A Study on the Reduced Differentials Transforms Approach for Nonlinear Partial Integro-Differential Equations

INTRODUCTION

Given a bounded interval in R for some

The significance of research into HIEs comes in the fact that many classes of differential and integrative equations are included in these instances as special cases. In the Krasnosel ski work, the subject of hybrid differential equations is included and is thoroughly examined on hybrid differential equations with various disturbances in the many publications. For the references within, see Burton, Dbage. This class of hybrid integrate differential equations involves the disturbances in various methods of the original integral differential equations. Dhage offers a precise categorization of various kinds of disturbances of integral differential equations, which may be handled using hybrid Dhage, Dhage, and Lai—Shmikantham fixed point theory. In this section, we begin with the basic theory of the second nonlinear hybrid, whole-difference mixed disruptive equations, and show some basic findings, such as integrated inequality, theorem of existence, maximum and minimum solutions. We note that the results of this chapter add to the theory of nonlinear normal integral and differential equations in principle and substantially.

The following hypothesis is often necessary. (Ao) The function is increasing in R for all We are first of all demonstrating the fundamental findings of hybrid, integral inequality. Theorem 1.1 Assume that the hypothesis (Ao) holds. Suppose that they exist such that And If one of the inequalities is strict and then or all PROOF. Suppose the unfairness is wrong. Then the set P is set to is non-empty. We can assume that without losing the generality for all t < t1. Assume that for Denote for As hypothesis (Ao) holds, it follows from that for all to < t < t1. The above relation further yields Little h < 0. We get the limit as h -» 0. That's why we receive The evidence is full. This is a contradiction.

In this section, we prove an existing result for the HIDE on a closed and bounded interval J = [to, to+a] under mixed Lip- schitz and compactness conditions on the nonlinearities involved in it. We place the HIDE in the space <7(J, M) of continuous real-valued functions defined on J and use a hybrid fixed point of Dhage. Define a supremum norm || • || in C(J,R) defined by is a space in Banach with the supremum standard above. The solution to the HDE is shown in the Banachspace by Dhage through the hybrid fixed point theorem. We provide certain preliminaries and definitions before specifying the fixed point theorem, which will then be utilized. Definition 1: A mapping if: is called a dominating function or, in short, D-function, if it is upper semi-continuous and non-decreasing function satisfying /A mapping /, is called D-Lipschitz if there is a D-function satisfying The details of different types of contractions appear in monographs of Dhage and Granas and Dugundji. There do exist D-functions and the commonly used D-functions are and etc. These D-functions have been widely used in the theory of nonlinear differential and integral equations for proving the existing results via fixed point methods. Definition 2: An operator Q on a Banach space E into itself is called compact if Q{E) is a relatively compact subset of E. Q is called bounded if, for any bounded subset S of E, Q(S) is a relatively compact subset of E. If Q is continuous and bounded, then it is called completely continuous on E. Theorem 1.2: Suppose that S is a closed, convex, and bounded subset of the Banach space E and let A: E —> E and B: S —E be two operators such that (a) A is nonlinear contraction, (b) B is compact and continuous, and (c) x — Ax + By for ally € S => x € S

(A1) There exists a constant L > 0 such that for all and Moreover, L < M. (A2) There exists a continuous function such that For all The following lemma is useful in the sequel. Lemma 1.1: Assume that hypothesis (AO) holds. Then for any continuous function h: J —> R, the function is a solution of the HIDE if and only if x satisfies the hybrid integral equation (HIE) PROOF. Let Assume first that x is a solution of the HIDE. By definition, x{t) — f(t,x(t)) is continuous on J, and so, differentiable there, where is integrable on J. Applying integration to from to t, we obtain the HIE on J. On the other hand, imagine that x meets HIE. Then we get the first equation in by straight differentiation. Again, t = to replace in returns Since the mapping In K for every t € J, mapping increases is injective in R, whence The lemma is therefore fully proven. Now we can demonstrate the following HIDE existence theorem at J. maximal and minimal solutions for the HIDE on J = [to, to + a]. We need the following definition in what follows. Definition 2: A solution r of the HIDE is said to be maximal if for any other solution x to the HIDE one has x(t) < r(t), for all t € J. Again, a solution p of the HIDE is said to be minimal if p{t) < x(t), for all where x is any solution of the HIDE existing on J. The case for the maximum solution is discussed mainly because the minimum solution is comparable and with suitable amendments may be reached with similar reasoning. Consider the following initial value issue of hiding, given an arbitrarily small real number e > 0, were, A HIDE existence theorem can be described as follows: Theorem 1.3 Assume that the (AO) by (A2) assumptions are in effect. The HIDE has a solution specified on J for any small integer e > 0. PROOF. Let/be such a declining series of good

for all and where is a solution of the HIDE, Defined on J. Since, by Theorems, is a reduction of positive real numbers, the limit exists. We show that the convergence is uniform on J. To finish, it is enough to prove that the sequence is equip continuous in C(J, R). Let be arbitrary. Then, where, and Since f is continuous with the compact set J x [—N, N], it is continuous uniformly there. Hence, Uniformly for all . Similarly, since the function p is continuous on compact set J. It is uniformly continuous and hence. Uniformly for all Therefore, from the above inequality (5.4.5), it follows that Uniformly for all Therefore, for The function r is hence a HIDE solution on J. Lastly, the inequality which follows for all Therefore the answer on J is maximum. This is the evidence.

An evaluation of the solution to HIDE-related integrated differential inequality is the main problem of integral inequalities. In this section, we demonstrate that maximum and minimum solutions are within the boundaries of integral inequality

Theorem 1.4 Assume that the hypotheses (Ao) through (A2) hold. Further, if there exists a function such that Then, for all Where r is J's highest HIDE solution. PROOF. Let e > 0 be small and random. The maximum solution is the HIDE by theorem Is consistent on J, and the r function is the highest possible HIDE solution on J. Therefore, we get

From the above inequality, it follows that Now we apply Theorem to the inequalities and conclude that for all This further because of limit implies that inequality holds on J. This completes the proof. Theorem 1.5 Assume that the hypotheses (Ao) through (A2) hold. Further, if there exists a function such that Then, for all where p is a minimal solution of the HIDE on J. Note that Theorem is useful to prove the boundedness and uniqueness of the solutions for the HIDE on J. A result in this direction is

The hybrid integrodifferential equations are a rich area for a variety of nonlinear ordinary as well as partial integrodifferential equations. Here, in this chapter, we have considered a very simple hybrid integro-differential equation involving two non linearies, however, a more complex hybrid integro-differential equation can also be studied on similar lines with appropriate modifications. Again, the results proved in this chapter are very fundamental in nature and therefore, all other problems for the hybrid integro-differential equation in question are still open. In a forthcoming paper we plan to prove some theoretical approximation results for the hybrid integro-differential equation.

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Research Scholar, Sunrise University, Alwar Rajasthan