Cyclic (Noncyclic) ϕ-condensing operator and its application to a system of differential equations

We establish a best proximity pair theorem for noncyclic ϕ-condensing operators in strictly convex Banach spaces by using a measure of noncompactness. We also obtain a counterpart result for cyclic ϕ-condensing operators in Banach spaces to guarantee the existence of best proximity points, and so, an extension of Darbo’s fixed point theorem will be concluded. As an application of our results, we study the existence of a global optimal solution for a system of ordinary differential equations

Investigating Automatic Static Analysis Results to Identify Quality Problems: an Inductive Study

Background: Automatic static analysis (ASA) tools examine source code to discover "issues", i.e. code patterns that are symptoms of bad programming practices and that can lead to defective behavior. Studies in the literature have shown that these tools find defects earlier than other verification activities, but they produce a substantial number of false positive warnings. For this reason, an alternative approach is to use the set of ASA issues to identify defect prone files and components rather than focusing on the individual issues. Aim: We conducted an exploratory study to investigate whether ASA issues can be used as early indicators of faulty files and components and, for the first time, whether they point to a decay of specific software quality attributes, such as maintainability or functionality. Our aim is to understand the critical parameters and feasibility of such an approach to feed into future research on more specific quality and defect prediction models. Method: We analyzed an industrial C# web application using the Resharper ASA tool and explored if significant correlations exist in such a data set. Results: We found promising results when predicting defect-prone files. A set of specific Resharper categories are better indicators of faulty files than common software metrics or the collection of issues of all issue categories, and these categories correlate to different software quality attributes. Conclusions: Our advice for future research is to perform analysis on file rather component level and to evaluate the generalizability of categories. We also recommend using larger datasets as we learned that data sparseness can lead to challenges in the proposed analysis proces

Multi-valued F-contractions and the solution of certain functional and integral equations

Wardowski [Fixed Point Theory Appl., 2012:94] introduced a new concept of contraction and proved a fixed point theorem which generalizes Banach contraction principle. Following this direction of research, we will present some fixed point results for closed multi-valued F-contractions or multi-valued mappings which satisfy an F-contractive condition of Hardy-Rogers-type, in the setting of complete metric spaces or complete ordered metric spaces. An example and two applications, for the solution of certain functional and integral equations, are given to illustrate the usability of the obtained results

Fixed Point Results for F-Contractive Mappings of Hardy-Rogers-Type

Recently,Wardowski introduced a new concept of contraction and proved a fixed point theorem which generalizes Banach contraction principle. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for self-mappings on complete metric spaces or complete ordered metric spaces. Moreover, an example is given to illustrate the usability of the obtained results

The existence of solutions for the modified (p(x), q(x))-Kirchhoff equation

We consider the Dirichlet problem Kp p(x) u(x) − ∆ Kq q(x) u(x) = f(x, u(x), ∇u(x)) in Ω, u = 0, driven by the sum of a p(x)-Laplacian operator and of a q(x)-Laplacian operator, both of them weighted by indefinite (sign-changing) Kirchhoff type terms. We establish the existence of weak solution and strong generalized solution, using topological tools (properties of Galerkin basis and of Nemitsky map). In the particular case of a positive Kirchhoff term, we obtain the existence of weak solution (= strong generalized solution), using the properties of pseudomonotone operators

Higher order evolution inequalities involving Leray-Hardy potential singular on the boundary

We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray-Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions

Impact of Common Property (E.A.) on Fixed Point Theorems in Fuzzy Metric Spaces

We observe that the notion of common property (E.A.) relaxes the required containment of range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. As a consequence, a multitude of recent fixed point theorems of the existing literature are sharpened and enriched

Nonexistence results for higher order fractional differential inequalities with nonlinearities involving Caputo fractional derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function