Concerning the Intermediate Point in the Mean Value Theorem
نویسندگان
چکیده
If the function f : I → R is differentiable on the interval I ⊆ R , then for each x,a ∈ I, according to the mean value theorem, there exists a number c(x) belonging to the open interval determined by x and a , and there exists a real number θ (x) ∈]0,1[ such that f (x)− f (a) = (x−a) f (1) (c(x)) and f (x)− f (a) = (x−a) f (1) (a+(x−a)θ (x)) . In this paper we shall study the differentiability of the functions c and θ in a neighbourhood of a. Mathematics subject classification (2000): 26A24.
منابع مشابه
On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings
In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...
متن کاملSOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS
In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating $gH$-derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for ...
متن کاملHigher order multi-point fractional boundary value problems with integral boundary conditions
In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed...
متن کاملSolvability of an impulsive boundary value problem on the half-line via critical point theory
In this paper, an impulsive boundary value problem on the half-line is considered and existence of solutions is proved using Minimization Principal and Mountain Pass Theorem.
متن کاملEXISTENCE OF FIXED POINTS OF CERTAIN CLASSES OF NONLINEAR MAPPINGS
In this study, we introduce the classes of $phi$-strongly pseudocontractive mappings in the intermediate sense and generalized $Phi$-pseudocontractive mappings in the intermediate sense and prove the existence of fixed points for those maps. The results generalise the results of several authors in literature including Xiang [Chang He Xiang, Fixed point theorem for generalized $Phi$-pseudocontra...
متن کامل