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In the present paper, we propose a new approach to solving a class of generalized split problems. This approach will open some new directions for research to solve the other split problems, for instance, the split common zero point problem and the split common fixed point problem. More precisely, we study the split common solution problem for monotone operator equations in real Hilbert spaces. To find a solution to this problem, we propose and establish the strong convergence of the two new iterative methods by using the Tikhonov regularization method. Meantime, we also study the stability of the iterative methods. Finally, two numerical examples are also given to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretization-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox. [ABSTRACT FROM AUTHOR]

First-order optimization methods have long been established as effective methods for solving large-scale unconstrained optimization problems. Over time, these methods have been extended to address the task of finding zeros of single-valued monotone maps within the framework of finite-dimensional Hilbert spaces. Notably, Abubakar et al. (Comput Appl Math 39(2):129, 2020) recently introduced a projection method that incorporates a convex combination of two distinct positive spectral coefficients to find the zeros of a single-valued monotone map. Motivated by their work, this paper generalises the Abubakar et al. technique to a more general setting of infinite-dimensional Hilbert spaces. Under mild assumptions, we establish weak convergence of the generated sequence of iterates. Furthermore, the extended method does not involve computing the resolvent of the monotone operator. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new unified splitting methods for monotone inclusion problems with three operators in real Hilbert spaces. These methods are based on the combination of Douglas-Rachford method and other methods, forward-backward-forward method and reflected-forward-backward method. The weak convergence of new algorithms under standard assumptions is established. We also give some numerical examples to demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The purpose of our paper is to present a few convergence results for the Krasnoselskii–Mann iteration and also the modified Krasnoselskii–Mann iteration used for approximating the fixed point of k-strictly pseudo contractive map in the Banach space, which can be derived from the corresponding convergence theorems in the class of non-expansive maps. Also, using the enrichment techniques for contractive type map T, i.e., the averaged operator T λ (u) = (1 - λ) u + λ T u , where λ ∈ (0 , 1 ] , we identified that pseudo-contractive and demi-contractive maps are an unsaturated class of mappings in the Banach space. Illustrative examples and numerical calculations are given to support the obtained results, which are in fact the generalizations of approximations results in the Banach space from the Hilbert space. [ABSTRACT FROM AUTHOR]

We propose a variation of the forward–backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward–backward splitting scheme and an accelerated O 1 n 2 -convergent forward–backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of approximating the common solution of monotone inclusions and demicontraction fixed point problems. Firstly, we present Tseng splitting method which incorporates the viscosity technique, new self-adaptive step size and double inertial extrapolations techniques for approximating the solution of the problem in the setting of Hilbert spaces. Unlike some double inertial methods recently studied by many authors, our algorithm does not require computation onto half space, containing the feasible set. The co-coerciveness condition that is often impose on the single-valued operator and the imposition of other stringent assumptions are not required in our method. The suggested method does not require any line search technique. The method uses a new non-monotonic step size which is allowed to increase from iteration to iteration. The step size embeds some relaxation parameters which also improve the rate of convergence of our method. The suggested method only needs one backward computation of the multi-valued operator at each iteration and one forward computation of the single-valued operator; a concept that has not been considered in several splitting methods for strongly convergence in the literature. We prove the strong convergence results of our method under some mild assumptions on the control parameters. We apply our main results to the solutions of several optimization problems. In three numerical experiments, the applicability and efficiency of our new method are compared with some well known methods in the existing literature. Our results improve, unify and generalize several well known results in the literature. [ABSTRACT FROM AUTHOR]

This work is concerned with the existence of mild solutions to nonlinear Fokker–Planck equations with fractional Laplace operator (- Δ) s for s ∈ 1 2 , 1 . The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diffusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean–Vlasov equations with Lévy noise, as well as the Markov property for their laws are proved. [ABSTRACT FROM AUTHOR]

We introduce a generalized proximal point algorithm, and perform a detailed convergence analysis with the focus on the case of degenerate metric. The degeneracy leads to a well-defined resolvent form restricted to a reduced dimensional space. This approach unifies the algorithmic structures of Douglas-Rachford splitting and other related operator splitting schemes. Various aspects of these algorithms, in particular, the convergence of Douglas-Rachford splitting in terms of the solution itself and Chambolle-Pock algorithm under the limit setting, are investigated or revisited by the unified degenerate proximal point analysis. [ABSTRACT FROM AUTHOR]

In this article, we propose a viscosity-type scheme for approximating a common solution of convex minimization problem, monotone vector field inclusion problem and fixed point problem involving multi-valued nonexpansive mapping in the framework of Hadamard spaces. We obtain a strong convergence theorem for the sequence generated therefrom to a solution of the problem. Furthermore, we apply our results to compute the Fréchet mean, find the mean of probabilities, minimize energy of measurable mappings and solve a problem of two-arm robotic motion control. Finally, we give numerical example to demonstrate the applicability of the method and also issue comparisons with some existing methods. Our results extend and complement some recent results in the literature. [ABSTRACT FROM AUTHOR]

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function f. We show that the generated sequences converge strongly to the minimal norm element from argmin f . We also show fast convergence for the potential energies f (x n) - min f and f (y n) - min f , where (x n) , (y n) are the sequences generated by our algorithm. Further we obtain fast convergence to zero of the discrete velocity and some estimates concerning the value of the gradient of the objective function in the generated sequences. Via some numerical experiments we show that we need both Tikhonov regularization terms in our algorithm in order to obtain the strong convergence of the generated sequences to the minimum norm minimizer of our objective function. [ABSTRACT FROM AUTHOR]

In this paper, we study a variational inequality problem which is defined over the the solution of multiple set split common fixed point problem for a finite family of generalized demimetric mappings. We present an accelerated steepest descent algorithm for solving this problem and then establish strong convergence of the proposed method under standard and mild conditions. We apply our proposed algorithm to solve some problems, including variational inequalities over multiple set split feasibility problem, variational inequalities over multiple set split zero point problem and the convex optimization problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the split equality Minty variational inequality problem in reflexive real Banach spaces. Then we construct a single projection inertial algorithm for solving the introduced problem. We establish a strong convergence result with the assumption that the mappings under consideration are Lipschitz continuous and quasimonotone. We give some specific applications of the main result and finally provide a numerical example to demonstrate the workability of our method. [ABSTRACT FROM AUTHOR]

By means of fixed point index theory for multivalued maps, we provide an analogue of the classical Birkhoff–Kellogg Theorem in the context of discontinuous operators acting on affine wedges in Banach spaces. Our theory is fairly general and can be applied, for example, to eigenvalues and parameter problems for ordinary differential equations with discontinuities. We illustrate in detail this fact for a class of second-order boundary value problem with deviated arguments and discontinuous terms. In a specific example, we explicitly compute the terms that occur in our theory. [ABSTRACT FROM AUTHOR]

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a fixed point problem for a finite family of relatively nonexpansive multivalued mappings, and a variational inequality problem. Our discussion encompasses the strong convergence of the proposed algorithm and highlights specific outcomes derived from our theorem. Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons. The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain. [ABSTRACT FROM AUTHOR]

In this article, we study a generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation. It is shown that the generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation is equivalent to a fixed point equation. We have proved that the generalized Yosida approximation operator is Lipschitz continuous. Finally, we prove an existence and convergence result for our problem.

This manuscript is devoted to constructing a novel iterative scheme and reckoning of fixed points for generalized contraction mappings in hyperbolic spaces. Also, we establish Δ\Delta and strong convergence results by the considered iteration under the class of mappings satisfying condition (E). Moreover, some qualitative results of the suggested iteration, like weak w2{w}^{2}-stability and data dependence results, are discussed. Furthermore, to test the efficiency and effectiveness of the proposed iteration, practical experiments are given. To support the theoretical results, illustrative examples are presented. Finally, our results improve and generalize several classical results in the literature of fixed point iterations.

In this paper, we study the variational inequality over the solution set of the split common solution problem with multiple output sets for monotone operator equations in real Hilbert spaces. We propose a new approach for solving this problem without depending on the norm of the transfer mappings. Some of its applications for solving the variational inequality over the solution set of the split common fixed point problem with multiple output sets and related problems are also mentioned. Finally, we give numerical examples to illustrate the performance of the proposed algorithm and compare it with several existing previous algorithms. [ABSTRACT FROM AUTHOR]

In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures. [ABSTRACT FROM AUTHOR]

Based on a degenerate proximal point analysis, we show that the Douglas-Rachford splitting can be reduced to a well-defined resolvent, but generally fails to be a proximal mapping. This extends the recent result of [Bauschke, Schaad and Wang. Math. Program. 2018;168:55–61] to more general setting. The related concepts and consequences are also discussed. In particular, the results regarding the maximal and cyclic monotonicity are instrumental for analysing many operator splitting algorithms. [ABSTRACT FROM AUTHOR]

The paper presents results on the solvability and parameter dependence for problems driven by weakly continuous potential operators with continuously differentiable and coercive potential. We provide a parametric version on the existence result to nonlinear equations involving coercive and weakly continuous operators. Applications address a variant of elastic beam equation. [ABSTRACT FROM AUTHOR]

We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with estimates for the non-asymptotic rate at which the residuals vanish. Strong and linear convergence are obtained in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and show that passing from one regime to the other is a continuous process in terms of the hypotheses on the parameters. Numerical illustrations are provided for an inertial primal-dual method and an inertial three-operator splitting algorithm, whose performance is superior to that of their non-inertial counterparts. [ABSTRACT FROM AUTHOR]

