  1 2t σ−1 + (1 − τ ) |f (τ, x(τ )) − f (τ, y(τ ))| dτ 2 (2 − ρ )Γ(σ) 0  ρ τ 2t σ−1 dτ (τ − r) (f (r, x(r)) − f (r, y(r)))dr + (2 − ρ2 )Γ(σ) 0 0 1   t 1 p σ−1 Γ(σ + 1) p ≤ (t − τ ) Θ(|x(τ ) − y(τ )| ) dτ 5p2 −3p+1 Γ(σ) 0 p2 2 1   1 p 2t σ−1 Γ(σ + 1) p + (1 − τ ) Θ(|x(τ ) − y(τ )| ) dτ 5p2 −3p+1 2 (2 − ρ )Γ(σ) 0 2 p 2  ρ τ 1 2t σ−1 Γ(σ + 1) p + (τ − r) (Θ(|x(r) − y(r)| )) p drdτ 5p2 −3p+1 2 (2 − ρ )Γ(σ) 0 0 2 p 2  t 1 1 σ−1 Γ(σ + 1) (t − τ ) (Θ(d(x, y))) p dτ ≤ 5p2 −3p+1 Γ(σ) 0 2 2 p  1 1 2t σ−1 Γ(σ + 1) + (1 − τ) (Θ(d(x, y))) p dτ 5p2 −3p+1 (2 − ρ2 )Γ(σ) 0 2 2 p  ρ τ 1 2t σ−1 Γ(σ + 1) + (τ − r) (Θ(d(x, y))) p drdτ, 5p2 −3p+1 (2 − ρ2 )Γ(σ) 0 0 2 2 p

which implies that, |Kx(t) − Ky(t)| ≤

Γ(σ + 1) 2

5p2 −3p+1 p2

1 Γ(σ) +

1

∗

(Θ(Ms (x, y))) p



2t (2 − ρ2 )

t

2t (2− ρ2 ) 0   ρ τ σ−1 (τ − r) drdτ σ−1

(t−τ )

0

0



1

dτ +

σ−1

(1−τ ) 0

dτ

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 2t 2t tσ 1 ρσ+1 + + 2) σ 2 ) σ(σ + 1) σ (2 − ρ (2 − ρ 2 

1 2t 2t Γ(σ + 1) 1 ρσ+1 ∗ σ sup t + + ≤ 5p2 −3p+1 (Θ(Ms (x, y))) p 2) 2 ) (σ+1) Γ(σ + 1) (2−ρ (2−ρ t∈(0,1) 2 p2 ≤

Γ(σ + 1)

= 2 < 2

5p2 −3p+1 p2

−3p2 +3p−1 p2 −3p2 +3p−1 p2

∗

1

∗

1

(Θ(Ms (x, y))) p

1 Γ(σ)

(Θ(Ms (x, y))) p 1

∗

(Ms (x, y)) p .

Thus for p > 1, we can write, |Kx(t) − Ky(t)|p ≤ 2

−3p2 +3p−1 p

Ms∗ (x, y).

(48)

Now, let us define ϕ, γ : [0, ∞) → [0, ∞) by ϕ(a) = a and γ(a) = Then clearly, ϕ is an altering distance function and γ is a non-decreasing continuous from the right such that ϕ(a) > γ(a) for all a > 0. Using inequality (48), we have,

p ϕ(s3 d(Kx, Ky)) = s3 d(Kx, Ky)  1 p 2a .

p

p

p

23p−3 sup |Kx(t) − Ky(t)| t∈I  p −3p2 +3p−1 3p−3 ∗ p ≤ 2 2 Ms (x, y) =

= 12 (Ms∗ (x, y))p = γ(Ms∗ (x, y)). Thus all the assumptions of Corollary 2.4 are satisfied and hence the operator K has a fixed point in X , consequently the non-linear fractional differential equation (44) has a unique solution in X . 

