New fixed point theorems on b -metric spaces with applications to coupled fixed point theory
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Journal of Fixed Point Theory and Applications
New fixed point theorems on b-metric spaces with applications to coupled fixed point theory Monica-Felicia Bota, Liliana Guran and Adrian Petru¸sel Abstract. Let (X, d) be a complete b-metric space endowed with a partial ´ c type operator. In this paper, order relation and f : X → X be a Ciri´ an extended study of the fixed point equation x = f (x), x ∈ X, is considered. As an application, coupled fixed point results are given in the same framework. Our results generalize some recent theorems in the literature. Mathematics Subject Classification. Primary 47H10; Secondary 54H25, 46T99. Keywords. Fixed point, b-metric space, coupled fixed point, ´ciri´c-type operator, Ulam–Hyers stability, well-posedness.
1. Introduction and preliminaries A well-known generalization of the concept of a metric space is that of a b-metric space (also called in some papers, quasi-metric space). Definition 1.1. (Bakhtin [1], Berinde [2], Czerwik [6]) Let X be a set and let s ≥ 1 be a given real number. A function d : X × X → R+ is said to be a b-metric if and only if for all x, y, z ∈ X the following conditions are satisfied: 1. d(x, y) = 0 if and only if x = y; 2. d(x, y) = d(y, x); 3. d(x, z) ≤ s · [d(x, y) + d(y, z)]. In this case, the pair (X, d) is called a b-metric space. Thus, the only difference is at the third axiom, where the right hand side is multiplied by a factor s ≥ 1. For s = 1, we obtain the axioms of the classical metric. The notion is important since, as we can see from the following examples, it is useful in some applications. 0123456789().: V,-vol
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Example. (see [2]) The space Lp [0, 1] (where 0 < p < 1) of all real func1 tions x(t), t ∈ [0, 1] such that 0 |x(t)|p dt < ∞, together with the functional 1 d(x, y) := ( 0 |x(t) − y(t)|p dt)1/p , is a b-metric space. Notice that s = 21/p . ∞ p Example. (see [2]) For 0 < p < 1, the set lp (R) := {(xn ) ⊂ R| n=1 |xn | < ∞ p p ∞} together with the functional d : l (R)×l (R) → R, d(x, y) := ( n=1 |xn − yn |p )1/p , is a b-metric space with coefficient s = 21/p > 1. Notice that the above result holds for the general case lp (X) with 0 < p < 1, where X is a Banach space. Classical notions in mathematical analysis get a similar form in this new context. For the below notions and related ones see Berinde [2], Cobza¸s, [5], Czerwik [6], Miculescu-Mihail [12]. See also [9] for other examples. Definition 1.2. Let (X, d) be a b-metric space. Then, a sequence (xn )n∈N in X is called: 1. Cauchy if and only if for all ε > 0 there exists n(ε) ∈ N such that for each n, m ≥ n(ε) we have d(xn , xm ) < ε. 2. Convergent if and only if there exists x ∈ X such that for all ε > 0 there exists n(ε) ∈ N such that for all n ≥ n(ε) we have d(xn , x) < ε. In this case, we write lim xn = x. n→∞
Definition 1.3. Let (X, d) be a b-metric space. Then, a subset Y of X is called: 1. Compact if and only if for every sequence of elements of Y there exists a subsequence that converges to an element of Y . 2. Closed if and only
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