Proceeding2562

390 การประชุมวิชาการระดับชาติมหาวิทยาลัยทักษิณ ครั้งที่ 29 ประจ�ำปี 2562 วิจัยและนวัตกรรมเพื่อการพัฒนาที่ยั่งยืน Introduction Fixed point theory in metric space has been found a lot of applications in different branches of Mathematics [1-8]. The concept of b-metric space as a generalization of a metric space is introduced by Bakhtin [1] and Czerwik [3]. Several researches have dealt with fixed point theories for various contraction mappings in b-metric spaces. Kamran et al., introduced a generalized b-metric spaces as an extension of b-metric spaces and gave an application for Fredholm integral equation. This pushed us to study the existence and uniqueness of a fixed point theory employs the Hardy and Rogers contraction mapping. Definition 1 [ 2, 6]. If M be a nonempty set and s  1 be a given real number. A function q: M  M is called a b-metric if the following conditions are satisfied: (b1) q(x,y) = 0 if and only if x=y; (b2) q(x,y) = q(y,x); (b3) q(x,z) = s[q(x,y) + q(y,z)] for all x, y, z  M. The pair (M, q) is called a b-metric space. Definition 2 [8]. Let M be a nonempty set and  :M  M  [1,  ). A function q  : M  M   is called an extended b-metric if the following conditions are satisfied: (eb1) q  (x,y) = 0 if and only if x = y; (eb2) q  (x,y) = q  (y,x); (eb3) q  (x,z) =  (x,z) [ q  (x,y) + q  (y,z)], for all x, y, z  M. The pair (M, q  ) is called an extended b-metric space or generalized b-metric space.. Remark : In the definition 2, If we let  (x,z) = s where s  1 then it is a b-metric space. Example 1 [8]. Let M={1, 2, 3}. Define  :M  M  [1,  ) and q  : M  M   as:  (x,z)= x+y+1; q  (1,1)= q  (2,2)= q  (3,3)=0; q  (1,2)= q  (2,1)=80, q  (1,3)= q  (3,1)=1000, q  (2,3)= q  (3,2)=600, we can show that (M, q  ) is an generalized b-metric space.