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The main purpose of this article is to present some results concerning Reich type contractions in the graph structure in the framework of recently introduced graphical b-metric spaces. Our results are significant extensions and generalizations of some pioneer results in the existing theory. Innovative examples along with directed graphs are propounded to support the newfangled results, making the established theory more comprehensible. Final section is devoted to apply our results to the existence of solutions of some nonlinear problems along with some open problems which may be fruitful for the further scope of the study.

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In this paper, we introduce some new classes of generalized -contractions and we establish certain fixed point results for such mappings in the setting of -metric-like spaces. Some examples will illustrate the results and the corresponding computer simulations are suggestive from the output point of view. A second purpose of the paper is to apply the abstract results in the study of the existence of a solution for an integral equation problem and for a boundary value problem related to a real life mathematical model, namely, the problem of conversion of solar energy to electrical energy. Our study is concluded with an open problem, related to an integrodifferential equation arising in the study of electrical and electronics circuit analysis.