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The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this study, we employ new results on the semi‐inner product, the adjoint operator, and the Schatten class of operators on a separable Banach space, which allow us to investigate a differential operator with unbounded operator‐valued coefficients. Specifically, we derive an asymptotic formula for the second regularized trace based on the asymptotic and spectral properties of the extended operator. For this purpose, we use the theory of continuous dense embeddings and known results regarding the regularized trace in Hilbert space. We conclude our paper by providing examples that support our findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the cycles and fixed point sets of compositions of resolvents using Attouch-Théra duality. We demonstrate that the cycles defined by the resolvent operators can be formulated in Hilbert space as solutions to a fixed point equation. Furthermore, we introduce the relationship between these cycles and the fixed point sets of the compositions of resolvents. [ABSTRACT FROM AUTHOR]

In real Hilbert spaces, given a single-valued Lipschitz continuous and monotone operator, we study generalized split feasibility problem (GSFP) over solution set of monotone variational inclusion problem. An inertia iterative method is proposed to solve this problem, by showing that the sequence generated by the iteration converges strongly to solution of GSFP. As against previous methods, our step size is chosen to be simple and not depending on norm of associated bounded linear map as well as Lipschitz constant of the single-valued operator. The obtained result was applied to study split linear inverse problem, precisely, the LASSO problem. Lastly, with the aid of numerical examples, we exhibited efficiency of our algorithm and its dominance over other existing schemes. [ABSTRACT FROM AUTHOR]

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results. [ABSTRACT FROM AUTHOR]

In the framework of a second‐order semilinear integro‐differential control system in Hilbert spaces, the paper provides sufficient conditions for proving the existence of optimal control. The Banach fixed point theorem is used to investigate the existence and uniqueness of mild solutions for the proposed problem. Additionally, it is shown that, under specific assumptions, there exists at least one optimal control pair for the Lagrange's problem as presented in the article. An example for validation is included in the paper to further support the theoretical findings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explore the split variational inclusion problem in Hilbert spaces. A splitting algorithm is constructed for solving the split variational inclusion with the help of self-adaptive techniques. Convergence analysis of the proposed algorithm is provided under additional conditions. [ABSTRACT FROM AUTHOR]

The research presented in this article focuses on studying a specific category of impulsive stochastic neutral integrodifferential systems in real Hilbert space. The first step in this process is to derive sufficient conditions for the existence and uniqueness of a mild solutions. We obtain our results using Banach's fixed point theorem and various techniques from stochastic analysis. In addition, an investigation of the exponential p-stability of the mild solutions is carried out. For an illustration of how our findings can be used, see the final section of this paper. [ABSTRACT FROM AUTHOR]

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper considers the approximate controllability of mild solutions for impulsive semilinear integrodifferential equations with statedependent delay in Hilbert spaces. We obtain our significant findings using Grimmer's resolvent operator theory and Schauder's fixed point theorem. We give an example at the end to ensure the compatibility of the results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Schrödinger equation coupled by the interface with a wave equation and with a boundary damping. The dissipation is acting on the wave equation through the Neumann boundary condition. We formulate the coupled system as an abstract evolution equation in an appropriate Hilbert space and use linear semigroup theory to show the well-posedness of the system. Then under some assumptions on the geometry of the spatial domain, we prove exponential stability of the solution. The proof of this result is based on a frequency domain approach which consists in verifying that the imaginary axis is included in the resolvent set of the system and analyzing the behavior of the resolvent operator of the system on the imaginary axis. The analysis of the resolvent is carried out by combining contradiction argument with the multipliers technique. This result extends Theorem 3.2 in [13] to multimensional spatial domains. [ABSTRACT FROM AUTHOR]

The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space H with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace X of H and its dual space X'. The three topological spaces X ⊂ H ⊂ X' are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron--Frobenius operators of the one-sided and twosided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on [0, 1] modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron--Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum. [ABSTRACT FROM AUTHOR]

We completely characterize the uniform well-posedness of the Dirichlet boundary value problem for degenerate second order abstract differential equations in Hilbert and Banach spaces. Our characterization is given solely in terms of spectral properties of the data and uniform boundedness properties of an appropriate resolvent operator. [ABSTRACT FROM AUTHOR]