Enriched multi-valued contraction mappings and enriched multi-valued nonexpansive mappings are introduced in the context of geodesic spaces, and their fixed points are analyzed in CAT(0) spaces. For the contraction mapping, the existence of a fixed point is established, and a scheme for approximating the point is provided. Conditions under which a sequence converges to a fixed point of the enriched multi-valued nonexpansive mapping are given, and the mapping is shown to exhibit a demiclosedness-type property. Furthermore, it is shown that the fixed point set of the mapping is closed and convex. Conditions under which a sequence converges to a Mann-type scheme are given, and an application to an equilibrium problem involving Nash non-cooperative games is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the investigation of the problem of finding a common element of the set of fixed points of a total uniformly L-Lipschitzian mapping and the set of solutions of a system of generalized nonlinear variational-like inclusions. To approximate such a point lying in the above two sets, by employing some total uniformly L-Lipschitzian mappings and the notion of P- η -proximal-point mapping, a new iterative scheme is constructed. We apply the notions of graph convergence and P- η -proximal-point mapping and establish a new equivalence relationship between graph convergence and proximal-point mappings convergence of a sequence of P- η -accretive mappings. As an application of the obtained equivalence relationship, the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common point of the above-mentioned two sets are proved. The final section is devoted to the investigation and analysis of the results given in [Nonlinear Anal. 67(2007), 917–929]. Some comments relating to them are pointed out. These results are new, and improve and generalize many known corresponding results. [ABSTRACT FROM AUTHOR]

Usurelu et al. (Int J Comput Math 98:1049–1068, 2021) presented stability and data dependence results for a TTP (Thakur–Thakur–Postolache) iteration algorithm associated with quasi-strictly contractive mappings and contraction mappings, respectively, but these results were subject to strong conditions on the parametric control sequences used in the TTP iteration algorithm. This article aims to expand those results conducting a thorough analysis of the convergence of TTP and S iteration algorithms and improve those results by removing the restrictions on the parametric control sequences. Additionally, a data dependence result for the TTP iteration algorithm of quasi-strictly contractive mappings is established and several collage theorems are introduced to offer new insights into the data dependence of fixed points of quasi-strictly contractive mappings and to solve related inverse problems. In order to exhibit the dependability and effectiveness of all the results discussed in this work, a multitude of numerical examples are furnished, encompassing both linear and nonlinear differential equations (DEs) and partial differential equations (PDEs). This work can be viewed as an important refinement and complement to the study by Usurelu et al. (Int J Comput Math 98:1049–1068, 2021). [ABSTRACT FROM AUTHOR]

This paper aims to propose a double inertial Mann-type extragradient algorithm using non-monotonic step size and establish a weak convergence result to find a common solution to the variational inequality problem involving pseudo-monotone mapping and fixed point problem for quasi-nonexpansive mapping. Moreover, we establish a strong convergence for the variational inequality involving strongly pseudo-monotone mapping. To illustrate our algorithm’s potential, we present several numerical experiments, including the Nash–Cournot oligopolistic market equilibrium model and fractional programming problem. We also employ performance profiles to discuss the efficiency of our algorithm over earlier established algorithms based on two performance metrics. [ABSTRACT FROM AUTHOR]

In this paper we obtain extension properties for monotone sublinear operators. The results obtained generalize those known for positive linear operators. Several examples illustrating the results obtained are given. [ABSTRACT FROM AUTHOR]

This paper proposes a new inertial subgradient extragradient method for solving equilibrium problems with pseudomonotone and Lipschitz-type bifunctions and fixed point problems for nonexpansive mappings in real Hilbert spaces. Precisely, we prove that the sequence generated by proposed algorithm converges strongly to a common solution of equilibrium problems and fixed point problems. We use an effective self-adaptive step size rule to accelerate the convergence process of our proposed iterative algorithm. Moreover, some numerical results are given to show the effectiveness of the proposed algorithm. The results obtained in this paper extend and improve many recent ones in the literature. [ABSTRACT FROM AUTHOR]

This paper attempts to prove the Lipschitz continuity of the resolvent operator associated with a (P , η) -accretive mapping and compute an estimate of its Lipschitz constant. This is done under some new appropriate conditions that are imposed on the parameter and mappings involved in it; with the goal of approximating a common element of the solution set of a system of generalized variational-like inclusions and the fixed point set of a total asymptotically nonexpansive mapping in the framework of real Banach spaces. A new iterative algorithm based on the resolvent operator technique is proposed. Under suitable conditions, we prove the strong convergence of the sequence generated by our proposed iterative algorithm to a common element of the two sets mentioned above. The final section is dedicated to investigating and analyzing the notion of a generalized H(.,.)-accretive mapping introduced and studied by Kazmi et al. (Appl Math Comput 217:9679–9688, 2011). In this section, we provide some comments based on the relevant results presented in their work. [ABSTRACT FROM AUTHOR]

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation. [ABSTRACT FROM AUTHOR]

The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality (DEHVI) which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space. First, by applying surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient, we show the nonempty of the solution set for the parabolic hemivariational inequality. Then, some topological properties of the solution set are established such as boundedness, closedness and convexity. Furthermore, we explore the upper semicontinuity of the solution mapping. Finally, we prove the solution set of the system (DEHVI) is nonempty and the set of all trajectories of (DEHVI) is weakly compact in C(I, X). [ABSTRACT FROM AUTHOR]

This paper provides an iterative construction for a common solution associated with the pseudomonotone equilibrium problems, fixed point problem of a finite family η -demimetric operators and the generalized split null point problem in Hilbert spaces. The sequence of approximants is a variant of the parallel shrinking extragradient algorithm with the inertial effect converging strongly to the optimal common solution under suitable set of control conditions. The viability of the approximants is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new Bregman extragradient method with a different line-search process for solving variational inequality problems in reflexive Banach spaces. Precisely, we prove that the sequence generated by our proposed iterative algorithm converges strongly to an element of the solution sets of variational inequality problems. Moreover, some numerical examples are given to show the effectiveness of the proposed algorithm. The results obtained in this paper extend and improve many recent ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a viscous-type extrapolation algorithm for finding a solution of the variational inequality problem and a fixed point constraint of quasi-nonexpansive mappings under the scope of real Hilbert spaces when the underlying cost operator is quasi-monotone. The method involves inertial viscosity approximation and a constructed self-adjustable step size condition that depends solely on the information of the previous step. We establish a strong convergence result of the proposed method under certain mild conditions on the algorithm parameters. Finally, to demonstrate the gain of our method, some numerical examples are presented in comparison with some related methods in literature. [ABSTRACT FROM AUTHOR]

We study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. In order to solve this problem, we introduce two new algorithms which are based on the inertial proximal point algorithm. We first establish a weak convergence theorem and a convergence rate for the first algorithm. Next, we also establish the strong convergence of sequences generated by the second algorithm. An application of our main theorems to solving the split minimum point problem with multiple output sets and a pertinent numerical example are also presented in Sects. 6 and 7, respectively. [ABSTRACT FROM AUTHOR]

Our study in this paper is focused on the split equality fixed-point problem with firmly quasi-non-expansive operators in infinite-dimensional Hilbert spaces. A self-adaptive simultaneous scheme is introduced, and its weak convergence is established under mild and standard assumptions. The new proposed scheme generalizes and extends some related works in the literature, and its simple structure makes it easy for implementation and numerical testing. Primary experiments presented in this paper, in finite- and infinite-dimensional spaces, emphasize their practical advantages over existing results. [ABSTRACT FROM AUTHOR]

This paper presents a projection and contraction method with a double inertial extrapolation step and self-adaptive step sizes to solve variational inequalities with quasi-monotonicity in real Hilbert spaces. Weak and strong convergence results are obtained under some mild conditions. We also give linear convergence results under a special case of our proposed method. Preliminary numerical results show that our proposed method is competitive with other related methods in the literature. [ABSTRACT FROM AUTHOR]

Surface-subsurface filtration transport with seawater intrusion phenomena are formulated and variationally analyzed, as coupled multimedia mixed pairs of free boundary interface problems. Physically, multidomain subsurface mixed velocity-pressure fractional Darcian flow models coupled with surface evolution Stokesian mixed flows are considered. Specifically, two-phase air-fresh water above the sea level and fresh water-seawater characterizations are considered. Internal boundary synchronizing transmission conditions of multidomain nonoverlapping decompositions are modeled in terms of variational Lagrangian dual subpotential maximal monotone inclusions. Similarly, filtration transport coupling interface transmission constraints are implemented by mass flux-velocity-pressure Lagrange dual multipliers as solutions of subpotential subdifferential equations. [ABSTRACT FROM AUTHOR]

The main purpose of the present paper is the characterization, in the finite dimensional setting, of weak and strong invariance of closed sets with respect to a differential inclusion governed by time-dependent maximal monotone operators and multi-valued perturbation, by the use of the corresponding Hamiltonians. [ABSTRACT FROM AUTHOR]

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space. [ABSTRACT FROM AUTHOR]

We introduce two new self-adaptive algorithms for solving the split feasibility problem with multiple output sets in Hilbert space. Our algorithms do not use the least squares method, and their step sizes do not depend on the norms of the transfer mappings. Furthermore, two relaxed iterative algorithms are introduced when the feasibility sets are sublevel sets of convex functions. [ABSTRACT FROM AUTHOR]