A classification Theorem for totally umbilical CR-submanifolds of a Kaehler manifold is proved. Introduction.Cft-submanifolds of a Kaehler manifold [1] being generalization of holomorphic and totally real submanifolds of a Kaehler manifold, has recently become subject of sufficient interest. Totally umbilical Ci?-submanifolds of a Kaehler manifold have been studied by A. Bejancu [3], Blair and Chen [4]. The purpose of this paper is to classify all totally umbilical C7?-submanifolds of a Kaehler manifold. In fact we prove the following theorem.THEOREM. Let M, (dimM^δ) be a complete simply connected totally umbilical CR-submanifold of a Raheler manifold M. Then M is one of the following * (i) Locally the Riemanman product of a holomorphic submamfold and a totally real submanifold of M (ii) totally real submanifold (iii) isometric to an ordinary sphere (iv) homothetic to a Sasakian manifold. The cases (iii) and (iv) occur when dim M is odd. Preliminaries.Let M be an m-dimensional Kaehler manifold with almost complex structure /. Then the curvature tensor R of M satisfies [11].

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.

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l(evwor&: Lettuce, powdery mildew, Erysiphe cichoruceurum, susceptibility Indroductipn Lettuce (Lactuca sutivu Linn.) is cultivated throughout India. Studies on host pathogen relationships of this crop have been carried out in U.S.A. and Europe (1,3,4). In Europe powdery mildew occurs sporadically on lettuce cultivars (1,3). Literature on the response of lettuce cultivars to powdery mildew (Erysiphe cichoruceurum) is limited (2). and Me-During 1992-94, twenty two commercial cultivars as follows; Avondefiance, Black seeded, Black seeded simpson, Californian wonder, Chinese Wong Bok, Elephants trunk, Green and Paris White, Honeydew, Hungarian wax, Himalyan grandeur, Iceberg, Italian sprouting, Lake Nyah, Maruraj, Pride of Fuji, Paris white cos, Suttons A-1, Suttons little gem, Suttons improved unrivalled, Suttons golden ball, Suttons white gem and Wayahead obtained from Sutton & Sons (India) Pvt. Ltd., 13-D Russel St, Calcutta, 700 071 and Cooper's Seed Co. 77/78 A, SahaneySujan Park, Pune, 41 1 040 were tested. The seeds were sown on 2 September 1992. Threeweek old seedlings of each cultivar were transplanted singly in 15 cm earthern pots containing a sterilised soiVcompost mixture (3:l). Trials were conducted in a glasshouse from the third week of September until February or March. The infected area of thirty leaves selected randomly from 15 replicate plants were assessed. Severity of infection was recorded using a 0-4 scale for assessments as follows: 0 = plants without powdery mildew infection; 1 = maximum leaf infection of 25%; 2 = 25-50% leaf infection; 3 = 50-75% leaf infection; 4 = 75-100% leaf infection. Powdery mildew infection occurred naturally in the glasshouse. &su! . &The results presented in Table 1, were recorded in 1992-93 when the powdery mildew infection was severe. Cultivars Suttons A-1 and Maruraj were the only cultivars which remained free of powdery mildew infection throughout the trial. Cultivars Black seeded, Black seeded simpson, Elephants trunk, Honeydew, Lake Nayh, Pride of Fuji and Suttons little gem showed sporadic occurrence of powdery mildew. Whereas cultivars Chinese Wong Bok, Green and paris white, Italian sprouting, Himalayan grandeur and Suttons white gem were severely infected by powdery mildew infection. In these cultivars nearly the whole plant was infected with powdery mildew.Observations in this trial support the results of earlier workers (1, 5), who have shown that there are resistant types among lettuce cultivars. Present studies therefore have shown preliminary results, further investigations should be carried out to determine lettuce resistance to E. cichoruceurum after artificial or natural infection. The cultivars showing some resistance could probably be utilised effectively in lettuce cultivation programmes where this pathogen causes heavy losses.-76 -

The main purpose of this paper is to find a common element in the solution set of equilibrium problem and fixed point problem of non-expansive mappings in the real Hilbert space with the help of normal S-iteration process. Also, under some acceptable assumptions, we prove the sequences induced by above stated process converge weakly to a point in the solution set of above stated problems. At the end, we give a numerical example to justify our work. The results studied in this work philosophize and boost some contemporary and known results in this direction.

In this paper, a perturbed iterative method for solving a generalized nonlinear quasi-variational inclusions, is presented and a convergence result which generalizes some known results in this field, is glven.In 1994, Hassouni and Moudafi [4], have introduced a perturbed method for solving a new class of variational inclusions and presented a convergence result. In 1996, Samir Adly [2], has studied a perturbed iterative method in order to approximate a solution for a general class of variational inclusions and proved the convergence of the iterative algorithm by using some fixed point theorems.The aim of this paper is, firstly to present a new iterative algorithm for solving a generalized nonlinear quasi-variational inclusions. Then we prove the convergence of this algorithm, by using the definition of multi-valued relaxed Lipschitz operators. Our result is more general than the one considered in [3,4,5,6,8,9,10] which motivated us for the present work.

In this paper, we introduce and study a class of generalized nonlinear vector quasi-variational- like inequalities with set-valued mappings in Hausdorff topological vector spaces which includes generalized nonlinear mixed variational-like inequalities, generalized vector quasi-variational-like inequalities, generalized mixed quasi-variational-like inequalities and so on. By means of fixed point theorem, we obtain existence theorem of solutions to the class of generalized nonlinear vector quasi-variational-like inequalities in the setting of locally convex topological vector spaces.

We introduce and study a new system of generalized variational inclusions involving -cocoercive and relaxed -cocoercive operators, which contain the systems of variational inclusions and the systems of variational inequalities, variational inclusions, and variational inequalities as special cases. By using the resolvent technique for the -cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces. An example is given to justify the main result. Our results can be viewed as a generalization of some known results in the literature.