The theoretical approach of random impulsive stochastic integrodifferential equations (RISIDEs) with finite delay, noncompact semigroups, and resolvent operators in Hilbert space is investigated in this article. Initially, a random impulsive stochastic integrodifferential system is proposed and the existence of a mild solution for the system is established using the Mönch fixed-point theorem and contemplating Hausdorff measures of noncompactness. Then, the stability results including a continuous dependence of solutions on initial conditions, exponential stability, and Hyers–Ulam stability for the aforementioned system are investigated. Finally, an example is proposed to validate the obtained results. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and the approximate controllability for a class of systems governed by neutral equations of second-order with infinite delay in infinite-dimensional Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. The discussion is based on the fundamental solution theory and Rothe's fixed point theorem. In addition, an example is given to illustrate our main conclusion. [ABSTRACT FROM AUTHOR]

This paper considers the approximate controllability of mild solutions for impulsive semilinear integrodifferential equations with statedependent delay in Hilbert spaces. We obtain our significant findings using Grimmer's resolvent operator theory and Schauder's fixed point theorem. We give an example at the end to ensure the compatibility of the results. [ABSTRACT FROM AUTHOR]

In the present paper, we address the global solvability and finite-time attractivity of strong solutions for an abstract Rayleigh–Stokes-type equation involving nonlinear perturbations in Hilbert spaces. Based on regularity estimates of resolvent operator, local estimates on Hilbert scales and fixed point arguments, we obtain some results on global existence and finite-time attractivity of strong solutions to our problem. [ABSTRACT FROM AUTHOR]

In this article, a nonlinear system of generalized ordered XOR-inclusion problems in Hilbert space is introduced and studied. Initially, we define the resolvent operator related to the (α , λ) -XOR-weak-ANODD multivalued mapping. Using the fixed point technique, we demonstrate the results of existence. In order to make the suggested system more realistic, we create the S-iterative algorithm and demonstrate that the sequence generated through this technique strongly converges with the proposed system's solution. One example is provided in support of the existence result as well. [ABSTRACT FROM AUTHOR]

In this article, we study a multi-step inertial forward-backward-half forward splitting algorithm to solve the solution of the sum of three operators with two of them being respectively, cocoercive and monotone-Lipschitz continuous. Notably, the convergence of the multi-step inertial forward-backward-half forward algorithm is established under the summability condition formulated on the basis of the iterative sequence in a Hilbert space setting. We also display two applications including problems that are: a composite monotone inclusion problem involving mixtures of linearly composed with parallel-sum type operators, and a composite convex optimization problem. As far as the resolvent of the composite operator is concerned, due to difficulty in calculation, then, the composite monotone inclusion problem is addressed by the proposed multi-step inertial algorithm together with the primal-dual idea. Experiments conducted on an image deblurring problem indicate the superiority and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the controllability of neutral stochastic integrodifferential evolution equations in Hilbert spaces. By employing the resolvent operator theory in the sense of Grimmer, stochastic analysis, and a fixed point approach, the sufficient conditions of exact controllability, for such a system, are established. We provide an example to illustrate the effectiveness of the proposed result. [ABSTRACT FROM AUTHOR]

The paper proposes some new iterative algorithms for solving a split variational inclusion problem involving maximally monotone multi-valued operators in a Hilbert space. The algorithms are constructed around the resolvent of operator and the regularization technique to get the strong convergence. Some stepsize rules are incorporated to allow the algorithms to work easily. An application of the proposed algorithms to split feasibility problems is also studied. The computational performance of the new algorithms in comparison with others is shown by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this study we shall introduce to the consideration a new type Sturm-Liouville problem, which differs from the standard problems in that it contain nondifferential term in the equation. Namely, the equation contains an abstract linear operator B, which may be non-selfadjoint. Moreover, the problem contains an additional transmission conditions at the finite number interior points of discontinuity and the spectral parameter λ appears not only in the equation, but also in the boundary conditions. It is not clear how to extend the known methods and results of classic Sturmian theory to our problem. The major difficulty lies in the existence and behavior the of eigenvalues in the complex plane. We developed an our own approaches for investigation of some important spectral properties of such type non-classical boundary value problems. At first we defined a new Hilbert spaces for operator-treatment of the considered problem. Then we established an isomorphism and coerciveness. We also established some important properties of the resolvent operator. Finally, we proved discreteness of the spectrum and derived asymptotic formulas for the eigenvalues. Note that, the obtained results are new even in the continuous case and/or when the boundary conditions do not contain the spectral parameter λ. [ABSTRACT FROM AUTHOR]

