In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are n...In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and <i>k</i>-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the <i>k</i>-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.展开更多
Some new systems of exponentially general equations are introduced and investigated, which can be used to study the odd-order, non-positive and nonsymmetric exponentially boundary value problems. Some important and in...Some new systems of exponentially general equations are introduced and investigated, which can be used to study the odd-order, non-positive and nonsymmetric exponentially boundary value problems. Some important and interesting results such as Riesz-Frechet representation theorem, Lax-Milgram lemma and system of absolute values equations can be obtained as special cases. It is shown that the system of exponentially general equations is equivalent to nonlinear optimization problem. The auxiliary principle technique is used to prove the existence of a solution to the system of exponentially general equations. This technique is also used to suggest some new iterative methods for solving the system of the exponentially general equations. The convergence analysis of the proposed methods is analyzed. Ideas and techniques of this paper may stimulate further research.展开更多
Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*, and C be a nonempty closed convex subset of E. Let {T(t) : t ≥ 0} be a nonexpansive semigroup on...Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*, and C be a nonempty closed convex subset of E. Let {T(t) : t ≥ 0} be a nonexpansive semigroup on C such that F :=∩t≥0 Fix(T(t)) ≠ 0, and f : C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as:{yn=αnxn+(1-αn)T(tn)xn,xn=βnf(xn)+(1-βn)yn{u0∈C,vn=anun+(1-an)T(tn)un,un+1=bnf(un)+(1-bn)vnWe prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix(T(t)), which is a unique solution in F to the following variational inequality:〈(I-f)q,j(q-u)〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51-60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751-757 (2007)].展开更多
It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes,such as nano-hydrodynamics,drop wise condensation,oceanography,earthquake and wind ripple in desert...It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes,such as nano-hydrodynamics,drop wise condensation,oceanography,earthquake and wind ripple in desert.Inspired and motivated by these facts,we use the variation of parameters method for solving system of nonlinear Volterra integro-differential equations.The proposed technique is applied without any discretization,perturbation,transformation,restrictive assumptions and is free from Adomian’s polynomials.Several examples are given to verify the reliability and efficiency of the proposed technique.展开更多
文摘In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and <i>k</i>-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the <i>k</i>-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.
文摘Some new systems of exponentially general equations are introduced and investigated, which can be used to study the odd-order, non-positive and nonsymmetric exponentially boundary value problems. Some important and interesting results such as Riesz-Frechet representation theorem, Lax-Milgram lemma and system of absolute values equations can be obtained as special cases. It is shown that the system of exponentially general equations is equivalent to nonlinear optimization problem. The auxiliary principle technique is used to prove the existence of a solution to the system of exponentially general equations. This technique is also used to suggest some new iterative methods for solving the system of the exponentially general equations. The convergence analysis of the proposed methods is analyzed. Ideas and techniques of this paper may stimulate further research.
基金Supported by the National Natural Science Foundation of China (Grant No. 10771050)supported by the Higher Education Commission, Pakistan, through Research Grant No. 1-29/HEC/HRD/2005/90
文摘Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*, and C be a nonempty closed convex subset of E. Let {T(t) : t ≥ 0} be a nonexpansive semigroup on C such that F :=∩t≥0 Fix(T(t)) ≠ 0, and f : C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as:{yn=αnxn+(1-αn)T(tn)xn,xn=βnf(xn)+(1-βn)yn{u0∈C,vn=anun+(1-an)T(tn)un,un+1=bnf(un)+(1-bn)vnWe prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix(T(t)), which is a unique solution in F to the following variational inequality:〈(I-f)q,j(q-u)〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51-60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751-757 (2007)].
基金This research is supported by the Visiting Professor Program of King Saud University,Riyadh,Saudi Arabia and Research grant No.KSU.-VPP.108.
文摘It is well known that nonlinear integro-differential equations play vital role in modeling of many physical processes,such as nano-hydrodynamics,drop wise condensation,oceanography,earthquake and wind ripple in desert.Inspired and motivated by these facts,we use the variation of parameters method for solving system of nonlinear Volterra integro-differential equations.The proposed technique is applied without any discretization,perturbation,transformation,restrictive assumptions and is free from Adomian’s polynomials.Several examples are given to verify the reliability and efficiency of the proposed technique.