Approximate solutions of impulsive integro-differential equations

Arabian Journal of Mathematics, Feb 2018

R. S. Jain, B. Surendranath Reddy, S. D. Kadam

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Approximate solutions of impulsive integro-differential equations

Approximate solutions of impulsive integro-differential equations R. S. Jain 0 B. Surendranath Reddy 0 S. D. Kadam 0 Mathematical Subject Classification 0 B. Surendranath Reddy E-mail: 0 R. S. Jain ( In this paper, we consider impulsive integro-differential equations in Banach space and we establish the bound on the difference between two approximate solutions. We also discuss nearness and convergence of solutions of the problem under consideration. The impulsive integral inequality of Grownwall type is used to obtain results. 1 Introduction In the recent decade, the study of impulsive integro-differential equations has become an important thrust area for many researchers across the world. Many real-life phenomena and processes are subject to short-term perturbations whose duration is negligible as compared to the whole phenomena. These problems mostly arise in medicine, economics, biological sciences, engineering, etc. Such type of problems can be modelled with impulsive integro-differential equations. Thus, many researchers [2,4,5,8–10,13] have opted for this research area and contributed to the development of the theory of impulsive differential equations. More information related to this can be found in monographs of Bainov and Simeonov [2] and Lkashmikantham et al. [8]. As it is difficult to provide explicit solutions to most of the physical problems, the method of approximate solution is the best analytic tool for such situation which provides the required information of solutions without finding the explicit solution. Tidke and Dhakne [14,15], Pachpatte [11], Pachpatte [12], Kucche et al. [7] used this technique to study the qualitative properties of solutions of different initial value problems. Since there is less information available in the literature about the approximate solutions of impulsive integro-differential equations, we apply this technique for the following impulsive integro-differential equation of the type: 0 x (t ) = Ax (t ) + f (t, x (t ), k(t, s)h(s, x (s))ds), t ∈ (0, T ], t = τk , k = 1, 2, . . . , m Δx (τk ) = Ik x (τk ), k = 1, 2, . . . , m, where A is the infinitesimal generator of strongly continuous semigroup of bounded linear operators {T (t )}t≥0 and Ik (k = 1, 2, . . . , m) are the linear operators acting in a Banach space X . Let k be a real-valued continuous function on [0, T ] × [0, T ] and the functions and f and h are given functions satisfying some assumptions. The impulsive moments τk are such that 0 ≤ τ0 < τ1 < τ2 < · · · < τm < τm+1 ≤ T , m ∈ N, Δx (τk ) = x (τk + 0) − x (τk − 0), where x (τk + 0) and x (τk − 0) are, respectively, the right and the left limits of x at τk . In [6], Kendre and Dhakne studied the existence, uniqueness, continuation and continuous dependence of solutions of IVP: x (t ) + Ax (t ) = f (t, x (t ), k(t, s)x (s)ds), t > t0, 0 t 0 s x PC([0,T ],X) = sup{ x (t ) : t ∈ [0, T ] \ {τ1, τ2, . . . , τm }}. Let X be a Banach space with the norm · . Let PC ([0, T ], X ) = {x : [0, T ] → X |x (t ) be piecewise continuous at t = τk , left continuous at t = τk , that is, x (τk−) = hl→im0+ x (τk − h) = x (τk ) and the right limit x (τk + 0) exists for k = 1, 2, . . . , m}. Clearly, PC ([0, T ], X ) is a Banach space with the supremum norm Definition 2.1 A function x ∈ PC ([0, T ], X ) satisfying the equations: x (t ) = T (t )x0 + T (t − s) f (s, x (s), k(s, τ )h(τ, x (τ )dτ )ds + T (t − τk )Ik x (τk ), t ∈ (0, T ], x (0) = x0 is said to be the mild solution of the initial value problem ( 1 )–( 3 ). Definition 2.2 Let xi ∈ PC ([0, T ], X ) (i = 1,2) be the function such that xi (t ) exists for each t ∈ [0, T ] and satisfies the inequality: using theory of analytic