Let ε > 0 be a small parameter. We consider the domain Ω ε : = Ω ∖ D ε , where Ω is an open domain in R n , and D ε is a family of small balls of the radius d ε = o (ε) distributed periodically with period ε. Let Δ ε be the Laplace operator in Ω ε subject to the Robin condition ∂ u ∂ n + γ ε u = 0 with γ ε ≥ 0 on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on d ε and γ ε , the operator Δ ε converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in Ω and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of L 2 → L 2 and L 2 → H 1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators. [ABSTRACT FROM AUTHOR]

In this paper we provide a new approach in the study of a variational-hemivariational inequality in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem. First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational-hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their convergence to the unique solution of the variational-hemivariational inequality. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of mild solutions for a fractional stochastic differential system derived by Atangana–Baleanu fractional derivative have been discussed in a Hilbert space. A new set of results is obtained with non-Lipschitz conditions of nonlinear functions using the well-known Picard-type iterations, basic concepts of fractional calculus, sectorial operator, and stochastic analysis techniques. At last, an application is discussed to show the applicability of the developed theory. [ABSTRACT FROM AUTHOR]

The objective of this article is to investigate a variant of ordered inclusion problems, namely, H˜ (·, ·)-compression (α, λ)-XOR-ordered NODSM mapping in real ordered Hilbert spaces by using graph convergence approach. An associated resolvent operator is defined and some of its properties are discussed. An equivalence between resolvent operator convergence and graph convergence is established. An existence result for H˜ (·, ·)-compression (α, λ)-XOR-ordered NODSM mapping is proved using XOR and XNOR operations technique. Moreover, an ordered inclusion problem in real ordered Hilbert space is considered and an iterative algorithm is presented. Also, the concept of graph convergence is applied to analyze the convergence of the proposed algorithm. Finally, the notion of graph convergence is clarified by an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of linear difference equations of fractional order 2 < α ≤ 3 in the sense of Riemann–Liouville. The explicit solution for this model is provided in terms of a fractional resolvent sequence which allows to write the solution to this equation as a variation of constant formula. We also characterize the existence and uniqueness of solutions in ℓ p (N 0 , X) spaces with X being a UMD-space in terms of the R-boundedness of the operator symbol of the model. Moreover, we are able to relax this condition in the case of Hilbert spaces. Finally, we illustrate our results with an example that involves the generator of a contraction semigroup. [ABSTRACT FROM AUTHOR]

This paper investigates approximate controllability of semilinear measure driven equations in Hilbert spaces. We focus on a specific category of nonlocal integrodifferential equations. We apply the theory of the resolvent operator in the sense of Grimmer, as well as the fixed point strategy and the theory of the Lebesgue-Stieljes integral, in the context of the space of regulated functions. In light of this, the prevalence of our findings is greater than that which is found in the literature. At last, an example is comprised that exhibits the significance of developed theory. [ABSTRACT FROM AUTHOR]

This paper presents approximate controllability results for impulsive stochastic integro-differential systems with statedependent delay in a Hilbert space. The use of the resolvent operator in the sense of Grimmer, as well as stochastic analysis techniques, yields a new set of results. Finally, an example is given to show how the theory that has been worked out can be put into practice. [ABSTRACT FROM AUTHOR]

This paper is concerned with the controllability results of neutral impulsive stochastic functional integrodifferential equations driven by a fractional Brownian motion with infinite delay in a real separable Hilbert space. The controllability results are obtained using stochastic analysis, the theory of resolvent operator in the sense of Grimmer and Krasnoselskii fixed point theorem. An example is provided to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

