Impulsive Partial Hyperbolic Functional Differential Inclusions
In this chapter, we shall present existence results for some classes of initial value problems for impulsive partial hyperbolic differential inclusions with fractional order.
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Impulsive Partial Hyperbolic Functional Differential Inclusions
6.1 Introduction In this chapter, we shall present existence results for some classes of initial value problems for impulsive partial hyperbolic differential inclusions with fractional order.
6.2 Impulsive Partial Hyperbolic Differential Inclusions 6.2.1 Introduction This section concerns the existence results to impulsive fractional order IVP for the system .c Dxr k u/.x; y/ 2 F .x; y; u.x; y//; if .x; y/ 2 Jk I k D 0; : : : ; m;
(6.1)
u.xkC ; y/ D u.xk ; y/ C Ik .u.xk ; y//; if y 2 Œ0; bI k D 1; : : : ; m;
(6.2)
u.x; 0/ D '.x/I x 2 Œ0; a; u.0; y/ D
.y/I y 2 Œ0; b;
(6.3)
where J0 D Œ0; x1 Œ0; b; Jk D .xk ; xkC1 I k D 1; : : : ; m; a; b > 0; 0 D x0 < x1 < < xm < xmC1 D a; F W J Rn ! P.Rn / is a compactvalued multivalued map, J D Œ0; a Œ0; b; P.Rn / is the family of all subsets of Rn ; Ik W Rn ! Rn ; k D 0; 1; : : : ; m are given functions and ' W Œ0; a ! Rn ; W Œ0; b ! Rn are given absolutely continuous functions with '.0/ D .0/: To define the solutions of (6.1)–(6.3), we shall consider the Banach space ˚ P C.J; Rn /D u W J ! Rn W there exist 0 D x0 < x1 < x2 < < xm < xmC1 Da
S. Abbas et al., Topics in Fractional Differential Equations, Developments in Mathematics 27, DOI 10.1007/978-1-4614-4036-9 6, © Springer Science+Business Media New York 2012
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6 Impulsive Partial Hyperbolic Functional Differential Inclusions
such that u.xk ; y/ and u.xkC ; y/ exist with u.xk ; y/ D u.xk ; y/I k D 0; : : : ; m; and u 2 C.Jk ; Rn /I k D 0; : : : ; m : n Definition 6.1. A function u 2 P C.J; Rn /\[m kD0 AC..xk ; xkC1 /.0; b; R / such 2 that its mixed derivative Dxy exists on Jk I k D 0; : : : ; m is said to be a solution of (6.1)–(6.3) if there exists a function f 2 L1 .J; Rn / with f .x; y/ 2 F .x; y; u.x; y// such that .c Dxr k u/.x; y/ D f .x; y/ on Jk I k D 0; : : : ; m and u satisfies conditions (6.2) and (6.3).
6.2.2 The Convex Case Now we are concerned with the existence of solutions for the problems (6.1)–(6.3) when the right-hand side is compact and convex valued. Theorem 6.2. Assume the following hypotheses hold: (6.2.1) F W Jk Rn ! Pcp;cv .Rn /; k D 0; : : : ; m; is a Carath´eodory multivalued map. (6.2.2) There exist p 2 L1 .J; RC / and W Œ0; 1/ ! .0; 1/ continuous and nondecreasing such that kF .x; y; u/kP p.x; y/ .kuk/ for .x; y/ 2 J; x ¤ xk ; k D 0; : : : ; m; and each u 2 Rn : (6.2.3) There exists l 2 L1 .J; RC / such that Hd .F .x; y; u/; F .x; y; u// l.x; y/ku uk for every u; u 2 Rn ; and d.0; F .x; y; 0// l.x; y/; a.e. .x; y/ 2 Jk ; k D 0; : : : ; m: (6.2.4) There exist constants ck , such that kIk .u/k ck , k D 1; : : : ; m for each u 2 Rn : (6.2.5) There exist constants ck , such that kIk .u/ Ik .u/k ck ku uk; for each u; u 2 Rn ; k D 1; : : : ; m: (6.2.6) There exists a number M > 0 such that M kk1 C 2
2p ar1 b r2 .M / kD1 ck C .r1 C 1/ .r2 C 1/
Pm
> 1;
(6.4)
where p D kpkL1 : Then the IVP (6.1)–(6.3) have at least one solution on J:
6.2 Impulsi
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