Behzad Djafari Rouhani

Common Fixed Point of Multivalued Generalized -Weak Contractive Mappings

Fixed point and coincidence results are presented for multivalued generalized -weak contractive mappings on complete metric spaces, where is a lower semicontinuous function with and for all . Our results extend previous results by Zhang and Song (2009), as well as by Rhoades (2001), Nadler (1969), and Daffer and Kaneko (1995).

Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space

Abstract The results in [9, 11, 12] for nonexpansive sequences are generalized to almost nonexpansive sequences in a Hilbert space and using this notion a direct proof of a result of H. Brezis and F. E. Browder [6] is given.

Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces

Abstract By suitably modifying the methods used by B. Djafari Rouhani (“Ergodic Theorems for Non-expansive Sequences in Hilbert Spaces and Related Problems,” Part I, pp. 1–76, Thesis, Yale University, and “A New Proof of the weak Convergence Theorems for Non-expansive Sequences and Curves in Hilbert Spaces,” preprint, 1984 ) and by B. Djafari Rouhani and S. Kakutani (“Ergodic Theorems for Non-expansive Non-linear operators in a Hilbert Space,” preprint, 1984 ) we study the asymptotic behaviour of the quasi-autonomous dissipative system du dt + Au ∋ ƒ , where A is a monotone operator in a Hilbert space H and ƒ ϵ L 1 ((0, +∞); H) .

Asymptotic Behavior of Solutions to Some Homogeneous Second-Order Evolution Equations of Monotone Type

We study the asymptotic behavior of solutions to the second-order evolution equation a.e.,, where is a maximal monotone operator in a real Hilbert space with nonempty, and and are real-valued functions with appropriate conditions that guarantee the existence of a solution. We prove a weak ergodic theorem when is the subdifferential of a convex, proper, and lower semicontinuous function. We also establish some weak and strong convergence theorems for solutions to the above equation, under additional assumptions on the operator or the function.

Asymptotic behavior for almost-orbits of a reversible semigroup of non-Lipschitzian mappings in a metric space

Let (M, ρ) be a metric space and τ a Hausdorff topology on M such that {M,τ } is compact. Let S be a right reversible semitopological semigroup and �={ T( s): s ∈ S} a representation of S as ρ-asymptotically nonexpansive type self-mappings of M and u a ρ-bounded almost-orbit of � . We study the τ -convergence of the net {u(s): s ∈ S} in M when the triplet {M,ρ,τ } satisfies various types of τ -Opial conditions. Our results extend and unify many previously known results.  2002 Elsevier Science (USA). All rights reserved.

Existence and asymptotic behaviour of solutions to first- and second-order difference equations with periodic forcing

By using previous results of Djafari Rouhani for non-expansive sequences in Refs (Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale University, Part I (1981), pp. 1–76; Djafari Rouhani, J. Math. Anal. Appl. 147 (1990), pp. 465–476; Djafari Rouhani, J. Math. Anal. Appl. 151 (1990), pp. 226–235), we study the existence and asymptotic behaviour of solutions to first-order as well as second-order difference equations of monotone type with periodic forcing. In the first-order case, our result extends to general maximal monotone operators, the discrete analogue of a result of Baillon and Haraux (Rat. Mech. Anal. 67 (1977), 101–109) proved for subdifferential operators. In the second-order case, our results extend among other things, previous results of Apreutesei (J. Math. Anal. Appl. 288 (2003), 833–851) to the non-homogeneous case, and show the asymptotic convergence of every bounded solution to a periodic solution.

Asymptotics of a General Second-Order Difference Equation and Approximation of Zeroes of Monotone Operators

ABSTRACT In this article, we prove several new ergodic, weak, and strong convergence theorems for solutions to the following general second-order difference equation where A is a maximal monotone operator in a real Hilbert space H and {cn} and {θn} are positive real sequences. We do not assume A−1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set of A being nonempty. These theorems provide new approximation results for zeroes of monotone operators, as well as significantly unify and extend previously known results by assuming much weaker conditions on the coefficients {cn} and {θn}.

Asymptotics of a Difference Equation and Zeroes of Monotone Operators

We prove several new weak and strong ergodic theorems, as well as weak and strong convergence theorems for solutions to the following second order difference equation: where A is a maximal monotone operator in a real Hilbert space H, and {c n } is a positive sequence of real numbers. We do not assume that A −1(0) ≠ ∅, and we prove among other things that the existence of solutions is in fact equivalent to the zero set of A being nonempty. These theorems provide new approximation results for zeros of monotone operators, as well as unify and extend previously known results in [2, 3, 7, 10, 18, 22, 23, 25] by considering much weaker conditions on the coefficients {c n }. In particular, our new strong ergodic theorem extends the results of [7] and [23, Theorem 3.3] for first-order difference equations, to the case of second order difference equations, and implies also a new strong convergence theorem.

Generalized Solutions of Multi-valued Monotone Quasi-variational Inequalities

An ill-posed quasi-variational inequality with multi-valued maps can be conveniently formulated as a parameter identification problem on the graph of a variational selection. Using elliptic regularization for parametric variational inequalities, it is possible to pose another parameter identification problem that gives a stable approximation procedure for the ill-posed problem. The results are quite general and are applicable to ill-posed variational inequalities, inverse problems, split-feasibility problem, among others.

Strong Convergence of Two Proximal Point Algorithms with Possible Unbounded Error Sequences

We consider a proximal point algorithm with errors for a maximal monotone operator in a real Hilbert space, previously studied by Boikanyo and Morosanu, where they assumed that the zero set of the operator is nonempty and the error sequence is bounded. In this paper, by using our own approach, we significantly improve the previous results by giving a necessary and sufficient condition for the zero set of the operator to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the zero set of the operator, without assuming the boundedness of the error sequence. We study also in a similar way the strong convergence of a new proximal point algorithm and present some applications of our results to optimization and variational inequalities.