The present paper considers the operator pencil A (λ) = A 0 + A 1 λ , where A 0 , A 1 ≠ 0 are bounded linear mappings between complex Hilbert spaces and A 0 is neither one-to-one nor onto. Assuming that 0 is an isolated singularity of A (λ) and that the image of A 0 is closed, certain operators are defined recursively starting from A 0 and A 1 and they are shown to provide a characterization of the image and null space of the operators in the principal part of the resolvent and of the logarithmic residues of A (λ) at 0. The relations with the classical results based on ascent and descent in [10] are discussed. In the special case of A 0 being Fredholm of index 0, the present results characterize the rank of the operators in the principal part of the resolvent, the dimension of the subspaces that define the ascent and descent, the partial multiplicities, and the algebraic multiplicity of A (λ) at 0. [ABSTRACT FROM AUTHOR]

In the paper, we characterize the polynomial stability of C 0 semigroup T (t) with generator A on Hilbert space H . Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for H . By the asymptotic relation of the real part and imaginary part of eigenvalues of A, we give the optimal decay rate of polynomial stability of T (t). Moreover, we give the zeros distribution of certain equations. As an application, we illustrate our general results by an acoustical system. [ABSTRACT FROM AUTHOR]

In this paper we examine the spectral analysis of a spatially non‐homogeneous Timoshenko beam mounted on the periphery of a rigid root rotating about its axis at a constant angular speed. The junction between the beam and the root is assumed to be elastically restrained and damped. The unbounded operator associated to the physical problem in the associated Hilbert space is non‐self‐adjoint and with a compact resolvent. We show that under some hypotheses on the physical properties of the beam, there exists a Riesz basis of root vectors of this unbounded operator. Furthermore, the solution of the initial value problem has an expansion in terms of this Riesz basis, uniform with respect to the time in a bounded interval. [ABSTRACT FROM AUTHOR]

We study in this work the optimal control and time optimal control problems for a semi‐linear integro‐differential evolution system with infinite delay in Hilbert space. We first study the existence and uniqueness result of mild solutions in space Xα for the considered system, and the main tools here are the theory of resolvent operators and fractional powers of operators and α−norm. In this way the obtained results are more general and can be applied to the systems involving terms having spatial derivatives. Then we discuss the Lagrange optimal control problem for the system via limit arguments. The time optimal control problem is also proposed and investigated deliberately for it. Finally, an example is provided to show the applications of the obtained results. [ABSTRACT FROM AUTHOR]

In the paper we investigate well-defined solvability of the Dirichlet problem for a second order partial operator-differential equation in a Hilbert space. First we prove a theorem on the isomorphism of the principle part of the given equation. Then it is proved that operator coefficients of the disturbed part of the equation can be chosen from a wider class so that theorems on solvability of a boundary value problem for complete equations hold and are easily verifiable in practical problems. This feature strongly distinguishes our research from the works, where the solvability conditions entail arbitrary smallness of the disturbed part of equations and they are expressed by means of restrictions on the resolvent growth of the appropriate operator beam. [ABSTRACT FROM AUTHOR]

In this paper, we apply the resolvent operator theory and an approximating technique to derive the existence and controllability results for nonlocal impulsive neutral integro‐differential equations with finite delay in a Hilbert space. To establish the results, we take the impulsive functions as a continuous function only, and we assume that the nonlocal initial condition is Lipschitz continuous function in the first case and continuous functions only in the second case. The main tools applied in our analysis are semigroup theory, the resolvent operator theory, an approximating technique, and fixed‐point theorems. Finally, we illustrate the main results with the help of two examples. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new inertial iterative method for solving split variational inclusion problems in real Hilbert spaces. We prove that the generated sequence converges weakly to the solution of the considered problem under some mild conditions. The major contributions of our results are: (ⅰ) to increase the rate of convergence of the method for solving split variational inclusion problem through the inertial extrapolation step, (ⅱ) to relax the choice of the inertial factor and show the inertial factor can be chosen greater than 1/3 unlike what is previously known before for inertial proximal point method in the literature (ⅲ) to show the numerical efficiency and superiority of our proposed method through some test example. [ABSTRACT FROM AUTHOR]

By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1) in a Hilbert space. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximally monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account. Using the concept of proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative of the resolvent operator associated to the maximally monotone parameterized variational inclusion. This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being itself a solution of a variational inclusion involving the semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual composite monotone inclusion is given. Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their derivatives belong to the derivative of the associated Kuhn–Tucker set. [ABSTRACT FROM AUTHOR]