4. Existence result for a solution to the system of non-linear integral equations In this section, we study an existence result for a unique common solution to the system of non-linear integral equations by using Corollary 2.2. Consider the system of nonlinear quadratic integral equations: ⎧ ⎫  1 ⎪ ⎪ ⎪ H(t, r)Φ1 (r, x(r))dr; ⎪ ⎨ x(t) = ⎬ 0 1 (49) ⎪ ⎪ ⎪ ⎪ ⎩ x(t) = H(t, r)Φ2 (r, x(r))dr, ⎭ 0

where H : [0, 1] × [0, 1] → [0, ∞) is continuous at t ∈ [0, 1] for every τ ∈ [0, 1] and measurable at τ ∈ [0, 1] for every t ∈ [0, 1] and f1 , f2 : [0, 1] × R → [0, ∞) are continuous functions. Define the operators K, L : X → X defined by, ⎧ ⎫  1 ⎪ ⎪ ⎪ H(t, r)Φ1 (r, x(r))dr; ⎪ ⎨ Kx(t) = ⎬ 0 1 (50) ⎪ ⎪ ⎪ ⎩ Lx(t) = ⎭ H(t, r)Φ2 (r, x(r))dr, ⎪ 0

where X = C([0, 1]) (the space of continuous functions defined on [0, 1]) with b-metric defined by (46) in Sect. 3.

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25

Theorem 4.1. Let K, L : X → X given by (50) be mappings for which the following conditions hold: (C1 ) H : [0, 1] × [0, 1] → [0, ∞) is continuous at t ∈ [0, 1] for every r ∈ [0, 1] and measurable at r ∈ [0, 1] for every t ∈ [0, 1] such that  1 q (H(t, r)) dr ≤ Ξ; 0

(C2 ) Φ1 , Φ2 : [0, 1] × R → [0, ∞) are continuous functions such that Φ2 > Φ1 and for all x, y ∈ R, p

|Φ1 (t, x)dr − Φ1 (t, y)| ≤ μ(t)Θ(|Lx − Ly|p ), where p > 1 and μ : [0, 1] → [0, ∞) is a continuous function satisfying  sup t∈[0,1]



1

μ(t)dr

<

2

−3p2 +3p−1 p

Ξp−1

0

,

and Θ is a continuous and non-decreasing function on [0, ∞) such that Θ(0) = 0 and Θ(t) < t for all t > 0; (C3 ) for all r, t ∈ [0, 1] and x ∈ X,     1  1   1 Φ2 r, dr = 0. H(t, r)Φ (r, x(r))dr −Φ H(t, r)Φ (r, x(r))dr r, 2 1 1 0

0

0

Then the system (49) of non-linear quadratic integral equations has a unique common solution. Proof. First, we will show that KX ⊆ LX such that LX is closed subspace of X. For this, let x ∈ X, then we have  1 Kx(t) = H(t, r)Φ1 (r, x(r))dr 0

 ≤

1

H(t, r)Φ2 (r, x(r))dr 0

= Lx(t),

(51)

that is Kx ≤ Lx for all x ∈ X and thus KX ⊆ LX. We have to show that LX is closed subspace of X. For this, let {Lxn } be any sequence in LX, where {xn } is a sequence in X such that xn → x ∈ X as n → ∞. Then  1  1 H(t, r)Φ2 (r, xn (r))dr − H(t, r)Φ2 (r, x(r))dr |Lxn (t) − Lx(t)| = 0  01 ≤ H(t, r) |Φ2 (r, xn (r)) − Φ2 (r, x(r))| dr. 0

(52) But xn → x ∈ X and Φ2 is continuous, so that |Φ2 (r, xn (r)) − Φ2 (r, x(r))| → 0 as n → ∞ and thus from (52), we can write Lxn → Lx ∈ LX as n → ∞. Hence LX is closed subspace of X.

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Next, we have to show that condition (35) holds. For this, choose p, q ∈ R older inequality, we have such that p1 + 1q = 1. Now, using the H¨  1 p  1 |Kx(t) − Ky(t)|p = H(t, r)Φ1 (r, x(r))dr − H(t, r)Φ1 (r, y(r))dr 0 0   1 1  1  1 p q p ≤ (H(t, r))q dr |Φ1 (r, x(r))dr − Φ1 (r, y(r))|p dr . 0

0

Applying condition (C1 ) of Theorem 4.1, we get   1 p p p |Kx(t) − Ky(t)| ≤ Ξ q |Φ1 (r, x(r))dr − Φ1 (r, y(r))| dr . 0

Applying condition (C2 ) of Theorem 4.1, we can write   1 p |Kx(t) − Ky(t)| ≤ Ξp−1 μ(t)Θ(|Lx − Ly|p )dr 0





1

≤ Ξp−1

μ(t)Θ(d(Lx, Ly))dr 0





1

≤Ξ

p−1

μ(t)Θ(Ns (x, y))dr 0





1

p−1

μ(t)dr Θ(Ns (x, y))

=Ξ

0 p−1 2

<Ξ <2

−3p2 +3p−1 p

Ξp−1

−3p2 +3p−1 p

Θ(Ns (x, y))

Ns (x, y).

