Abstract
In this paper, we introduce and study a new system of general nonconvex variational inclusions involving four different nonlinear operators (SGNVI) and prove the equivalence between the SGNVI and a fixed point problem. By using this equivalent formulation, we establish the existence and uniqueness theorem for solution of the SGNVI. We use the foregoing equivalent alternative formulation and two nearly uniformly Lipschitzian mappings S1 and S2 to suggest and analyze some new two-step projection iterative algorithms for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping Q =( S1, S2), which is the unique solution of the SGNVI. Further, the convergence analysis of the suggested iterative algorithms under suitable conditions is studied. MSC: Primary 47H05; secondary 47J20; 49J40; 90C33
Highlights
The variational principle, the origin of which can be traced back to Fermat, Newton, Leibniz, Bernoulli, Euler, and Lagrange, has been one of the major branches of mathematical and engineering sciences for more than two centuries
The variational inequalities theory constituted a significant and novel extension of the variational principles and describe a broad spectrum of interesting and fascinating developments involving a link among various fields of mathematics, physics, economics, equilibrium, financial, optimization, regional, and engineering sciences
We prove the equivalence between the SGNVI and a fixed point problem, and by this equivalent formulation, we discuss the existence and uniqueness of solution of the SGNVI
Summary
The variational principle, the origin of which can be traced back to Fermat, Newton, Leibniz, Bernoulli, Euler, and Lagrange, has been one of the major branches of mathematical and engineering sciences for more than two centuries. Some works in this direction for finding the solutions of the variational inequalities/inclusion problems and the fixed points of the nearly uniformly Lipschitzian mappings have been done, see, for example, [ , , ]. In , Clarke et al [ ] introduced and studied a new class of nonconvex sets, called proximally smooth sets; subsequently, Poliquin et al in [ ], investigated the aforementioned set under the name of uniformly prox-regular sets. These have been successfully used in many nonconvex applications in areas such as optimizations, economic models, dynamical systems, differential inclusions, etc.
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