semigroups and fractional power of operators. The problem of existence, uniqueness and other basic properties of IVP ( 1 )–( 3 ) and their special forms have been studied by several authors using different methods such as Banach fixed point theorem, semigroup approach, progressive contractions, etc. See [3,10,13]. Our aim is to find the bound on the difference between two approximate solutions, nearness, convergence and continuous dependence of solutions on parameters of mild solutions of IVP ( 1 )–( 3 ). The paper is organised as follows: Sect. 2 consists of preliminaries and hypotheses. In Sect. 3, we establish the bound on the difference between two approximate solutions, nearness and convergence properties of solutions and, finally, we give continuous dependence of solutions on parameters and functions involved therein. 2 Preliminaries and hypotheses for a given constant εi ≥ 0, where it is considered that the initial and impulsive conditions, are satisfied. Then, xi (t ) are called i -approximate solutions to the IVP ( 1 )–( 3 ). xi (t ) − Axi (t ) − f (t, xi (t ), k(t, s)h(s, xi (s)ds) ≤ εi , xi (0) = x 0i, Δxi (τk ) = Ik xi (τk ) 0 0 ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) Lemma 2.3 [1] Assume the following inequality holds for t ≥ t0: u(t) ≤ a(t) + b(t, s)u(s)ds + t0 t t ⎛ s ⎝ t0 t0 ⎞ k(t, s, τ )u(τ )dτ ⎠ ds + t0<τk<t βk (t)u(tk ) where, u, a ∈ PC([t0, ∞), R+), a is nondecreasing, b(t, s) and k(t, s, τ ) are continuous and non negative functions for t, s, τ ≥ t0 and are nondecreasing with respect to t, βk (t)(k ∈ N) are nondecreasing for t ≥ t0. Then, for t ≥ t0, the following inequality holds: u(t) ≤ a(t) (1 + βk (t))ex p ⎝ t0<τk<t ⎛ t t0 ⎞ b(t, s)ds⎠ + t s Now, we introduce the following hypotheses: (H1) Let f : [0, T ] × X × X → X and h : [0, T ] × X → X be continuous functions such that there exist continuous nondecreasing functions p : [0, T ] → R+ = [0, ∞) and q : [0, T ] → R+ and f (t, ψ, x) − f (t, φ, y) ≤ p(t)( ψ − φ + x − y ), h(t, ψ) − h(t, φ) ≤ q(t)( ψ − φ ), for every t ∈ [0, T ], ψ ∈ X and x ∈ X . (H2) Let Ik : X → X be functions such that there exist positive constants Lk satisfying Ik (x) − Ik (y) ≤ Lk x − y , x, y ∈ X, k = 1, 2, . . . , m. In this paper, we consider that there exists a constant K0 > 0 such that T (t) ≤ K0. Also since k : [0, T ] × [0, T ] → R is a continuous function on compact set [0, T ] × [0, T ], there exists a constant L > 0 such that |k(t, s)| ≤ L, for 0 ≤ s ≤ t ≤ T . Let R(t) = max{ p(t), Lq(t), h(t)} and R∗ = sup{R(t) : t ∈ [0, T ]}. 3 Main results Theorem 3.1 Suppose that the hypotheses (H1) and (H2) hold. If x1(t) and x2(t) are εi approximate solutions of Eq. ( 1 ) with Conditions ( 5 ) and ( 6 ) such that (x01 − x0 ) ≤ δ, where δ is a nonnegative constant, then the 2 following inequality holds: x1(t) − x2(t) ≤ [(ε1 + ε2)t + K0δ] 0<τk<t (1 + K0 Lk ) exp K0 R∗T + K0(R∗)2 T 2 . 2 Proof Let xi (i = 1, 2) be approximate solutions of Eq. ( 1 ) with Conditions ( 5 ) and ( 6 ). Then we get xi (t) − Axi (t) − f (t, xi (t), k(t, s)h(s, xi (s)ds) ≤ εi . ( 7 ) 0 t 0 ξ 0 s Taking t = ξ in ( 7 ) and integrating with respect to ξ from 0 to t, we obtain 0 t 0 t − 0<τk<t 0 t T (t − τk )Ik xi (τk ) . εi dξ ≥ xi (ξ ) − Axi (ξ ) − f (ξ, xi (ξ ), k(ξ, s)h(s, xi (s))ds) dξ ≥ xi (t) − T (t)xi (0) − T (t − s) f (s, xi (s), k(s, τ )h(τ, xi (τ ))dτ )ds Using the inequalities u1 − v1 ≤ u1 + v1 and | u1 − v1 | ≤ u1 − v1 , we get (ε1 + ε2)t ≥ x1(t) − T (t)x01 − 0t T (t − s) f (s, x1(s), 0s + x2(t) − T (t)x02 − 0t T (t − s) f (s, x2(s), 0s ≥−[x01t(tT)(−t−x2s()t[)]f −(s,[xT1((ts))(,x010−skx(02s),]τ )h(τ, x1(τ )dτ ) − f (s, x2(s), 0s − T (t − τk )[Ik x1(τk ) − Ik x2(τk )] Using hypotheses (H1) and (H2), we get k(s, τ )h(τ, x1(τ ))dτ )ds − T (t − τk )Ik x1(τk ) k(s, τ )h(τ, x2(τ ))dτ )ds − T (t − τk )Ik x2(τk ) 0<τk<t 0<τk<t k(s, τ )h(τ, x2(τ ))dτ )] k(s, τ )h(τ, x2(τ ))dτ ) (ε1 + ε2)t ≥ x1(t) − x2(t) − T (t) (x01 − x02) − 0 s t + 0 |k(s, τ )|[ h(τ, x1(τ )) − h(τ, x2(τ )) dτ )] ds − t 0<τk<ts T (t − s) p(t) x1(s) − x2(s) T (t − τk ) Ik x1(τk ) − Ik x2(τk ) ≥ x1(t) − x2(t) − K0δ − 0 K0 p(t) x1(s) − x2(s) + 0 Lq(τ ) x1(τ ) − x2(τ ) dτ ds − K0 Lk x1(τk ) − x2(τk ) . 