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In this paper we study implicit obstacle problems driven by a nonhomogenous differential operator, called double phase operator, and a multivalued term which is described by Clarke’s generalized gradient. Based on a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis, we prove the existence of at least one solution.

Operator-splitting methods convert optimization and inclusion problems into fixed-point equations; when applied to convex optimization and monotone inclusion problems, the equations given by operator-splitting methods are often easy to solve by standard techniques. The hard part of this conversion, then, is to design nicely behaved fixed-point equations. In this paper, we design a new, and thus far, the only nicely behaved fixed-point equation for solving monotone inclusions with three operators; the equation employs resolvent and forward operators, one at a time, in succession. We show that our new equation extends the Douglas-Rachford and forward-backward equations; we prove that standard methods for solving the equation converge; and we give two accelerated methods for solving the equation.

In this paper, an iterative algorithm is introduced to solve the split common fixed point problem for asymptotically nonexpansive mappings in Hilbert spaces. The iterative algorithm presented in this paper is shown to possess strong convergence for the split common fixed point problem of asymptotically nonexpansive mappings although the mappings do not have semi-compactness. Our results improve and develop previous methods for solving the split common fixed point problem. MSC: 47H09, 47J25.

In the article, we introduce a new concept of contraction and prove a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature. The article includes an example which shows the validity of our results, additionally there is delivered numerical data which illustrates the provided example. MSC: 47H10; 54E50[PUBLICATION ABSTRACT]

By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

In this paper, we extend the result of Wardowski (Fixed Point Theory Appl. 2012:94, 2012) by applying some weaker conditions on the self map of a complete metric space and on the mapping F, concerning the contractions defined by Wardowski. With these weaker conditions, we prove a fixed point result for F-Suzuki contractions which generalizes the result of Wardowski. MSC: 74H10, 54H25.

In this manuscript, based on extended b-metric spaces, we propose a new structure of cyclic mapping, that is, BSS-type cyclic mapping. The fixed-point theorem for BSS-type cyclic mapping is established. A concrete example of BSS-type cyclic mapping is given to show the reasonability and correctness of the obtained results. The result proposed by this manuscript extends known conclusions in some references.

A newly proposed p-Laplacian nonperiodic boundary value problem is studied in this research paper in the form of generalized Caputo fractional derivatives. The existence and uniqueness of solutions are fully investigated for this problem using some fixed point theorems such as Banach and Schauder. This work is supported with an example to apply all obtained new results and validate their applicability.

In this article, we introduce an inertial projection and contraction algorithm by combining inertial type algorithms with the projection and contraction algorithm for solving a variational inequality in a Hilbert space H. In addition, we propose a modified version of our algorithm to find a common element of the set of solutions of a variational inequality and the set of fixed points of a nonexpansive mapping in H. We establish weak convergence theorems for both proposed algorithms. Finally, we give the numerical experiments to show the efficiency and advantage of the inertial projection and contraction algorithm.

We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results.