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In this paper, we analyze the existence of nontrivial p- k i -convex radial solutions for a coupled system of p- k i -Hessian equations S k 1 (D (| D u 1 | p - 2 D u 1)) = λ 1 g 1 (| x |) f 1 (- u 2) in B , S k 2 (D (| D u 2 | p - 2 D u 2)) = λ 2 g 2 (| x |) f 2 (- u 3) in B , ⋮ S k n - 1 (D (| D u n - 1 | p - 2 D u n - 1 )) = λ n - 1 g n - 1 (| x |) f n - 1 (- u n) in B , S k n (D (| D u n | p - 2 D u n)) = λ n g n (| x |) f n (- u 1) in B , u i = 0 , on ∂ B , i ∈ I n = { 1 , 2 , ⋯ , n } , where λ i > 0 are parameters and B is the unit ball in R N (N ≥ 2) . In addition, the asymptotic behaviors of nontrivial p- k i -convex radial solutions on the parameter λ i (i ∈ I n) are also studied using the eigenvalue theory. This is probably the first time that a system of p- k i -Hessian equations has been studied by employing this technique. New nonexistence conclusions are also derived in this article. As an application, we present several new sufficient conditions for the existence and asymptotic behavior of nontrivial p- k i -convex radial solutions for the power-type coupled system of p- k i -Hessian equations. [ABSTRACT FROM AUTHOR]

The paper aims to explore the impact of composite operators containing a few physical quantities on the gravitational and electromagnetic fields, studying the influencing factors and physical properties of octonion field equations. Maxwell first utilized the quaternions and vector terminology to describe the electromagnetic fields. The octonions can be used to simultaneously describe the physical quantities of electromagnetic and gravitational fields, including the octonion field potential, field strength, field source, linear momentum, angular momentum, torque and force. In the octonion spaces, the field strength and quaternion operator are able to combine together to become one composite operator, making an important contribution to the field equations. Similarly, the field potential can also form some composite operators with the quaternion operator, and they have a certain impact on the field equations. Furthermore, other physical quantities can also be combined with the quaternion operator to form several composite operators. In these composite operators, multiple physical quantities can also have a certain impact on the field equations. In other words, the field strength does not occupy a unique central position, compared to other physical quantities in these composite operators. In the field theories described by the octonions, various field equations can be derived from the application of different composite operators. According to different composite operators, it is able to infer the field equations when the field strength or/and field potential make a certain contribution, and it is also possible to deduce several new field equations when the remaining physical quantities play a certain role. This further deepens the understanding of the physical properties of field equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove the weak convergence of the iterates generated by the nonlinearly preconditioned forward-backward splitting method for the sum of a maximally hypermonotone operator A and a hypercocoercive operator B under several suitable conditions. We provide several choices of the nonlinear preconditioners for solving nonlinearly composed inclusions. In particular, the backward-forward splitting method is recovered by the nonlinearly preconditioned forward-backward splitting method with a special choice of the nonlinear preconditioner. [ABSTRACT FROM AUTHOR]

Let T1, T2 be two singular integral operators with nonsmooth kernels introduced by Duong and McIntosh. In this paper, by establishing certain bi‐sublinear sparse dominations, the authors obtain some quantitative bounds on Lp(Rn,w) with p∈(1,∞) and w∈Ap(Rn) for the composite operator T1T2. Some weighted weak type endpoint estimates are also given. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel splitting method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. The weak convergence the method is proved in real Hilbert spaces. Applying the proposed method to composite monotone inclusions involving parallel sums yields a new primal-dual splitting which is different from the existing methods. Connections to existing works are clearly stated. We also provide an application of the proposed method to the image denoising by the total variation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on L p (R d , w) , p ∈ (1 , ∞) , which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint p = 1 , the L log L weighted weak-type bound is also obtained, which has interests of its own in the theory of rough singular integral even in the unweighted case. [ABSTRACT FROM AUTHOR]

In this article, we investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in the literature for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive operator introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi-Fejér-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models. [ABSTRACT FROM AUTHOR]

We propose an inertial forward–backward splitting algorithm to compute a zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal–dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a stochastic forward-backward-forward splitting algorithm and prove its almost sure weak convergence in real separable Hilbert spaces. Applications to composite monotone inclusion and minimization problems are demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we provide a splitting method for finding a zero of the sum of a maximally monotone operator, a Lipschitzian monotone operator, and a normal cone to a closed vector subspace of a real Hilbert space. The problem is characterised by a simpler monotone inclusion involving only two operators: the partial inverse of the maximally monotone operator with respect to the vector subspace and a suitable Lipschitzian monotone operator. By applying the Tseng's method in this context, we obtain a fully split algorithm that exploits the whole structure of the original problem and generalises partial inverse and Tseng's methods. Connections with other methods available in the literature are provided, and the flexibility of our setting is illustrated via applications to some inclusions involving $$m$$ maximally monotone operators, to primal-dual composite monotone inclusions, and to zero-sum games. [ABSTRACT FROM AUTHOR]

We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic structure of the problem by activating explicitly the cocoercive operator in the first step, and taking advantage of a vector space decomposition in the second step. The second step of the first method is a Douglas–Rachford iteration involving the maximally monotone operator and the normal cone. In the second method, it is a proximal step involving the partial inverse of the maximally monotone operator with respect to the vector subspace. Connections between the proposed methods and other methods in the literature are provided. Applications to monotone inclusions with finitely many maximally monotone operators and optimization problems are examined. [ABSTRACT FROM PUBLISHER]

We propose a splitting algorithm for solving a coupled system of primal-dual monotone inclusions in real Hilbert spaces. The proposed algorithm has a structure identical to that of the forward-backward algorithm with variable metric. The operators involved in the problem formulation are used separately in the sense that single-valued operators are used individually and approximately in the forward steps and multi-valued operators are used individually via their generalization resolvent in the backward steps. The weak convergence of the algorithm proposed is proved. Applications to coupled system of monotone inclusions in duality and minimization problems, and multi-dictionary signal representation are demonstrated. [ABSTRACT FROM AUTHOR]

We propose a variable metric forward–backward splitting algorithm and prove its convergence in real Hilbert spaces. We then use this framework to derive primal-dual splitting algorithms for solving various classes of monotone inclusions in duality. Some of these algorithms are new even when specialized to the fixed metric case. Various applications are discussed. [ABSTRACT FROM AUTHOR]