A General Study on Differential and Integral Inequalities

The dominating branch of mathematics has been analysis and inequalities constitute a very significant component of the analysis. In the area of mathematics, the significance of inequalities has long been acknowledged. In the 18th and 19th centuries, renowned mathematicians laid the mathematical foundations of the theory of inequality. In the years that followed, the impact of inequalities was enormous, and many prominent mathematicians including H were drawn to the topic. However, in the 20th century, it was the pioneering work of G H. Hardy, J. E. Littlewoods, and G. Polya, which was published in 1934, that led to the subject's growth as a branch in contemporary mathematics. Then, after years of significant and productive study has been conducted on the subject of inequalities and their use in many areas of math. Despite the long history of this study, the topic of disparities is always interesting and remains a very dynamic difficult field. Indeed, it is reasonable to give the users rather than the creators of inequality the primary driving force for the fast growth of this area. It has a magnificent theory of inequality, a strong technical approach, and a very deep effect on science and technology. It is recognized to be extremely complicated and not straightforward in most of the dynamic systems in the cosmos. These dynamic systems are therefore controlled by non-linear differential and comprehensive equations. The study of these nicht linear differential equations is the core or major portion of the nonlinear analysis. The subject of non-linear differential and integrated equations is amongst the major mathematical problems in contemporary science and technology and the theory of such equations has called most of the world's leading mathematicians into question. Even the theory of inequalities does not escape the realm of differential and integral equations. In mathematical literature, the theory of difference and integrated inequalities has long been established and these inequalities are used again as an instrument for dealing with many qualitative issues in the solutions of the corresponding differing and integral equations. There are various methods to address nonlinear differential and integrated equations such as the approximation method and theoretical methodology for the operator, etc. A nonlinear equation is normally approached by a linear equation in approximation techniques, which may be explicitly solved using the known method, and the solution of the linear equation is used as an approximation to the nonlinear differential solution. However, it must be mentioned that it is not always possible to approximate all non-linear differential equations with linear differential equations and therefore this procedure has certain restrictions. The creation of an abstract space sub-set, mapping into that by a non-linear operator expressing the differential equation, is a very necessary method for another operator, such as fixed point theorems. This needs the skills of a person who is not always feasible in the area of nonlinear functional analysis. Therefore, the strong tools to analyze non-linear differential and integral equations for various features of the solutions are differential and integral equality. which may be real or complex. Then the triplet (X; +, •) is a vector or linear space, where + : X x X —> X and • : T x X -» X are two binary operations called addition and scaler multiplication in X such that, 1. (X, +) is an abelian or commutative group. 2. [(a • β) . γx] = [a • (β. γx)]. (Associativity) 3. {a + β)x = ax + β and a{x + y) = ax + ay. (Distributive laws) 4. 1 • x = x.

A mapping T: X - X is called linear if Is a linear operator called the linear integral operator of the functions of C(J, R)? The details of linear operators appear in Lusternik and Sobolev and Kregzig. An operator T: X —> X is nonlinear if it is not linear. The theory of nonlinear operators is well developed and there are plenty in numbers. The details are given in Granas and Dugundji and Heikkila and Lakshmikantham. Similarly, a function f (t, x) defined on J x R into R is called linear in x if the map x -f(t, x) satisfies two conditions of linearity in the variable x for each t G J. The function (£, x) = ax is linear in x. Again, a function f (t, x) defined on J x 1 into R is called nonlinear in x if it is not linear for each t e J. There are ample examples of nonlinear functions available in the literature. A simple example is (£, x) = t + x which is

Certain basic comparison results, that concerned with estimating a function that satisfies a differential inequality by the external solutions of the corresponding differential equation. Sometimes, it is enough to have the differential inequality satisfied relative to only Dini derivatives. We adopt the following notation for Dini derivatives:

Where, , U'+ denotes the right derivative (t). Such a link is also marked by u' (t when D u(t) = _u. (t).

