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In this paper, the optimal control problem is governed by weak coupled parabolic PDEs and involves pointwise state and control constraints. We use measure theory method for solving this problem. In order to use the weak solution of problem, first problem has been transformed into measure form. This problem is reduced to a linear programming problem. Then we obtain an optimal measure which is approximated by a finite combination of atomic measures. We find piecewise-constant optimal control functions which are an approximate control for the original optimal control problem.

In this paper, we propose to define the concept of family of local atoms and then we generalize this concept to the atomic system for operator in Banach spaces by using semi-inner product. We also give a characterization of atomic systems leading to obtain new frames. In addition, a reconstruction formula is obtain. Next, some new results are established. The characterization of atomic systems allows us to state some results for sampling theory in semi-inner product reproducing kernel Banach spaces. Finally, having used frame operator in Banach spaces, new perturbation results are established

A generalization of the quotient integral formula is presented and some of its properties are investigated. Also the relations between two function spaces related to the spacial homogeneous spaces are derived by using general quotient integral formula. Finally our results are supported by some examples.

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We introduce the notion of regularized quasi-semigroup of bounded linear operators on Banach spaces and its infinitesimal generator, as a generalization of regularized semigroups of operators.After some examples of such quasi-semigroups, the properties of this family of operators will be studied. Also some applications of regularized quasi-semigroups in the abstract evolution equations will be considered. Next some elementary perturbation results on regularized quasisemigroups will be discussed.4 R s, t ≤ M s t , for some continuous increasing mapping M : 0, ∞ → 0, ∞ .Also regularized semigroups and their connection with abstract Cauchy problems are introduced in 7 and have been studied in 8-12 and many other papers. Abstract and Applied AnalysisWe mention that if C ∈ B X is an injective operator, then a one-parameter family {T t } ≥0 ⊆ B X is called a C-semigroup if for any s, t ≥ 0 it satisfies T s t C T s T t and T 0 C.In this paper we are going to introduce regularized quasi-semigroups of operators.In Section 2, some useful examples are discussed and elementary properties of regularized quasi-semigroups are studied.In Section 3 regularized quasi-semigroups are applied to find solutions of the abstract evolution equations. Also perturbations of the generator of regularized quasi-semigroups are also considered in this section. Our results are mainly based on the work of Barcenas and Leiva 1 . Regularized Quasi-SemigroupsSuppose X is a Banach space and {K s, t } s,t≥0 is a two-parameter family of operators in B X . This family is called commutative if for any r, s, t, u ≥ 0,3 {K s, t } s,t≥0 is strongly continuous, that is, lim s,t → s 0 ,t 0 K s, t x − K s 0 , t 0 x 0, x ∈ X; 2.2 4 there exists a continuous and increasing function M : 0, ∞ → 0, ∞ , such that for any s, t > 0, K s, t ≤ M s t .For a C-quasi-semigroups {K s, t } s,t≥0 on Banach space X, let D be the set of all x ∈ X for which the following limits exist in the range of C: 2.3Now for x ∈ D and s ≥ 0, defineCx t . 2.4 {A s } s≥0 is called the infinitesimal generator of the regularized quasi-semigroup {K s, t } s,t≥0 . Somewhere we briefly apply generator instead of infinitesimal generator. Abstract and Applied Analysis 3 Here are some useful examples of regularized quasi-semigroups. Example 2.2. Let {T t } t≥0 be an exponentially bounded strongly continuous C-semigroup on Banach space X, with the generator A. Then K s, t : T t , s,t ≥ 0, 2.5 defines a C-quasi-semigroup with the generator A s A, s ≥ 0, and so D D A . Example 2.3. Let X BUC R , the space of all bounded uniformly continuous functions on R with the supremum-norm. Define C, K s, t ∈ B X , bySubmit your manuscripts at

Let H{(i = 1,2,..., n), be closed operators in a Banach space X. The generalised initial value problemof the abstract Cauchy problem is studied. We show that the uniqueness of solution u :• X of this n-abstract Cauchy problem is closely related to Co-n-parameter semigroups of bounded linear operators on X. Also as another application of Cb-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

In this note we introduce the notions of generalized probabilistic metric spaces and generalized Menger probabilistic metric spaces. After making our elementary observations and proving some basic properties of these spaces, we are going to prove some fixed point result in these spaces.