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Starting from two extensions of the Banach contraction principle due to Ćirić (1974) and Wardowski (2012), in the present paper we introduce the concepts of Ćirić type <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ψ</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula>-contraction and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ψ</mi> <mi>F</mi> </mrow> </semantics> </math> </inline-formula>-quasicontraction on a metric space and give some sufficient conditions under which the respective mappings are Picard operators. Some known fixed point results from the literature can be obtained as particular cases.