Integral ΦΦ-type contractions and existence of continuous solutions for nonlinear integral inclusions

In this paper, we discuss a general common fixed point theorem of integral ΦΦ-type for two pairs of weakly compatible mappings satisfying certain integral type implicit relations in symmetric spaces by using the notion of a pair of mappings satisfying property (E.A). Our main result improves and extends the recent results of Aliouche [A. Aliouche, A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type, J. Math. Anal. Appl. 322 (2006) 796–802], Pathak, Tiwari and Khan [H.K. Pathak, Rakesh Tiwari, M.S. Khan, A common fixed point theorem satisfying implicit relation, Appl. Math. E-Notes, 7 (2007) 222–228] and several other known results. Subsequently, a multi-valued version of Krasnoselski, fixed point theorem in a cone is used to discuss the existence of C[0,T]C[0,T] and Lp[0,T]Lp[0,T] solutions to nonlinear integral inclusion View the MathML sourcey(t)∈∫0Tk(t,s)[a(s)g(s,y(s))+f(s,y(s),y′(s))]ds which improves the recent result of Agarwal, Meehan and O’Regan [R.P. Agarwal, M. Meehan, D. O’Regan, Positive LpLp and continuous solutions for Fredholm integral inclusions, in: Set Valued Mappings with Applications in Nonlinear Analysis, Gordon & Breach, Amsterdam, SIMAA 4 (2002) 1–9].