We present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner–Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that L 2 -based Sobolev spaces are not suitable Hilbert space choices. Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon. We introduce L 2 -based Gevrey estimates for the wave equation that are intimately connected to the existence of conserved quantities along null infinity and the event horizon. We relate this new framework to the traditional interpretation of quasinormal frequencies as poles of the meromorphic continuation of a resolvent operator and obtain new quantitative results in this setting. [ABSTRACT FROM AUTHOR]

Some norm inequalities are presented for non- selfadjoint differential operators. First, the resolvent of the introduced non-selfadjoint differential operator using m-sectorial operators in the Hilbert space H = L 2 (Ω) is established. As application of this new result, the resolvent of the considered operator in the Hilbert space H ℓ = L 2 (Ω) ℓ is obtained utilizing some analytic techniques and diagonalizable way. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a system of variational inclusions called system of nonlinear variational inclusion problem involving A-monotone mappings in real Hilbert spaces. By means of resolvent operator technique, we suggest an resolvent iterative algorithm for finding the approximate solution of this system and discuss the convergence criteria of the sequences generated by the resolvent iterative algorithm under some suitable conditions. [ABSTRACT FROM AUTHOR]

In this paper, we develop an iterative algorithm for obtaining common so- lutions to the Cayley inclusion problem and the set of fixed points of a non-expansive mapping in Hilbert spaces. A numerical example is given for the justification of our claim. [ABSTRACT FROM AUTHOR]

In this paper, two main results on the uniform stability of a resolvent family { R h (t) } t ≥ 0 , depending on a parameter h are presented. First, we discuss a GGP type theorem on the resolvent family { R h (t) } t ≥ 0 and give some sufficient conditions on the uniform stability of { R h (t) } t ≥ 0 . Then we prove that under some suitable conditions, the weak L p -stability of { R h (t) } t ≥ 0 implies its uniform stability. Our results both essentially generalize previous work on the uniform stability of a family of C 0 -semigroups { T h (t) } t ≥ 0 and a resolvent family { R (t) } t ≥ 0 , without depending on the parameter h. Examples are also given to illustrate our results. [ABSTRACT FROM AUTHOR]

Non-self-adjoint operators have many applications, including quantum and heat equations. On the other hand, the study of these types of operators is more difficult than that of self-adjoint operators. In this paper, our aim is to study the resolvent and the spectral properties of a class of non-self-adjoint differential operators. So we consider a special non-self-adjoint elliptic differential operator (Au)(x) acting on Hilbert space and first investigate the spectral properties of space H 1 = L 2 Ω 1 . Then, as the application of this new result, the resolvent of the considered operator in ℓ -dimensional space Hilbert H ℓ = L 2 Ω ℓ is obtained utilizing some analytic techniques and diagonalizable way. [ABSTRACT FROM AUTHOR]

We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent. [ABSTRACT FROM AUTHOR]

Two new iterative algorithms are introduced for solving the split common null point problem (SCNPP). The first algorithm is shown to converge weakly and the second is shown to converge strongly to a solution of SCNPP in a Hilbert space. The algorithms are forward methods (i.e, involving no computation of the resolvent of a monotone operator). Another feature of the algorithms is that the selection of the step-sizes does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT (0) spaces stands in opposed to the monotonicity defined earlier in CAT (0) spaces by Khatibzadeh and Ranjbar [J. Aust. Math. Soc. 103(1), 70–90 (2017).] and Chaipunya and Kumam [Optimization 66(10), 1647–1665 (2017).]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [AMS/IP Stud. Adv. Math., vol. 8, pp. 1–47. Amer. Math. Soc., Providence, RI (1998)], Mayer [Comm. Anal. Geom. 6(2), 199–253 (1998).], Stojkovic [Adv. Calc. Var. 5(1), 77–126 (2012).], and Bačák [Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. De Gruyter, Berlin (2014).] for proper, convex, lower semicontinuous functions in the context of complete CAT (0) spaces, and also by Iwamiya and Okochi [Nonlinear Anal. 54(2), 205–214 (2003).] for monotone vector fields in the context of Hadamard manifolds. [ABSTRACT FROM AUTHOR]