This article gives a really vital and curiously inequality on Jain-Saraswat’s functional discrimination in terms of the Hellinger Bhattacharya by taking into thoughtL-Lipschitzian functions. Encourage, we outlined few results utilizing inferred with numerical confirmation.

We consider some typical optimal control problems for a nonlinear model of longitudinal vibrations in a viscoelastic rod. In trying to follow the usual pattern of showing that every innmizing sequence of controls contains a subsequence suitably converging to an optimal control, we confront the severe technical diiculty that the constitutive function cannot be uniformly Lipschitzian in its argum...

In this work we introduce for extended real valued functions, defined on a Banach space X, the concept of K directionally Lipschitzian behavior, where K is a bounded subset of X. For different types of sets K (e.g., zero, singleton, or compact), the K directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally Lipschitzian, directionally Lipschitzian, or co...

We show that the variational inequality V I(C,A) has a unique solution for a relaxed (γ, r)-cocoercive, μ-Lipschitzian mapping A : C → H with r > γμ, where C is a nonempty closed convex subset of a Hilbert space H . From this result, it can be derived that, for example, the recent algorithms given in the references of this paper, despite their becoming more complicated, are not general as they ...

Consider the variational inequality V I(C, F ) of finding a point x∗ ∈ C satisfying the property 〈Fx∗, x − x∗〉 ≥ 0 for all x ∈ C, where C is a nonempty closed convex subset of a real Hilbert space H and F : C → H is a nonlinear mapping. If F is boundedly Lipschitzian and strongly monotone, then we prove that V I(C, F ) has a unique solution and iterative algorithms can be devised to approximate...

In this paper, we consider the existence of multiple periodic solutions for the problem du , . , — +Lu = g(u) + h, r>0, dt "(0) = u(T), where L is a uniformly strongly elliptic operator with domain D(L) = Hjf(Q), g: R —► R is a continuous mapping, T > 0 and h: (0,T) —> HTM(Q.) is a measurable function.

and Applied Analysis 3 to a fixed point of the mapping T , which is also a solution of VI 1.5 defined on the set of fixed points of T . As direct consequences, we obtain the unique minimum-norm fixed point of T . Namely, we find the unique solution of the quadratic minimization problem: ‖x̃‖ min{‖x‖ : x ∈ Fix T }. 2. Preliminaries and Lemmas Throughout this paper, when {xn} is a sequence inH, xn...

is called the modulus of (uniform) continuity of f . The mapping f is said to be uniformly continuous if Ω f (d) → 0 as d ↓ 0. In this case the modulus of continuity is a subadditive monotone continuous function. The definition of Ω f implies that f (Br(x)) ⊂ BΩ f (r)( f (x)). (By Bρ(y) and Bρ(y) we denote, respectively, the open and the closed ball of radius ρ, centered at y.) One important cl...

Lipschitz self mappings of metric spaces appear in many branches of mathematics. In this paper we introduce a modification of the Lipschitz condition which takes into account not only the mapping itself but also the behaviour of a finite number of its iterates. We refer to such mappings as mean Lipschitzian. The study of this new class of mappings seems potentially interesting and leads to some...