• Computational Structural Engineering
• /
• v.4 no.2
• /
• pp.87-94
• /
• 1991

In this paper, a mixed variational principle applicable to the linear elasticity of inhomogeneous anisotropic materials is presented. For derivation of the general variational principle, a systematic procedure for the variational formulation of linear coupled boundary value problems developed by Sandhu et al. is employed. Consistency condition of the field operators with the boundary operators results in explicit inclusion of boundary conditions in the governing functional. Extensions of admissible state function spaces and specialization to a certain relation in the general governing functional lead to the desired mixed variational principle. In the physical sense, the present variational principle is analogous to the Reissner's recent formulation obtained by applying Lagrange multiplier technique followed by partial Legendre transform to the classical minimum potential energy principle. However, the present one is more advantageous for the application to the general anisotropic materials since Reissner's principle contains an implicit function which is not easily converted to an explicit form.

• Korean Journal of Mathematics
• /
• v.23 no.4
• /
• pp.537-555
• /
• 2015

This paper is concerned with the existence of solutions to the following fractional differential inclusion $$\{-{\frac{d}{dx}}\(p_0D^{-{\beta}}_x(u^{\prime}(x)))+q_xD^{-{\beta}}_1(u^{\prime}(x))\){\in}{\partial}F_u(x,u),\;x{\in}(0,1),\\u(0)=u(1)=0,$$ where $_0D^{-{\beta}}_x$ and $_xD^{-{\beta}}_1$ are left and right Riemann-Liouville fractional integrals of order ${\beta}{\in}(0,1)$ respectively, 0 < p = 1 - q < 1 and $F:[0,1]{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is locally Lipschitz with respect to the second variable. Due to the general assumption on the constants p and q, the problem does not have a variational structure. Despite that, here we study it combining with an iterative technique and nonsmooth critical point theory, we obtain an existence result for the above problem under suitable assumptions. The result extends some corresponding results in the literatures.

• Communications of the Korean Mathematical Society
• /
• v.28 no.2
• /
• pp.243-267
• /
• 2013

This paper is dedicated to study a new class of general nonlinear random A-maximal $m$-relaxed ${\eta}$-accretive (so called (A, ${\eta}$)-accretive [49]) equations with random relaxed cocoercive mappings and random fuzzy mappings in $q$-uniformly smooth Banach spaces. By utilizing the resolvent operator technique for A-maximal $m$-relaxed ${\eta}$-accretive mappings due to Lan et al. and Chang's lemma [13], some new iterative algorithms with mixed errors for finding the approximate solutions of the aforesaid class of nonlinear random equations are constructed. The convergence analysis of the proposed iterative algorithms under some suitable conditions are also studied.

• Nonlinear Functional Analysis and Applications
• /
• v.26 no.2
• /
• pp.411-432
• /
• 2021

In this paper, an inertial Halpern-type forward backward iterative algorithm for approximating solution of a monotone inclusion problem whose solution is also a fixed point of some nonlinear mapping is introduced and studied. Strong convergence theorem is established in a real Hilbert space. Furthermore, our theorem is applied to variational inequality problems, convex minimization problems and image restoration problems. Finally, numerical illustrations are presented to support the main theorem and its applications.

• Communications of the Korean Mathematical Society
• /
• v.25 no.2
• /
• pp.313-325
• /
• 2010

General models for the relaxed proximal point algorithm using the notion of relative maximal accretiveness (RMA) are developed, and then the convergence analysis for these models in the context of solving a general class of nonlinear inclusion problems differs significantly than that of Rockafellar (1976), where the local Lipschitz continuity at zero is adopted instead. Moreover, our approach not only generalizes convergence results to real Banach space settings, but also provides a suitable alternative to other problems arising from other related fields.

• Nonlinear Functional Analysis and Applications
• /
• v.27 no.4
• /
• pp.895-902
• /
• 2022

The aim of this paper is to construct a new method for finding the zeros of the sum of two maximally monotone mappings in Hilbert spaces. We will define a simple function such that its set of zeros coincide with that of the sum of two maximal monotone operators. Moreover, we will use the Newton-Raphson algorithm to get an approximate zero. In addition, some illustrative examples are given at the end of this paper.

• Journal of the Chungcheong Mathematical Society
• /
• v.21 no.2
• /
• pp.147-155
• /
• 2008

In this paper, we consider proximal point algorithms based on ($A,{\eta}$)-monotone mappings in the framework of Hilbert spaces. Since ($A,{\eta}$)-monotone mappings generalize A-monotone mappings, H-monotone mappings and many other mappings, our results improve and extend the recent ones announced by [R.U. Verma, Rockafellars celebrated theorem based on A-maximal monotonicity design, Appl. Math. Lett. 21 (2008), 355-360] and [ R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877-898] and some others.