0<τk<t Let u(t) = x1(t) − x2(t) . Then, we have (≥ε1 u+(tε)2−)t K0δ − 0t K0 R(t)u(s)ds − 0t 0s K0 R(t)R(τ )u(τ )dτ ds − t Applying Lemma 2.3, we get u(t) ≤ [(ε1 + ε2)t + K0δ] 0<τk<t (1 + K0 Lk ) exp 0<τk<t 0t K0 R(t)ds + 0t 0s K0 R(t)R(τ )dτ ds 0t K0 R∗ds + 0t 0s K0(R∗)2dτ ds Remark The inequality obtained in ( 8 ) establishes the bound on the difference between the two approximate solutions of Eqs. ( 1 )–( 3 ). If x1(t ) is a solution of Eq. ( 1 ) with x (t ) = x01, then we have ε1 = 0 and, from ( 8 ), we see that x2(t ) → x1(t ) as ε2 → 0 and δ → 0. Moreover, if we put ε1 = ε2 = 0 and x0 = x02 in ( 8 ), then 1 the uniqueness of the solutions of ( 1 )–( 3 ) is established. Consider the impulsive IVP ( 1 )–( 3 ), along with the following initial value problem: 0 t y (t ) = Ay(t ) + f¯ t, y(t ), k(t, s)h(s, y(s))ds , t ∈ [0, T ], where k, h are as given in ( 1 )–( 3 ) , f¯ : [0, T ] × X × X → X, I¯k : X → X. Theorem 3.2 Suppose that the functions f, k, h in ( 1 )–( 3 ) satisfy the hypotheses (H1) and (H2) and there exist nonnegative constants 3, δk such that 2 . Proof Using the facts that x (t ) and y(t ) are, respectively, the solutions of the initial value problem ( 1 )–( 2 ) and ( 9 )–( 11 ) and hypotheses (H1) and (H2), we get ( 8 ) ( 9 ) ( 10 ) ( 11 ) ( 12 ) ( 13 ) ( 14 ) ( 15 ) Let x (t ) and y(t ) be, respectively, solutions of the initial value problem ( 1 )–( 3 ) and ( 9 )–( 11 ) on [0,T]. Then, the following inequality holds: x − y K0 p(t )Lq(τ ) x (τ ) − y(τ ) dτ ds Let u(t ) = x (t ) − y(t ) u(t ) = x (t ) − y(t ) Now, applying the inequality given in Lemma 2.3, we get K0 p(t )Lq(τ )u(τ )dτ ds + This completes the proof. x − y ≤ K0[ x0 − y0 + 3t + δk ] x − y 0<τk <t where k, h are as given in ( 1 ), and fn : [0, T ] × X × X → X is a sequence in X. As an immediate consequence of the above theorem, we have the following corollary: Corollary 3.3 Suppose that the functions f, k, h in ( 1 )–( 3 ) satisfy the hypotheses (H1) and (H2) and there exist nonnegative constants n, δn, δkn such that f (t, φ, x ) − fn(t, φ, x ) ≤ n, x0 − yn0 ≤ δn. Ik φ (τk ) − Iknφ (τk ) ≤ δkn, with n → 0, δn → 0, δkn → 0 as n → ∞. If x (t ) and yn(t ), n = 1, 2, . . . are, respectively, solutions of the initial value problems ( 1 )–( 3 ) and (17)–(19) on (0,T], then yn(t ) → x (t )as n → ∞ on (0, T]. Remark The result obtained in this corollary provides sufficient conditions to ensure that the solutions of the initial value problem (17)–(19) will converge to solutions of the initial value problem ( 1 )–( 3 ). Here, we will study the continuous dependence of the solutions of IVP ( 1 )–( 3 ) on parameters and functions involved in them. Consider the following IVP: x (t ) = Ax (t ) + f (t, x (t ), k(t, s)h(s, x (s))ds, δ2), t ∈ (0, T ], t = τk , k = 1, 2, . . . , m, (23) 2 , and y (t ) = Ay(t ) + f (t , y(t ), k(t , s)h(s, y(s))d s, δ3), t ∈ (0, T ], t = τk , k = 1, 2, . . . , m, where f : [0, T ] × X × X × R → X, δ2 and δ3 are real parameters. Corollary 3.4 Assume the hypotheses (H1) and (H2) hold. Let f : [0, T ] × X × X × R → X be a function satisfying f (t , ψ, x , δ) − f (t , φ , y, δ ) ≤ h(t )( ψ − φ + x − y + δ − δ ), ψ, φ , x , y ∈ X, δ, δ ∈ R. If x (t ) and y(t ) are solutions of Eqs. (23)–(25) and (26)–(28), then x − y B ≤ K0[ x0 − y0 Proof It is an easy consequence of our main result, so we have omitted the proof. K0 R∗T + K0( R∗)2 T 2 2 . Acknowledgements One of the authors Ms. S. D. Kadam would like to acknowledge DST-INSPIRE, New Delhi, for providing the INSPIRE fellowship. 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R. S. Jain, B. Surendranath Reddy, S. D. Kadam. Approximate solutions of impulsive integro-differential equations, Arabian Journal of Mathematics, 2018, 1-7, DOI: 10.1007/s40065-018-0200-1