PROOF. The condition is necessary. Let us prove that it is sufficient. Assume first that D+m(t) > 0 on (a, b). Which implies D+m(C) < 0. This is a contradiction and therefore the proof is complete. Lemma 1.2 Let v,w ε C([to,T],R) For certain fixed derivatives of Dini, PROOF. Define the function It then follows, from the assumption, that Thus, m(t) does not rise int [to, T] by Lemma. Consequently, And the lemma is proved. , then the common value is called the derivative of the function u and we denote it by du/dt, and the function u is called differentiable or derivable on J. The following result is sometimes useful while establishing the nonlinear Lipschitz condition of a differentiable function on the domain of its definition. Theorem 1.1 If the function f defined during a shutdown period is actually evaluated [a, b] (a) Continuous on [a, b], and (b) Derivable on the open interval (a, b),

The existence theorem together with the extension result implies the following. Theorem 1.2 Let g ε C (E, R), where E = J x R is an open (t, u)-set in R2 and (to, uo) ε E. Then, the IVP Have external solutions (that is, minimal and maximal solutions) which can be extended up to the boundary of E. The next lemma is useful in certain situations.

Let J = [a, b] be a closed and bounded interval in R, the set of real numbers and let x: J —> R be a bounded function. Let P = {x0,x1,….,xn } be a partition of J, where a = x0 R such that

And

Theorem 1.9 Assume that Hypothesis (H1) holds. Then every solution of the quadratic integral equation defined on J is bounded. PROOF. The given integral equation is

For every t μ J in which M(t) is an ongoing function in J. Because every continuous function reaches a maximum in M at a closed and bounded interval. Therefore, the number M* = sup M (t) t ε J exists, hence |x(t)| < M* for all t ε J. As a result, every solution of the given integral equation is bounded on J.

Fixed item theory is an essential subject in nonlinear functional analysis that offers some strong tools for nonlinear equations research. Banach (see Agarwal et.al.), Granas and Dugundji, and Krasnoselskii have started the systematic study of the non-linear equations using the fixed point theory. For some of the initial values issues of nonlinear ordinary differential and integrated differential equations, we utilized certain known and newly-established fixed point theorems for non-lingerie operators in this research. Definition 1: Let X e an abstract space and let T: X —> X be an operator. A variable £ ε X is called a fixed point of the operator T if T ε = ε and A fixed point affirming the presence of a fixed point.

(1) Geometric methods, (2) Topological methods, and (3) Algebraic methods.

Theorem 1.10 Let (E,d) be a complete metric space and T: E ↔ E is a contraction mapping, that is, for every x,y ε E,

Using the above-mentioned fixed point theorem not only creates a unique existence but also solutions for non-linear differential and integrated equations. The aforementioned technique should be used only if specific Lipschitz conditions are satisfied by the nonlinearity involved in the nonlinear equations. The Banach contraction concept gives the conventional iteration or following methods an abstract framework. Topological fixed point theory: The topological fixed o point theory for completely continuous operators is generally used in the study of solutions for differential and integral equations. We recall a few definitions from Banach spaces. Let T: X —> X be a operator. T is called compact if T(X) is a compact subset of X. T is termed completely limited when T(B) is a completely limited X-sub-set, where B is a limited X-set. Finally, when the operator is continuous and fully limited by x, T is termed continuous. Note that any compact operator on X is completely bound, while the opposite is not always true, although both concepts correspond with the bounded X sub-sets. The applicable fixed-point theorem is,

Schauder's fixed point theorem, which we will present next, is another fixed point finding with Theorem 1.12 Let E be a Banach space and B ⊂ E, a convex, closed bounded set. IfT:E—>E is a constant user so that T{B) to B and T is rather compact, T has a fixed point.

The main difference from a standard is that |x| = 0 doesn't always mean x = 0. Theorem 1.13 Let F be a Frechet space whose distance function is constructed utilizing a sufficient,

The above fixed-point theorems have several versions useful in applications. The topological methods are purely existential and only applicable if the nonlinearity involved in the nonlinear equations satisfy certain compactness type conditions.

Theorem 1.14Let a and b be two elements of X such that a < b and let [a, b] be an order interval in the ordered Banach space X. If Q: [a, b] —» [a, b] Be a fully continuous and non-diminished operator and the

In this study, we have discussed abstract analysis, derivatives and differential inequalities, integration and integral inequalities, nonlinear analysis. Which concludes in integral inequalities that the number M* = sup M (t) t ε J exists, hence |x(t)| < M* for all t ε J. As a result, every solution of the given integral equation is bounded on J. In non-linear analysis, the algebraic fixed-point method yields the theoretical successive approximations. results for the existence of solutions. It also gives qualitative information about the solutions for the equations.

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Research Scholar, Sunrise University, Alwar Rajasthan