Now, let us define ϕ, γ : [0, ∞) → [0, ∞) by ϕ(a) = ap and γ(a) = 12 ap , then clearly, ϕ is an altering distance function and γ is a non-decreasing continuous from the right such that ϕ(a) > γ(a) for all a > 0. Using last inequality, we have

p ϕ(s3 d(Kx, Ky)) = s3 d(Kx, Ky) p p

23p−3 sup |Kx(t) − Ky(t)| t∈[0,1]  p −3p2 +3p−1 3p−3 p ≤ 2 2 Ms∗ (x, y) =

−1

= (2 p Ms∗ (x, y))p = 12 (Ms∗ (x, y))p = γ(Ms∗ (x, y)). Finally, to show that the pair (K, L) is weakly compatible. we have  1   1  LKx(t) − KLx(t) = L H(t, r)Φ (r, x(r))dr − K H(t, r)Φ (r, x(r))dr 1 2 0 0  1   1  H(t, r)Φ2 r, H(t, r)Φ1 (r, x(r))dr dr = 0 0   1   1 − H(t, r)Φ1 r, H(t, r)Φ2 (r, x(r))dr dr 0

0

Fixed point theorems in b-metric spaces

 =

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25

   1  H(t, r) Φ2 r, H(t, r)Φ1 (r, x(r))dr 0 0   1  − Φ1 r, H(t, r)Φ2 (r, x(r))dr dr . 1

(53)

0

If Kx = Lx for some x ∈ X, then we have 1 1 H(t, r)Φ1 (r, x(r))dr = 0 H(t, r)Φ2 (r, x(r))dr, 0 for all t ∈ [0, 1]. Thus (54) becomes  1    LKx(t) − KLx(t) = H(t, r) Φ2 r,

 H(t, r)Φ2 (r, x(r))dr 0 0    1 − Φ1 r, H(t, r)Φ1 (r, x(r))dr dr 0   1   1 ≤ H(t, r) H(t, r)Φ2 (r, x(r))dr Φ2 r, 0 1

  − Φ1 r,

0

1

0

 H(t, r)Φ1 (r, x(r))dr dr

for all t ∈ [0, 1]. From (C3 ), we get |LKx(t) − KLx(t)| ≤ 0, that is, LKx(t) = KLx(t) for all t ∈ [0, 1]. Therefore, LKx = KLx whenever Kx = Lx, that is, the pair (K, L) is weakly compatible. Thus all the assumptions of Corollary 2.2 are satisfied. Hence, the maps K and L have a unique common fixed point in X and consequently, the system (49) has a unique common solution in X. Acknowledgements The authors wish to thank the editor and anonymous referees for their comments and suggestions, which helped to improve this paper. Compliance with ethical standards Author contributions All authors read and approved the final manuscript. All author contribute equally to the writing of this manuscript. Conflict of interest The authors declare that they have no competing interests.

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[23] Sintunavarat, W.: Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations. Rev. Real Acad. Cienc. Exactas Fisicas Nat. Ser. A Mat. 110, 585–600 (2016) [24] Yamaod, O., Sintunavarat, W.: Fixed point theorems for (α, β)-(ψ, ϕ)contractive mapping in b-metric spaces with some numerical results and applications. J. Nonlinear Sci. Appl. 9(1), 22–34 (2016) [25] Yamaod, O., Sintunavarat, W., Cho, Y.J.: Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces. Open Math. 14(1), 128–145 (2016) Mian Bahadur Zada and Muhammad Sarwar Department of Mathematics University of Malakand Chakdara Dir (L) Pakistan e-mail: [email protected] Muhammad Sarwar e-mail: [email protected] Cemil Tunc Department of Mathematics, Faculty of Sciences Yuzuncu Yil University Kampus Van Turkey e-mail: [email protected]