Abstract
Based on the concepts of contractive conditions due to Suzuki (Suzuki, T., A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society, 2008, 136, 1861–1869) and Jleli (Jleli, M., Samet, B., A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014, 8 pages), our aim is to combine the aforementioned concepts in more general way for set valued and single valued mappings and to prove the existence of best proximity point results in the context of b-metric spaces. Endowing the concept of graph with b-metric space, we present some best proximity point results. Some concrete examples are presented to illustrate the obtained results. Moreover, we prove the existence of the solution of nonlinear fractional differential equation involving Caputo derivative. Presented results not only unify but also generalize several existing results on the topic in the corresponding literature.
Highlights
Metric fixed point theory progressed a lot after the classical result due to Banach [1], known as the Banach contraction principle and it states that “Every contractive self mapping on a complete metric space has a unique fixed point”
Hancer et al [15] introduced the notion of multi-valued θ-contraction mapping as follows: Let ( X, d) be a metric space and T : X → CB( X ) a multivalued mapping
Existence Results in b-Metric Space Endowed with Graph
Summary
Metric fixed point theory progressed a lot after the classical result due to Banach [1], known as the Banach contraction principle and it states that “Every contractive self mapping on a complete metric space has a unique fixed point”. Liu et al [14] proved some fixed point results for θ-type contraction and θ-type Suzuki contraction in complete metric spaces. Hancer et al [15] introduced the notion of multi-valued θ-contraction mapping as follows: Let ( X, d) be a metric space and T : X → CB( X ) a multivalued mapping. Several researchers studied fixed point theory for single-valued and set-valued mappings in b-metric spaces (see [2,3,5,6,16,17,18] and references therein). The aim of this paper is to define multivalued Suzuki type (α, θ)-contraction and prove the existence of best proximity point results in the setting of b-metric spaces. We show the existence of solution of nonlinear fractional differential equation
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