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We study and investigate a convex minimization problem of the sum of two convex functions in the setting of a Hilbert space. By assuming the Lipschitz continuity of the gradient of the function, many optimization methods have been invented, where the stepsizes of those algorithms depend on the Lipschitz constant. However, finding such a Lipschitz constant is not an easy task in general practice. In this work, by using a new modification of the linesearches of Cruz and Nghia [ 7 ] and Kankam et al. [ 14 ] and an inertial technique, we introduce an accelerated algorithm without any Lipschitz continuity assumption on the gradient. Subsequently, a weak convergence result of the proposed method is established. As applications, we apply and analyze our method for solving an image restoration problem and a regression problem. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

In this work, we study the split null point problem and the fixed point problem in Hilbert spaces. We introduce a self-adaptive algorithm based on the viscosity approximation method without prior knowledge of the operator norm for finding a common solution of the considered problem for maximal monotone mappings and demicontractive multivalued mappings. A strong convergence result of our proposed algorithm is established under some suitable conditions. Some convergence results for the split feasibility problem and the split minimization problem are consequences of our main result. Finally, we also give numerical examples for supporting our main result.

Over the past decades, the split inverse problem has been widely studied, and one of the objectives of those researches is to invent some efficient algorithms for approximating solutions. Most of those algorithms depend on the norms of bounded linear operators; however, the calculation of the operator norms is not an easy task in general practice. In this article, we study and investigate the split fixed point problem for multi-valued mappings in Hilbert spaces. We introduce a self-adaptive algorithm without prior knowledge of the operator norm for two demicontractive multi-valued mappings, and establish a strong convergence theorem of the proposed method under some suitable conditions. Our main result in this paper generalizes and improves many results in the literature.

In this paper, we consider and investigate a convex minimization problem of the sum of two convex functions in a Hilbert space. The forward-backward splitting algorithm is one of the popular optimization methods for approximating a minimizer of the function; however, the stepsize of this algorithm depends on the Lipschitz constant of the gradient of the function, which is not an easy work to find in general practice. By using a new modification of the linesearches of Cruz and Nghia [Optim. Methods Softw. 31:1209–1238, 2016] and Kankam et al. [Math. Methods Appl. Sci. 42:1352–1362, 2019] and an inertial technique, we introduce an accelerated viscosity-type algorithm without any Lipschitz continuity assumption on the gradient. A strong convergence result of the proposed algorithm is established under some control conditions. As applications, we apply our algorithm to solving image and signal recovery problems. Numerical experiments show that our method has a higher efficiency than the well-known methods in the literature.

In this article, we consider the split common fixed point problem for two infinite families of multivalued mappings in real Hilbert spaces. We introduce an algorithm based on the viscosity method for solving the split common fixed point problem for two infinite families of multivalued demicontractive mappings. We establish a strong convergence result under some suitable conditions. As applications, we also apply our main result to the split variational inequality problem and the split common null point problem. Finally, we give the numerical example for supporting our main theorem.

In this article, we consider a split common fixed point and null point problem which includes the split common fixed point problem, the split common null problem and other problems related to the fixed point problem and the null point problem. We introduce an algorithm for studying the split common fixed point and null problem for demicontractive operators and maximal monotone operators in real Hilbert spaces. We establish a strong convergence result under some suitable conditions and reduce our main result to above-mentioned problems. Moreover, we also apply our main results to the split equilibrium problem. Finally, we give numerical results to demonstrate the convergence of our algorithms.

The purpose of this research is to introduce a regularized algorithm based on the viscosity method for solving the proximal split feasibility problem and the fixed point problem in Hilbert spaces. A strong convergence result of our proposed algorithm for finding a common solution of the proximal split feasibility problem and the fixed point problem for nonexpansive mappings is established. We also apply our main result to the split feasibility problem, and the fixed point problem of nonexpansive semigroups, respectively. Finally, we give numerical examples for supporting our main result.

In this paper, we consider the split null point problem and the fixed point problem for multivalued mappings in Hilbert spaces. We introduce a Halpern-type algorithm for solving the problem for maximal monotone operators and demicontractive multivalued mappings, and establish a strong convergence result under some suitable conditions. Also, we apply our problem of main result to other split problems, that is, the split feasibility problem, the split equilibrium problem, and the split minimization problem. Finally, a numerical result for supporting our main result is also supplied.