In this paper, we look into a new concept of accretive mappings called αβ-H((.,.),(.,.))-mixed accretive mappings in Banach spaces. We extend the concept of proximal-point mappings connected with generalized m-accretive mappings to the αβ-H((.,.),(.,.))-mixed accretive mappings and discuss its characteristics like single-valuable and Lipschitz continuity. Some illustration are given in support of αβ-H((.,.),(.,.))-mixed accretive mappings. Since proximal point mapping is a powerful tool for solving variational inclusion. Therefore, As an application of introduced mapping, we construct an iterative algorithm to solve variational inclusions and show its convergence with acceptable assumptions.

We investigate a new class of cocoercive operators named generalized -cocoercive operators in Hilbert spaces. We prove that generalized -cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized -cocoercive operators. Some examples are given to justify the definition of generalized -cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized -cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

We investigate a class of new -mixed cocoercive operators in Hilbert spaces. We extend the concept of resolvent operators associated with -cocoercive operators to the -mixed cocoercive operators and prove that the resolvent operator of -mixed cocoercive operator is single valued and Lipschitz continuous. Some examples are given to justify the definition of -mixed cocoercive operators. Further, by using resolvent operator technique, we discuss the approximate solution and suggest an iterative algorithm for the generalized mixed variational inclusions involving -mixed cocoercive operators in Hilbert spaces. We also discuss the convergence criteria for the iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

The reconstruction technique and stability of various well-known cosmological parameters of f(T, T _G ) gravity are investigated in this paper, where T and T _G denote the torsion scalar and Gauss-Bonnet invariant torsion component. In this model, we consider Barrow holographic dark energy (BHDE) as candidates of IR cut-offs and develop the corresponding field equations. In our analysis, we have examined two well-known IR cutoffs Hubble and GrandaOliveros (GO) cutoffs. Our findings indicate a transition from a deceleration phase to an accelerated phase in the Universe for both cutoff scenarios, aligning with recent observational data. A reconstruction method is introduced to analyze the parameters linked to BHDE within the framework of viscous cosmology. Our investigation revealed that the power-law cosmology effectively aligns with the observational datasets. By combining the Markov Chain Monte Carlo (MCMC) technique with Bayesian analysis and likelihood function, we were able to determine the model parameters H ₀ = 69.6 ± 1.8 km/s/Mpc; q = − 0.281 ± 0.034 and M = 23.846 ± 0.010. We have solved the modified Einstein’s field equations by taking the bulk-viscosity component ξ = ξ ₀ + ξ ₁ H. Additionally, we have examined the energy conditions to assess the consistency of our model.

This work is concerned with the new notion called generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mapping that is the sum of two symmetric accretive mappings. It is an extension of the generalized $\alpha \beta$-$H(., .)$-accretive mapping. The proximal point mapping linked to the generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings is defined, and some of its characteristics are discussed. As an application of the new proximal point mapping, we consider a set-valued variational inclusion problem in q-uniformly smooth Banach spaces. Further, we propose an iterative scheme connected with ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-proximal point mapping to find the solution of a variational inclusion problem and discuss its convergence criteria under appropriate assumptions. Some examples are constructed in support of generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings.