Partial Hadamard Fractional Integral Equations

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1 Advances in Dynamical Syems and Applications ISSN , Volume, Number 2, pp (25) Partial Hadamard Fractional Integral Equations Saïd Abbas University of Saïda Laboratory of Mathematics P.O. Box 38, 2 Saïda, Algeria abbasmsaid@yahoo.fr Mouffak Benchohra University of Sidi Bel-Abbès Laboratory of Mathematics P.O. Box 89, Sidi Bel-Abbès 22, Algeria and King Abdulaziz University Department of Mathematics, Faculty of Science P.O. Box 823, Jeddah 2589, Saudi Arabia benchohra@univ-sba.dz Johnny Henderson Baylor University Department of Mathematics Waco, Texas USA Johnny Henderson@baylor.edu Abract This paper deals with the exience and uniqueness of solutions for a class of partial integral equations via Hadamard s fractional integral, by applying some fixed point theorems. AMS Subject Classifications: 34A8, 34K5. Keywords: Functional integral equation, Hadamard partial fractional integral, exience, solution, fixed point. Received July 6, 25; Accepted September 6, 25 Communicated by John Graef

2 98 Saïd Abbas, Mouffak Benchohra, and Johnny Henderson Introduction The fractional calculus represents a powerful tool in applied mathematico udy many problems from different fields of science and engineering, with many break-through results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, atiical mechanics, arophysics, cosmology and bioengineering [, 7]. There has been a significant development in ordinary and partial fractional differential and integral equations in recent years; see the monographs of Abbas et al. [3,4], Kilbas et al. [2], Miller and Ross [3], and the papers of Abbas et al. [,2,5], Benchohra et al. [6], Vityuk et al. [8, 9], and the referenceherein. In [7], Butzer et al. inveigated properties of the Hadamard fractional integral and derivative. In [8], they obtained the Mellin transform of the Hadamard fractional integral and differential operators, and in [5], Pooseh et al. obtained expansion formulas of the Hadamard operators in terms of integer order derivatives. Many other intereing properties of those operators and others are summarized in [6] and the references therein. This article deals with the exience and uniqueness of solutiono the following Hadamard partial fractional integral equation of the form u(x, y) = µ(x, y) ( log x ) r ( log y ) r2 f(s, t, u(s, t)) ; if (x, y) J, Γ(r )Γ(r 2 ) (.) where J := [, a] [, b], a, b >, r, r 2 >, µ : J R, f : J R R are given continuous functions. We present two results for the integral equation (.). The fir one is based on Banach s contraction principle and the second one on the nonlinear alternative of Leray Schauder type. This paper initiatehe udy of Hadamard integral equations of two independent variables. 2 Preliminaries In this section, we introduce notations, definitions, and preliminary facthat are used throughout this paper. We let C := C(J, R) be the Banach space of continuous functions u : J R with the norm u C = sup (x,y) J u(x, y), and L (J, R) be the Banach space of functions u : J R that are Lebesgue integrable with norm u L = a b u(x, y) dydx.

3 Partial Hadamard Fractional Integral Equations 99 Definition 2. (See [, 2]). The Hadamard fractional integral of order q > for a function g L ([, a], R), is defined as ( H Ig)(x) r = x ( log x ) q g(s) Γ(q) s s ds, where Γ( ) ihe Euler gamma function. Definition 2.2. Let r, r 2, σ = (, ) and r = (r, r 2 ). For w L (J, R), define the Hadamard partial fractional integral of order r by the expression ( H Iσw)(x, r x y ( y) = log x ) r ( log y ) r2 w(s, t). Γ(r )Γ(r 2 ) Theorem 2.3 (See [9]; Nonlinear alternative of Leray Schauder type). Let X be a Banach space and C a nonempty convex subset of X. Let U a nonempty open subset of C with U and T : U C be a continuous and compact operator. Then, either (a) T has fixed points, or (b) There exi u U and λ (, ) with u = λt (u). Set J := {(x, y, s) : s x a, y [, b]}, J := {(x, y, s, t) : s x a, t y b}, D := x, D 2 := y and D D 2 := 2 x y. In the sequel we will make use of the following variant of the inequality for two independent variables due to Pachpatte. Lemma 2.4 (See [4]). Let w C(J, R ), p, D p C(J, R ), q, D q, D 2 q, D D 2 q C(J, R ) and c > be a conant. If w(x, y) c x p(x, y, s)w(s, y)ds for (x, y) [, a] [, b], then ( x w(x, y) ca(x, y) exp where in which and Q(x, y) = x y A(x, y) = exp(q(x, y)), [ p(s, y, s) B(x, y) = q(x, y, x, y)a(x, y) y x D 2 q(x, y, x, t)a(x, t)dt s x y q(x, y, s, t)w(s, t), ) B(s, t), ] D p(s, y, ξ)dξ ds, D q(x, y, s, y)a(s, y)ds D D 2 q(x, y, s, t)a(s, t).

4 Saïd Abbas, Mouffak Benchohra, and Johnny Henderson From the above lemma and when p, we get the following lemma. Lemma 2.5. Let w C(J, R ), q, D q, D 2 q, D D 2 q C(J, R ) and c > be a conant. If for (x, y) J, then where 3 Main Results w(x, y) c ( x w(x, y) c exp B(x, y) = q(x, y, x, y) y D 2 q(x, y, x, t)dt q(x, y, s, t)w(s, t), y x ) B(s, t), D q(x, y, s, y)ds D D 2 q(x, y, s, t). Let us art by defining what we mean by a solution of the integral equation (.). Definition 3.. A function u C is said to be a solution of (.) if u satisfies equation (.) on J. Further, we present conditions for the exience and uniqueness of a solution of the equation (.). Theorem 3.2. Assume If (H ) For any u, v C and (x, y) J, there exis k > such that f(x, y, u) f(x, y, v) k u v C. L := k(log a)r Γ( r )Γ( r 2 ) then there exis a unique solution for the equation (.) on J. <, (3.) Proof. Transform the integral equation (.) into a fixed point equation. Consider the operator N : C C defined by: (N u)(x, y) = µ(x, y) Γ(r )Γ(r 2 ) ( log x ) r ( log y ) r2 f(s, t, u(s, t)). (3.2)

5 Partial Hadamard Fractional Integral Equations Let v, w C. Then, for (x, y) J, we have x y (Nv)(x, y) (Nw)(x, y) log x r y log Γ(r )Γ(r 2 ) f(s, t, u(s, t)) f(s, t, v(s, t)) x y log x r y log Γ(r )Γ(r 2 ) Consequently, k u v C k(log a)r Γ( r )Γ( r 2 ) v w C. N(v) N(w) C L v w C. r 2 r 2 By (3.), N is a contraction, and hence N has a unique fixed point by Banach s contraction principle. Theorem 3.3. Assume that the following hypothesis holds: (H 2 ) There exi functions p, p 2 C(J, R ) such that for any u R and (x, y) J. f(x, y, u) p (x, y) p 2 (x, y) u(x, y), Then the integral equation (.) has at lea one solution defined on J. Proof. Consider the operator N defined in (3.2). We shall show that the operator N is continuous and completely continuous. Step. N is continuous. Let {u n } be a sequence such that u n u in C. Let η > be such that u n C η. Then (Nu n )(x, y) (Nu)(x, y) log x s r y log Γ(r )Γ(r 2 ) t f(s, t, u n(s, t)) f(s, t, u(s, t)) x y log x r y log Γ(r )Γ(r 2 ) r 2 r 2 sup (s,t) J f(s, t, u n (s, t)) f(s, t, u(s, t)) (log a)r Γ( r )Γ( r 2 ) f(,, u n (, )) f(,, u(, )) C.

6 2 Saïd Abbas, Mouffak Benchohra, and Johnny Henderson From Lebesgue s dominated convergence theorem and the continuity of the function f, we get (Nu n )(x, y) (Nu)(x, y) as n. Step 2. N maps bounded sets into bounded sets in C. Indeed, it is enough show that, for any η >, there exis a positive conant l such that, for each u B η = {u C : u C η }, we have N(u) C l. Set p i = From (H 2 ), for each (x, y) J, we have (N u)(x, y) µ(x, y) sup p i (x, y); i =, 2. (x,y) J Γ(r )Γ(r 2 ) log x s p (s, t) p 2 (s, t) u C µ (log a)r Γ( r )Γ( r 2 ) (p p 2η ) := l. r y log t r 2 Hence N(u) C l. Step 3: N maps bounded sets into equicontinuous sets in C. Let (x, y ), (x 2, y 2 ) (, a] (, b], x < x 2, y < y 2, B η be a bounded set of C as in Step 2, and let u B η. Then, (Nu)(x 2, y 2 ) (Nu)(x, y ) µ(x, y ) µ(x 2, y 2 ) x y [ log x 2 r y 2 log r 2 log x r y ] log r 2 Γ(r )Γ(r 2 ) f(s, t, u(s, t)) x2 y2 log x 2 r y 2 log r 2 f(s, t, u(s, t)) Γ(r )Γ(r 2 ) x y x y2 log x 2 r y 2 log r 2 f(s, t, u(s, t)) Γ(r )Γ(r 2 ) y x2 y log x 2 r y 2 log r 2 f(s, t, u(s, t)). Γ(r )Γ(r 2 ) x

7 Partial Hadamard Fractional Integral Equations 3 Thus, (Nu)(x 2, y 2 ) (Nu)(x, y ) µ(x, y ) µ(x 2, y 2 ) x y [ log x 2 r y 2 log r 2 log x r y ] log r 2 Γ(r )Γ(r 2 ) p p 2η x2 y2 log x 2 r y 2 log r 2 p p 2η Γ(r )Γ(r 2 ) x y x y2 log x 2 r y 2 log r 2 p p 2η Γ(r )Γ(r 2 ) y x2 y log x 2 r y 2 log r 2 p p 2η Γ(r )Γ(r 2 ) x p p 2η Γ( r )Γ( r 2 ) [2(log y 2 ) r 2 (log x 2 log x ) r 2(log x 2 ) r (log y 2 log y ) r 2 (log x ) r (log y ) r 2 (log x 2 ) r (log y 2 ) r 2 2(log x 2 log x ) r (log y 2 log y ) r 2 ]. As x x 2 and y y 2, the right-hand side of the above inequality tendo zero. As a consequence of Steps to 3 together with the Arzela Ascoli theorem, we can conclude that N is continuous and completely continuous. Step 4. (A priori bounds) We now show that there exis an open set U C with u λn(u), for λ (, ) and u U. Let u C be such that u = λn(u) for some < λ <. Thus, for each (x, y) J, u(x, y) = λµ(x, y) λ Γ(r )Γ(r 2 ) This impliehat, for each (x, y) J, we have u(x, y) µ(x, y) Γ(r )Γ(r 2 ) ( log x ) r ( log y p (s, t) p 2 (s, t) u(s, t) µ p (log a) r Γ( r )Γ( r 2 ) p 2 Γ(r )Γ(r 2 ) log x s log x s ) r2 f(s, t, u(s, t)). r y log t r 2 r y log r 2 u(s, t). t

8 4 Saïd Abbas, Mouffak Benchohra, and Johnny Henderson Thus, for each (x, y) J, we get where and u(x, y) µ p (log a) r Γ( r )Γ( r 2 ) p 2 log x r y log r 2 u(s, t) Γ(r )Γ(r 2 ) c q(x, y, s, t) u(s, t), c := µ p (log a) r Γ( r )Γ( r 2 ), p 2 q(x, y, s, t) := log x r y log r 2. Γ(r )Γ(r 2 ) From Lemma 2.5, we obtain ( x u(x, y) c exp where B(x, y) = q(x, y, x, y) y x D 2 q(x, y, x, t)dt y ) B(s, t), D q(x, y, s, y)ds p 2 xyγ(r )Γ(r 2 ) (log x)r (log y) r 2. D D 2 q(x, y, s, t) Hence ( ) p 2 u(x, y) c exp Γ(r )Γ(r 2 (log s)r (log t) r2 ( p c exp 2 (log a) r ) Γ( r )Γ( r 2 ) := R. Set U = {u C : u < R }. By our choice of U, there is no u U such that u = λn(u), for λ (, ). As a consequence of the nonlinear alternative of Leray Schauder type [9], we deduce that N has a fixed point u in U which is a solution to our equation (.).

9 Partial Hadamard Fractional Integral Equations 5 4 An Example As an application of our results we consider the following partial Hadamard integral equation of the form u(x, y) = µ(x, y) where and with ( log x ) r ( log y ) r2 f(s, t, u(s, t)) ; (x, y) [, e] [, e], (4.) Γ(r )Γ(r 2 ) r, r 2 >, µ(x, y) = x y 2 ; (x, y) [, e] [, e], f(x, y, u(x, y)) = cu(x, y) ; (x, y) [, e] [, e], exy2 c := e4 2 Γ( r )Γ( r 2 ). For each u, u R and (x, y) [, e] [, e] we have f(x, y, u(x, y)) f(x, y, u(x, y)) c e 4 u u C. Hence condition (H ) is satisfied with k = c. We shall show that condition (3.) holds e4 with a = b = e. Indeed, k(log a) r Γ( r )Γ( r 2 ) = c e 4 Γ( r )Γ( r 2 ) = 2 <. Consequently, Theorem 3.2 impliehat the integral equation (4.) has a unique solution defined on [, e] [, e]. References [] S. Abbas and M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (29), [2] S. Abbas and M. Benchohra, Fractional order integral equations of two independent variables, Appl. Math. Comput. 227 (24), [3] S. Abbas, M. Benchohra and G.M. N Guérékata, Topics in Fractional Differential Equations, Springer, New York, 22. [4] S. Abbas, M. Benchohra and G.M. N Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 25.

10 6 Saïd Abbas, Mouffak Benchohra, and Johnny Henderson [5] S. Abbas, M. Benchohra and A. N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Frac. Calc. Appl. Anal. 5 (22), [6] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Exience results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (28), [7] P. L. Butzer, A. A. Kilbas, and J. J. Trujillo. Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269 (22), 27. [8] P. L. Butzer, A. A. Kilbas, and J. J. Trujillo. Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 27 (22), 5. [9] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 23. [] J. Hadamard, Essai sur l étude des fonctions données par leur développment de Taylor, J. Pure Appl. Math. 4 (8) (892), 86. [] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2. [2] A. A. Kilbas, Hari M. Srivaava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amerdam, 26. [3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 993. [4] B. G. Pachpatte, Monotone methods for syems of nonlinear hyperbolic problems in two independent variables, Nonlinear Anal. 3 (997), [5] S. Pooseh, R. Almeida, and D. Torres. Expansion formulas in terms of integerorder derivatives for the Hadamard fractional integral and derivative. Numer. Funct. Anal. Optim. 33 (3) (22), [6] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 993. [7] V. E. Tarasov, Fractional dynamics: Application of Fractional Calculuo Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2. [8] A. N. Vityuk, On solutions of hyperbolic differential inclusions with a nonconvex right-hand side. (Russian) Ukran. Mat. Zh. 47 (995), no. 4, ; translation in Ukrainian Math. J. 47 (995), no. 4, (996).

11 Partial Hadamard Fractional Integral Equations 7 [9] A. N. Vityuk and A. V. Golushkov, Exience of solutions of syems of partial differential equations of fractional order, Nonlinear Oscil. 7 (24),